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--- address: | $^1$SUBATECH, Laboratoire de Physique Subatomique et des Technologies Associées\ University of Nantes - IN2P3/CNRS - Ecole des Mines de Nantes 4 rue Alfred Kastler, F-44072 Nantes, Cedex 03, France author: - 'Aman D. Sood$^1$' - 'Ch. Hartnack$^1$' - Jörg Aichelin$^1$ title: '$K^{+}$ and $K^{-}$ potentials in hadronic matter can be observed' --- Introduction {#introduction .unnumbered} ============ One key question in the analysis of sub-threshold kaon production is how to obatin information on the properties of strange mesons in dense nuclear matter [@Aichelin:1986ss]. The principal problem for extracting precise information on these properties is, however, that almost all observables depend simultaneously not only on the $K^-$ potential but also on several other input quantities which are only vaguely known eg. life time of $\Delta$ and in-medium modification of the cross section. The situation were much better if experiment provides an observable which depends on the $K$ potentials only and which is not spoiled by other little or unknown quantities. Here we aim show that the ratio of the $K^+$ and $K^-$ momentum spectra at small momentum in light systems can be such an observable. 0.5cm ![\[fig1\] Logarithmic ratio of $p_{T}$ spectra of $K^+$ and $K^-$ for different strengths of potential. Various lines are explained in the figure.[]{data-label="fig2"}](fig_1.eps "fig:"){width="6cm"} In order to study this observable and in order to make sure that it does not depend on other input quantities we have separated the $K^-$ into 2 classes (by tracing back $K^-$ to its corresponding anti strange partner $K^+$).\ (a) $K^-$ coming directly from reactions like $BB \to BB K^+ K^-$ called direct contribution and abbreviated in the figures (Dir)\ (b) $K^-$ coming from $\pi Y$ or $BY\to K^-$ abbreviated in the figures by Y.\ For the present study we use IQMD model the details of which are described in ref. [@hartphysrep]. Results and Discussion {#results-and-discussion .unnumbered} ====================== In fig. \[fig2\] we display the logarithm of the ratio of the $p_{T}$ spectra of $K^+$ and $K^-$. Top, middle, and bottom panels show this ratio for all $K^-$, for the directly produced $K^-$ and for those produced in secondary collisions, respectively. Different lines are for different strengths of potential which we vary by multiplying the K potential by a constant factor x. The total yields depend on the choice of x. The ratio is nearly constant without KN potential (x=0, dotted magenta line). When we switch on the potential the slope of the ratio changes very sharply in the low momentum region and decreases with increasing strength of the potential, whereas it remains nearly constant in the high momentum region. Comparing top and middle panel, we see that the influence of those $K^-$ (which come from secondary collisions) on the spectral form at small $p_t$ is not essential. This means that this ratio is almost exclusively sensitive to the potential and does not depend on the little or unknown cross sections. 0.5cm ![\[fig1\] Same as fig. \[fig2\] but only $K^-$N potential is varied for a fixed $K^+$N potential. []{data-label="fig3"}](fig_2.eps "fig:"){width="6cm"} -0.3cm 0.5cm ![\[fig1\] Density at which the finally observed kaons are produced. The various panels are explained in the text.[]{data-label="fig4"}](fig_3.eps "fig:"){width="6cm"} -0.3cm Fig. \[fig3\] presents as well the ratio of the $K^+$ and $K^-$ spectra but this time $K^+$N potential is taken as given by the theoretical predictions (x =1) whereas for the $K^-$ we vary the potential assuming that the $K^+$ potentials can be determined by other means. This time we have chosen a linear scale. We observe, as expected, that the dependence of the slope on the $K^-$ potential becomes weaker as compared to a variation of both potentials but still varies by a factor of two and is hence a measurable quantity. This ratio depends on the $K^-$N potential only and presents therefore the possibility to measure directly the strength of the $K^-$N potential. It is therefore the desired ’smoking gun’ signal to determine experimentally the $K^-$ potentials in matter at finite densities. It is interesting to see at which density the kaons are produced which are finally seen in the detector. This is displayed in fig. \[fig4\]. On the left (right) hand side we display as a function of $p_T$ the average density at which the $K^+$ ($K^-$) are produced which are finally seen in the detectors. The top panel shows the density for all events in which a $K^+$ and a $K^-$ is produced, the middle part that for those events in which the $K^+$ and a $K^-$ are produced simultaneously and the bottom part for those events in which the $K^-$ is produced in a secondary collision. Independent of the potential the kaons are produced at densities around normal nuclear matter density. The density for the directly produced kaons is slightly lower than that of the other events because the higher the density the higher is also the probability that the $K^-$ is reabsorbed in a $\Lambda$. Acknowledgments {#acknowledgments .unnumbered} =============== This work has been supported by a grant from Indo-French Centre for the Promotion of Advanced Research (IFCPAR) under project no 4104-1. [50]{} J. Aichelin and C. M. Ko, Phys. Rev. Lett.  [**55**]{}, 2661 (1985); S. W. Huang et al., Prog. Part. Nucl. Phys. [**30,**]{} 105 (1993); ibid. Phys. Lett. B [**298,**]{} 41 (1993). C. Hartnack et al., Phys. Rep.- to be published \[arXiv:nucl-th/1106.2083\].

--- author: - 'Bryan A. Plummer, Kevin J. Shih, Yichen Li, Ke Xu, Svetlana Lazebnik, , Stan Sclaroff, , Kate Saenko' bibliography: - 'egbib.bib' title: 'Revisiting Image-Language Networks for Open-ended Phrase Detection' ---

--- abstract: 'We strengthen our previous results [@lmzoll] regarding the moduli spaces of Zoll metrics and Zoll projective structures on $S^2$. In particular, we describe a concrete, open condition which suffices to guarantee that a totally real embedding ${\mathbb R\mathbb P}^2\hookrightarrow {\mathbb C\mathbb P}_2$ arises from a unique Zoll projective structure on the $2$-sphere. Our methods ultimately reflect the special role such structures play in the initial value problem for the $3$-dimensional Lorentzian Einstein-Weyl equations.' author: - 'Claude LeBrun[^1]  and L.J. Mason[^2]' date: 'February 11, 2010' title: 'Zoll Metrics, Branched Covers, and Holomorphic Disks' --- A [*Zoll metric*]{} on a smooth manifold $M$ is a Riemannian metric $g$ whose geodesics are all simple closed curves of equal length. This terminology commemorates the fundamental contribution of Otto Zoll [@zoll], who exhibited an infinite-dimensional family of such metrics on $M=S^2$. It is easy to prove [@beszoll] that a manifold admitting Zoll metrics is compact and has finite fundamental group, so the only two-dimensional candidates for $M$ are $S^2$ and ${\mathbb R\mathbb P}^2$; conversely, the standard metrics on both of these surfaces are obviously Zoll. However, Green’s proof [@grezoll] of the Blaschke conjecture shows that, after rescaling, every Zoll metric on ${\mathbb R\mathbb P}^2$ is actually a pull-back of the standard one via some diffeomorphism. By contrast, Zoll’s examples show that the situation for the $2$-sphere is fundamentally different. Indeed, in the decade following Zoll’s work, Funk [@funk] gave a formal-power-series argument indicating that, modulo isometries and rescalings, the general Zoll perturbation of the standard metric on $S^2$ depends on one [odd]{} function $f: S^2 \to {\mathbb R}$. However, a rigorous proof of Funk’s conjectural picture was only found half a century later, when Victor Guillemin [@guillzoll] brought the power of Nash-Moser implicit function theorems to bear on the problem. More recently, twistor techniques have given us new insights into global aspects of the problem. Indeed, the present authors have elsewhere shown [@lmzoll] that Zoll surfaces can in principle be completely understood in terms of families of holomorphic disks in ${\mathbb C\mathbb P}_2$. These same techniques are also naturally adapted to the study of more general [*Zoll projective structures*]{}. Recall that a projective structure is by definition an equivalence class $[\nabla ]$ of affine connections $\nabla$ on a manifold $M$, where two connections are declared to be equivalent iff they have the same geodesics, considered as [unparameterized]{} curves. A projective structure is said to be Zoll iff its geodesics (again, as unparameterized curves) are all embedded circles. It can then be shown [@grogro; @lmzoll] that a Riemannian metric $g$ on a compact surface $M$ is Zoll iff the equivalence class $[\nabla ]$ of its Levi-Civita connection is a Zoll projective structure. A compact surface $M$ can admit a Zoll projective structure $[\nabla ]$ iff it is diffeomorphic to $S^2$ or ${\mathbb R\mathbb P}^2$; and, as in the Riemannian case, any Zoll projective structure on ${\mathbb R\mathbb P}^2$ is actually the standard one, pulled back via some self-diffeomorphism of ${\mathbb R\mathbb P}^2$. Our proof of this last assertion [@lmzoll] hinged on the fact that the complex structure of ${\mathbb C\mathbb P}_2$ is unique [@yau] up to biholomorphism. We now summarize our previous results [@lmzoll] regarding the the case of $M=S^2$. Given a smooth Zoll projective structure $[\nabla ]$ on $M$, its space of unoriented geodesics $N\approx {\mathbb R\mathbb P}^2$ has a natural embedding in ${\mathbb C\mathbb P}_2$ as a totally real submanifold, in a manner which is completely determined up to a projective linear transformation; for example, the usual projective structure induced by the standard “round” metric corresponds to a “real linear” embedding ${\mathbb R\mathbb P}^2\hookrightarrow {\mathbb C\mathbb P}_2$. Each point $x\in M$ determines an embedded holomorphic disk $\Delta_x\subset{\mathbb C\mathbb P}_2$ with $\partial \Delta_x\subset N$, and the relative homology class $[\Delta_x]$ of any such disk generates $H_2 ({\mathbb C\mathbb P}_2, N; {\mathbb{Z}})\approx {\mathbb{Z}}$. These disks meet $N$ only along their boundaries, and their interiors foliate ${\mathbb C\mathbb P}_2- N$. The family of disks $\Delta_x$ moreover sweeps out an entire connected component in the moduli space of holomorphic disks $(D^2, \partial D^2) \to ({\mathbb C\mathbb P}_2, N)$. If the family of disks $\{ \Delta_x~|~x\in M\}$ is known, the projective structure $[\nabla ]$ can then be completely reconstructed; namely, given a point $z\in N$, the set $${\mathfrak C}_z = \{ x\in M~|~ z\in \partial \Delta_z \}$$ is a geodesic of $[\nabla ]$, and every geodesic arises in this way. The construction proceeds by first creating an abstract complex surface, and then showing that it must be biholomorphic to ${\mathbb C\mathbb P}_2$. In the process, the bundle of orientation-compatible almost-complex structures over $M=S^2$ is identified with the complement ${\mathbb C\mathbb P}_2 - N$ of the relevant totally real submanifold $N$. If there is an orientation-compatible complex structure ${\zap J}$ on $M$ which is parallel with respect to some torsion-free connection $\triangledown \in [\nabla ]$, then the graph of ${\zap J}$ becomes a holomorphic curve ${\mathcal Q}\subset {\mathbb C\mathbb P}_2 -N$. For homological reasons, this curve must be a non-singular conic, and so may be put in the standard form $$\label{conic} z_1^2 + z_2^2 + z_3^2 =0$$ by making a suitable choice of homogeneous coordinates on ${\mathbb C\mathbb P}_2$. Notice that this happens precisely when there is a conformal structure $[g]$ on $M$ for which $\triangledown$ is a compatible Weyl connection. If there is actually a Zoll metric $g$ with Levi-Civita connection $\triangledown \in [\nabla ]$, then the totally real submanifold $N\subset {\mathbb C\mathbb P}_2$ is moreover [*Lagrangian*]{} with respect to the sign-ambiguous symplectic form $\Omega = \Im m ~\Upsilon$ on ${\mathbb C\mathbb P}_2 - {\mathcal Q}$, where $$\label{oops} \Upsilon = \pm ~ \frac{z_1 ~dz_2\wedge dz_3 + z_2 ~dz_3 \wedge dz_1 + z_3~ dz_1\wedge dz_2}{ {\sqrt{(z_1^2 + z_2^2 + z_3^2)^3}} } ~ .$$ In the converse direction, one would like to assert that the totally real submanifold $N\subset {\mathbb C\mathbb P}_2$ can be chosen essentially arbitrarily, and that each such choice uniquely determines a Zoll projective structure $[\nabla ]$ on $M=S^2$. But while our previous results in this direction may have been conceptually suggestive, they were technically crude in important respects. Indeed, using an elementary inverse-function theorem argument, we merely showed in [@lmzoll] that every $N\subset {\mathbb C\mathbb P}_2$ which is $C^{2k+5}$ close to the standard “real linear” ${\mathbb R\mathbb P}^2$ in the topology actually arises from a $C^k$ Zoll projective structure $[\nabla ]$, and that this projective structure is unique among those that are close to the standard “round” projective structure. By contrast, the rest of the story was quite clean; the choice of a reference conic ${\mathcal Q}\subset {\mathbb C\mathbb P}_2$ disjoint from such an $N$ then gives rise to a conformal structure $[g]$ on $M=S^2$ for which the Zoll projective structure $[\nabla ]$ is represented by a unique $[g]$-compatible Weyl connection $\triangledown \in [\nabla ]$, and this Weyl connection is the Levi-Civita connection of a Zoll metric $g\in [g]$ iff $N$ is Lagrangian with respect to the sign-ambigious symplectic form $\Omega$. Still, it must be admitted that our previous results remain esthetically unsatisfactory in two essential ways: we neither provided an effective condition on $N\subset {\mathbb C\mathbb P}_2$ sufficient for the existence of an associated family of holomorphic disks, nor proved the uniqueness of this family when it does exist. The present article will address these issues by proving global existence and uniqueness results for holomorphic disks; see Theorems \[snap\], \[crackle\], and \[pop\] below. For the sake of clarity, our discussion is set almost entirely in the smooth ($C^\infty$) context. While our present methods certainly afford us this luxury, the interested reader may nonetheless wish to verify that most of our arguments can in fact be carried out with much less regularity. The reader may also find it interesting to compare and contrast our uniqueness results with the rather different ones found in [@rochon]. We now begin by fixing the standard non-singular conic $\mathcal Q\subset {\mathbb C\mathbb P}_2$ given by (\[conic\]). Of course, any two non-singular conics are actually projectively equivalent, but our conventional choice of $\mathcal Q$ has the nice additional feature that it is manifestly invariant under an anti-holomorphic involution $$\begin{aligned} {\mathfrak c}: {\mathbb C\mathbb P}_2 & \longrightarrow & {\mathbb C\mathbb P}_2 \label{conjugate} \\ ~[ z_1 , z_2 , z_3] &\longmapsto & [\bar{z}_1:\bar{z}_2:\bar{z}_3] \nonumber \end{aligned}$$ whose fixed-point set is disjoint from $\mathcal Q$. This fixed-point set will henceforth be called the [*standard*]{} ${\mathbb R\mathbb P}^2 \subset {\mathbb C\mathbb P}_2$. Next, notice that a projective line ${\mathcal A} \subset {\mathbb C\mathbb P}_2$ is tangent to ${\mathcal Q}$ iff it is given by $$a_1 z_1 + a_2 z_2 + a_3 z_3 =0$$ for an element $[a_1:a_2:a_3]$ of the dual projective plane ${\mathbb C\mathbb P}_2^*$ satisfying $$\label{dual} a_1^2 + a_2^2 + a_3^2 =0.$$ When this happens, the point of tangency is then given by $$[z_1:z_2:z_3]= [a_1:a_2:a_3].$$ Also notice that if $p= [p_1:p_2:p_3]$ belongs to the complement of $ {\mathcal Q}$ in ${\mathbb C\mathbb P}_2 $, there are always exactly two tangent lines of $\mathcal Q$ which pass through $p$; indeed, the incidence equation $$a_1 p_1 + a_2 p_2 + a_3 p_3 =0$$ for $[a_1:a_2:a_3]$ describes a line in the dual projective plane ${\mathbb C\mathbb P}_2^*$ which is [*not*]{} tangent to the dual conic (\[dual\]), and which therefore, by Bézout’s theorem, must meet the dual conic in two distinct points. The standard ${\mathbb R\mathbb P}^2 \subset {\mathbb C\mathbb P}_2$ is an example of a totally real submanifold. If $Z$ is a complex manifold with integrable almost-complex structure $J$, recall that a differentiable submanifold $S\subset Z$ is said to be (maximally) [*totally real*]{} if $TZ|_S = TS \oplus J(TS)$. We now introduce a special class of totally real surfaces in ${\mathbb C\mathbb P}_2$ that will be central to our discussion. A compact connected smoothly embedded $2$-manifold $N \subset {\mathbb C\mathbb P}_2$ will be called a [*docile surface*]{} if - $N$ is a totally real submanifold of $ {\mathbb C\mathbb P}_2$; - $N$ is disjoint from the conic ${\mathcal Q}$ defined by (\[conic\]); and - $N$ is transverse to each tangent projective line (\[dual\]) of the conic ${\mathcal Q}$. For example, the standard ${\mathbb R\mathbb P}^2 \subset {\mathbb C\mathbb P}_2$ is docile. Indeed, the transversality condition is satisfied in this case because a projective line is tangent to this standard ${\mathbb R\mathbb P}^2$ iff it is invariant under complex conjugation, whereas the two tangent lines of $\mathcal Q$ which pass through a real point $p \in {\mathbb R\mathbb P}^2$ are, by contrast, interchanged by the anti-holomorphic involution $\mathfrak c$ defined above in (\[conjugate\]). Since the condition of docility is obviously open in the $C^1$ topology, any small perturbation of the standard ${\mathbb R\mathbb P}^2$ will also be a docile surface. In the converse direction, we have the following result: \[primo\] Let $N\subset {\mathbb C\mathbb P}_2$ be a docile surface. Then $N$ is diffeomorphic to ${\mathbb R\mathbb P}^2$, and is isotopic to the standard ${\mathbb R\mathbb P}^2\subset {\mathbb C\mathbb P}_2$ through a family of other docile surfaces. The argument will proceed by systematically exploiting the map $$\begin{aligned} \Pi: {\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1 &\to& {\mathbb C\mathbb P}_2\\ ([u_1:u_2], [v_1:v_2])&\mapsto& [i(u_1v_1+u_2v_2):u_1v_1-u_2v_2:u_1v_2+u_2v_1]~,\end{aligned}$$ which is a $2$-to-$1$ branched cover, ramified over the conic $\mathcal Q$. Indeed, notice that the diagonal $\mathcal D$, explicitly given by $u_1v_2 - v_1 u_2 =0$, is sent bijectively to the conic $z_1^2+z_2^2+z_3^2=0$. Moreover, each factor line $u= \mbox{const}$ or $v= \mbox{const}$ is sent isomorphically to a tangent line $\mathcal A$ of $\mathcal Q$, and every tangent line conversely occurs in each of these families. Since $\Pi$ has degree two, and since exactly two tangent lines pass through each $p\in {\mathbb C\mathbb P}_2 - {\mathcal Q}$, it follows that $\Pi$ is actually unramified away from ${\mathcal Q}$. We note in passing that the “anti-diagonal” $\overline{\mathcal D}$, defined by $[u_1: u_2 ] = [ -\bar{v}_2:\bar{v}_1]$, is therefore a $2$-to-$1$ cover of the standard ${\mathbb R\mathbb P}^2 \subset {\mathbb C\mathbb P}_2$ defined by $z_j=\bar{z}_j$, $j=1,2,3$. Now suppose that $N\subset {\mathbb C\mathbb P}_2$ is a docile surface, and let $\tilde{N}= \Pi^{-1}(N)$ be its pre-image in ${\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1$. Since $N \cap {\mathcal Q}= \varnothing$, it follows that $\tilde{N}$ is a smooth compact surface in ${\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1$, and the induced map $\tilde{N} \to N$ is a two-to-one submersion. In particular, $\tilde{N}$ has at most two connected components. On the other hand, the transversality hypothesis guarantees that $\tilde{N}$ is transverse to each factor line $u= \mbox{const}$ or $v= \mbox{const}$, so each factor projection $$\varpi_j|_{\tilde{N}}: \tilde{N} \to {\mathbb C\mathbb P}_1,~~~j= 1, 2,$$ is a submersion, and hence a covering map on each connected component. Since ${\mathbb C\mathbb P}_1$ is simply connected, both projections are therefore diffeomorphisms on each connected component, and each component of $\tilde{N}$ is therefore the graph of some diffeomorphism $\varphi : {\mathbb C\mathbb P}_1 \to {\mathbb C\mathbb P}_1$. Moreover, since $\tilde{N}\cap {\mathcal D}= \varnothing$, the relevant $\varphi$ cannot have fixed points, and is therefore homotopic to the antipodal map $$\begin{aligned} {\mathfrak a} : {\mathbb C\mathbb P}_1 &\longrightarrow & {\mathbb C\mathbb P}_1 \\ ~[ z_1 : z_2 ] & \longmapsto & [ - \bar{z}_2 : \bar{z}_1 ]\end{aligned}$$ via the familiar geometric construction of pushing $\varphi (z)$ away from $z$ along great circles. Consequently, each such diffeomorphism $\varphi$ has degree $-1$, and so is orientation-reversing. Thus, if $\tilde{N}$ were disconnected, it would have to be the union of two disjoint graphs of orientation-reversing diffeomorphisms $\varphi_1, \varphi_2 : {\mathbb C\mathbb P}_1 \to {\mathbb C\mathbb P}_1$. Since these two graphs would be disjoint, we would have $\varphi_1(u) \neq \varphi_2(u)$, and hence $(\varphi_2^{-1}\circ \varphi_1) (u) \neq u$, for all $u \in {\mathbb C\mathbb P}_1$. Thus $\varphi_2^{-1}\circ \varphi_1$ would be fixed-point-free, and hence would also have have degree $-1$. But since $$\deg (\varphi_2^{-1}\circ \varphi_1) = (\deg \varphi_2)^{-1}(\deg \varphi_1) = (-1)^2 = + 1,$$ this is clearly a contradiction. Hence $\tilde{N}$ is connected, and therefore the graph of a single fixed-point-free, orientation-reversing diffeomorphism $\varphi : {\mathbb C\mathbb P}_1 \to {\mathbb C\mathbb P}_1$. However, by construction, $\tilde{N}= \Pi^{-1}(N )$ is invariant under the holomorphic involution $$\begin{aligned} \varrho : {\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1 &\longrightarrow & {\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1\\ (u,v) & \longmapsto & (v, u)\end{aligned}$$ and every $(u ,\varphi (u))$ must therefore also be expressible as $(\varphi (v), v)$. Thus $\varphi= \varphi^{-1}$, and $\varphi^2 = \mbox{id}$. By projection to the first factor, the action of the non-trivial deck transformation on $\tilde{N}$ can therefore be identified with the action of $\varphi$ on ${\mathbb C\mathbb P}_1$, and the quotient $N$ of $\tilde{N}$ by this deck transformation can therefore be identified with ${\mathbb C\mathbb P}_1/\langle \varphi \rangle$. But this is a smooth compact surface with fundamental group ${\mathbb{Z}}_2$, and so necessarily diffeomorphic to ${\mathbb R\mathbb P}^2= {\mathbb C\mathbb P}_1 / \langle {\mathfrak a} \rangle$. Lifting some diffeomorphism ${\psi}_0: {\mathbb C\mathbb P}_1 / \langle {\mathfrak a} \rangle \to {\mathbb C\mathbb P}_1/\langle \varphi \rangle$ to an orientation-preserving diffeomorphism $\psi : {\mathbb C\mathbb P}_1 \to {\mathbb C\mathbb P}_1$ of universal covers now shows that we must have $$\varphi = \psi \circ {\mathfrak a} \circ \psi^{-1}$$ for some orientation-preserving diffeomorphism $\psi$ of ${\mathbb C\mathbb P}_1$. This conclusion can now be reverse-engineered. Indeed, given any orientation preserving diffeomorphism $\psi : {\mathbb C\mathbb P}_1 \to {\mathbb C\mathbb P}_1$, the graph of the orientation-reversing involution $\varphi = \psi \circ {\mathfrak a} \circ \psi^{-1}$ necessarily projects, via $\Pi$, to a docile surface in ${\mathbb C\mathbb P}_2$. However, the group of orientation-preserving diffeomorphisms of ${\mathbb C\mathbb P}_1$ is connected, since it acts transitively on the (connected) space of conformal metrics on $S^2$, with (connected) isotropy subgroup $PSL(2, {\mathbb{C}})$. It therefore follows that the space of docile surfaces in ${\mathbb C\mathbb P}_2$ is also connected. In particular, any given docile surface $N$ may be smoothly deformed, via a family of docile surfaces, into the standard ${\mathbb R\mathbb P}^2\subset {\mathbb C\mathbb P}_2$. \[secondo\] Let $N\subset {\mathbb C\mathbb P}_2$ be any docile surface. Then the homomorphism $H_2 ({\mathbb C\mathbb P}_2 , N) \to {\mathbb{Z}}$ given by homological intersection with $[{\mathcal Q}] \in H_2 ({\mathbb C\mathbb P}_2 - N)$ is an isomorphism. In particular, $H_2 ({\mathbb C\mathbb P}_2 , N) \cong {\mathbb{Z}}$. Let us first observe that there is a homeomorphism $\Psi: {\mathbb C\mathbb P}_2\to {\mathbb C\mathbb P}_2$ which sends $N$ to ${\mathbb R\mathbb P}^2$, and $\mathcal Q$ to itself. Indeed, let $\psi : {\mathbb C\mathbb P}_1 \to {\mathbb C\mathbb P}_1$ be a diffeomorphism such that $\tilde{N}$ is the graph of $\psi\circ {\mathfrak a} \circ \psi^{-1}$, and then define $\tilde{\Psi} : ({\mathbb C\mathbb P}_1\times {\mathbb C\mathbb P}_1) \to ({\mathbb C\mathbb P}_1\times {\mathbb C\mathbb P}_1)$ to be $\psi \times \psi$. Then $\tilde{\Psi}$ send $\bar {\mathcal D}$ to $\tilde{N}$, and ${\mathcal D}$ to itself, while commuting with the involution $\varrho$ given by $\varrho (u, v) \mapsto (v, u)$. Since $\varrho$ acts transitively on the fibers of $\Pi$, and since $\Pi : {\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1 \to {\mathbb C\mathbb P}_2$ is a quotient map, there is a unique homeomorphism $\Psi : {\mathbb C\mathbb P}_2 \to {\mathbb C\mathbb P}_2$ such that $\Pi \circ \tilde{\Psi} = \Psi \circ \Pi$. Moreover, this homeomorphism $\Psi$ then sends $N$ to ${\mathbb R\mathbb P}_2$, and $\mathcal Q$ to $\mathcal Q$, as promised. Thus $H_2 ({\mathbb C\mathbb P}_2 , N ) \cong H_2 ({\mathbb C\mathbb P}_2 , {\mathbb R\mathbb P}^2 )$, and we only really need to check the claim when $N = {\mathbb R\mathbb P}^2$. However, ${\mathbb C\mathbb P}_2$ is the union of a $2$-disk bundle $X\to {\mathbb R\mathbb P}^2$ and a $2$-disk bundle $Y \to {\mathcal Q}$, identified along their boundaries. (This can be checked by hand [@lmzoll], but it can more elegantly be deduced [@grozi] from the fact that there is a cohomogeneity-one action of $SO(3)$ on ${\mathbb C\mathbb P}_2$, with exceptional orbits ${\mathbb R\mathbb P}^2$ and $\mathcal Q$.) Thus $H_2 ({\mathbb C\mathbb P}_2 , {\mathbb R\mathbb P}^2)\cong H_2 ({\mathbb C\mathbb P}_2 , X)$ by homotopy equivalence, whereas $H_2 ({\mathbb C\mathbb P}_2 , X) \cong H_2 (Y , \partial Y )$ by excision, and $H_2 (Y, \partial Y)\cong H^2 (Y)$ by Lefschetz-Poincaré duality. Since ${\mathcal Q}$ is a deformation retract of $Y$, it follows that $H_2 ({\mathbb C\mathbb P}_2 , N )\cong H^2 ({\mathcal Q}) = {\mathbb{Z}}$. Moreover, since $[\mathcal Q]$ generates $H_2 (Y)\cong {\mathbb{Z}}$, Lefschetz-Poincaré duality guarantees that homological intersection with $[\mathcal Q]$ is an isomorphism $H_2({\mathbb C\mathbb P}_2, N)\to {\mathbb{Z}}$. The homeomorphism $\Psi$ constructed above will typically fail to be smooth along $\mathcal Q$. However, it is not difficult to modify $\Psi$ to produce a self-diffeomorphism of ${\mathbb C\mathbb P}_2$ with all the properties in question. We leave this exercise as a challenge to the interested reader. \[trio\] Let $N \subset {\mathbb C\mathbb P}_2$ be any docile surface, and let $\tilde{N}\subset {\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1$ be its inverse image under $\Pi$. Then ${\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1$ admits a $\varrho$-invariant Kähler metric $h$ for which $\tilde{N}$ is Lagrangian. This metric can be chosen so that its Kähler form $\omega$ represents $2\pi c_1 ({\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1 )$ in deRham cohomology, and if $N$ is smoothly varied through a family of other docile surfaces, a corresponding family of such Kähler metrics can moreover be chosen so as to depend smoothly on the given parameters. The construction is a variant of one used in [@lmew Lemma 2]. Express $\tilde{N}$ as the graph of a smooth orientation-reversing involution $\varphi : {\mathbb C\mathbb P}_1 \to {\mathbb C\mathbb P}_1$. Let $\alpha$ be the area form of the standard unit-sphere metric on $S^2={\mathbb C\mathbb P}_1$. Then the area form $$\check{\omega} = \frac{\alpha - \varphi^*\alpha}{2}$$ satisfies $\varphi^*\check{\omega} = -\check{\omega}$, and is the Kähler form of a unique Kähler metric $\check{h}$ on ${\mathbb C\mathbb P}_1$. We now define a Kähler metric on ${\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1$ by setting $$h= \varpi_1^*\check{h} + \varpi_2^*\check{h}$$ where $\varpi_j : {\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1 \to {\mathbb C\mathbb P}_1$, $j=1,2$, are the factor projections. Since the restriction of the associated Kähler form $\omega$ to the graph of $\varphi$ is $$(\mbox{id}\times \varphi)^* ( \varpi_1^*\check{\omega} + \varpi_2^*\check{\omega}) = \check{\omega} + \varphi^*\check{\omega} = \check{\omega}- \check{\omega}=0 ,$$ the graph $\tilde{N}$ is Lagrangian with respect to $\omega$. Moreover, $h$ is invariant under the interchange of the factors of ${\mathbb C\mathbb P}_1\times {\mathbb C\mathbb P}_1$, and the Kähler class of $h$ is obviously given by $[\omega]= 2\pi c_1 ({\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1)$, since $\omega$ has integral $4\pi$ on either factor ${\mathbb C\mathbb P}_1$. Finally, this construction can be uniformly applied to a smooth family $N_t$ of docile surfaces by replacing $\varphi$ with a corresponding of smooth family $\varphi_t$ of smooth involutions, and the corresponding family $h_t$ of Kähler metrics will then manifestly depend smoothly on the parameter $t$. Let $D^2$ denote the closed unit disk in ${\mathbb{C}}$, and let $Z$ be any complex manifold. A continuous map $f: D^2 \to Z$ will be called a [*parameterized holomorphic disk*]{} in $Z$ if $f$ is holomorphic in the open unit disk $\mathring{D}^2 = D^2 - \partial D^2$. If, in addition, $f (\partial D) \subset W$ for a specified subset $W\subset Z$, we will sometimes say that $f$ is a [parameterized holomorphic disk]{} in $(Z,W)$. \[discus\] Let $N \subset {\mathbb C\mathbb P}_2$ be any docile surface, and suppose that $f$ is a parameterized holomorphic disk in $({\mathbb C\mathbb P}_2 , N )$ whose relative homology class $[f]$ generates $H_2({\mathbb C\mathbb P}_2, N) \cong {\mathbb{Z}}$. Then $f$ is a smooth embedding, $f(D^2 )$ meets $N$ only along $f(\partial D^2)$, and $f(D^2)$ meets $\mathcal Q$ transversely, in a single point. Because $f|_{\partial D^2}$ takes values in the totally real submanifold $N$, and the latter is assumed to be a submanifold of class $C^{\infty}$, the holomorphic map $f$ must actually be smooth up to the boundary [@alibaro; @chirka]. Since we have also assumed that $[f]$ generates $H_2({\mathbb C\mathbb P}_2, N) \cong {\mathbb{Z}}$, its homological intersection number with ${\mathcal Q}$ must be $1$ by Lemma \[secondo\], and the disk $f$ can therefore only geometrically intersect $\mathcal Q$ transversely, in a single point, since every geometric intersection of distinct holomorphic curves makes a contribution with positive multiplicity toward their total homological intersection number. Now, because $\Pi : {\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1\to {\mathbb C\mathbb P}_2$ is a $2$-to-$1$ branched cover, ramified only at $\mathcal Q$, path-lifting of the null-homotopic circles $f(re^{2i\theta})$ in ${\mathbb C\mathbb P}_2 - {\mathcal Q}$ allows us to construct a continuous lift $\tilde{f}: D^2 \to {\mathbb C\mathbb P}_1\times {\mathbb C\mathbb P}_1$ with $\Pi(\tilde{f}(\zeta))= f(\zeta^2)$. Since this lift is moreover holomorphic away from the origin, it then follows that $\tilde{f}$ is also holomorphic across the origin by the Riemann removable singularities theorem. We thus obtain a parameterized holomorphic disk $\tilde{f}: D^2 \to {\mathbb C\mathbb P}_1\times {\mathbb C\mathbb P}_1$ which sends $\partial D^2$ to $\tilde{N} = \Pi^{-1} (N)$ and $0$ to the unique $\tilde{p}\in {\mathcal D}$ with $\Pi (\tilde{p})= p$, while satisfying $\tilde{f}(-\zeta) = \varrho (\tilde{f}(\zeta))$, where $\varrho$ is once again the involution $(u,v)\mapsto (v,u)$ of ${\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1$. Also note that, given any $\tilde{f}$ satisfying these properties, one may conversely construct a disk $f$ in ${\mathbb C\mathbb P}_2$ with the desired properties by setting $f(\zeta ) = \Pi (\tilde{f}(\pm \sqrt{\zeta}))$, since this well-defined map is obviously continuous, and is holomorphic away from the origin. Now, given a holomorphic disk $f$ representing the generator of $H_2({\mathbb C\mathbb P}_2, N)$, the associated disk $\tilde{f}$ will then represent the generator of $H_2 ({\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1 , \tilde{N})\cong {\mathbb{Z}}$. Indeed, since $\tilde{N}$ is homotopic to the anti-diagonal $\overline{\mathcal D}$ in $({\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1)- {\mathcal D}$, the long exact sequence $$\cdots \to H_2 (\tilde{N}) \to H_2 ({\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1 ) \to H_2 ({\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1 , \tilde{N}) \to H_1 (\tilde{N}) \to \cdots$$ implies that the diagonal class $[\mathcal D]$ will necessarily represent twice the generator of $H_2 ({\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1 , \tilde{N})$. However, $\Pi (\mathcal D )$ is the degree-$2$ holomorphic curve $\mathcal Q$, so $\Pi_* : H_2 ({\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1 , \tilde{N}) \to H_2 ({\mathbb C\mathbb P}_2 , N )$ is therefore the homomorphism ${\mathbb{Z}}\to {\mathbb{Z}}$ given by multiplication by $2$. But $\Pi_* ([\tilde{f}])= 2 [f]$ by construction, so it follows that $[\tilde{f} ]$ is indeed the generator of $H_2 ({\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1 , \tilde{N})$, as claimed. In particular, since $\tilde{N}$ is Lagrangian with respect the Kähler form $\omega$ of the metric $h$ constructed in Lemma \[trio\], we must have $$\int_{D^2} \tilde{f}^* \omega = \frac{1}{2} \int_{\mathcal D} \omega = \frac{1}{2} \int_{\mathcal D} 2\pi c_1 = \frac{1}{2} 2\pi (4) = 4\pi ~.$$ Next, by making precise an argument previously sketched in [@lmew Lemma 3], we will show that $\tilde{f}$ must be a topological embedding. Indeed, consider the abstract oriented $2$-sphere obtained by taking the double $D^2 \cup \overline{D^2}$, where the two copies of the disk are identified along the boundary, $D^2$ is given the usual orientation coming from the unit disk in ${\mathbb{C}}$, and $\overline{D^2}$ is given the opposite orientation. Given $f$ as above, we can then construct a continuous map $F: D^2 \cup \overline{D^2}\to {\mathbb C\mathbb P}_1$, defined to equal $\varpi_1 \circ\tilde{f}$ on $D^2$, and to equal $\varphi^{-1}\circ \varpi_2 \circ\tilde{f}$ on $\overline{D^2}$. Note that ${F}$ is actually smooth when restricted to either $D^2$ or $\overline{D^2}$, and that it is orientation-preserving at every regular point of the interior of either $2$-disk hemisphere. The complex dilatation $\mu$ of $F$ is bounded by that of $\varphi^{-1}$, so $F$ is consequently quasi-regular in the sense of [@rickman]. However, via the measurable Riemann mapping theorem, this implies [@lehvir section VI.2.3] that $F= G\circ H$ for some quasi-conformal homeomorphism $H: D^2 \cup \overline{D^2}\to {\mathbb C\mathbb P}_1$ and some holomophic map $G: {\mathbb C\mathbb P}_1 \to {\mathbb C\mathbb P}_1$. However, equipping ${\mathbb C\mathbb P}_1$ and ${\mathbb C\mathbb P}_1\times {\mathbb C\mathbb P}_1$ respectively with the Kähler forms $\check{\omega}$ and $\omega$ used in the proof of Lemma \[trio\], we have $$\int_{D^2 \cup \overline{D^2}}F^*\check{\omega}= \int_{D^2}\tilde{f}^*(\varpi_1^*\check{\omega}+ \varpi_2^* \omega_2) = \int_{D^2}\tilde{f}^*\omega = 4\pi = \int_{{\mathbb C\mathbb P}_1}\check{\omega}$$ and it therefore follows that the piece-wise smooth map ${F}$ has degree $1$. The holomorphic map $G: {\mathbb C\mathbb P}_1 \to {\mathbb C\mathbb P}_1$ is therefore a biholomorphism, and $F$ is therefore a quasi-conformal homeomorphism. In particular, $\varpi_j\circ \tilde{f}$ must be a homeomorphism for $j=1,2$, and $(\varpi_1 \circ\tilde{f}) (\mathring{D}^2)$ must be disjoint from $(\varphi^{-1}\circ \varpi_2 \circ\tilde{f})(\mathring{D}^2)$. In particular, the graph $\tilde{N}$ of $\varphi$ is disjoint from $\tilde{f} (\mathring{D}^2)$, and hence ${N}$ is disjoint from ${f} (\mathring{D}^2)$, too. To finish the proof, we will now show that the smooth maps $$\varpi_j\circ \tilde{f}: D^2 \to {\mathbb C\mathbb P}_1, ~~ j=1,2,$$ are actually smooth embeddings. Of course, since we already know that they are homeomorphisms, and since they are manifestly holomorphic on the interior $\mathring{D}^2$ of the disk, it only remains to show that these maps are immersions along $\partial D^2$. However, since $\tilde{N}\subset {\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1$ is a smooth, totally real submanifold, and since $\tilde{f} (\partial D^2) \subset \tilde{N}$, it follows [@alibaro] that the non-constant holomorphic map $\tilde{f}$ cannot be constant to infinite order at any boundary point. Moreover, since $\tilde{N}$ is the graph of a diffeomorphism $\varphi: {\mathbb C\mathbb P}_1\to {\mathbb C\mathbb P}_1$, neither of the factor maps $\varpi_j \circ \tilde{f}$, $j=1,2$, can be constant to infinite order at a boundary point, either. Thus, setting ${{\textcyr c}}= \varpi_1 \circ \tilde{f}$, letting $\zeta_0\in \partial D^2$ be any boundary point of the disk, and equipping ${\mathbb C\mathbb P}_1$ with a local complex coordinate $z$ centered at ${{\textcyr c}}(\zeta_0)$, there must be an integer $k\geq 1$ such that $${{\textcyr c}}(\zeta) = a (\zeta-\zeta_0)^k + O(|\zeta-\zeta_0|^{k+1})$$ for some $a\neq 0$, since the $C^{\infty}$ function ${{\textcyr c}}$ satisfies the Cauchy-Riemann equations up to the boundary. If $k\geq 2$, it therefore follows that, in the fixed coordinate system, ${{\textcyr c}}(\mathring{D}^2)$ must contain a punctured half-disk $$0< |z | \leq \epsilon , ~~ \Re e (e^{-i\theta} z) \geq 0$$ because the boundary of a wedge-shaped subregion $$|\zeta -\zeta_0| < \delta , ~~ \left|\arg (\zeta-\zeta_0) -\theta\right| < \frac{\pi }{3k}$$ of $D^2$ is mapped by the smooth homeomorphism ${{\textcyr c}}$ to a piece-wise $C^1$ Jordan curve with internal break-angle $4\pi/3 > \pi$ at ${{\textcyr c}}(\zeta_0)$. However, the same argument can also be applied to the holomorphic map $\varpi_2 \circ \tilde{f}$; thus, if $k\geq 2$, even after composing with the orientation-reversing diffeomorphism $\varphi^{-1}$, the boundary of an analogous wedge-shaped region would still be sent to a piece-wise $C^1$ Jordan curve whose internal break angle would have absolute value $> \pi$ at the origin, and the image of $\mathring{D}^2$ under the smooth map ${\mbox{{\renewcommand\rmdefault{wncyr}\renewcommand\sfdefault{wncyss}\renewcommand\encodingdefault{OT2}\normalfont \selectfont}sh}}= \varphi^{-1} \circ \varpi_2 \circ \tilde{f}$ would therefore also contain a punctured half-disk $$0< |z | \leq \varepsilon , ~~ \Re e (e^{-i\vartheta} z) \geq 0 .$$ Since any two such punctured half-disks must meet, $k\geq 2$ thus implies that $${{\textcyr c}}(\mathring{D}^2) \cap {\mbox{{\renewcommand\rmdefault{wncyr}\renewcommand\sfdefault{wncyss}\renewcommand\encodingdefault{OT2}\normalfont \selectfont}sh}}(\mathring{D}^2) \neq \varnothing ,$$ contradicting the previously established fact that $F$ is injective. Hence $k=1$, and, since $z_0\in \partial D^2$ is arbitrary, ${{\textcyr c}}$ and ${\mbox{{\renewcommand\rmdefault{wncyr}\renewcommand\sfdefault{wncyss}\renewcommand\encodingdefault{OT2}\normalfont \selectfont}sh}}$ are both smooth embeddings $D^2 \hookrightarrow {\mathbb C\mathbb P}_1$. The parameterized holomorphic disk $\tilde{f}$ in $({\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1, \tilde{N})$ is therefore smoothly embedded, and the parameterized holomorphic disk $f$ in $({\mathbb C\mathbb P}_2, N )$ is therefore smoothly embedded, too. If $L \to D^2$ is a complex line bundle over the disk, and if $\ell \to S^1$ is a real line sub-bundle of $L|_{\partial D^2}$, recall [@mcsalt] that the Maslov index ${\operatorname{ind}}(L,\ell )$ is obtained by trivializing $L$, viewing $\ell $ as a map $\partial D^2 \to {\mathbb R\mathbb P}^1$, and declaring ${\operatorname{ind}}(L,\ell )$ to be the winding number of this map; the resulting integer is independent of the trivialization of $L$, and amounts [@lebrsb] to the first Chern class of the double of $(L,\ell )$. More generally, if $V\to D^2$ is a rank-$r$ complex vector bundle, and if ${\zap v} \to S^1$ is a rank-$r$ real sub-bundle of $V|_{\partial D^2}$, the Maslov index ${\operatorname{ind}}(V, {\zap v})$ is defined to be the Maslov index of the associated line-bundle pair $(\Lambda^rV, \Lambda^r{\zap v})$. If $ Z$ is a complex manifold and $ W\subset Z$ is a totally real submanifold, the [*total Maslov index*]{} of a parameterized holomorphic disk ${\zap f}$ in $( Z, W)$ is defined to be ${\operatorname{ind}}({\zap f}^*T Z, ({\zap f}|_{\partial D^2})^*T W)$. If the disk happens to be embedded, with image ${\Delta}= {\zap f}(D^2)$, we also define the [*normal Maslov index*]{} of ${\Delta}$ to be ${\operatorname{ind}}(N, {\zap n})$, where $N=T Z/T{\Delta}$ is the normal bundle of the disk, and ${\zap n}= T W/T\partial {\Delta}$ is the relative normal bundle of its boundary. Since the Maslov index is additive, in the sense that $${\operatorname{ind}}(V_1 \oplus V_2, {\zap v}_1 \oplus {\zap v}_2) = {\operatorname{ind}}(V_1, {\zap v}_1) + {\operatorname{ind}}(V_2, {\zap v}_2),$$ and because ${\operatorname{ind}}(TD^2 , T\partial D^2 )= 2$, the total Maslov index of an embedded holomorphic disk equals its normal Maslov index plus two. \[masl\] Let $N \subset {\mathbb C\mathbb P}_2$ be a docile surface, and let $f$ be a parameterized holomorphic disk in $({\mathbb C\mathbb P}_2 , N )$ whose relative homology class represents the generator of $H_2({\mathbb C\mathbb P}_2, N)$. Then $f$ has total Maslov index $3$, and its image $f(D^2)$ has normal Maslov index $1$. Given such a disk $f$, let $\tilde{f} : D^2 \hookrightarrow {\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1$ be the branched lifting constructed in the proof of Proposition \[discus\]. Since the embedded holomorphic disk $\tilde{f}(D^2)$ is the graph of a diffeomorphism between domains in ${\mathbb C\mathbb P}_1$ with smooth boundary, it is transverse to the first factor of $T{\mathbb C\mathbb P}_1 \times T{\mathbb C\mathbb P}_1$, and its normal bundle can therefore be identified with the restriction of $T{\mathbb C\mathbb P}_1$ to its first-factor projection ${{\textcyr c}}(D^2) \subset {\mathbb C\mathbb P}_1$; this simultaneously identifies the relative normal bundle of the boundary with $(J\circ {{\textcyr c}}_*) (T\partial D^2)$. Taking an affine chart that contains ${{\textcyr c}}(D^2)$, setting $\gamma = {{\textcyr c}}|_{\partial D^2}$, and systematically exploiting the coordinate trivialization of $T{\mathbb{C}}$, the normal Maslov index of $\tilde{f}(D^2)$ therefore equals the winding number of $$(i\gamma^\prime)^2: S^1 \to {\mathbb{C}}-\{0\}.$$ Since $\gamma$ is isotopic to the standard circle $e^{i\theta} \mapsto e^{i\theta}$ in ${\mathbb{C}}$, $(i\gamma^\prime)^2$ is homotopic to $e^{i\theta} \mapsto e^{2i\theta}$, and the normal Maslov index of $\tilde{f}(D^2)$ consequently equals $2$. On the other hand, $\tilde{f}(D^2)$ is a $2$-to-$1$ branched over of $f(D^2)$, so the corresponding winding number for $f$ is only half as large, and the normal Maslov index of ${f}(D^2)$ therefore equals $1$. The total Maslov index of $f$ is therefore $3$, as claimed. \[snap\] Let $N \subset {\mathbb C\mathbb P}_2$ be any docile surface, and let $p\in {\mathcal Q}$ be any point of the reference conic (\[conic\]). Then there is a holomorphic disk in $({\mathbb C\mathbb P}_2 , N)$ which passes through $p$ and represents the generator of $H_2({\mathbb C\mathbb P}_2, N) \cong {\mathbb{Z}}$. Moreover, this disk is unique, modulo reparameterizations. By Lemma \[primo\], there exists a smooth family of docile surfaces $N_t$, $t\in [0,1]$, such that $N_1=N$, and such that $N_0$ is the standard linear ${\mathbb R\mathbb P}^2\subset {\mathbb C\mathbb P}_2$. For the docile surface $N_0={\mathbb R\mathbb P}^2$, we can construct such a disk by using the complex conjugation map $\mathfrak c$ defined by (\[conjugate\]). Indeed, since ${\mathfrak c}$ acts freely on ${\mathcal Q}$, the points $p$ and ${\mathfrak c}(p)$ are distinct, and hence joined by a unique projective line ${\mathbb C\mathbb P}_1\subset {\mathbb C\mathbb P}_2$. Since $p$ and ${\mathfrak c}(p)$ are interchanged by ${\mathfrak c}$, this projective line must be invariant under complex conjugation ${\mathfrak c}$, and it is therefore the complexification of a unique real projective line ${\mathbb R\mathbb P}^1\subset {\mathbb R\mathbb P}^2$. This ${\mathbb R\mathbb P}_1$ divides the complex projective line ${\mathbb C\mathbb P}_1$ into two disks, exactly one of which contains the given point $p\in {\mathcal Q}$; and since this disk meets $\mathcal Q$ transversely in a single point, its relative homology class generates $H_2 ({\mathbb C\mathbb P}_2 , {\mathbb R\mathbb P}^2)$. Conversely, the only disk with these properties is this half-projective-line. Indeed, if ${\Delta}$ is a disk in $({\mathbb C\mathbb P}_2 , {\mathbb R\mathbb P}^2)$ which passes through $p\in {\mathcal Q}$ and represents the generator of $H_2 ({\mathbb C\mathbb P}_2 , {\mathbb R\mathbb P}^2)$, then ${\Delta}\cup {\mathfrak c}({\Delta})$ is a holomorphic curve by the reflection principle, and, by Bézout’s theorem, has degree one because it intersects the conic ${\mathcal Q}$ in two points. Thus, any holomorphic disk representing the generator of $H_2({\mathbb C\mathbb P}_2, {\mathbb R\mathbb P}^2)$ is one hemisphere of a $\frak c$-invariant complex projective line, with boundary a real projective line ${\mathbb R\mathbb P}^2 \subset {\mathbb R\mathbb P}^2$; and there is exactly one such hemisphere containing $p$. We have thus proved both existence and uniqueness for the “model” docile surface $N_0={\mathbb R\mathbb P}^2$. We will now apply the continuity method to obtain an appropriate disk for each $t\in [0,1]$. Let ${\zap E}\subset [0,1]$ be the set of $t$ for which such a disk exists; our goal is to show that ${\zap E}=[0,1]$. We already know that ${\zap E}\neq \varnothing$, since $0\in {\zap E}$. Because $[0,1]$ is connected, it therefore suffices to show that ${\zap E}$ is both open and closed. To show that ${\zap E}$ is open, we appeal to the perturbation theory of holomorphic disks. Indeed, suppose that such a disk ${\Delta}$ exists for a certain value $\tau$ of $t$. By Proposition \[discus\], ${\Delta}$ is smoothly embedded, and by Proposition \[masl\], its normal Maslov index is $1$. The double of the normal bundle of ${\Delta}$, in the sense used in [@lebrsb Theorem 3], is therefore the ${\mathcal O}(1)$ line bundle over ${\mathbb C\mathbb P}_1= {\Delta}\cup \overline{{\Delta}}$, and, since $H^1({\mathbb C\mathbb P}_1, {\mathcal O}(1))=0$, the disk ${\Delta}$ is Fredholm regular. The moduli space $M_\tau$ of nearby holomorphic disks in $({\mathbb C\mathbb P}_2, N_\tau)$ is therefore smooth, with tangent space $$T_{\tiny {\Delta}}M_\tau=H^0_{{\mathbb R}} ({\mathbb C\mathbb P}_1 , {\mathcal O}(1))$$ where the right-hand-side denotes the real-linear subspace of $H^0({\mathbb C\mathbb P}_1 , {\mathcal O}(1))$ consisting of sections which are real along ${\mathbb R\mathbb P}^1\subset {\mathbb C\mathbb P}_1$. Now observe that evaluation at two distinct points $p,q\in {\mathbb C\mathbb P}_1$ gives rise to an isomorphism $H^0({\mathbb C\mathbb P}_1 , {\mathcal O}(1))\to {\mathbb{C}}^2$, and that evaluation at a single point $p\in {\mathbb C\mathbb P}_1-{\mathbb R\mathbb P}^1$ therefore gives us a real-linear isomorphism $H^0_{{\mathbb R}} ({\mathbb C\mathbb P}_1 , {\mathcal O}(1))\to {\mathbb{C}}$, as may be seen by setting $q=\bar{p}$. Since ${\Delta}$ is transverse to $\mathcal Q$ at $p\in \mathring{{\Delta}}$, it follows that the map $\varkappa_\tau : M_\tau\to \mathcal Q$ obtained by sending a disk to its intersection with $\mathcal Q$ has maximal rank at ${\Delta}$, and so is a diffeomorphism between a neighborhood of ${\Delta}\in M_\tau$ and a neighborhood of $p\in {\mathcal Q}$. This shows that, at least locally, ${\Delta}$ is the only disk in the family passing through the chosen point $p\in {\mathcal Q}$. Moreover, the Fredholm regularity of ${\Delta}$ guarantees that, for all $t$ in a neighborhood of $\tau$, there is a corresponding $2$-parameter family $M_t$ of embedded holomorphic disks in $({\mathbb C\mathbb P}_2, N_t)$, and the corresponding intersection map $\varkappa_t: M_t \to {\mathcal Q}$ is a local diffeomorphism onto a neighborhood of $p$. Consequently, for values of $t$ in an interval about $\tau$, there is a unique smooth family ${\Delta}_t$ of such holomorphic disks passing through $p$ with ${\Delta}_\tau={\Delta}$. In particular, ${\zap E}\subset [0,1]$ is open. To show that ${\zap E}$ is closed, we will now use a Gromov compactness argument. Indeed, suppose that $t_j$ is a sequence of values of $t\in [0,1]$ for which there exist a corresponding sequence of holomorphic disks ${\Delta}_{t_j}$ in $({\mathbb C\mathbb P}_2, N_{t_j})$, each of which intersects ${\mathcal Q}$ transversely in the single point $p$; moreover, suppose that the numbers $t_j$ converge to some $\tau \in [0,1]$. Let $\tilde{{\Delta}}_{t_j}= \Pi^{-1}({\Delta}_{t_j})$ be the corresponding ramified lifts to ${\mathbb C\mathbb P}_1\times {\mathbb C\mathbb P}_1$. Each of these disks then meets the diagonal ${\mathcal D}= \Pi^{-1}({\mathcal Q})$ transversely in the unique point $\tilde{p}$ corresponding to $p$. Consequently, each one represents the generator of $H_2 ({\mathbb C\mathbb P}_1\times {\mathbb C\mathbb P}_1, {\mathcal D})$. For $t\in [0,1]$, we now endow ${\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1$ with the smooth family of $\varrho$-invariant Kähler forms $\omega_t$ constructed in Lemma \[trio\]; this family is chosen so that $\tilde{N}_t$ is Lagrangian with respect to $\omega_t$, and all these forms $\omega_t$ belong to the same cohomology class, with respect to which each factor ${\mathbb C\mathbb P}_1$’s has area $4\pi$. By Moser stablity and the Weinstein tubular neighborhood theorem for Lagrangian submanifolds, the triples $({\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1, {\tilde{N}}_t , \omega_t)$ are thus all symplectomorphic to the standard model $({\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1, {\tilde{N}}_0 , \omega_0)$, allowing us to think of the $\tilde{{\Delta}}_{t_j}$ as being a sequence of disks in $({\mathbb C\mathbb P}_1 \times {\mathbb C\mathbb P}_1, {\tilde{N}}_0 , \omega_0)$ which are pseudo-holomorphic with respect to a sequence of $\omega_0$-compatible almost-complex structures. By Gromov compactness [@grofrau; @groye], this sequence therefore has a subsequence which converges to a (possibly singular) pseudo-holomorphic curve ${\zap X}$ in the same relative homology class, where ${\zap X}$ is a union of holomorphic ${\mathbb C\mathbb P}_1$’s and at most one holomorphic disk in $({\mathbb C\mathbb P}_1\times {\mathbb C\mathbb P}_1 , {\tilde{N}}_{\tau })$. Moreover, since ${\zap X}$ is the Gromov limit of a sequence of $\varrho$-invariant curves through $\tilde{p}$, it must also be $\varrho$-invariant and pass through $\tilde{p}$. However, by construction, $$\omega_{\tau }= \varpi_1^* \check{\omega}_{\tau }+ \varpi_2^* \check{\omega}_{\tau }$$ for some area form $ \check{\omega}_{\tau }$ on ${\mathbb C\mathbb P}_1$. Since ${\zap X}$ is invariant under the factor-switching involution $\varrho$ and represents the generator in $H_2({\mathbb C\mathbb P}_1\times {\mathbb C\mathbb P}_1 , {\tilde{N}}_{\tau })$, $$2\int_{\zap X} \varpi_1^*\check{\omega}_{\tau } = \int_{\zap X} \varpi_1^*\check{\omega}_{\tau }+ \int_{\zap X} \varpi_2^*\check{\omega}_{\tau } = \int_{\zap X} \omega_{\tau }= 4\pi .$$ The area, with multiplicities, of the projection of ${\zap X}$ to either factor will thus be $2\pi$, or in other words, half the area of the entire sphere. In particular, neither factor projection ${\zap X}\to {\mathbb C\mathbb P}_1$ can be onto. Consequently, ${\zap X}$ cannot have any compact irreducible components, and is therefore a disk $\tilde{{\Delta}}_{\tau }$. Since $\tilde{{\Delta}}_{\tau }$ is $\varrho$-invariant, it must, moreover, be the ramified lift of a disk ${\Delta}_{\tau }$ in $({\mathbb C\mathbb P}_2, N_{\tau })$. Because $\tilde{{\Delta}}_{\tau }$ meets the diagonal $\mathcal D$ transversely in the single point $\tilde{p}$, the disk ${\Delta}_{\tau }$ consequently meets $\mathcal Q$ transversely in the single point $p$. In particular, ${\Delta}_{\tau }$ passes through $p$ and represents the generator of $H_2({\mathbb C\mathbb P}_2, N_{\tau })$. Thus $\tau \in {\zap E}$, and ${\zap E}$ is therefore closed. Since ${\zap E}\subset [0,1]$ has now been shown to be non-empty, open, and closed, the connectedness of the interval implies that ${\zap E}=[0,1]$. In particular, $1\in {\zap E}$, so there exists a holomorphic disk in $({\mathbb C\mathbb P}_2 , N)$ which passes through $p$ and represents the generator of $H_2({\mathbb C\mathbb P}_2, N)$, as claimed. It only remains for us to show that this disk is in fact unique. To prove uniqueness, we apply the continuity method in reverse. Suppose that ${\Delta}^\prime$ is any holomorphic disk in $({\mathbb C\mathbb P}_2 , N)$ which passes through $p$ and represents the generator of $H_2({\mathbb C\mathbb P}_2, N)$. By the same disk-perturbation argument as before, there is a unique smooth family ${\Delta}_t^\prime$, $t\in (\tau, 1]$, of holomorphic disks in $({\mathbb C\mathbb P}_2, N_t)$, meeting $\mathcal Q$ transversely in the single point $p$, such that ${\Delta}_1^\prime= {\Delta}^\prime$ and such that $\tau \in [0,1)$ is minimal. The above compactness argument then allows one to construct a limit disk ${\Delta}_\tau$, and perturbations of this disk allow one to extend the family across $t=\tau$. Since $\tau$ was taken to be as small as possible, this is a contradiction unless $\tau=0$. We thus actually obtain such a family of disks ${\Delta}_t^\prime$ for $t\in [0,1]$. However, ${\Delta}_0^\prime$ is then a disk in $({\mathbb C\mathbb P}_2 , {\mathbb R\mathbb P}^2)$ which meets $\mathcal Q$ transversely in the single point $p$, and we have already observed that this implies that ${\Delta}_0^\prime$ is one hemisphere of the projective line joining $p$ to ${\mathfrak c}(p)$. Thus, if there were two disks ${\Delta}$ and ${\Delta}^\prime$ in $({\mathbb C\mathbb P}_2 , N)$, each meeting $\mathcal Q$ transversely in the single point $p$, both could be evolved backwards in $t$ to obtain the same disk. However, the process of evolving forward in $t$ starting with an initial disk at $t=0$ necessarily yields a unique final disk when $t=1$, so it follows that ${\Delta}^\prime={\Delta}$. We have thus succeeded in establishing uniqueness, and our proof is therefore complete. For clarity’s sake, it is worth emphasizing that the above perturbation argument is carried out on the level of unparameterized embedded disks. For each unparameterized embedded disk, there is of course an $SL(2, {\mathbb R})$’s worth of different parameterizations. The interested reader is invited to double-check our perturbation argument using the more popular machinery of parameterized disks and the total Maslov index [@forst; @mcsalt]. From this perspective, the moduli space of parameterized disks in $({\mathbb C\mathbb P}_2 , N_t)$ near a given one will be $5$-dimensional, and the moduli space of parameterized disks passing through $p$ will be $3$-dimensional. One can furthermore specialize the parameterization by requiring that $0\in D^2$ be sent to $p$, and the resulting parameterized disk $f: D^2\to {\mathbb C\mathbb P}_2$ will then be unique modulo rotations $f(\zeta) \rightsquigarrow f(e^{i\phi}\zeta)$. \[crackle\] Let $N \subset {\mathbb C\mathbb P}_2$ be any docile surface, and let $M$ denote the moduli space of all holomorphic disks in $({\mathbb C\mathbb P}_2, N)$ which represent the generator of $H_2 ({\mathbb C\mathbb P}_2, N)\cong {\mathbb{Z}}$. Then $M$ is diffeomorphic to $S^2$. The interiors of these disks foliate ${\mathbb C\mathbb P}_2 - N$, and the intersection pattern of their boundaries defines a unique Zoll projective structure $[\nabla]$ on $M$. Moreover, the reference conic $\mathcal Q$ induces a specific conformal structure $[g]$ on $M$, and there is a unique $\triangledown \in [\nabla]$ which is a Weyl connection for the conformal class $[g]$. Theorem \[snap\] asserts that, through each $p\in {\mathcal Q}$, there is a unique holomorphic disk representing the generator of $H_2({\mathbb C\mathbb P}_2, N)$; moreover, the proof of this theorem shows the map $\varkappa : M\to {\mathcal Q}$, obtained by sending a disk to its intersection with the conic, to be a local diffeomorphism. Hence $\varkappa$ is actually a diffeomorphism, and $M\approx {\mathcal Q}\approx S^2$. For each $x\in M$, let ${\Delta}_x\subset {\mathbb C\mathbb P}_2$ be the embedded holomorphic disk it represents, and set $${\mathcal F} = \{ (x,y) \in M \times {\mathbb C\mathbb P}_2 ~|~ y\in {\Delta}_x\}.$$ Then the projection $(x,y) \mapsto x$ makes ${\mathcal F}$ into a smooth family of disks ${\zap p} : {\mathcal F}\to M$. Let ${\mathcal B}$ be the Melrose blow-up of ${\mathbb C\mathbb P}_2$ along $N$, which is the manifold-with-boundary obtained from ${\mathbb C\mathbb P}_2$ by replacing each point of $N$ with the unit circle bundle in the normal bundle of $N$; let ${\zap b} : {\mathcal B}\to {\mathbb C\mathbb P}_2$ be canonical projection, which we shall call the [*Melrose blow-down*]{}. Since each of the disks ${\Delta}_x$ is smoothly embedded, with $\partial {\Delta}_x\subset N$, the tautological smooth projection ${\zap q}: {\mathcal F}\to {\mathbb C\mathbb P}_2$ given by $(x,y)\mapsto y$ can be lifted to a smooth map $\hat{\zap q}: {\mathcal F}\to {\mathcal B}$, with ${\zap q} = {\zap b}\circ \hat{\zap q}$, by sending each boundary point to the radial derivative of ${\zap q}$ relative to the corresponding disk. However, since each disk ${\Delta}_x$ has normal Maslov index $1$, the normal bundle of ${\Delta}_x$ can be trivialized in such a manner that elements of the tangent space $T_xM$ are represented by some holomorphic function $\varsigma : D^2\to {\mathbb{C}}$ of the form $$\varsigma (\zeta) = a + \bar{a} \zeta$$ for $a\in {\mathbb{C}}$ arbitrary. For $a\neq 0$, such a variation has a zero only at the boundary of the disk, and at this zero, its radial derivative is non-zero. Thus the derivative of $\hat{\zap q}$ has maximal rank everywhere. Since the manifolds-with-boundary in question are compact and connected, it follows that $\hat{\zap q}$ is a covering map. However, $\mathcal B$ diffeomorphic to the complement of a tubular neighborhood of $N$, and ${\mathcal Q}\hookrightarrow {\mathcal B}$ is therefore a homotopy equivalence by the proof of Lemma \[secondo\]. Thus ${\mathcal B}$ is simply connected, and $\hat{\zap q}$ is therefore a diffeomorphism. In particular, it induces a diffeomorphism between the interiors of ${\mathcal F}$ and $\mathcal B$, so the interiors of the disks ${\Delta}_x$ foliate ${\mathbb C\mathbb P}_2 - N$, as claimed. For any $y\in {\mathcal B}$ and $z={\zap b}(y)$, consider the pull-back map ${\zap b}^*: \Lambda^{1,0}_z \to \Lambda^1_y\otimes {\mathbb{C}}$. If $y$ is an interior point of $\mathcal B$, this is obviously injective, because ${\zap b}$ is a local diffeomorphism near $y$. However, it remains injective even if $y$ is a boundary point. To see this, notice that, when $y\in \partial {\mathcal B}$, the kernel of the pull-back map $\Lambda^{1}_z \otimes {\mathbb{C}}\to \Lambda^1_y\otimes {\mathbb{C}}$ is $1$-dimensional, and spanned by a real co-vector. Since $\Lambda^{1,0}_z\subset \Lambda^{1}_z \otimes{\mathbb{C}}$ contains no non-zero real vectors, it follows that $\Lambda^{1,0}_z \to \Lambda^1_y\otimes {\mathbb{C}}$ is injective, as claimed. Because ${\zap q} = {\zap b}\circ \hat{\zap q}$, the annihilator of ${\zap b}^*(\Lambda^{1,0})$ therefore corresponds, via the diffeomorphism $\hat{\zap q}$, to a rank-$2$ sub-bundle $\textcyr{D} \subset T_{\mathbb{C}}{\mathcal F}$, explicitly given by the kernel of ${\zap q}^{1,0}_* : T_{\mathbb{C}}{\mathcal F} \to T^{1,0}{\mathbb C\mathbb P}_2$. This sub-bundle is involutive on the interior of ${\mathcal F}$, and so, by continuity, is involutive even along $\partial \mathcal F$. However, since the derivative of ${\zap b}$ has rank $3$ at every boundary point, the same is true of ${\zap q}$, and $\textcyr{D}= \ker {\zap q}^{1,0}_*$ therefore contains a real direction at every point of $\partial {\mathcal F}$. It also contains the $(0,1)$-tangent space of the fiber disks of ${\zap p} : {\mathcal F}\to M$, so we can apply [@lmzoll Lemma 4.6] to the double ${\mathcal F}\cup \overline{\mathcal F}$, exactly as in the proof of [@lmzoll Theorem 4.7]. Thus, there is a unique projective connection $[\nabla]$ on $\mathcal M$ for which the closed curves ${\mathfrak C}_z$, defined for $z\in N$ by $${\mathfrak C}_z = {\zap p} ( {\zap q}^{-1} [\{ z\} ]) = \{ x\in \mathcal M~|~ z\in \partial \Delta_x\},$$ are the geodesics of $[\nabla]$. Moreover, since each disk-boundary $\partial\Delta_x$ is an embedded $S^1\subset N$, no fiber of ${\zap p}|_{\partial {\mathcal F}}$ can meet a fiber of ${\zap q}|_{\partial {\mathcal F}}$ in more than one point. Hence the ${\mathfrak C}_z\subset {\mathfrak M}$ are all simple closed curves , and $[\nabla]$ is therefore a Zoll projective structure on $M\approx S^2$. Finally, there is [@lmzoll footnote 4, p. 514] a unique choice of $\triangledown\in [\nabla]$ which is a Weyl connection for the conformal structure on $M$ induced by its identification with $\mathcal Q$. \[pop\] Let $N \subset {\mathbb C\mathbb P}_2$ be a docile surface, and let $( [g], \triangledown)$ be the Weyl structure on $M\approx S^2$ whose existence is guaranteed by Theorem \[crackle\]. Then $\triangledown$ is the Levi-Civita connection of a Zoll metric $g\in [g]$ iff $N\subset {\mathbb C\mathbb P}_2 - N$ is Lagrangian with respect to the sign-ambiguous symplectic form $\Omega = \Im m \Upsilon$, where $\Upsilon$ is defined by equation (\[oops\]). When this happens, there is a unique such $g$ whose closed geodesics all have length $2\pi$. This normalization is moreover equivalent to requiring that $(S^2, g)$ have total area $4\pi$. By [@lmzoll Theorem 4.8], the Lagrangian condition is equivalent to the requirement that $\triangledown$ be the Levi-Civita connection of a Zoll metric $g$. When this happens, $g$ is then of course determined up to an multiplicative constant, since $\triangledown$ and $[g]$ are already known. We can moreover fix this scaling constant by specifying the length of some (and hence every) closed geodesic. However, it is also known [@lmzoll Theorem 2.15] that the geodesic flow of $g$ is differentiably conjugate to that of the standard metric $g_0$ on $S^2$. Hence Weinstein’s theorem [@weinstein] predicts that the total area of $(S^2, g)$ will coincide with that of $(S^2, g_0)$ iff the closed geodesics of $g$ and $g_0$ have identical lengths. Theorem \[crackle\] considerably clarifies our understanding of the return journey from totally real surfaces $N\subset {\mathbb C\mathbb P}_2$ to Zoll projective structure $[\nabla ]$ on $S^2$. As long as $N$ is docile with respect to some non-singular conic $\mathcal Q$, then there is a unique Zoll projective structure $[\nabla ]$ on $S^2$ to which $N$ corresponds via the construction of [@lmzoll]. However, because docility is an open condition, a surface $N$ which is docile with respect to a particular conic $\mathcal Q$ will also be docile with respect to a $5$-complex-parameter family of nearby conics ${\mathcal Q}^\prime$. Thus, every Zoll projective structure $[\nabla]$ arising from Theorem \[crackle\] can actually be represented by a $10$-real-parameter family of Weyl connections $\triangledown$, each compatible with a different conformal structure. However, moving ${\mathcal Q}^\prime$ far enough will always result in conics with respect to which a given $N$ will fail to be docile. For this reason, docility is best understood in terms of Weyl connections rather than projective structures. One motivation for the present article stems from the fact that a Weyl structure $(S^2, [g], \triangledown)$ with all geodesics geometrically closed may be treated as time-symmetric Cauchy data for a Lorentzian Einstein-Weyl structure on $S^2 \times {\mathbb R}$. From the twistor perspective, the latter arises as the moduli space of holomorphic disks in ${\mathbb C\mathbb P}_1\times {\mathbb C\mathbb P}_1$ with boundaries in $\tilde{N}= \Pi^{-1}(N)$. When $N$ is docile with respect to the branch locus ${\mathcal Q}$ of $\Pi$, our results from [@lmew] guarantee long-time existence of the solution, and allow one to directly interpret the orientation-reversing involution $\varphi : {\mathbb C\mathbb P}_1\to {\mathbb C\mathbb P}_1$ in terms of the scattering of light-rays in the resulting space-time. In this context, it would be fascinating to obtain a better understand the scattering map $\varphi$ directly in terms of the initial-value surface. An attack on this initial-value problem by more direct methods might therefore offer new insights into the geometry of Zoll manifolds. Of course, the most classical aspects of our present subject pertain not to Zoll projective structures, but rather to Zoll metrics. Since the conformal structure is intrinsically part of the geometry in this setting, it is completely natural in this context to fix a conic ${\mathcal Q}\subset {\mathbb C\mathbb P}_2$ once and for all. Our previous work in [@lmzoll] showed that every Zoll metric on $S^2$ gives rise to a Lagrangian surface $N\subset {\mathbb C\mathbb P}_2-{\mathcal Q}$, one might perhaps hope that this totally real submanifold would always turn out to be docile with respect to $\mathcal Q$. However, our own calculations have revealed that this isn’t even true in the axisymmetric case. Thus, while we hope that the present paper offers some interesting advances in the theory of Zoll metrics, further new ideas and results will be needed in order, for example, to determine whether the space of Zoll metrics on $S^2$ is connected. The first author warmly thanks Chris Bishop and Dennis Sullivan for their helpful explanations of various relevant aspects of the theory of quasi-conformal mappings. [10]{} , [*Unique continuation and regularity at the boundary for holomorphic functions*]{}, Duke Math. J., 61 (1990), pp. 635–653. , [*Manifolds [A]{}ll of [W]{}hose [G]{}eodesics [A]{}re [C]{}losed*]{}, Springer-Verlag, Berlin, 1978. , [*Regularity of the boundaries of analytic sets*]{}, Mat. Sb. (N.S.), 117(159) (1982), pp. 291–336, 431. , [*Analytic disks with boundaries in a maximal real submanifold of [${\bf C}^2$]{}*]{}, Ann. Inst. Fourier (Grenoble), 37 (1987), pp. 1–44. , [*Gromov convergence of pseudoholomorphic disks*]{}, J. 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[Authors’ addresses:]{} [Mathematics Department, SUNY, Stony Brook, NY 11794, USA\ Mathematical Institute, 24–29 St. Giles, Oxford OX1 3LB, UK ]{} [^1]: Supported in part by NSF grant DMS-0905159. [^2]: Supported in part by FP6 Marie Curie RTN [*ENIGMA*]{} (MRTN-CT-2004-5652).

--- abstract: 'Let $G$ be a transitive permutation group on a finite set of size at least $2$. By a well known theorem of Fein, Kantor and Schacher, $G$ contains a derangement of prime power order. In this paper, we study the finite primitive permutation groups with the extremal property that the order of every derangement is an $r$-power, for some fixed prime $r$. First we show that these groups are either almost simple or affine, and we determine all the almost simple groups with this property. We also prove that an affine group $G$ has this property if and only if every two-point stabilizer is an $r$-group. Here the structure of $G$ has been extensively studied in work of Guralnick and Wiegand on the multiplicative structure of Galois field extensions, and in later work of Fleischmann, Lempken and Tiep on $r''$-semiregular pairs.' address: - 'T.C. Burness, School of Mathematics, University of Bristol, Bristol BS8 1TW, UK' - 'H.P. Tong-Viet, Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242, USA' author: - 'Timothy C. Burness' - 'Hung P. Tong-Viet' title: Primitive permutation groups and derangements of prime power order --- Introduction {#s:intro} ============ Let $G$ be a transitive permutation group on a finite set $\Omega$ of size at least $2$. An element $x \in G$ is a *derangement* if it acts fixed-point-freely on $\Omega$. An easy application of the orbit-counting lemma shows that $G$ contains derangements (this is originally a classical theorem of Jordan [@Jordan]), and we will write $\Delta(G)$ for the set of derangements in $G$. Note that if $H$ is a point stabilizer, then $x$ is a derangement if and only if $x^G \cap H$ is empty, where $x^G$ denotes the conjugacy class of $x$ in $G$, so we have $$\label{e:delta} \Delta(G) = G \setminus \bigcup_{g \in G}H^g.$$ The existence of derangements in transitive permutation groups has interesting applications in number theory and topology (see Serre’s article [@Serre], for example). Various extensions of Jordan’s theorem on the existence of derangements have been studied in recent years. For example, if $\delta(G) = |\Delta(G)|/|G|$ denotes the proportion of derangements in $G$, then a theorem of Cameron and Cohen [@CC] states that $\delta(G) {\geqslant}|\Omega|^{-1}$, with equality if and only if $G$ is sharply $2$-transitive. More recently, Fulman and Guralnick have established the existence of an absolute constant ${\epsilon}>0$ such that $\delta(G)>{\epsilon}$ for any simple transitive group $G$ (see [@FG1; @FG2; @FG3; @FG4]). This latter result confirms a conjecture of Boston et al. [@Boston] and Shalev. The study of derangements with special properties has been another major theme in recent years. By a theorem of Fein et al. [@FKS], $\Delta(G)$ contains an element of prime power order (their proof requires the classification of finite simple groups), and this result has important number-theoretic applications. For instance, it implies that the relative Brauer group of any finite extension of global fields is infinite. In most cases, $\Delta(G)$ contains an element of prime order, but there are some exceptions, such as the $3$-transitive action of the smallest Mathieu group ${\rm M}_{11}$ on $12$ points. The transitive permutation groups with this property are called *elusive* groups, and they have been investigated by many authors; see [@CGJKKMN; @Giudici; @GiuKel], for example. In this paper, we are interested in the permutation groups with the special property that *every* derangement is an $r$-element (that is, has order a power of $r$) for some fixed prime $r$. One of our main motivations stems from a theorem of Isaacs et al. [@IKLM], which describes the finite transitive groups in which every derangement is an involution; by [@IKLM Theorem A], such a group is either an elementary abelian $2$-group, or a Frobenius group with kernel an elementary abelian $2$-group. In [@BDS], this result is used to classify the finite groups whose irreducible characters vanish only on involutions. It is natural to consider the analogous problem for odd primes, and more generally for prime powers. As noted in [@IKLM], it is easy to see that such a generalization will involve a wider range of examples. For instance, if $p$ is an odd prime then every derangement in the affine group ${\rm ASL}_{2}(p) = {\rm SL}_{2}(p){:}p^2$ (of degree $p^2$) has order $p$ (if $p=2$, the derangements have order $2$ or $4$). Our first result is a reduction theorem. \[t:main1\] Let $G$ be a finite primitive permutation group such that every derangement in $G$ is an $r$-element for some fixed prime $r$. Then $G$ is either almost simple or affine. Our next result, Theorem \[t:main2\] below, describes all the almost simple primitive groups that arise in Theorem \[t:main1\]. Notice that in Table \[tab:main\], we write ${\rm P}_{1}$ for a maximal parabolic subgroup of ${\rm L}_{2}(q)$ or ${\rm L}_{3}(q)$, which can be defined as the stabilizer of a $1$-dimensional subspace of the natural module (similarly, ${\rm P}_{2}$ is the stabilizer of a $2$-dimensional subspace). In addition, we define $${\mathcal{E}}(G) = \{|x| \,:\, x \in \Delta(G)\}.$$ \[t:main2\] Let $G$ be a finite almost simple primitive permutation group with point stabilizer $H$. Then every derangement in $G$ is an $r$-element for some fixed prime $r$ if and only if $(G,H,r)$ is one of the cases in Table \[tab:main\]. In particular, every derangement has order $r$ if and only if $|\mathcal{E}(G)|=1$. \[r:isom\] $$\begin{array}{lllll} \hline G & H & r & {\mathcal{E}}(G) & \mbox{Conditions} \\ \hline {\rm L}_{3}(q) & {\rm P}_1, {\rm P}_{2} & r & r & q^2+q+1 = (3,q-1)r \\ & & r & r, r^2 & q^2+q+1 = 3r^2 \\ {\rm \Gamma L}_2(q) & {{\mathbf {N}}}_{G}({\rm D}_{2(q+1)}) & r & r & \mbox{$r=q-1$ Mersenne prime} \\ {\rm \Gamma L}_{2}(8) & {{\mathbf {N}}}_{G}({\rm P}_1), {{\mathbf {N}}}_{G}({\rm D}_{14}) & 3 & 3,9 & \\ {\rm PGL}_{2}(q) & {{\mathbf {N}}}_{G}({\rm P}_{1}) & 2 & 2^i, \, 1 {\leqslant}i {\leqslant}e+1 & \mbox{$q=2^{e+1}-1$ Mersenne prime} \\ {\rm L}_2(q) & {\rm P}_1 & r & r^i, \, 1 {\leqslant}i {\leqslant}e & q=2r^e-1 \\ & {\rm P}_1, {\rm D}_{2(q-1)} & r & r & \mbox{$r=q+1$ Fermat prime} \\ & {\rm D}_{2(q+1)} & r & r & \mbox{$r=q-1$ Mersenne prime} \\ {\rm L}_{2}(8) & {\rm P}_{1}, {\rm D}_{14} & 3 & 3,9 & \\ {\rm M}_{11} & {\rm L}_{2}(11) & 2 & 4,8 & \\ \hline \end{array}$$ Now let us turn our attention to the affine groups that arise in Theorem \[t:main1\]. In order to state Theorem \[t:main3\] below, we need to introduce some additional terminology. Let ${\mathbb{F}}$ be a field and let $V$ be a finite dimensional vector space over ${\mathbb{F}}$. Let $H {\leqslant}{\rm GL}(V)$ be a finite group and let $r$ be a prime. Recall that $x \in H$ is an *$r'$-element* if the order of $x$ is indivisible by $r$. Following Fleischmann et al. [@FLT], the pair $(H,V)$ is said to be *$r'$-semiregular* if every nontrivial $r'$-element of $H$ has no fixed points on $V\setminus\{0\}$ (equivalently, no nontrivial $r'$-element of $H$ has eigenvalue $1$ on $V$). \[t:main3\] Let $G = HV {\leqslant}{\rm AGL}(V)$ be a finite affine primitive permutation group with point stabilizer $H = G_0$ and socle $V = (\mathbb{Z}_{p})^k$, where $p$ is a prime and $k {\geqslant}1$. Then every derangement in $G$ is an $r$-element for some fixed prime $r$ if and only if $r=p$ and the pair $(H,V)$ is $r'$-semiregular. Let $G=HV$ be an affine group as in Theorem \[t:main3\] and notice that $(H,V)$ is $r'$-semiregular if and only if every two-point stabilizer in $G$ is an $r$-group. As a special case, observe that if $G$ is a Frobenius group then every two-point stabilizer is trivial and it is clear that every derangement in $G$ has order $r$. Therefore, it is natural to focus our attention on the non-Frobenius affine groups arising in Theorem \[t:main3\], which correspond to $r'$-semiregular pairs $(H,V)$ such that $r$ divides $|H|$. In this situation, Guralnick and Wiegand [@GW Section 4] obtain detailed information on the structure of $H$, which they use to investigate the multiplicative structure of finite Galois field extensions. Similar results were established in later work of Fleischmann et al. [@FLT]. We refer the reader to the end of Section \[s:affine\] for further details (see Propositions \[p:flt1\] and \[p:flt2\]). Transitive groups with the property in Theorem \[t:main1\] arise naturally in several different contexts. For instance, let us recall that the existence of a derangement of prime power order in any finite transitive permutation group implies that the relative Brauer group $B(L/K)$ of any finite extension $L/K$ of global fields is infinite. More precisely, let $L = K({\alpha})$ be a separable extension of $K$, let $E$ be a Galois closure of $L$ over $K$, and let $\Omega$ be the set of roots in $E$ of the minimal polynomial of ${\alpha}$ over $K$. Then the $r$-primary component $B(L/K)_r$ is infinite if and only if the Galois group ${\rm Gal}(E/K)$ contains a derangement of $r$-power order on $\Omega$ (see [@FKS Corollary 3]). In this situation, it follows that the relative Brauer group $B(L/K)$ has a unique infinite primary component if and only if every derangement in ${\rm Gal}(E/K)$ is an $r$-element for some fixed prime $r$. In a different direction, our property arises in the study of permutation groups with *bounded movement*. To see the connection, let $G {\leqslant}{\rm Sym}(\Omega)$ be a transitive permutation group of degree $n$ and set $$m = \max\{|\Gamma^x \setminus \Gamma| \, :\, \Gamma \subseteq \Omega,\, x \in G\} \in \mathbb{N},$$ where $\Gamma^x = \{\gamma^x \,:\, \gamma \in \Gamma\}$. Following Praeger [@Praeger], we say that $G$ has *movement* $m$. If $G$ is not a $2$-group and $n = \lfloor 2mp/(p-1) \rfloor$, where $p{\geqslant}5$ is the least odd prime dividing $|G|$, then $p$ divides $n$ and every derangement in $G$ has order $p$ (see [@HKKP Proposition 4.4]). Moreover, the structure of these groups is described in [@HKKP Theorem 1.2]. Some additional related results are established by Mann and Praeger in [@MP]. For instance, [@MP Proposition 2] states that if $G$ is a transitive $p$-group, where $p=2$ or $3$, then every derangement in $G$ has order $p$ only if $G$ has exponent $p$. It is still not known whether or not the same conclusion holds for *any* prime $p$ (see [@MP p.905]), although [@HKKP Proposition 6.1] does show that the exponent of such a group is bounded in terms of $p$ only. \[r:prime\] *Let $G = HV {\leqslant}{\rm AGL}(V)$ be a finite affine primitive permutation group as above, and assume that every derangement in $G$ is an $r$-element for some fixed prime $r$. Let $P$ be a Sylow $r$-subgroup of $G$ and set $K = H \cap P$. As explained in Proposition \[p:prime\], $P$ is a transitive permutation group on $P/K$ such that $\mathcal{E}(G) = \mathcal{E}(P)$, so $\mathcal{E}(G) = \{r\}$ if and only if $\mathcal{E}(P) = \{r\}$, and we will show that $\mathcal{E}(P) = \{r\}$ if and only if $P$ has exponent $r$ (see Theorem \[c:prime\]).* There is also a connection between our property and $2$-coverings of abstract groups. First notice that Jordan’s theorem on the existence of derangements is equivalent to the well known fact that no finite group $G$ can be expressed as the union of $G$-conjugates of a proper subgroup (see ). However, it may be possible to express $G$ as the union of the $G$-conjugates of two proper subgroups; if $H$ and $K$ are proper subgroups such that $$G = \bigcup_{g \in G}H^g \cup \bigcup_{g \in G}K^g,$$ then $G$ is said to be *$2$-coverable* and the pair $(H,K)$ is a *$2$-covering* for $G$. This notion has been widely studied in the context of finite simple groups. For instance, Bubboloni [@B] proves that ${\mathrm{A}}_n$ is $2$-coverable if and only if $5 {\leqslant}n {\leqslant}8$, and similarly ${\rm L}_{n}(q)$ is $2$-coverable if and only if $2 {\leqslant}n {\leqslant}4$ (see [@BL]). We refer the reader to [@BLW] and [@Pellegrini] for further results in this direction. The connection between $2$-coverable groups and the property in Theorem \[t:main1\] is transparent. Indeed, if $G$ is a transitive permutation group with point stabilizer $H$, then every derangement in $G$ is an $r$-element (for some fixed prime $r$) if and only if $(H,K)$ is a $2$-covering for $G$, where $K$ is a Sylow $r$-subgroup of $G$. Finally, some words on the organisation of this paper. In Section \[s:prel\] we record several preliminary results that we will need in the proofs of our main theorems. The proof of Theorem \[t:main1\] is given in Section \[s:red\], and the almost simple groups are handled in Section \[s:as\], where we prove Theorem \[t:main2\]. Finally, in Section \[s:affine\] we turn to affine groups and we establish Theorem \[t:main3\]. *Notation.* Our group-theoretic notation is standard, and we adopt the notation of Kleidman and Liebeck [@KL] for simple groups. For instance, $${\rm PSL}_{n}(q) = {\rm L}_{n}^{+}(q) = {\rm L}_{n}(q),\;\; {\rm PSU}_{n}(q) = {\rm L}_{n}^{-}(q) = {\rm U}_{n}(q).$$ If $G$ is a simple orthogonal group, then we write $G = {\rm P\Omega}_{n}^{{\epsilon}}(q)$, where ${\epsilon}=+$ (respectively $-$) if $n$ is even and $G$ has Witt defect $0$ (respectively $1$), and ${\epsilon}=\circ$ if $n$ is odd (in the latter case, we also write $G = \Omega_n(q)$). Following [@KL], we will sometimes refer to the *type* of a subgroup $H$, which provides an approximate description of the group-theoretic structure of $H$. For integers $a$ and $b$, we use $(a,b)$ to denote the greatest common divisor of $a$ and $b$. If $p$ is a prime number, then we write $a=a_p \cdot a_{p'}$, where $a_p$ is the largest power of $p$ dividing $a$. Finally, if $X$ is a finite set, then $\pi(X)$ denotes the set of prime divisors of $|X|$. *Acknowledgements.* This work was done while the second author held a position at the CRC 701 within the project C13 ‘The geometry and combinatorics of groups’, and he thanks B. Baumeister and G. Stroth for their assistance. Part of the paper was written during the second author’s visit to the School of Mathematics at the University of Bristol and he thanks the University of Bristol for its hospitality. Burness thanks R. Guralnick for helpful comments. Both authors thank an anonymous referee for suggesting several improvements to the paper, including a simplified proof of Proposition \[p:alt\] and a proof of Theorem \[c:prime\]. Preliminaries {#s:prel} ============= In this section we record several preliminary results that will be useful in the proofs of our main theorems. Let $H$ be a proper subgroup of a finite group $G$ and set $$\Delta_H(G) = G \setminus \bigcup_{g \in G}H^g.$$ Notice that if $G$ is a transitive permutation group with point stabilizer $H$, then $\Delta(G)=\Delta_H(G)$ is the set of derangements in $G$ (see ). It will be convenient to define the following property: $$\mbox{\emph{Every element in $\Delta_H(G)$ is an $r$-element for some fixed prime $r$.} \label{e:star} \tag{$\star$}}$$ \[lem:power order\] Let $H$ be a proper subgroup of a finite group $G$. If holds, then - $\pi(G) = \pi(H) \cup \{r\}$; and - ${{\mathbf {C}}}_G(x)$ is an $r$-group for every $x \in\Delta_H(G)$. If $s \in \pi(G) \setminus \pi(H)$, then $\Delta_H(G)$ contains an $s$-element, so (i) follows. Now consider (ii). Let $x\in\Delta_H(G)$ and assume $s\neq r$ is a prime divisor of $|{{\mathbf {C}}}_G(x)|$. Let $y\in{{\mathbf {C}}}_G(x)$ with $|y|=s$ and let $z=xy=yx$, so $z^s=x^s$ and ${\langle}x {\rangle}{\leqslant}{\langle}z {\rangle}$. Then $z \in \Delta_H(G)$, but this is incompatible with property . \[lem:normal\] Let $H$ be a proper subgroup of a finite group $G$, let $N$ be a normal subgroup of $G$ such that $G=NH$, and let $K$ be a proper subgroup of $N$ containing $H\cap N$. Then $\Delta_K(N)\subseteq \Delta_H(G)$. Let $x\in \Delta_K(N)$ and assume that $x\not\in\Delta_H(G)$. Then $x^g\in H$ for some $g\in G$. Since $g\in G=NH$, we may write $g=nh$ for some $n\in N$ and $h\in H$. Then $x^g=(x^n)^h\in H$ which implies that $x^n\in H^{h^{-1}}=H$. Since both $x$ and $n$ are in $N$, we deduce that $x^n \in H\cap N{\leqslant}K$, contradicting the fact that $x\in \Delta_K(N)$. \[r:pgraph\] The next result is a special case of [@GMS Lemma 3.3]. \[l:gms\] Let $G$ be a finite permutation group and let $N$ be a transitive normal subgroup of $G$ such that $G/N = {\langle}Ng {\rangle}$ is cyclic. Then $Ng \cap \Delta(G)$ is empty if and only if every element of $Ng$ has a unique fixed point. We will also need several number-theoretic lemmas. Given a positive integer $n$ we write $n_2$ for the largest power of $2$ dividing $n$. In addition, recall that $(a,b)$ denotes the greatest common divisor of the positive integers $a$ and $b$. The following result is well known. \[l:nt\] Let $q {\geqslant}2$ be an integer. For all integers $n,m {\geqslant}1$ we have $$\begin{aligned} (q^n-1,q^m-1) & = q^{(n,m)}-1 \\ (q^n-1,q^m+1) & = \begin{cases} q^{(n,m)}+1 & \mbox{if } 2m_2{\leqslant}n_2\\ (2,q-1) & \mbox{otherwise} \end{cases} \\ (q^n+1,q^m+1) & = \begin{cases} q^{(n,m)}+1 & \mbox{if } m_2 = n_2\\ (2,q-1) & \mbox{otherwise} \end{cases}\end{aligned}$$ Let $q=p^f$ be a prime power, let $e {\geqslant}2$ be an integer and let $r$ be a prime dividing $q^e-1$. We say that $r$ is a *primitive prime divisor* (ppd for short) of $q^e-1$ if $r$ does not divide $q^{i}-1$ for all $1 {\leqslant}i <e$. A classical theorem of Zsigmondy [@Zsig] states that if $e {\geqslant}3$ then $q^e-1$ has a primitive prime divisor unless $(q,e)=(2,6)$. Primitive prime divisors also exist when $e=2$, provided $q$ is not a Mersenne prime. Note that if $r$ is a ppd of $q^e-1$ then $r \equiv 1 {\allowbreak\mkern4mu({\operator@font mod}\,\,e)}$. Also note that if $n$ is a positive integer, then $r$ divides $q^n-1$ if and only if $e$ divides $n$. If a pdd of $q^e-1$ exists, then we will write $\ell_{e}(q)$ to denote the largest pdd of $q^e-1$. Note that $\ell_e(q)>e$. \[lem:primepower\] Let $r,s$ be primes, and let $m,n$ be positive integers. If $r^m+1=s^n$, then one of the following holds: - $(r,s,m,n) = (2,3,3,2)$; - $(r,n) = (2,1)$, $m$ is a $2$-power and $s=2^m+1$ is a Fermat prime; - $(s,m) = (2,1)$, $n$ is a prime and $r=2^n-1$ is a Mersenne prime. This is a straightforward application of Zsigmondy’s theorem [@Zsig]. For completeness, we will give the details. First assume that $m=1$, so $r=s^n-1$ is a prime. If $s$ is odd, then $r$ is even, so $r=2$ and $s^n=3$, which implies that $n=1$ and $s=3$. This case appears in (ii). Now assume $s=2$, so $r=2^n-1$ is prime. It follows that $n$ must also be a prime and thus $r$ is a Mersenne prime. This is (iii). For the remainder, we may assume that $m {\geqslant}2$. Notice that $r^{2m}-1=s^n(r^m-1)$. If $(m,r)= (3,2)$, then $s^n=2^3+1=3^2$ and thus $(s,n)=(3,2)$ as in (i). Now assume that $(m,r)\neq (3,2)$. By Zsigmondy’s theorem [@Zsig], the ppd $\ell_{2m}(r)$ exists and divides $r^{2m}-1=s^n(r^m-1)$, but not $r^m-1$, hence $s=\ell_{2m}(r)>2m{\geqslant}4$. Therefore $s {\geqslant}5$ is an odd prime and $r^m=s^n-1$ is even, so $r=2$. We now consider three cases. If $n=1$, then $s=r^m+1=2^m+1$ is an odd prime, which implies that $m$ is a $2$-power as in case (ii). Next assume that $n= 2$. Here $2^m=s^2-1=(s-1)(s+1)$ and thus $s-1=2^a$ and $s+1=2^b$ for some positive integers $a$ and $b$. Then $2^b-2^a=(s+1)-(s-1)=2$ and thus $2^{b-1}=2^{a-1}+1$, which implies that $(a,b)=(1,2)$, so $s=3$ and thus $m=3$. Therefore, $(r,s,m,n)=(2,3,3,2)$ as in case (i). Finally, let us assume that $n{\geqslant}3$. Now $2^m=s^n-1$ and Zsigmondy’s theorem implies that the ppd $\ell_{n}(s)>n{\geqslant}3$ exists and divides $2^m$, which is absurd. \[lem:numerical\] Let $q$ be a prime power and let $(a,{\epsilon}), (b,\delta) \in \mathbb{N} \times \{\pm 1\}$, where $b>a{\geqslant}2$ and $(a,{\epsilon}) \neq (2,-1)$. Let $N=(q^a+{\epsilon})(q^b+\delta)$. Then one of the following holds: - $N$ has two distinct prime divisors that do not divide $q^2-1$; - $(a,{\epsilon}) = (2,1)$, $(b,\delta) = (4,-1)$ and $q^2+1 = (2,q-1)r^e$ for some prime $r$ and positive integer $e$; - $q=3$, $(a,{\epsilon}) = (2,1)$ and $(b,\delta) = (3,1)$; - $q=2$, $(a,{\epsilon}) = (3,1)$ and $2^b+\delta$ is divisible by at most two distinct primes, one of which is $3$; - $q=2$, $a=3$ and $(b,\delta) = (6,-1)$. There are four cases to consider, according to the possibilities for the pair $({\epsilon},\delta)$. First assume that $({\epsilon},\delta)=(1,1)$. Suppose that neither $(a,q)$ nor $(b,q)$ is equal to $(3,2)$. Then the primitive prime divisors $\ell_{2a}(q)$ and $\ell_{2b}(q)$ exist, and they both divide $N$. Moreover, these primes are distinct since $2a<2b$, and neither of them divides $q^2-1$ since $2b>2a{\geqslant}4$. If $(a,q)=(3,2)$ then $b{\geqslant}4$, $N = 3^2(2^b+1)$ and either (i) or (iv) holds. If $(b,q)=(3,2)$, then $a=2$, $N=3^2\cdot 5$ and (iii) holds. Next suppose that $({\epsilon},\delta)=(-1,-1)$, so $a {\geqslant}3$. If neither $(a,q)$ nor $(b,q)$ is equal to $(6,2)$, then $N$ is divisible by the distinct primes $\ell_a(q)$ and $\ell_b(q)$, neither of which divide $q^2-1$. If $(a,q)=(6,2)$, then $N=3^2\cdot 7(2^b-1)$ is divisible by $7$ and $\ell_b(2)>b {\geqslant}7$. Finally, suppose that $(b,q)=(6,2)$, so $N=3^2\cdot 7(2^a-1)$ and $3{\leqslant}a{\leqslant}5$. It is easy to check that (i) holds if $a=4$ or $5$, and that (v) holds if $a=3$. Now assume that $({\epsilon},\delta)=(1,-1)$. If $(a,q)=(3,2)$ then (i) or (iv) holds, so we may assume that $(a,q)\neq (3,2)$. If $(b,q)=(6,2)$ then $N=3^2\cdot 7(2^a+1)$, $a \in \{2,4,5\}$ and (i) holds. In each of the remaining cases, the primitive prime divisors $\ell_{2a}(q)$ and $\ell_b(q)$ exist, and they divide $N$, but not $q^2-1$. Clearly, if $b\neq 2a$ then these two primes are distinct and (i) holds, so let us assume that $b=2a$, so $N=(q^a+1)^2(q^a-1)$. If $(a,q)=(6,2)$ then (i) holds. If $(a,q)\neq (6,2)$ and $a {\geqslant}3$ then we can take the primitive prime divisors $\ell_a(q)$ and $\ell_{2a}(q)$, so once again (i) holds. Finally, if $a=2$ and $b=4$ then $N = (q^2-1)(q^2+1)^2$ and either (i) or (ii) holds. Finally, let us assume that $({\epsilon},\delta)=(-1,1)$. Here we may assume that $a {\geqslant}3$. If $(a,q)\neq (6,2)$ then take $\ell_a(q)$ and $\ell_{2b}(q)$, otherwise $N=3^2\cdot 7(2^b+1)$ is divisible by $7$ and $\ell_{2b}(2)$. In both cases, (i) holds. \[l:ppower\] Let $q$ be a prime power and let $N$ be one of the integers in Table \[tab:int\], where ${\epsilon}=\pm 1$. Then $N$ is a prime power if and only if $({\epsilon},q)$ is one of the cases recorded in the second column of the table. $$\begin{array}{ll} \hline N & ({\epsilon},q) \\ \hline (q^6-1)/(7,q-{\epsilon}) & \mbox{none} \\ (q^6-1)/(q-{\epsilon})(6,q-{\epsilon}) & (-,2) \\ (q^5-{\epsilon})/(6,q-{\epsilon}) & (+,2), (+,3), (+,7), (-,2), (-,5) \\ (q^4-1)/(5,q-{\epsilon}) & \mbox{none}\\ (q^4-1)/(q-{\epsilon})(4,q-{\epsilon}) & (-,2), (-,3) \\ (q^3-{\epsilon})/(4,q-{\epsilon}) & (+,2), (+,3), (+,5), (-,2), (-,3) \\ (q^3-1)(q+1)/(5,q-{\epsilon}) & \mbox{none}\\ \hline \end{array}$$ This is entirely straightforward. For example, suppose that $N = (q^5-1)/(6,q-1)$. Let $d = (6,q-1)$ and suppose that $N = r^e$ for some prime number $r$ and positive integer $e$. Then $r = \ell_{5}(q)$ and $$(q-1)(q^4+q^3+q^2+q+1) = dr^e.$$ Since $r$ does not divide $q-1$, we must have $q-1=d$ and thus $q-1 \in \{1,2,3,6\}$. If $q=4$ then $N = 341 = 11\cdot 31$ is not a prime power, but one checks that $N$ is a prime power if $q \in \{2,3,7\}$. The other cases are very similar. We will also need the following result, which follows from a theorem of Nagell [@Nagell]. \[l:nagell\] Let $q = p^f$ be a prime power and let $r$ be a prime. - If $e$ is a positive integer such that $q^2+q+1 = r^e$, then $q \not\equiv 1 {\allowbreak\mkern4mu({\operator@font mod}\,\,3)}$ and $e=1$. - If $e$ is a positive integer such that $q^2+q+1 = 3r^e$, then $q \equiv 1 {\allowbreak\mkern4mu({\operator@font mod}\,\,3)}$ and $e \in \{1,2\}$. - If $q^2+q+1 = (3,q-1)r^e$ for some positive integer $e$, then either $(q,r,e) = (4,7,1)$, or $f=3^a$ for some integer $a {\geqslant}0$. Parts (i) and (ii) follow from [@Nagell]. For (iii), let $d=(3,q-1)$ and write $f=3^a m$ with $(3,m)=1$ and $a{\geqslant}0$. We may assume that $q \neq 4$. Seeking a contradiction, suppose that $m>1$. Notice that $$r^e=\frac{p^{3^{a+1}m}-1}{d(p^{3^am}-1)}.$$ Since $q \neq 4$, the ppd $\ell_{3f}(p)$ exists and divides $q^2+q+1$, so $r=\ell_{3f}(p)$. Let $s=\ell_{3^{a+1}}(p)$. Since $f=3^am$ is indivisible by $3^{a+1}$, it follows that $(s,q-1)=1$, so $s$ does not divide $d(q-1)$ and thus $s$ divides $r^e$, so $r=s$. But $m>1$, so $3f>3^{a+1}$ and thus $r \neq s$. This is a contradiction and the result follows. *By a theorem of van der Waall [@Waall], the Diophantine equation $x^2+x+1 = 3y^2$ has infinitely many integer solutions; the smallest nontrivial solution is $(x,y) = (313,181)$. Here $x$ and $y$ are both primes, and another solution in the primes is $(x,y) = (2288805793,1321442641)$.* A reduction theorem {#s:red} =================== The following theorem reduces the study of primitive permutation groups with property to almost simple and affine groups. \[thm:reduction\] Let $G{\leqslant}{\mathrm{Sym}}(\Omega)$ be a primitive permutation group with point stabilizer $H$. If holds, then either - $G$ is almost simple; or - $G=HN$ is an affine group with socle $N \cong (\mathbb{Z}_{r})^k$ for some integer $k {\geqslant}1$. Moreover, if (ii) holds and $|H|$ is indivisible by $r$, then $G$ is a Frobenius group with kernel $N$ and complement $H$. Let $N$ be a minimal normal subgroup of $G$, so $N\cong S_1\times S_2\times\cdots\times S_k$, where $S_i\cong S$ for some simple group $S$ and integer $k{\geqslant}1$. Then $G=HN$ and $N$ is transitive on $\Omega$. Let us assume that holds. First assume that $H\cap N=1$, so $N$ is regular and every nontrivial element in $N$ is a derangement. If $N$ is abelian, then we are in case (ii). Moreover, if $|H|$ is indivisible by $r$, then $N$ is a Sylow $r$-subgroup of $G$ and thus $\Delta(G)\subseteq N$. In this situation, [@BTV Lemma 4.1] implies that $G$ is a Frobenius group with kernel $N$ and complement $H$. Now, if $N$ is nonabelian then $S$ is a nonabelian simple group and thus $|S|$ is divisible by at least three distinct primes, whence $S$ (and thus $N$) contains derangements of distinct prime orders, which is incompatible with property . For the remainder, we may assume that $H\cap N$ is nontrivial. It follows that $N \cong S^k$, where $S$ is a nonabelian simple group and $k{\geqslant}1$. If $k=1,$ then $G$ is almost simple and (i) holds. Therefore, we may assume that $k{\geqslant}2$. Let $T{\leqslant}N$ be a maximal subgroup of $N$ containing $H\cap N$. By Lemma \[lem:normal\], we have $\Delta_T(N) \subseteq \Delta_H(G)$. Since $k{\geqslant}2$, there exist integers $i$ and $j$ such that $1{\leqslant}i<j {\leqslant}k$ and $L:=S_i\times S_j\not{\leqslant}T$. By relabelling the $S_{\ell}$, if necessary, we may assume that $L = S_1 \times S_2$. Now $L{\trianglelefteqslant}N$, so $N=TL$ and thus $$\label{e:deltak} \Delta_K(L)\subseteq \Delta_T(N) \subseteq\Delta_H(G)=\Delta(G),$$ where $K$ is a maximal subgroup of $L$ containing $L\cap T$. Therefore, every derangement of $L=S_1\times S_2$ on the right cosets $L/K$ is an $r$-element. By [@Thevenaz Lemma 1.3], there are essentially two possibilities for $K$; either $K$ is a diagonal subgroup of the form $\{(s,\phi(s)) \,:\, s\in S_1\}$ for some isomorphism $\phi:S_1 {\rightarrow}S_2$, or $K$ is a standard maximal subgroup, i.e., $K=S_1\times K_2$ or $K_1\times S_2$, where $K_i<S_i$ is maximal. In the diagonal case, every element in $L$ of the form $(s,1)$ with $1\neq s\in S_1$ is a derangement on $L/K$. Clearly, this situation cannot arise. Now assume $K$ is a standard maximal subgroup. Without loss of generality, we may assume that $K=K_1\times S_2$, where $K_1$ is maximal in $S_1$. Let $s\in S_1$ be a derangement on $S_1/K_1$ of prime power order, say $p^e$ for some prime $p$ and integer $e{\geqslant}1$ (such an element exists by the main theorem of [@FKS]). Since $|\pi(S)|{\geqslant}3$, choose $t\in S_2$ of prime order different from $p$. Then $(s,t) \in L$ is a derangement on $L/K$ of non-prime power order, so once again we have reached a contradiction. This completes the proof of Theorem \[t:main1\]. Almost simple groups {#s:as} ==================== In this section we prove Theorem \[t:main2\]. We fix the following notation. Let $r$ be a prime and let $G {\leqslant}{\mathrm{Sym}}(\Omega)$ be an almost simple primitive permutation group with socle $G_0$ and point stabilizer $H$. Set $H_0 = H \cap G_0$ and let $M$ be a maximal subgroup of $G_0$ containing $H_0$. As before, let $\Delta(G)$ be the set of derangements in $G$, and let ${\mathcal{E}}(G)$ be the set of orders of elements in $\Delta(G)$. By Lemma \[lem:normal\], we have $$\label{e:g0} \Delta_{M}(G_0) \subseteq \Delta_{H_0}(G_0) \subseteq \Delta_{H}(G) = \Delta(G).$$ Recall that if $X$ is a finite set, then $\pi(X)$ denotes the set of prime divisors of $|X|$. Let us assume that holds, so every derangement in $G$ is an $r$-element, for some fixed prime $r$. Clearly, every derangement of $G_0$ on $\Omega$ is also an $r$-element. Now, if $s \in \pi(G_0) \setminus \pi(M)$ then every nontrivial $s$-element in $G_0$ is a derangement, so $\pi(G_0)=\pi(M)$ or $\pi(M) \cup \{r\}$. In particular, if we set $\pi_0:=\pi(G_0)\setminus\pi(M)$, then $|\pi_0|{\leqslant}1$. Sporadic groups {#ss:sporadic} --------------- \[p:spor\] Theorem \[t:main2\] holds if $G_0$ is a sporadic group or the Tits group. First assume that $G_0$ is not the Monster. The maximal subgroups of $G_0$ are available in [@GAP], and it is easy to identify the cases $(G_0,M)$ with $|\pi_0| {\leqslant}1$. For the reader’s convenience, the cases that arise are listed in Table \[Tab:sporadic\]. We now consider each of these cases in turn. With the aid of [@GAP], we can compute the permutation character $\chi=1_M^{G_0}$, and we observe that $$\Delta_M(G_0) = \{x \in G_0 \,:\, \chi(x)=0\}.$$ In this way, we deduce that property holds if and only if $(G_0,M)=({\rm M}_{11},{\rm L}_2(11))$. Here $\pi(M) = \pi(G_0)$, $G = {\rm M}_{11}$, $H = {\rm L}_{2}(11)$ and ${\mathcal{E}}(G) = \{4,8\}$. This case is recorded in Table \[tab:main\]. Now assume $G = \mathbb{M}$ is the Monster. As noted in [@BW; @NW], there are $44$ conjugacy classes of known maximal subgroups of $\mathbb{M}$ (these subgroups are conveniently listed in [@BW Table 1], together with ${\rm L}_2(41)$). Moreover, it is known that any additional maximal subgroup of $\mathbb{M}$ is almost simple with socle ${\rm L}_2(13),{\rm U}_3(4),{\rm U}_3(8)$ or ${}^2{\rm B}_2(8)$. It is routine to check that $|\pi_0|{\geqslant}2$ in each of these cases. $$\begin{array}{lll} \hline G_0 & M & \pi_0 \\ \hline {\rm M}_{11}& {\mathrm{A}}_6.2_3,{\mathrm{S}}_5&11\\ & {\rm L}_2(11)&-\\ {\rm M}_{12}& {\rm M}_{11},{\rm L}_{2}(11)&-\\ & {\mathrm{A}}_6.2^2, 2\times{\mathrm{S}}_5&11\\ {\rm M}_{22}& {\mathrm{A}}_7, {\rm L}_3(4)&11\\ & {\rm L}_2(11)&7\\ {\rm M}_{23}& {\rm M}_{22}&23\\ {\rm M}_{24}& {\rm M}_{23}&-\\ & {\rm M}_{22}.2 &23\\ {\rm J}_{2}& {\rm L}_3(2).2, {\rm U}_3(3) &5\\ & 3.{\mathrm{A}}_6.2_2, 2^{1+4}{:}{\mathrm{A}}_5, {\mathrm{A}}_4\times{\mathrm{A}}_5,{\mathrm{A}}_5\times {\rm D}_{10},5^2{:}{\rm D}_{12},{\mathrm{A}}_5&7\\ {\rm J}_{3}& {\rm L}_2(16).2&19\\ & {\rm L}_2(19)&17\\ {\rm Co}_{1}& 3.{\rm Suz}.2&23\\ & {\rm Co}_2,{\rm Co}_3,2^{11}{:}{\rm M}_{24}&13\\ {\rm Co}_{2}& {\rm M}_{23}&-\\ & {\rm McL},{\rm HS}.2,{\rm U}_6(2).2,2^{10}{:}{\rm M}_{22}.2 &23\\ {\rm Co}_{3}& {\rm M}_{23}&-\\ & {\rm McL}.2,{\rm HS}&23\\ {\rm Fi}_{22}& 2.{\rm U}_6(2),2^{10}{:}{\rm M}_{22}&13\\ & \Omega_7(3)&11\\ {\rm Fi}'_{24}& {\rm Fi}_{23}&29\\ {\rm HS}& {\rm M}_{22}&-\\ & {\rm U}_3(5).2,{\rm L}_3(4).2_1,{\mathrm{S}}_8&11\\ &{\rm M}_{11}&7\\ {\rm McL}& {\rm M}_{22}&-\\ & {\rm U}_4(3),{\rm U}_3(5),{\rm L}_3(4).2_2, 2.{\mathrm{A}}_8, 2^4{:}{\mathrm{A}}_7&11\\ &{\rm M}_{11}&7\\ {\rm Suz}&{\rm G}_2(4)&11\\ {\rm He}& {\rm Sp}_4(4).2&7\\ &2^2.{\rm L}_3(4).{\mathrm{S}}_3, 3.{\mathrm{S}}_7&17\\ {\rm HN}&2.{\rm HS}.2,{\mathrm{A}}_{12}&19\\ {\rm O'N}&{\rm J}_1&31\\ {\rm Ru}&(2^2\times {}^2{\rm B}_2(8)){:}3 &29\\ &{\rm L}_2(29)&13\\ {}^2{\rm F}_4(2)'&{\rm L}_2(25)&-\\ &{\rm L}_3(3).2&5\\ &{\mathrm{A}}_6.2^2,5^2{:}4{\mathrm{A}}_4&13\\ \hline \end{array}$$ Alternating groups {#ss:alt} ------------------ \[p:alt\] Theorem \[t:main2\] holds if $G_0 = {\mathrm{A}}_n$ is an alternating group. If $n< 12$ then the result can be checked directly using [@GAP]; the only cases $(G,H)$ with property are the following: $$({\mathrm{A}}_6, 3^2{:}4), \, ({\mathrm{A}}_5, {\rm D}_{10}), \, ({\mathrm{A}}_5, {\mathrm{A}}_4), \, ({\mathrm{A}}_5, {\rm S}_{3}),$$ which are recorded in Table \[tab:main\] as $$({\rm L}_{2}(9),{\rm P}_{1}),\, ({\rm L}_{2}(4),{\rm D}_{10}),\, ({\rm L}_{2}(4),{\rm P}_{1}),\, ({\rm L}_{2}(4),{\rm D}_{6})$$ respectively (see Remark \[r:isom\]). For the remainder, we may assume that $n {\geqslant}12$. Seeking a contradiction, let us assume that there is a fixed prime $r$ such that every derangement in $G$ is an $r$-element. Let $s$ be a prime such that $n/2<s<n-2$ and let $x \in G_0$ be an $s$-cycle (such a prime exists by *Bertrand’s postulate*). Since ${{\mathbf {C}}}_G(x)$ is not an $r$-group, Lemma \[lem:power order\](ii) implies that $x$ is not a derangement and thus $H$ contains $s$-cycles. By applying a well known theorem of Jordan (see [@Wielandt Theorem 13.9]), we deduce that $H$ is either intransitive or imprimitive, and we can rule out the latter possibility since $s$ divides $|H|$. Therefore, $H$ is the stabilizer of a $k$-set for some $k$ with $1<k<n/2$. Suppose $n$ is even and let $x_i \in G_0$ be an element with cycles of length $i$ and $n-i$ for $i \in \{3,5,7\}$. Then at least two of the $x_i$ are derangements, so we have reached a contradiction. Now assume $n$ is odd. An $n$-cycle does not fix a $k$-set, so $n$ must be an $r$-power. Therefore, any element with cycles of length $(n-1)/2$, $(n-1)/2$ and $1$ must fix a $k$-set (since its order is not an $r$-power), so $k=1$ or $(n-1)/2$. It follows that any element with cycles of length $2,3$ and $n-5$ is a derangement, and this final contradiction completes the proof of the proposition. Exceptional groups {#ss:ex} ------------------ Now let us assume that $G_0$ is a simple exceptional group of Lie type over $\mathbb{F}_{q}$, where $q=p^f$ and $p$ is a prime. For $x \in G_0$, let $\mathcal{M}(x)$ be the set of maximal subgroups of $G_0$ containing $x$. We will write $\Phi_i$ for the $i$-th cyclotomic polynomial evaluated at $q$, so $q^n - 1 = \prod_{d|n}\Phi_d$. Recall that if $e {\geqslant}2$ and $q^e-1$ has a primitive prime divisor, then we use the notation $\ell_{e}(q)$ to denote the largest such divisor of $q^e-1$. \[p:ex\] Theorem \[t:main2\] holds if $G_0$ is a simple exceptional group of Lie type. Recall the notational set-up introduced at the beginning of Section \[s:as\]: $H$ is a point stabilizer in $G$, and $H_0 = H \cap G_0$. In view of , in order to show that does not hold we may assume that $G=G_0$. Seeking a contradiction, suppose that every derangement in $G$ is an $r$-element, for some fixed prime $r$. We will consider each possibility for $G$ in turn. *Case 1.* $G={}^2{\rm B}_2(q)$, with $q=2^{2m+1}$ and $m{\geqslant}1$. Let $\Phi_4'=q+\sqrt{2q}+1$ and $\Phi_4''=q-\sqrt{2q}+1$ (note that $\Phi_4'\Phi_4'' = q^2+1$). By inspecting [@BPS Table II], [@GM1 Table 6] and [@GM2 Table 1], we see that $G$ has two cyclic maximal tori $T_i={\langle}x_i{\rangle}$, $i=1,2$, of order $\Phi_4'$ and $\Phi_4''$, respectively, such that $|{{\mathbf {N}}}_{G}(T_i)/T_i|=4$, $(|x_1|,|x_2|)=1$ and ${\mathcal{M}}(x_i)=\{{{\mathbf {N}}}_{G}(T_i)\}$. Since no maximal subgroup of $G$ can contain conjugates of both $x_1$ and $x_2$, it follows that $x_i^{G}\cap H$ is empty for some $i=1,2$. Therefore, $x_i\in\Delta(G)$ and thus $|x_i|$ is a power of $r$. Let $j=3-i$. Then $|x_j|$ is indivisible by $r$, so $H$ contains a conjugate of $x_j$ and thus $H={{\mathbf {N}}}_{G}(T_j)$ is the only possibility (up to conjugacy). Now $G$ has a cyclic maximal torus of order $q-1$, so let $x \in G$ be an element of order $q-1{\geqslant}7$. Since $|H|$ is indivisible by $q-1$, it follows that $x \in \Delta(G)$. But $r$ does not divide $q-1$, so we have reached a contradiction. *Case 2.* $G = {}^2{\rm G}_2(q)$, with $q=3^{2m+1}$ and $m{\geqslant}1$. This is very similar to the previous case. Here we take two cyclic maximal tori $T_i={\langle}x_i{\rangle}$, $i=1,2$, of order $\Phi_{6}' = q+\sqrt{3q}+1$ and $\Phi_{6}''= q-\sqrt{3q}+1$, respectively, such that $|{{\mathbf {N}}}_{G}(T_i)/T_i|=6$, $(|x_1|,|x_2|)=1$ and ${\mathcal{M}}(x_i)=\{{{\mathbf {N}}}_{G}(T_i)\}$. Note that $\Phi_{6}'\Phi_{6}'' = q^2-q+1$. By arguing as in Case 1, we deduce that $|x_i|$ is a power of $r$ and $H={{\mathbf {N}}}_{G}(T_j)$ for some distinct $i,j$. Let $x \in G$ be an element of order $9$ (see part (2) in the main theorem of [@Ward], for example). Since $|H|$ is indivisible by $9$, it follows that $x$ is a derangement, but this is a contradiction since $r \neq 3$. *Case 3.* $G = {}^2{\rm F}_4(q)$, with $q=2^{2m+1}$ and $m {\geqslant}1$. Again, we proceed as in Case 1. Here $G$ has two cyclic maximal tori $T_i={\langle}x_i{\rangle}$, $i=1,2$, where $$\begin{aligned} |T_1| & = \Phi_{12}' = q^2 + \sqrt{2q^3}+q+\sqrt{2q}+1 \\ |T_2| & = \Phi_{12}'' = q^2 - \sqrt{2q^3}+q-\sqrt{2q}+1\end{aligned}$$ and $|{{\mathbf {N}}}_{G}(T_i)/T_i|=12$, $(|x_1|,|x_2|)=1$ and ${\mathcal{M}}(x_i)=\{{{\mathbf {N}}}_{G}(T_i)\}$. Note that $\Phi_{12}'\Phi_{12}'' = q^4-q^2+1$. As in Case 1, we see that $|x_i|$ is a power of $r$ and $H={{\mathbf {N}}}_{G}(T_j)$ for some distinct $i,j$. Let $x \in G$ be an element of order $\ell_4(q)$. Since $|H|$ is indivisible by $\ell_4(q)$, it follows that $x \in \Delta(G)$, but this is a contradiction since $r \neq \ell_4(q)$. *Case 4.* $G = {\rm E}_8(q)$. Again, we can proceed as in the previous cases, working with cyclic maximal tori $T_1,T_2$ and an element $x \in G$ of order $\ell_{24}(q)$, where $$\begin{aligned} |T_1| & =\Phi_{15} = q^8-q^7+q^5-q^4+q^3-q+1 \\ |T_2| & =\Phi_{30} = q^8+q^7-q^5-q^4-q^3+q+1\end{aligned}$$ and $|{{\mathbf {N}}}_{G}(T_i)/T_i|=30$, $i=1,2$. We omit the details (note that $\ell_{24}(q)\in \pi(G)\setminus \pi({{\mathbf {N}}}_{G}(T_i))$). *Case 5.* $G = {}^3{\rm D}_4(q)$. As indicated in [@GM1 Table 6], $G$ has a maximal torus $T={\langle}x{\rangle}$ of order $\Phi_{12} = q^4-q^2+1$ such that $|{{\mathbf {N}}}_{G}(T)/T|=4$ and ${\mathcal{M}}(x)=\{{{\mathbf {N}}}_{G}(T)\}$. Suppose that $x \not\in \Delta(G)$. Then $x^{G}\cap H$ is non-empty, and without loss of generality we may assume that $x \in H$ and thus $H={{\mathbf {N}}}_{G}(T)$. If $q=2$ then $|H|=52$ and $|\pi(G)\setminus\pi(H)|=2$, so we must have $q>2$. Let $y_i \in G$ ($i=1,2$) be elements of order $\ell_i:=\ell_{m_i}(q){\geqslant}5$, where $m_1=3$ and $m_2=6$. Since $|H|$ is indivisible by $\ell_1$ and $\ell_2$, it follows that $y_1,y_2 \in \Delta(G)$. But this is a contradiction since $\ell_1,\ell_2$ are distinct primes. Now assume that $x\in\Delta(G)$, so $|x|=\Phi_{12}$ is a power of $r$. If $q=2$ then $r=13$ and $H$ must contain elements of order $7,8,9,14,18,21$ and $28$, but no maximal subgroup of $G$ has this property (see [@ATLAS], for example). Therefore, $q>2$. Following [@GM2 p.698], let $y \in G$ be an element of order $\Phi_3$ such that $|{{\mathbf {C}}}_{G}(y)|$ divides $\Phi_3^2$ and $$\mathcal{M}(y)=\{G_2(q),{\rm PGL}_3(q),(\Phi_6\circ {{\mathrm {SL}}}_3(q)).2d,\Phi_3^2.{{\mathrm {SL}}}_2(3)\},$$ where $d = (3,\Phi_3)$. Now $(\Phi_{12},\Phi_3) = 1$, so $y \not\in \Delta(G)$ and thus we may assume that $H \in \mathcal{M}(y)$. Let $z \in G$ be an element of order $\Phi_1\Phi_2\Phi_6 = (q^2-1)(q^2-q+1)$. Then $|H|$ is indivisible by $|z|$, so $z \in \Delta(G)$. But this is a contradiction since $(\Phi_{12}, \Phi_1\Phi_2\Phi_6) = 1$. *Case 6.* $G = {}^2{\rm E}_6(q)$. Let $d=(3,q+1)$. As indicated in [@GM1 Table 6] and [@GM2 Table 1], $G$ has two cyclic maximal tori $T_i={\langle}x_i{\rangle}$, $i=1,2$, of order $\Phi_{18}/d$ and $\Phi_6\Phi_{12}/d$, respectively. Then $(|x_1|,|x_2|)=1$ and $${\mathcal{M}}(x_1)=\{ {{\mathrm {SU}}}_3(q^3).3 \},\;\; {\mathcal{M}}(x_2)=\left\{\begin{array}{ll} \{\Phi_6.{}^3{\rm D}_4(q).3/d\} & \mbox{if $q>2$} \\ \{\Phi_6.{}^3{\rm D}_4(2), {\rm F}_4(2), {\rm Fi}_{22}\} & \mbox{if $q=2$.} \end{array}\right.$$ No maximal subgroup of $G$ contains both $x_1$ and $x_2$ (see [@LPS Table 10.5]), so $x_i \in \Delta(G)$ for some $i$, and thus $|x_i|$ is a power of $r$. First assume that $q=2$, so $|x_1|=19$, $|x_2|=13$ and thus $r \in \{13,19\}$. If $r=13$, then $H$ contains a conjugate of $x_1$, so $H = {{\mathrm {SU}}}_3(8).3$ is the only option, but this is not possible since $|\pi(G) \setminus \pi(H)| = 4$. Similarly, if $r=19$ then $H \in {\mathcal{M}}(x_2)$ must contain elements of order $11,13$ and $17$, but it is easy to check that this is not the case. Now assume that $q>2$. Let $x \in G$ be an element of order $\ell_{10}(q)$. Both $|{{\mathrm {SU}}}_3(q^3).3|$ and $|\Phi_6.{}^3{\rm D}_4(q).3/d|$ are indivisible by $\ell_{10}(q)$, so $x \in \Delta(G)$. However, this is not possible since $\ell_{10}(q)$ and $|x_i|$ are coprime. *Case 7.* $G = {\rm G}_2(q)$, $q {\geqslant}3$. We can use [@GAP] to rule out the cases $q {\leqslant}5$, so we may assume that $q {\geqslant}7$. First assume that $q=7$. By inspecting [@GM1 Table 6] and [@GM2 Table 1], we see that $G$ has two cyclic maximal tori $T_i={\langle}x_i{\rangle}$, $i=1,2$, of order $\Phi_6 = 43$ and $\Phi_3 = 57$, respectively, with ${\mathcal{M}}(x_1)=\{{{\mathrm {SU}}}_3(7).2\}$ and ${\mathcal{M}}(x_2)=\{{{\mathrm {SL}}}_3(7).2\}$. From [@LPS Table 10.5], it follows that $x_i\in\Delta(G)$ for some $i$, so $H$ contains a conjugate of $x_j$, where $j=3-i$. Therefore, $H = {{\mathrm {SL}}}_3^{\epsilon}(7).2$ for some ${\epsilon}=\pm$. As noted in [@KS Table A.7], $G$ contains elements of order $7^2+7=56$ and $7^2+7+1=57$. Now ${{\mathrm {SU}}}_3(7).2$ contains no element of order $57$, and ${{\mathrm {SL}}}_3(7).2$ has no element of order $56$. Therefore, $G$ always contains a derangement of non-prime power order, which is a contradiction. For the remainder, we may assume that $q> 7$. We use the set-up in [@FMP Section 5.7]. Choose a $4$-tuple $(k_1,k_2,k_3,k_6)$ such that $(k_1,k_2)=1$, $k_i$ divides $\Phi_i$ for $i \in \{1,2\}$, $k_3=\Phi_3/(3,\Phi_3)$ and $k_6=\Phi_6/(3,\Phi_6)$. Note that the numbers $k_1,k_2,k_3$ and $k_6$ are pairwise coprime. Let $y_1\in G$ be an element of order $k_6$, and fix a regular semisimple element $y_2\in G$ of order $k_1$. Similarly, fix $z_i\in G$, $i=1,2$, where $|z_1|=k_3$ and $z_2$ is a regular semisimple element of order $k_2$. From [@LPS Table 10.5], it follows that either $y_1$ or $z_1$ is a derangement. Suppose that $y_1\in\Delta(G)$. Then $H$ contains a conjugate of $z_1$, so [@FMP Lemma 5.27] implies that $H = {{\mathrm {SL}}}_3(q).2$ is the only possibility. If $H$ also contains a conjugate of $z_2$, then $H=G$ by [@FMP Corollary 5.28], a contradiction. Therefore $z_2\in\Delta(G)$, but once again we reach a contradiction since $(k_2,k_6)=1$. An entirely similar argument applies if $z_1\in\Delta(G)$. *Case 8.* $G \in \{{\rm E}_{6}(q), {\rm E}_7(q)\}$. First assume that $G = {\rm E}_{7}(q)$. Let $d=(2,q-1)$. As in [@FMP Section 5.2], let $y_1,y_2\in G$ be elements of order $\Phi_{18}$ and $\Phi_2\Phi_{14}/d = (q^7+1)/d$, respectively, and let $z_1,z_2\in G$ be elements of order $\Phi_9$ and $\Phi_1\Phi_7/d = (q^7-1)/d$, respectively. From [@FMP Corollary 5.6], we deduce that $y_i,z_j\in\Delta(G)$ for some $i,j \in \{1,2\}$. However, it is easy to check that $(|y_i|,|z_j|) = 1$ for all $i,j$, so this is a contradiction. The case $G = {\rm E}_6(q)$ is entirely similar, using [@FMP Corollary 5.11] and elements $y_i,z_i \in G$ with $|y_1| = \Phi_9/d$, $|y_2| = \Phi_4$, $|z_1| = \Phi_3\Phi_{12}$ and $|z_2| = \Phi_5$ (where $d = (3,q-1)$). *Case 9.* $G = {\rm F}_4(q)$. For $q>2$, we can proceed as in Case 8, using the information in [@FMP Section 5.5]. The reader can check the details. Now assume that $q=2$. By inspecting [@GM1 Table 6] and [@GM2 Table 1], we see that $G$ has two cyclic maximal tori $T_i={\langle}x_i{\rangle}$, $i=1,2$, of order $\Phi_{12} = 13$ and $\Phi_8=17$, respectively, such that ${\mathcal{M}}(x_1)=\{{}^3{\rm D}_4(2).3,{}^2{\rm F}_4(2), {\rm L}_4(3).2_2\}$ and ${\mathcal{M}}(x_2)=\{{\rm Sp}_8(2)\}$. Therefore, $r \in \{13,17\}$. If $r=13$, then $H$ contains a conjugate of $x_2$, so $H = {\rm Sp}_{8}(2)$. However, [@ATLAS] indicates that $G$ has an element of order $28$, but ${\rm Sp}_8(2)$ does not, so this case is ruled out. Therefore, $r=17$ and $H$ contains a conjugate of $x_1$, so $H\in {\mathcal{M}}(x_1)$. However, in each case one can check that $H$ does not contain an element of order $30$, but $G$ does. This final contradiction eliminates the case $G = {\rm F}_{4}(q)$. This completes the proof of Proposition \[p:ex\]. Classical groups {#ss:class} ---------------- In order to complete the proof of Theorem \[t:main2\], we may assume that $G_0$ is a classical group over $\mathbb{F}_{q}$. Due to the existence of certain exceptional isomorphisms involving low-dimensional classical groups (see [@KL Proposition 2.9.1], for example), and in view of our earlier work in Sections \[ss:sporadic\], \[ss:alt\] and \[ss:ex\], we may assume that $G_0$ is one of the groups listed in Table \[tab:gps\]. $$\begin{array}{ll} \hline G_0 & \mbox{Conditions} \\ \hline {\rm L}_{n}(q) & n {\geqslant}2,\, (n,q) \neq (2,2), (2,3), (2,4), (2,5), (2,9), (3,2), (4,2) \\ {\rm U}_{n}(q) & n {\geqslant}3,\, (n,q) \neq (3,2) \\ {\rm PSp}_{n}(q) & \mbox{$n {\geqslant}4$ even, $(n,q) \neq (4,2), (4,3)$} \\ {\rm P\Omega}_{n}^{{\epsilon}}(q) & n {\geqslant}7 \\ \hline \end{array}$$ We will focus initially on the low-dimensional classical groups with socle ${\rm L}_{2}(q)$ and ${\rm L}_{3}^{{\epsilon}}(q)$, which require special attention. As before, if a primitive prime divisor of $q^e-1$ exists, then $\ell_e(q)$ denotes the largest such prime divisor (as noted in Section \[s:prel\], if $e {\geqslant}2$, then $\ell_e(q)$ exists unless $(q,e) = (2,6)$, or $e=2$ and $q$ is a Mersenne prime). \[lem:L2even\] Theorem \[t:main2\] holds if $G = {\rm L}_2(q)$ and $q$ is even. Write $q=2^f$, where $f {\geqslant}3$ (since ${\rm L}_{2}(4) \cong {\mathrm{A}}_5$, we may assume that $f {\geqslant}3$). The maximal subgroups of $G$ were originally classified by Dickson [@Dickson] (also see [@BHR Tables 8.1 and 8.2]); the possibilities for $H$ are as follows: - $H = (\mathbb{Z}_{2})^f{:}\mathbb{Z}_{q-1} = {\rm P}_1$ is a maximal parabolic subgroup of $G$; - $H = {\rm D}_{2(q\pm 1)}$; - $H = {\rm L}_{2}(q_0)$ with $q=q_0^e$, where $e$ is a prime and $q_0 \neq 2$. The case $f=3$ can be handled using [@GAP], and we find that holds if and only if $(H,r,{\mathcal{E}}(G))$ is one of the following (recall that ${\mathcal{E}}(G)$ denotes the set of orders of derangements in $G$): $$({\rm P}_1, 3,\{3,9\}),\; ({\rm D}_{18}, 7, \{7\}),\; ({\rm D}_{14}, 3, \{3,9\}).$$ For the remainder, we may assume that $f {\geqslant}4$. Note that a Sylow $2$-subgroup of $G$ is self-centralizing and elementary abelian. In particular, if $x \in G$ then either $|x|=2$, or $|x|$ divides $q \pm 1$. Also note that $G$ contains elements of order $q\pm 1$, and it has a unique class of involutions. *Case 1.* $H = {\rm P}_1$. We claim that holds if and only if $r=q+1$ is a Fermat prime. To see this, first observe that $|G:H|=q+1$ and $|H| = q(q-1)$ are relatively prime, so any element $x \in G$ of order $q+1$ is a derangement. Therefore, if holds then $q+1=r^e$ for some $e{\geqslant}1$, and thus Lemma \[lem:primepower\] implies that $f$ is a $2$-power and $e=1$ (so $r=q+1$ is a Fermat prime). For the converse, suppose that $q+1$ is a Fermat prime. We need to show that every derangement in $G$ has order $r=q+1$. Let $y \in \Delta(G)$, so $|y|$ divides $2$ or $q\pm 1$. But $q+1=r$ is a prime, so either $|y| \in \{2,r\}$ or $|y|$ divides $q-1$. Every involution has fixed points since $G$ has a unique class of involutions, so $|y|>2$. If $|y|$ divides $q-1$, then $y$ belongs to a maximal torus that is $G$-conjugate to the subgroup $\mathbb{Z}_{q-1}<H$. Again, this implies that $y$ has fixed points. Therefore, $|y|=r$ is the only possibility and the result follows. *Case 2.* $H = {\rm D}_{2(q\pm 1)}$. The case $H = {\rm D}_{2(q-1)}$ is identical to the previous one, and the same conclusion holds. A very similar argument also applies if $H = {\rm D}_{2(q+1)}$. Here any element of order $q-1$ is a derangement and by applying Lemma \[lem:primepower\] we deduce that holds if and only if $r=q-1$ is a Mersenne prime. *Case 3.* $H = {\rm L}_{2}(q_0)$, where $q=q_0^e$, $e$ prime, $q_0 \neq 2$. Finally, observe that subfield subgroups are easily eliminated since elements of order $q \pm 1$ are derangements. \[lem:L2ext\] Theorem \[t:main2\] holds if $G_0 = {\rm L}_2(q)$ and $q$ is even. As before, write $q=2^f$, where $f {\geqslant}3$. In view of Lemma \[lem:L2even\], we may assume that $$G = G_0. {\langle}\phi {\rangle}{\leqslant}{\rm \Gamma L}_{2}(q) = {\rm Aut}(G_0),$$ where $\phi$ is a nontrivial field automorphism of $G_0$, so the order of $\phi$ divides $f$. The case $f=3$ can be handled directly, using [@GAP] for example. Here $G = {\rm \Gamma L}_{2}(8)$ and we find that holds if and only if $(H,r,{\mathcal{E}}(G))$ is one of the following: $$({{\mathbf {N}}}_{G}({\rm P}_1), 3,\{3,9\}),\; ({{\mathbf {N}}}_{G}({\rm D}_{18}), 7, \{7\}),\; ({{\mathbf {N}}}_{G}({\rm D}_{14}), 3, \{3,9\}).$$ For the remainder, we may assume that $f {\geqslant}4$. Since $G_0 \not{\leqslant}H$, we have $G=G_0H$. Set $H_0 = H \cap G_0$ and note that $H_0$ is a maximal subgroup of $G_0$ (see [@BHR Table 8.1]). As in , we have $\Delta_{H_0}(G_0) \subseteq \Delta_H(G)$, whence Lemma \[lem:L2even\] implies that holds only if one of the following holds: - $H_0={\rm P}_1$, $r=q+1$ is a Fermat prime; - $H_0={\rm D}_{2(q+1)}$, $r=q-1$ is a Mersenne prime; - $H_0={\rm D}_{2(q-1)}$, $r=q+1$ is a Fermat prime. We consider each of these cases in turn. *Case 1.* $H_0={\rm D}_{2(q+1)}$, $r=q-1$ is a Mersenne prime. Here $f {\geqslant}5$ is a prime, so $G = {\rm \Gamma L}_{2}(q) = G_0. {\langle}\phi {\rangle}$ and $H=H_0.{\langle}\phi {\rangle}$ is the only possibility, where $\phi$ has order $f$. Note that $${{\mathbf {C}}}_G(\phi) = {\rm L}_{2}(2) \times {\langle}\phi {\rangle}\cong {\mathrm{S}}_3 \times \mathbb{Z}_{f},$$ so if $x\in G$ then either $|x|\in \{2,r,f,2f,3f\}$, or $|x|$ divides $q+1$. We claim that ${\mathcal{E}}(G)=\{r\}$. Note that ${\langle}\phi {\rangle}$ is a Sylow $f$-subgroup of $G$. Let $y \in G$ be a nontrivial element. If $|y| \in \{2,f\}$, or if $|y|$ divides $q+1$, then $y$ is conjugate to an element of $H$ and thus $y$ has fixed points. Next suppose that $|y|=kf$ and $k \in \{2,3\}$. Then $|y^k|=f$ and thus $y^k$ is $G$-conjugate to $\phi^i$ for some $1 {\leqslant}i<f$. Without loss of generality, we may assume that $y^k=\phi$, so $y \in{{\mathbf {C}}}_G(\phi)$. Since $|H|=2(q+1)f$, $H$ has a Sylow $2$-group $R={\langle}u{\rangle}\cong \mathbb{Z}_{2}$ and a normal $2$-complement $V{\langle}\phi {\rangle}$ of order $(q+1)f$, where $V \cong \mathbb{Z}_{q+1}$. Since $\phi$ normalizes $H_0=VR$, we deduce that $\phi$ centralizes $R$. Now $q+1$ is divisible by $3$, so $V$ has a unique subgroup of order $3$, say ${\langle}x{\rangle}$. Then the involution $u$ inverts $x$, and $\phi$ centralizes $x$ since $|\phi|=f {\geqslant}5$ is odd. Thus ${\mathrm{S}}_3\cong {\langle}u,x {\rangle}{\leqslant}{{\mathbf {C}}}_{G_0}(\phi)$, which implies that ${{\mathbf {C}}}_G(\phi)={\langle}u,x {\rangle}\times {\langle}\phi {\rangle}{\leqslant}H$. Therefore, $y \in H$. We conclude that every derangement in $G$ has order $r$, as required. In the two remaining cases, $r=q+1$ is a Fermat prime and $f=2^m$ for some integer $m {\geqslant}2$. In both cases, we claim that does not hold. In order to see this, we may assume that the index of $G_0$ in $G$ is a prime number, which in this case implies that $|G:G_0|=2$, so $G = G_0.{\langle}\phi {\rangle}$ and $\phi$ is an involutory field automorphism of $G_0$. Indeed, if $G_0 {\trianglelefteqslant}G_1{\trianglelefteqslant}G$ then $G=HG_1$ and Lemma \[lem:normal\] implies that $\Delta_L(G_1)\subseteq\Delta(G)$ for any subgroup $L$ of $G_1$ containing $G_1\cap H$. *Case 2.* $H_0={\rm D}_{2(q-1)}$, $r=q+1$ is a Fermat prime. By the above comments, we may assume that $G = G_0.{\langle}\phi {\rangle}$ and $H={\rm D}_{2(q-1)}.{\langle}\phi {\rangle}$, where $\phi$ has order $2$. Note that ${{\mathbf {C}}}_G(\phi)={\rm L}_2(2^{f/2}) \times {\langle}\phi{\rangle}$. Since ${{\mathbf {C}}}_G(\phi)$ does not contain a Sylow $2$-subgroup of $G$, we deduce that the Sylow $2$-subgroups of $G$ are nonabelian. Therefore $G$ contains an element $z$ of order $4$. However, the Sylow $2$-subgroups of $H$ are isomorphic to $C_2 \times C_2$, so $z\in\Delta(G)$. We conclude that $G$ contains derangements of order $r$ and $4$, so does not hold. *Case 3.* $H_0={\rm P}_1$, $r=q+1$ is a Fermat prime. Finally, let us assume that $H = {{\mathbf {N}}}_G({\rm P}_1) = {\rm P}_{1}.{\langle}\phi {\rangle}= H_0.{\langle}\phi {\rangle}$, where $|\phi|=2$. As above, we have ${{\mathbf {C}}}_G(\phi)={\rm L}_2(2^{f/2})\times {\langle}\phi{\rangle}$, so ${{\mathbf {C}}}_G(\phi)$ contains an element of order $2(q_0+1)$, where $q_0 = 2^{f/2}$. We claim that $H$ does not contain such an element. Seeking a contradiction, suppose $x \in H$ has order $2(q_0+1)$. Since $H=H_0\cup H_0\phi$ and $H_0={\rm P}_1$ has no element of order $2(q_0+1)$, we deduce that $x\in H_0\phi$ and we may write $x=u\phi$ with $u\in H_0$. In terms of matrices (and a suitable basis for the natural ${\rm L}_{2}(q)$-module), we have $$u=\left(\begin{array}{cc}\lambda & a \\ 0 & \lambda^{-1} \end{array}\right)$$ where $\lambda,a\in{\mathbb{F}}_{q}$ and $\lambda\neq 0$. Then $x^2=(u\phi)(u\phi)=uu^\phi$ has order $q_0+1$. We may assume that $\phi$ is the standard field automorphism of order $2$ with respect to this basis, so $$x^2=uu^\phi=\left(\begin{array}{cc}\lambda & a \\ 0 & \lambda^{-1} \end{array}\right)\left(\begin{array}{cc}\lambda^{q_0} & a^{q_0} \\ 0 & \lambda^{-q_0} \end{array}\right)=\left(\begin{array}{cc}\lambda^{1+q_0} & b \\ 0 & \lambda^{-1-q_0} \end{array}\right)$$ with $b=\lambda^{q_0}+a\lambda^{-q_0}$. Since $x^2$ has order $q_0+1$ we deduce that $\lambda^{2(q_0+1)}=1$, which implies that $\lambda^{q_0+1}=1$ since ${\mathbb{F}}_q$ has characteristic $2$. Therefore $$x^2=\left(\begin{array}{cc}1 & b \\ 0 & 1 \end{array}\right)$$ has order $q_0+1$, which is absurd. This justifies the claim, and we deduce that $\Delta(G)$ contains elements of order $2(q_0+1)$. In particular, does not hold. \[lem:L2odd\] Theorem \[t:main2\] holds if $G_0 = {\rm L}_2(q)$ and $q$ is odd. Write $q=p^f$, where $p$ is an odd prime. In view of the isomorphisms ${\rm L}_{2}(5) \cong {\mathrm{A}}_5$ and ${\rm L}_{2}(9) \cong {\mathrm{A}}_6$, we may assume that $q {\geqslant}7$ and $q \neq 9$. The case $q=7$ can be checked directly using , and we find that holds if and only if $(G,H,r,{\mathcal{E}}(G))$ is one of the following: $$({\rm L}_{2}(7), {\rm P}_{1}, 2,\{2,4\}),\; ({\rm L}_{2}(7), {\mathrm{S}}_4, 7,\{7\}),\; ({\rm PGL}_{2}(7), {{\mathbf {N}}}_{G}({\rm P}_{1}), 2,\{2,4,8\}).$$ For the remainder, we may assume that $q {\geqslant}11$. *Case 1.* $G=G_0$. First assume that $G = {\rm L}_{2}(q)$. The maximal subgroups of $G$ are well known (see [@BHR Tables 8.1 and 8.2]); the possibilities for $H$ are as follows: - $H = (\mathbb{Z}_{p})^f{:}\mathbb{Z}_{(q-1)/2} = {\rm P}_1$ is a maximal parabolic subgroup of $G$; - $H = {\rm D}_{q-{\epsilon}}$, where $q {\geqslant}13$ if ${\epsilon}=1$; - $H = {\rm L}_{2}(q_0)$, where $q=q_0^e$ for some odd prime $e$; - $H = {\rm PGL}_{2}(q_0)$, where $q=q_0^2$; - $H={\mathrm{A}}_5$, where $q \equiv \pm 1 {\allowbreak\mkern4mu({\operator@font mod}\,\,10)}$ and either $q=p$, or $q=p^2$ and $p \equiv \pm 3 {\allowbreak\mkern4mu({\operator@font mod}\,\,10)}$; - $H={\mathrm{A}}_4$, where $q =p \equiv \pm 3 {\allowbreak\mkern4mu({\operator@font mod}\,\,8)}$ and $q \not\equiv \pm 1 {\allowbreak\mkern4mu({\operator@font mod}\,\,10)}$; - $H={\mathrm{S}}_4$, where $q=p \equiv \pm 1 {\allowbreak\mkern4mu({\operator@font mod}\,\,8)}$. Note that $G$ contains elements of order $(q \pm 1)/2$, and a unique conjugacy class of involutions. If $H$ is a subfield subgroup (as in (c) or (d) above), then it is clear that any element of order $(q \pm 1)/2$ is a derangement, so property does not hold in this situation. Similarly, it is straightforward to handle the cases $H \in \{{\mathrm{A}}_5, {\mathrm{A}}_4, {\mathrm{S}}_4\}$. For example, suppose $H = {\mathrm{A}}_5$, so $q \equiv \pm 1 {\allowbreak\mkern4mu({\operator@font mod}\,\,10)}$ and either $q=p$, or $q=p^2$ and $p \equiv \pm 3 {\allowbreak\mkern4mu({\operator@font mod}\,\,10)}$. Note that every nontrivial element of $H$ has order $2,3$ or $5$. If $q {\geqslant}19$ then any element of order $(q \pm 1)/2$ is a derangement; if $q=11$, then elements of order $6$ are derangements. The cases $H = {\mathrm{A}}_4$ and ${\mathrm{S}}_4$ are just as easy. If $H = {\rm D}_{q - 1}$ then any element in $G$ of order $p$ or $(q+1)/2$ is a derangement, and the dihedral groups of order $q+1$ can be eliminated in a similar fashion. Finally, let us assume that $H = {\rm P}_{1} = (\mathbb{Z}_{p})^f{:}\mathbb{Z}_{(q-1)/2}$, so $|H| = q(q-1)/2$ and $|G:H|=q+1$. We claim that holds if and only if $q=2r^e-1$ for some positive integer $e$. First observe that any element of order $(q+1)/2$ is a derangement, so if holds then $q=2r^e-1$ for some $e \in \mathbb{N}$. For the converse, suppose that $q=2r^e-1$. We claim that $${\mathcal{E}}(G) = \{r^i \,:\, 1 {\leqslant}i {\leqslant}e\}.$$ Since $|H|$ is indivisible by $r$, the inclusion $\{r^i \,:\, 1 {\leqslant}i {\leqslant}e\} \subseteq {\mathcal{E}}(G)$ is clear. To see that equality holds, let $y \in G$ be a nontrivial element, and suppose that $|y|$ is divisible by a prime $s \neq r$. Since a Sylow $p$-subgroup of $G$ is self-centralizing, it follows that either $|y| = p$, or $|y|=2$ and $r$ is odd, or $|y|$ is a divisor of $(q-1)/2$. In the first two cases, it is clear that $y$ has fixed points, so let us assume that $|y|$ divides $(q-1)/2$. Then $y$ is conjugate to an element of the maximal torus $\mathbb{Z}_{(q-1)/2} < H$, so once again $y$ has fixed points. This justifies the claim. *Case 2.* $G \neq G_0$. To complete the proof of the lemma, we may assume that $G \neq G_0$, $q {\geqslant}11$ and $H_0 = H \cap G_0 = {\rm P}_{1}$, in which case holds only if $q=2r^e-1$ for some positive integer $e$ (note that $H \cap G_0$ is a maximal subgroup of $G_0$). There are several possibilities for $G$. First assume that $G = {\rm PGL}_{2}(q)$, so $H = (\mathbb{Z}_{p})^f{:}{\mathbb{Z}}_{q-1}$. We claim that holds if and only if $r=2$ and $q=2^{e+1}-1$ is a Mersenne prime. As above, any element of order $(q+1)/2$ is a derangement. Now $G$ also contains elements of order $q+1$, and they are also derangements. Therefore, if holds then $r=2$ is the only possibility, so $p^f+1 = 2^{e+1}$ and Lemma \[lem:primepower\] implies that $q=p=2^{e+1}-1$ is a Mersenne prime. For the converse, suppose that $q=p=2^{e+1}-1$ is a Mersenne prime. We claim that $${\mathcal{E}}(G) = \{2^i \,:\, 1 {\leqslant}i {\leqslant}e+1\}.$$ As above, any involution in $G_0$ is a derangement, and so is any element in $G$ of order $2^i$ with $1< i {\leqslant}e+1$ since $|H|_2 = 2$, hence $\{2^i \,:\, 1 {\leqslant}i {\leqslant}e+1\} \subseteq {\mathcal{E}}(G)$. To see that equality holds, suppose that $y \in G$ has order divisible by an odd prime. Then either $|y|=p$, or $y$ is conjugate to an element of the maximal torus $\mathbb{Z}_{q-1} < H$; in both cases, $y$ has fixed points. The result follows. To complete the proof of the lemma, we may assume that $G = G_0.{\langle}\phi {\rangle}$ or $G_0.{\langle}\delta\phi {\rangle}$, where $\phi$ is a nontrivial field automorphism of $G_0$ of order $e$ (so $e$ divides $f$) and $\delta = {\rm diag}(\omega_1,\omega_2) \in {\rm PGL}_{2}(q)$ (modulo scalars) is a diagonal automorphism of $G_0$. Recall that $(q+1)/2 = r^e$ for some prime $r$ and positive integer $e$. Our goal is to show that does not hold. First observe that $r$ is odd. Indeed, if $r=2$ then $p^f+1 = 2^{e+1}$ and thus Lemma \[lem:primepower\] implies that $f=1$, which is false. Next we claim that $f$ is a $2$-power. To see this, first assume that $f$ is odd and $p=2^t-1$ is a Mersenne prime. Then $r^e=(p^f+1)/2$ is divisible by $(p+1)/2=2^{t-1}$, but $r$ is odd so this is not possible. For the general case, suppose that $f=2^am$ where $a {\geqslant}0$ and $m>1$ is odd (and we may assume that $a>0$ if $p$ is a Mersenne prime). We now proceed as in the proof of Lemma \[l:nagell\](iii). We have $$r^e = \frac{q^2-1}{2(q-1)} = \frac{p^{2^{a+1}m}-1}{2(p^{2^{a}m}-1)}$$ and thus $r = \ell_{2f}(p)$. Set $s=\ell_{2^{a+1}}(p)$ (note that $s$ exists since $a>0$ if $p$ is a Mersenne prime). Now $f = 2^am$ is indivisible by $2^{a+1}$, so $(s,q-1)=1$ and thus $s$ does not divide $2(q-1)$. Therefore, $r=s$ is the only possibility, but this is a contradiction since $2f=2^{a+1}m>2^{a+1}$. This justifies the claim. Therefore, in order to show that does not hold, we may assume that $|G:G_0|=2$. Write $G = G_0 \cup G_0\gamma$. If we identify $\Omega$ with the set of $1$-dimensional subspaces of the natural ${\rm L}_{2}(q)$-module, then $\phi$ and $\delta\phi$ fix the $1$-spaces ${\langle}(1,0) {\rangle}$ and ${\langle}(0,1) {\rangle}$. Therefore, Lemma \[l:gms\] implies that the coset $G_0\gamma$ contains derangements. But every element in this coset has even order, which is incompatible with property . To summarize, we have now established the following result. (Note that the case appearing in the final row of Table \[tab:l2\] is recorded as $(G,H) = ({\rm L}_{3}(2), {\rm P}_{1})$ in Table \[tab:main\].) \[p:l2\] Let $G$ be a finite almost simple primitive permutation group with point stabilizer $H$ and socle ${\rm L}_{2}(q)$, where $q {\geqslant}4$ and $q \neq 5$. Then holds if and only if $(G,H,r)$ is one of the cases in Table \[tab:l2\]. $$\begin{array}{lllll} \hline G & H & r & {\mathcal{E}}(G) & \mbox{Conditions} \\ \hline {\rm \Gamma L}_2(q) & {{\mathbf {N}}}_{G}({\rm D}_{2(q+1)}) & r & r & \mbox{$r=q-1$ Mersenne prime} \\ {\rm \Gamma L}_{2}(8) & {{\mathbf {N}}}_{G}({\rm P}_1), {{\mathbf {N}}}_{G}({\rm D}_{14}) & 3 & 3,9 & \\ {\rm PGL}_{2}(q) & {{\mathbf {N}}}_{G}({\rm P}_{1}) & 2 & 2^i, \, 1 {\leqslant}i {\leqslant}e+1 & \mbox{$q=2^{e+1}-1$ Mersenne prime} \\ {\rm L}_2(q) & {\rm P}_1 & r & r^i, \, 1 {\leqslant}i {\leqslant}e & q=2r^e-1 \\ & {\rm P}_1, {\rm D}_{2(q-1)} & r & r & \mbox{$r=q+1$ Fermat prime} \\ & {\rm D}_{2(q+1)} & r & r & \mbox{$r=q-1$ Mersenne prime} \\ {\rm L}_{2}(9) & {\rm P}_{1} & 5 & 5 & \\ {\rm L}_{2}(8) & {\rm P}_{1}, {\rm D}_{14} & 3 & 3,9 & \\ {\rm L}_{2}(7) & {\mathrm{S}}_4 & 7 & 7 & \\ \hline \end{array}$$ \[lem:L3\] Theorem \[t:main2\] holds if $G_0 = {\rm L}_3(q)$. Set $d=(3,q-1)$ and note that $G_0$ contains elements of order $(q^2+q+1)/d$ and $(q^2-1)/d$. We may assume that $q {\geqslant}3$ since ${\rm L}_{3}(2) \cong {\rm L}_{2}(7)$. If $3 {\leqslant}q {\leqslant}7$, then we can use to verify the desired result; we find that holds if and only if $G = {\rm L}_{3}(q)$, $H \in \{{\rm P}_{1},{\rm P}_{2}\}$ and $r = (q^2+q+1)/d$, in which case ${\mathcal{E}}(G) = \{r\}$ (note that $(q^2+q+1)/d$ is a prime number for all $q \in \{3,4,5,7\}$). For the remainder, we will assume that $q {\geqslant}8$. In particular, note that $(q^2-1)/d$ is not a prime power (indeed, it is easy to check that $(q^2-1)/d$ is a prime-power if and only if $q=3$ or $7$). *Case 1.* $G=G_0$. First assume that $G = {\rm L}_{3}(q)$. The possibilities for $H$ are given in [@BHR Tables 8.3 and 8.4]. We can immediately eliminate any subgroup $H$ that does not contain an element of order $(q^2-1)/d$, so this implies that $H$ is either a maximal parabolic subgroup, or $H = {\rm SO}_{3}(q)$ (with $q$ odd). Suppose that $H$ is a maximal parabolic subgroup. Without loss of generality, we may assume that $H = {\rm P}_{1}$ (the actions of $G$ on $1$-spaces and $2$-spaces are permutation isomorphic), so $|H| = q^3(q-1)(q^2-1)/d$. We claim that $G$ has property if and only if one of the following holds: $$\label{e:ab} \begin{array}{ll} \mbox{(a)} & \mbox{$d=1$ and $q^2+q+1 = r$; or} \\ \mbox{(b)} & \mbox{$d=3$ and $q^2+q+1 \in \{3r,3r^2\}$.} \end{array}$$ To see this, first notice that any element $x \in G$ of order $(q^2+q+1)/d$ is a derangement. Therefore, if holds then $(q^2+q+1)/d = r^e$ for some positive integer $e$, and by applying Lemma \[l:nagell\] we deduce that (a) or (b) holds. Conversely, suppose that (a) or (b) holds. We claim that $${\mathcal{E}}(G) = \left\{\begin{array}{ll} \{r,r^2\} & \mbox{if $d=3$ and $q^2+q+1 = 3r^2$} \\ \{r\} & \mbox{otherwise.} \end{array}\right.$$ To see this, we use the fact that the action of $G$ on $1$-spaces is doubly transitive, so the corresponding permutation character has the form $1_H^G = 1+\chi$ for some irreducible character $\chi \in {\rm Irr}(G)$ of degree $q(q+1)$. By inspecting the character table of $G$ (see [@FS Table 2], for example), we see that $\chi(x)=-1$ if and only if $x$ has order $r$ (or $r^2$ if $d=3$ and $q^2+q+1 = 3r^2$). This justifies the claim. Now assume that $H = {\rm SO}_{3}(q)$, so $q$ is odd. Here elements of order $(q^2+q+1)/d$ are derangements, and so is any unipotent element with Jordan form $[J_2,J_1]$ (where $J_i$ denotes a standard unipotent Jordan block of size $i$). Therefore, does not hold in this situation. *Case 2.* $G \neq G_0$. To complete the proof of the lemma, we may assume that $G \neq G_0$ and $q {\geqslant}8$. Let $M$ be a maximal subgroup of $G_0$ containing $H_0 := H \cap G_0$. From the analysis in Case 1, we may assume that $M = {\rm P}_{1}$, in which case $H_0$ is either equal to ${\rm P}_{1}$, or it is a non-maximal subgroup of type ${\rm P}_{1,2}$ (a Borel subgroup of $G_0$) or ${\rm GL}_{2}(q) \times {\rm GL}_{1}(q)$. We can quickly eliminate the latter two possibilities. For instance, if $H_0$ is a Borel subgroup then $\Delta_{H_0}(G_0)$ contains all elements of order $(q^2-1)/d$, so does not hold (see ). Similarly, if $H_0$ is of type ${\rm GL}_{2}(q) \times {\rm GL}_{1}(q)$ then $\Delta_{H_0}(G_0)$ contains elements of order $(q^2+q+1)/d$, and also unipotent elements with Jordan form $[J_3]$. Therefore, we may assume that $H_0 = {\rm P}_{1}$, with $q {\geqslant}8$. To show that does not hold, we may as well assume that we are in one of the two cases (a) and (b) in above (otherwise the conclusion is clear). Note that the condition $H_0 = {\rm P}_{1}$ implies that $G {\leqslant}{\rm \Gamma L}_{3}(q)$ (that is, $G$ does not contain a graph automorphism). Also note that we may identify $\Omega$ with the set of $1$-dimensional subspaces of the natural ${\rm L}_{3}(q)$-module. Note that $r>3$. First assume that $G = {\rm PGL}_{3}(q)$, so $d=3$ since we are assuming that $G \neq G_0$. Here $G$ has a cyclic maximal torus ${\langle}x {\rangle}$ of order $q^2+q+1$. Then $x$ is a derangement and thus does not hold since $q^2+q+1$ is not a prime power (note that $(q^2+q+1)_3=3$). For the remainder, we may assume that $q=p^f$ and $f {\geqslant}2$ (also recall that $q {\geqslant}8$). In view of , Lemma \[l:nagell\](iii) implies that $f$ is a $3$-power. To deduce that does not hold, we may assume that $|G:G_0|$ is a prime number. Since $G {\leqslant}{\rm \Gamma L}_{3}(q)$ and $f$ is a $3$-power, we may assume that $|G:G_0|=3$ and thus $G=G_0.{\langle}\phi {\rangle}$ or $G_0.{\langle}\delta\phi{\rangle}$, where $\phi$ is a field automorphism of order $3$ and $\delta$ is an appropriate diagonal automorphism ${\rm diag}(\omega_1, \omega_2,\omega_3) \in {\rm PGL}_{3}(q)$ (modulo scalars). In both cases, the result follows by applying Lemma \[l:gms\]. For example, $\delta\phi$ has more than one fixed point on $\Omega$, so Lemma \[l:gms\] implies that the coset $G_0\delta\phi$ contains derangements, none of which has $r$-power order. In view of this final contradiction, we conclude that does not hold if $G \neq G_0$. \[lem:U3\] Theorem \[t:main2\] holds if $G_0 = {\rm U}_3(q)$. Set $d=(3,q+1)$ and observe that $G_0$ contains elements of order $(q^2-q+1)/d$ and $(q^2-1)/d$. Note that $(q^2-1)/d$ is a prime power if and only if $q \in \{3,5\}$. In order to show that does not hold, we may assume that $G = G_0$. The cases $q \in \{3,4,5\}$ can be handled directly, using , so for the remainder we will assume that $q {\geqslant}7$. Let $V$ be the natural $G_0$-module, and let ${\rm P}_{1}$ (respectively ${\rm N}_{1}$) be the $G_0$-stabilizer of a $1$-dimensional totally isotropic (respectively, non-degenerate) subspace of $V$. Note that ${\rm N}_{1}$ is a subgroup of type ${\rm GU}_{1}(q) \times {\rm GU}_{2}(q)$. We can immediately rule out any subgroup $H$ that does not contain elements of order $(q^2-1)/d$, which means that we may assume $H$ is of type ${\rm P}_{1}$, ${\rm N}_{1}$ or $O_{3}(q)$ ($q$ odd). In all three cases, elements of order $(q^2-q+1)/d$ are derangements. In addition, if $H = {\rm N}_{1}$ (respectively, ${\rm SO}_{3}(q)$) then unipotent elements with Jordan form $[J_3]$ (respectively, $[J_2,J_1]$) are derangements. Finally, suppose that $H = {\rm P}_{1}$. Let $\omega \in \mathbb{F}_{q^2}$ be an element of order $q+1$ and set $x = {\rm diag}(1,\omega,\omega^{-1}) \in G$ (modulo scalars) with respect to an orthonormal basis for $V$. Then $x$ does not fix a totally isotropic $1$-space, whence $x$ is a derangement of order $q+1$. Having handled the low-dimensional groups, we are now in a position to complete the proof of Theorem \[t:main2\] for linear and unitary groups. \[lem:Ln\] Theorem \[t:main2\] holds if $G_0 = {\rm L}_n^{{\epsilon}}(q)$. We may assume that $n {\geqslant}4$. Set $d=(n,q-{\epsilon})$ and $e=(q-{\epsilon})d$. Let $V$ be the natural $G_0$-module. Let ${\rm P}_{i}$ be the $G_0$-stabilizer of a totally isotropic $i$-dimensional subspace of $V$ (so ${\rm P}_{i}$ is a maximal parabolic subgroup of $G_0$, and we can take any $i$-space if ${\epsilon}=+$). Similarly, if ${\epsilon}=-$ then let ${\rm N}_{i}$ denote the $G_0$-stabilizer of an $i$-dimensional non-degenerate subspace of $V$ (so ${\rm N}_{i}$ is of type ${\rm GU}_{i}(q) \times {\rm GU}_{n-i}(q)$). In order to show that does not hold, we may assume that $G=G_0$. There are several cases to consider. *Case 1.* $n=2m$ and $m {\geqslant}4-{\epsilon}$ is odd. First assume that $m {\geqslant}5$. As in the proof of [@BTV Proposition 3.11], let $x \in G$ be an element of order $(q^{m+2}-{\epsilon})(q^{m-2}-{\epsilon})/e$. Then $|x|$ is not a prime power (see Lemma \[lem:numerical\]), and [@GK Table II] indicates that $x$ is a derangement unless one of the following holds: - ${\epsilon}=+$ and $H={\rm P}_{m-2}$ (or ${\rm P}_{m+2}$); - ${\epsilon}=-$ and $H = {\rm N}_{m-2}$. In (a), any element of order $\ell_{n}(q)$ or $\ell_{n-1}(q)$ is a derangement, and elements of order $\ell_{n}(q)$ and $\ell_{2(n-1)}(q)$ are derangements in case (b). Now assume $m=3$, so $({\epsilon},n)=(+,6)$. Let $x \in G$ be an element of order $(q^6-1)/e$, which is not a prime power by Lemma \[l:ppower\]. Here $x$ is a *Singer element*, and the main theorem of [@Ber] implies that $x$ is a derangement, unless $H$ is a field extension subgroup, so we have reduced to the case where $H$ is of type ${\rm GL}_{3}(q^2)$ or ${\rm GL}_{2}(q^3)$. In this situation, elements of order $\ell_{5}(q)$ are derangements, and so are unipotent elements with Jordan form $[J_2,J_1^4]$. *Case 2.* $n=2m$ and $m {\geqslant}3-{\epsilon}$ is even. First assume that $m {\geqslant}4$. Let $x \in G$ be an element of order $(q^{m+1}-{\epsilon})(q^{m-1}-{\epsilon})/e$. Then Lemma \[lem:numerical\] implies that $|x|$ is not a prime power, and from [@GK Table II] we deduce that $x$ is a derangement unless one of the following holds: - ${\epsilon}=+$ and $H={\rm P}_{m-1}$ (or ${\rm P}_{m+1}$); - ${\epsilon}=-$ and $H= {\rm N}_{m-1}$. To deal with these cases, we can repeat the argument in Case 1. Now assume $m=2$, so $({\epsilon},n)=(+,4)$. By applying the main theorem of [@Ber], we deduce that elements of order $(q^4-1)/e$ are derangements unless $H$ is a field extension subgroup of type ${\rm GL}_{2}(q^2)$. Moreover, since $(q^4-1)/e$ is not a prime power (see Lemma \[l:ppower\]), we can assume that $H$ is of type ${\rm GL}_{2}(q^2)$. Here elements of order $\ell_{3}(q)$ and unipotent elements with Jordan form $[J_2,J_1^2]$ are derangements. *Case 3.* ${\epsilon}=+$, $n=2m+1$ and $m {\geqslant}2$. If $G = {\rm L}_{11}(2)$, then any element of order $2^{11}-1 = 23 \cdot 89$ is a derangement, unless $H$ is a field extension subgroup of type ${\rm GL}_{1}(2^{11})$, in which case elements of order $2^{10}-1$ are derangements. For the remainder, we may assume that $(n,q) \neq (11,2)$. Let $x \in G$ be an element of order $(q^{m+1}-1)(q^{m}-1)/e$. By Lemmas \[lem:numerical\] and \[l:ppower\], $|x|$ is not a prime power, so we may assume that $H = {\rm P}_{m}$ (see [@GK Table II]). If $m {\geqslant}3$, then elements of order $\ell_{n}(q)$ or $\ell_{n-2}(q)$ are derangements. Similarly, if $m=2$ then we can take elements of order $\ell_{5}(q)$ or $\ell_{4}(q)$. *Case 4.* ${\epsilon}=-$, $n=2m+1$ and $m {\geqslant}4$. Fix $x \in G$, where $$|x| = \left\{\begin{array}{ll} (q^{m+1}+1)(q^{m}-1)/e & \mbox{$m$ even} \\ (q^{m+2}+1)(q^{m-1}-1)/e & \mbox{$m$ odd.} \end{array}\right.$$ By Lemma \[lem:numerical\], $|x|$ is not a prime power, and [@GK Table II] indicates that $x$ has fixed points only if $H$ stabilizes a subspace $U$ of $V$ with $\dim U {\geqslant}2$. Therefore, we may assume that $H$ has this property, in which case any element of order $\ell_{2n}(q)$ or $\ell_{n-1}(q)$ is a derangement. *Case 5.* ${\epsilon}=-$ and $n \in \{4,5,6,7\}$. First assume that $n=7$. Let $x \in G$ be an element of order $(q^6-1)/d$. Since $|x|$ is not a prime power, by inspecting the list of maximal subgroups of $G$ (see [@BHR Tables 8.37 and 8.38]) it follows that we can assume that $H \in \{{\rm P}_{3}, {\rm N}_{1}, {\rm SO}_{7}(q)\}$. In all three cases, any element of order $\ell_{14}(q)$ is a derangement. Similarly, elements of order $\ell_{10}(q)$ are derangements, unless $H = {\rm N}_{1}$, in which case any unipotent element with Jordan form $[J_7]$ is a derangement. The case $n=5$ is entirely similar. Next assume that $n=6$. For now, let us assume that $q \not\in \{2,5\}$. Let $x \in G$ be an element of order $(q^5+1)/d$. Then $|x|$ is not a prime power (see Lemma \[l:ppower\]) and $H = {\rm N}_{1}$ is the only maximal subgroup of $G$ containing such an element (see [@GK p.767]). Now, if $H = {\rm N}_{1}$ then any element of order $(q^6-1)/e$ is a derangement of non-prime power order. Suppose that $n=6$ and $q \in \{2,5\}$. The case $q=2$ can be checked directly, using for example, so let us assume that $q=5$. Let $x \in G$ be an element of order $(5^6-1)/e = 434$. By inspecting the list of maximal subgroups of $G$ (see [@BHR Tables 8.26 and 8.27]), we deduce that $x$ is a derangement unless $H$ is of type ${\rm P}_{3}$, ${\rm GL}_{3}(5^2)$ or ${\rm GU}_{2}(5^3)$, so we may assume that $H$ is one of these subgroups, in which case any element of order $\ell_{10}(5)=521$ is a derangement. Suppose that $H = {\rm P}_{3}$. Fix an orthonormal basis for $V$ and let $y = {\rm diag}(1,1,\omega,\omega^{-1},\omega^{2},\omega^{-2})$ (modulo scalars), where $\omega \in \mathbb{F}_{25}$ is an element of order $6$. Then $y$ is a derangement. Similarly, if $H$ is of type ${\rm GL}_{3}(5^2)$ or ${\rm GU}_{2}(5^3)$, then any unipotent element with Jordan form $[J_2,J_1^4]$ is a derangement. This eliminates the case $G = {\rm U}_{6}(5)$. A very similar argument applies if $n=4$. Here the cases $q \in \{2,3\}$ can be checked directly, so let us assume that $q {\geqslant}4$. Let $x \in G$ be an element of order $(q^3+1)/d$. Then $|x|$ is not a prime power (see Lemma \[l:ppower\]) and again we reduce to the case $H = {\rm N}_{1}$ (see [@GK p.767]). We can now take any element of order $(q^4-1)/e$, which will be a derangement of non-prime power order. Next, we turn our attention to symplectic groups. Let $G_0 = {\rm PSp}_n(q)$ be a symplectic group with natural module $V$. As before, we will write ${\rm P}_{i}$ (respectively, ${\rm N}_{i}$) for the $G_0$-stabilizer of an $i$-dimensional totally isotropic (respectively, non-degenerate) subspace of $V$. We will also use $n = m \perp (n-m)$ to denote an orthogonal decomposition of $V$ of the form $V = V_1 \perp V_2$, where $V_1$ is a non-degenerate $m$-space. Further, we will say that a semisimple element $x \in G_0$ is of *type* $m \perp (n-m)$ if it fixes such an orthogonal decomposition of $V$, acting irreducibly on $V_1$ and $V_2$. Similar notation is used in [@BGK; @BTV; @GK]. To begin with, we will assume that $n {\geqslant}6$; the special case $G_0 = {\rm PSp}_{4}(q)$ will be handled separately in Lemma \[lem:Sp4\]. \[lem:Sp\] Theorem \[t:main2\] holds if $G_0 = {\rm PSp}_n(q)$ and $n {\geqslant}6$. Set $d=(2,q-1)$ and write $n=2m$ with $m {\geqslant}3$. As before, we may assume that $G=G_0$. *Case 1.* $m$ odd. The case $(n,q) = (6,2)$ can be handled directly (using , for example), so let us assume that $(n,q) \neq (6,2)$. Let $x \in G$ be an element of order $(q^{m}+1)/d$. If $q$ is even, then Lemma \[lem:primepower\] implies that $|x|$ is not a prime power, and it is easy to see that the same conclusion also holds if $q$ is odd. By the main theorem of [@Ber], $x$ is a derangement unless one of the following holds: - $H$ is a field extension subgroup of type ${\rm Sp}_{n/k}(q^k)$ for some prime divisor $k$ of $n$; - $q$ is even and $H = O_{n}^{-}(q)$. In (a), elements of order $\ell_{n-2}(q)$ are derangements, and so are unipotent elements with Jordan form $[J_2,J_1^{n-2}]$. Similarly, if (b) holds then elements of order $\ell_{m}(q)$ are derangements, and so are semisimple elements of type $(n-2) \perp 2$ and order $(q^{m-1}+1)(q+1)$. *Case 2.* $m {\geqslant}6$ even. First assume that $q$ is odd. Let $x \in G$ be a semisimple element of type $(n-4) \perp 4$, so $$|x| = \left\{\begin{array}{ll} q^{m-2}+1 & \mbox{if $m \equiv 0 {\allowbreak\mkern4mu({\operator@font mod}\,\,4)}$} \\ (q^{m-2}+1)(q^{2}+1)/2 & \mbox{if $m \equiv 2 {\allowbreak\mkern4mu({\operator@font mod}\,\,4)}$.} \end{array}\right.$$ Clearly, if $m \equiv 2 {\allowbreak\mkern4mu({\operator@font mod}\,\,4)}$ then $|x|$ is divisible by $\ell_{4}(q)$ and $\ell_{n-4}(q)$, so $|x|$ is not a prime power. The same conclusion also holds if $m \equiv 0 {\allowbreak\mkern4mu({\operator@font mod}\,\,4)}$ (see Lemma \[lem:primepower\]). By [@BGK Proposition 5.10], we may assume that $H$ is of type ${\rm N}_{4}$ or ${\rm Sp}_{n/2}(q^2)$. In both cases, elements of order $\ell_{n-2}(q)$ are derangements. In addition, unipotent elements with Jordan form $[J_n]$ (respectively, $[J_2,J_1^{n-2}]$) are derangements if $H$ is of type ${\rm N}_{4}$ (respectively, ${\rm Sp}_{n/2}(q^2)$). Now assume that $q$ is even. Let $x \in G$ be a semisimple element of type $(n-2) \perp 2$ and order $q^{m-1}+1$. Now Lemma \[lem:primepower\] implies that $|x|$ is not a prime power, and by applying the main theorem of [@GPPS] we deduce that $x$ is a derangement unless $H \in \{{\rm N}_{2},O_{n}^{+}(q)\}$. In both cases, elements of order $\ell_{n}(q)$ are derangements. Also, unipotent elements with Jordan form $[J_n]$ are derangements if $H = {\rm N}_{2}$. Now, if $H=O_{n}^{+}(q)$ then let $y \in G$ be a block-diagonal element of the form $y=[y_1,y_2]$ (with respect to an orthogonal decomposition $n = (n-2) \perp 2$), where $y_1 \in {\rm Sp}_{n-2}(q)$ has order $\ell_{m-1}(q)$ and $y_2 \in {\rm Sp}_{2}(q)$ has order $q+1$. Then $y$ is a derangement and the result follows. *Case 3.* $m = 4$. The case $q = 2$ can be checked directly, so we may assume that $q {\geqslant}3$. If $q$ is even, then we can repeat the relevant argument in Case 2. Now assume $q$ is odd. Let $x \in G$ be a semisimple element of type $6 \perp 2$ and order $q^3+1$. By Lemma \[lem:primepower\], $|x|$ is not a prime power. The maximal subgroups of $G$ are listed in [@BHR Tables 8.48 and 8.49], and we deduce that $x$ is a derangement unless $H$ is of type ${\rm N}_{2}$, ${\rm GU}_{4}(q)$ or ${\rm L}_{2}(q^3)$ (in the terminology of [@BHR; @KL], the latter possibility is an almost simple irreducibly embedded subgroup in the collection $\mathcal{S}$). In each of these exceptional cases, any element of order $(q^4+1)/2$ is a derangement. In addition, if $H={\rm N}_{2}$ then unipotent elements with Jordan form $[J_8]$ are derangements. Similarly, if $H$ is of type ${\rm GU}_{4}(q)$ or ${\rm L}_{2}(q^3)$ then elements with Jordan form $[J_2,J_1^6]$ are derangements. \[lem:Sp4\] Theorem \[t:main2\] holds if $G_0 = {\rm PSp}_n(q)$. We may assume that $n=4$. The result can be checked directly if $q {\leqslant}7$, so let us assume that $q {\geqslant}8$. First assume that $q$ is odd. In terms of an orthogonal decomposition $4=2\perp 2$, let $x =[x_1,x_2] \in G$ (modulo scalars) be an element of order $p(q+1)$, where $x_1 \in {\rm Sp}_{2}(q)$ is a unipotent element of order $p$, and $x_2 \in {\rm Sp}_{2}(q)$ is irreducible of order $q+1$. By inspecting the list of maximal subgroups of $G$ (see [@BHR Tables 8.12 and 8.13]), we deduce that $x$ is a derangement unless $H$ is of type ${\rm P}_{1}$ or ${\rm Sp}_{2}(q) \wr {\rm S}_2$. In both of these cases, any element of order $\ell_4(q)$ is a derangement. Similarly, unipotent elements with Jordan form $[J_4]$ are derangements if $H$ is of type ${\rm Sp}_{2}(q) \wr {\rm S}_2$. Finally, suppose that $H = {\rm P}_{1}$. Now ${\rm Sp}_{2}(q)$ has precisely $\varphi(q+1)/2 {\geqslant}2$ distinct classes of elements of order $q+1$ (where $\varphi$ is the Euler totient function); if $y_1, y_2 \in {\rm Sp}_{2}(q)$ represent distinct classes, then $y = [y_1,y_2] \in G$ (modulo scalars) is a derangement since it does not fix a totally isotropic $1$-space. Now assume $q$ is even. As above, let $x \in G$ be an element of order $2(q+1)$. The maximal subgroups of $G$ are listed in [@BHR Table 8.14], and we see that $x$ is a derangement unless $H$ is of type ${\rm P}_{1}$, ${\rm Sp}_{2}(q) \wr S_2 \cong O_{4}^{+}(q)$ or $O_{4}^{-}(q)$. For $H = {\rm P}_{1}$, we can repeat the argument in the $q$ odd case, so let us assume that $H = O_{4}^{{\epsilon}}(q)$. If ${\epsilon}=+$ then any element of order $\ell_4(q)$ is a derangement, and we can also find derangements of order $4$ (with Jordan form $[J_4]$), since there are two conjugacy classes of such elements in $G$, but only one in $H$. Finally, if ${\epsilon}=-$ then we can find derangements of order $2$ (with Jordan form $[J_2^2]$; these are $a_2$-type involutions in the sense of Aschbacher and Seitz [@AS]), and also derangements of order $q+1$ of the form $[y_1,y_2]$ as above. To complete the proof of Theorem \[t:main2\], we may assume that $G_0 = {\rm P\Omega}_{n}^{{\epsilon}}(q)$ is an orthogonal group, where $n {\geqslant}7$. The low-dimensional groups with $n \in \{7,8\}$ require special attention. We extend our earlier notation for orthogonal decompositions by writing $m^{\pm}$ to denote a non-degenerate $m$-space of type $\pm$ (when $m$ is even). Similarly, we write ${\rm N}_{m}^{\pm}$ for the $G_0$-stabilizer of such a subspace of the natural $G_0$-module $V$. If $q$ is even, we will also adopt the standard Aschbacher-Seitz notation for involutions (see [@AS]). \[lem:O7\] Theorem \[t:main2\] holds if $G_0 = \Omega_7(q)$. We may assume that $G=G_0$. The case $q=3$ can be checked directly, so we may assume that $q {\geqslant}5$ (recall that $q$ is odd). Let $x \in G$ be an element of order $(q^3+1)/2$, which is not a prime power. By [@BGK Proposition 5.20], $x$ is a derangement unless $H = {\rm N}_{6}^{-}$, in which case any element of order $\ell_3(q)$ is a derangement, and so are unipotent elements with Jordan form $[J_7]$. \[lem:O8+\] Theorem \[t:main2\] holds if $G_0 = {\rm P\Omega}_8^{+}(q)$. As usual, we may assume that $G=G_0$. Let $V$ be the natural module for $G_0$. The case $q=2$ can be checked directly, using . Next suppose that $q=3$. Let $x \in G$ be an element of order $20$, fixing a decomposition of $V$ of the form $8 = 4^{-}\perp 4^{-}$. As indicated in [@BGK Table 3], $x$ is a derangement unless the type of $H$ is one of the following: $${\rm P}_{4}, O_7(3), O_{4}^{-}(3) \wr {\rm S}_2, {\rm GU}_{4}(3), {\rm Sp}_{4}(3) \otimes {\rm Sp}_{2}(3)$$ where $O_7(3)$ is irreducible and ${\rm P}_{4}$ is the stabilizer in $G$ of a maximal totally singular subspace of $V$. By considering elements of order $14$, we can immediately eliminate the cases ${\rm P}_{4}$, $O_{4}^{-}(3) \wr S_2$ and ${\rm Sp}_{4}(3) \otimes {\rm Sp}_{2}(3)$. Similarly, $G$ contains derangements of order $15$ if $H$ is of type ${\rm GU}_{4}(3)$. Finally, suppose that $H$ is an irreducible subgroup of type $O_7(3)$. To see that does not hold, we may replace $H$ by a conjugate $H^{\tau}$, where $\tau \in {\rm Aut}(G)$ is an appropriate triality graph automorphism such that $H^{\tau}$ is the stabilizer in $G$ of a non-degenerate $1$-space. For this reducible subgroup, elements with Jordan form $[J_2^4]$ are derangements, and so are elements $y \in G$ of order $5$ of the form $y = \hat{y}Z$, where $Z = Z(\Omega_{8}^{+}(3))$ and $C_V(\hat{y})$ is trivial (the eigenvalues of $\hat{y}$ (in $\mathbb{F}_{3^4}$) are the nontrivial fifth roots of unity, each occurring with multiplicity $2$). For the remainder, we may assume that $q {\geqslant}4$. Let $x \in G$ be an element of order $(q^3+1)/(2,q-1)$, fixing an orthogonal decomposition $8 = 6^{-} \perp 2^{-}$. Then $|x|$ is not a prime power, and $x$ is a derangement unless $H$ is of type ${\rm N}_{2}^{-}$ or ${\rm GU}_{4}(q)$ (see [@GK p.767]). In both of these cases, elements of order $\ell_3(q)$ are derangements. Similarly, if $q$ is odd then unipotent elements with Jordan form $[J_7,J_1]$ are also derangements. Finally, if $q$ is even and $H$ is of type ${\rm N}_{2}^{-}$ (respectively, ${\rm GU}_{4}(q)$) then unipotent elements with Jordan form $[J_4^2]$ (respectively, $[J_2^2,J_1^4]$; $c_2$-involutions in the terminology of [@AS]) are derangements. The result follows. \[lem:O8-\] Theorem \[t:main2\] holds if $G_0 = {\rm P\Omega}_8^{-}(q)$. Again, we may assume that $G=G_0$. If $q {\leqslant}3$ then we can use to verify the result, so let us assume that $q {\geqslant}4$. The maximal subgroups of $G$ are listed in [@BHR Tables 8.52 and 8.53]. By considering elements of order $\ell_{8}(q)$ and $\ell_{6}(q)$, we can eliminate subfield subgroups, together with the reducible subgroups of type ${\rm P}_{2}$, ${\rm P}_{3}$, ${\rm N}_{2}^{-}$, ${\rm N}_{3}$ and ${\rm N}_{4}^{+}$. Similarly, elements of order $\ell_{8}(q)$ and $\ell_{4}(q)$ are derangements if $H$ is a non-geometric subgroup of type ${\rm L}_{3}^{{\epsilon}}(q)$. Therefore, to complete the proof, we may assume that $H$ is either a field extension subgroup of type $O_{4}^{-}(q^2)$, or a reducible subgroup of type ${\rm P}_{1}$, ${\rm N}_{2}^{+}$, $O_7(q)$ ($q$ odd) or ${\rm Sp}_{6}(q)$ ($q$ even). If $H$ is of type $O_{4}^{-}(q^2)$, then elements of order $\ell_{6}(q)$ are derangements, as well as unipotent elements with Jordan form $[J_{7},J_{1}]$ if $q$ is odd, and unipotent elements with Jordan form $[J_2^2,J_1^4]$ ($a_2$-type involutions) if $q$ is even. Similarly, if $H = {\rm P}_{1}$ or ${\rm N}_{2}^{+}$ then elements of order $\ell_{8}(q)$ and $\ell_{3}(q)$ are derangements (note that an element of order $\ell_3(q)$ fixes a $2^{-}$-space, but not a $2^{+}$-space). Finally, suppose $H$ is of type $O_7(q)$ ($q$ odd) or ${\rm Sp}_{6}(q)$ ($q$ even). In both cases, elements of order $\ell_8(q)$ are derangements. In addition, there are derangements with Jordan form $[J_5,J_3]$ ($q$ odd) and $[J_4^2]$ ($q$ even). \[lem:O\] Theorem \[t:main2\] holds if $G_0 = {\rm P\Omega}_n^{{\epsilon}}(q)$. We may assume that $G=G_0$ and $n {\geqslant}9$. We have three cases to consider. *Case 1.* $G_0 = {\rm P\Omega}_{n}^{+}(q)$ and $n {\geqslant}10$. Write $n=2m$ and first assume that $m$ is odd. Let $x \in G$ be an element of order $(q^{(m-1)/2}+1)(q^{(m+1)/2}+1)/(4,q-1)$, fixing an orthogonal decomposition of the form $(m+1)^{-} \perp (m-1)^{-}$. Then Lemma \[lem:numerical\] implies that $|x|$ is not a prime power, so by [@BGK Proposition 5.13] we may assume that $H = {\rm N}_{m-1}^{-}$. In this situation, elements of order $\ell_{n-2}(q)$ are derangements, and so are unipotent elements with Jordan form $[J_{n-1},J_{1}]$ ($q$ odd) or $[J_{n-2},J_2]$ ($q$ even). A similar argument applies if $m$ is even. Here we take an element $x \in G$ of order $(q^{(m-2)/2}+1)(q^{(m+2)/2}+1)/(4,q-1)$, fixing a decomposition $(m+2)^{-} \perp (m-2)^{-}$. Then $|x|$ is not a prime power, and [@BGK Proposition 5.14] implies that $x$ is a derangement unless $H$ is of type ${\rm N}_{m-2}^{-}$ or $O_{n/2}^{+}(q^2)$. In the former case, we complete the argument as above, so let us assume that $H$ is of type $O_{n/2}^{+}(q^2)$. Any element of order $\ell_{n-2}(q)$ is a derangement, and so are unipotent elements with Jordan form $[J_{n-1},J_{1}]$ if $q$ is odd. Finally, if $q$ is even then $a_2$-type involutions are derangements. *Case 2.* $G_0 = {\rm P\Omega}_{n}^{-}(q)$ and $n {\geqslant}10$. Again, write $n=2m$. First assume that $m {\geqslant}11$. Let $x \in G$ be an element of order $${\rm lcm}(q^{m-5}+1,q^3+1,q^2+1)/(2,q-1)$$ fixing a decomposition $(n-10)^{-} \perp 6^{-} \perp 4^{-}$. Then $|x|$ is not a prime power, and [@BGK Proposition 5.16] implies that $x$ is a derangement unless $H$ is of type ${\rm N}_{4}^{-}$, ${\rm N}_{6}^{-}$ or ${\rm N}_{10}^{+}$. In each of these cases, it is clear that elements of order $\ell_{n}(q)$ and $\ell_{n-2}(q)$ are derangements. Next suppose that $m \in \{5,6,7,9,10\}$. Let $x \in G$ be an irreducible element of order $(q^m+1)/(2,q-1)$. We claim that $|x|$ is not a prime power (here we require $m \neq 8$). If $q$ is even, this follows immediately from Lemma \[lem:primepower\], so let us assume that $q$ is odd. Suppose $m=5$ and $q^5+1 = 2r^e$ for some prime $r$ and positive integer $e$. Then $(q+1)(q^4-q^3+q^2-q+1) = 2r^e$ and $r = \ell_{10}(q)$. Therefore, $q+1=2$ is the only possibility, which is absurd. Similarly, if $m=6$ and $q^6+1 = (q^2+1)(q^4-q^2+1) = 2r^e$, then $r=\ell_{12}(q)$ and $q^2+1=2$, which is not possible. The other cases are entirely similar. Now, by the main theorem of [@Ber], $x$ is a derangement unless $H$ is a field extension subgroup of type $O_{n/k}^{-}(q^k)$ ($k$ a prime divisor of $n$, $n/k {\geqslant}4$ even) or ${\rm GU}_{n/2}(q)$ ($n/2$ odd). In both cases, elements of order $\ell_{n-2}(q)$ are derangements. In addition, there are unipotent derangements; take $[J_{n-1},J_1]$ if $q$ is odd, an $a_2$-involution if $q$ is even and $H$ is of type $O_{n/k}^{-}(q^k)$, and a $c_2$-involution if $q$ is even and $H$ is of type ${\rm GU}_{n/2}(q)$. Finally, let us assume that $m=8$. As in [@GK Table II], let $x \in G$ be an element of order ${\rm lcm}(q^5+1,q^2+1,q+1)/(2,q-1)$, fixing an orthogonal decomposition of the form $10^{-} \perp 4^{-} \perp 2^{-}$. Note that $|x|$ is divisible by $\ell_{10}(q)$ and $\ell_4(q)$, so it is not a prime power. As indicated in [@GK Table II], $x$ is a derangement unless $H$ is of type ${\rm N}_{2}^{-}$, ${\rm N}_{4}^{-}$ or ${\rm N}_{6}^{+}$. In each of these cases, elements of order $\ell_{16}(q)$ and $\ell_{14}(q)$ are derangements. *Case 3.* $G_0 = \Omega_{n}(q)$ and $n {\geqslant}9$ is odd. Write $n=2m+1$ and note that $q$ is odd. First assume $m$ is odd. Let $x \in G$ be an element of order $${\rm lcm}(q^{(m+1)/2}+1, q^{(m-1)/2}+1)/2 = (q^{(m+1)/2}+1)(q^{(m-1)/2}+1)/4,$$ fixing an orthogonal decomposition $(m+1)^{-} \perp (m-1)^{-} \perp 1$. Note that $|x|$ is divisible by $\ell_{m+1}(q)$ and $\ell_{m-1}(q)$, so $|x|$ is not a prime power. Let $H$ be a maximal subgroup of $G$ containing $x$. By carefully applying the main theorem of [@GPPS], we deduce that $H \in \{{\rm N}_{m+1}^{-}, {\rm N}_{m-1}^{-}, {\rm N}_{2m}^{+}\}$. For example, the order of $x$ rules out subfield subgroups and imprimitive subgroups of type $O_1(q) \wr {\mathrm{S}}_{n}$ (see [@BGK Remark 5.1(i)]), and the dimensions of the irreducible constituents of $x$ are incompatible with field extension subgroups of type $O_{n/k}(q^k)$. Now, if $H$ is one of these reducible subgroups, then elements of order $\ell_{n-1}(q)$ and $\ell_{n-3}(q)$ are derangements. The result follows. A similar argument applies if $m$ is even. Here we take $x \in G$ to be an element of order $${\rm lcm}(q^{(m+2)/2}+1, q^{(m-2)/2}+1)/2 = (q^{(m+2)/2}+1)(q^{(m-2)/2}+1)/4,$$ fixing a decomposition $(m+2)^{-} \perp (m-2)^{-} \perp 1$. We claim that $|x|$ is not a prime power. This is clear if $m {\geqslant}6$, or if $m=4$ and $q$ is not a Mersenne prime, since $|x|$ is divisible by $\ell_{m \pm 2}(q)$. Suppose that $m=4$ and $q$ is a Mersenne prime. If $q=3$ then $|x|=28$ and the claim holds, and if $q>3$ then $|x|$ is divisible by $2$ and $\ell_6(q)$. This justifies the claim. Using [@GPPS] one can check that the only maximal subgroups of $G$ containing $x$ are of type ${\rm N}_{m+2}^{-}$, ${\rm N}_{m-2}^{-}$ or ${\rm N}_{2m}^{+}$, so we may assume that $H$ is one of these subgroups. Here we observe that elements of order $\ell_{n-1}(q)$ are derangements, and so are unipotent elements with Jordan form $[J_n]$. This completes the proof of Theorem \[t:main2\]. Affine groups {#s:affine} ============= Let $G$ be a finite primitive permutation group. By Theorem \[t:main1\], if holds then $G$ is either almost simple or affine. In the previous section, we determined all the almost simple examples, and we now turn our attention to the affine groups with property . Our main aim is to prove Theorem \[t:main3\]. Let $G = HV {\leqslant}{\rm AGL}(V)$ be a finite affine primitive permutation group with point stabilizer $H = G_0$ and socle $V = (\mathbb{Z}_{p})^k$. As an abstract group, $G$ is a semidirect product of $V$ by $H$. Therefore, we will begin our analysis by studying the structure of a general semidirect product $G=H \ltimes N$ with property , so $G$ is a finite group, $H$ is a proper subgroup and $N$ is a normal subgroup of $G$ such that $G = HN$ and $H \cap N = 1$. We will need some additional notation. If $K$ is a subgroup of $G$ and $g\in G$, then we set $$[K,g] = \{ [k,g]=k^{-1}g^{-1}kg \,:\, k \in K\}.$$ We also write $K^*$ for the set of all nontrivial elements of $K$. \[lem:general facts\] Let $G=H \ltimes N$. The following hold: - ${{\mathbf {C}}}_G(x)={{\mathbf {C}}}_H(x){{\mathbf {C}}}_N(x)$ for all $x\in H$. - If $K{\leqslant}H$, then $K\cap K^n={{\mathbf {C}}}_K(n)$ for all $n \in N^*$. - If $N$ is abelian, then $\Delta_H(G)=\{tv \, : \, t\in H,v\in N\setminus [N,t]\}$. - If property holds, then $N$ is an $r$-group. First consider part (i). The result is clear if $x=1$, so assume that $x\in H^*$. The inclusion ${{\mathbf {C}}}_H(x){{\mathbf {C}}}_N(x) \subseteq {{\mathbf {C}}}_G(x)$ is clear. Conversely, suppose that $g=hn\in{{\mathbf {C}}}_G(x)$ where $h\in H$, $n\in N$. Then $hnx=xhn$. Multiplying both sides by $(xh)^{-1}=h^{-1}x^{-1}$, we obtain $$hnxh^{-1}x^{-1}=(xh)n(xh)^{-1}$$ which implies that $$(hnh^{-1})(hxh^{-1}x^{-1})=(xh)n(xh)^{-1}.$$ Since $n\in N{\trianglelefteqslant}G$ and $h,x\in H$, we deduce that $$hxh^{-1}x^{-1}=(hn^{-1}h^{-1})(xh)n(xh)^{-1}\in H\cap N=1$$ so $h\in {{\mathbf {C}}}_H(x)$. Since $hnx=xhn=hxn$, we deduce that $nx=xn$ and thus $n\in{{\mathbf {C}}}_N(x)$. Therefore, $g=hn\in{{\mathbf {C}}}_H(x){{\mathbf {C}}}_N(x)$ and part (i) follows. For part (ii), let $K{\leqslant}H$ and let $n\in N^*$. Assume that $y\in K\cap K^n$. Then $y=k^n\in K$ for some $k\in K$, so $$k^{-1}y=k^{-1}n^{-1}kn=(k^{-1}n^{-1}k)n\in K\cap N=1,$$ which implies that $kn=nk$ and $y=k$, or equivalently $y \in {{\mathbf {C}}}_K(n)$. Therefore, $K\cap K^n {\leqslant}{{\mathbf {C}}}_K(n)$. Conversely, if $y\in {{\mathbf {C}}}_K(n)$ then $y\in K$ and $y=n^{-1}yn\in K^n$, so $y\in K\cap K^n$ and thus ${{\mathbf {C}}}_K(n){\leqslant}K\cap K^n$. The result follows. Now consider part (iii). Assume that $N$ is abelian. Set $$\Gamma:= \{tv \, : \, t\in H,v\in N\setminus [N,t]\}.$$ First we claim that $\Gamma \subseteq \Delta_H(G)$. Let $g\in \Gamma$, say $g=hn$ with $h\in H$ and $n\in N\setminus [N,h]$. Seeking a contradiction, suppose that $g \not\in \Delta_H(G)$. Then $g\in H^t$ for some $t\in G$. Since $t\in G = HN$, we may write $t=h_1m_1$ with $h_1\in H$ and $m_1\in N$. It follows that $g\in H^t=H^{m_1}$, so $m_1gm_1^{-1}\in H$. Let $m:=m_1^{-1}\in N$. Then $m^{-1}gm=m^{-1}hnm=h^mn\in H$ (note that $nm=mn$ since $N$ is abelian) and thus $h^{-1}h^mn=[h,m]n\in H$. We also have $[h,m]n=(h^{-1}m^{-1}h)mn\in N$, so $[h,m]n\in H\cap N=1$ and we deduce that $n=[m,h]\in [N,h]$, contradicting our choice of $n$. We have now shown that $\Gamma \subseteq \Delta_H(G)$. Conversely, suppose that $g=hn\in \Delta_H(G)$ with $h\in H$, $n\in N$. We claim that $n \in N\setminus [N,h]$. Seeking a contradiction, suppose that $n \in [N,h]$, say $n=[m,h]$ for some $m\in N$. Then $m^{-1}(hn)m=h$, or equivalently $g^m\in H$, which is a contradiction. Finally, let us turn to part (iv). If $x\in N^*$ then $x^G\subset N$, so $x^G\cap H \subseteq N \cap H=1$ and thus $x^G\cap H=\emptyset$ since $x\neq 1$. Therefore $N^*\subseteq \Delta_H(G)$. In particular, if every element of $\Delta_H(G)$ is an $r$-element (for some fixed prime $r$), then every element of $N$ is also an $r$-element and thus $N$ is an $r$-group. \[lem:equivalence\] Let $G=H\ltimes N$, where $N$ is an $r$-group for some prime $r$. Then the following are equivalent: - Property holds. - ${{\mathbf {C}}}_H(n)=H\cap H^n$ is an $r$-group for all $n\in N^*$. - ${{\mathbf {C}}}_N(x)=1$ for every nontrivial $r'$-element $x \in H$. In other words, every nontrivial $r'$-element of $H$ induces a fixed-point-free automorphism of $N$ via conjugation. First we will show that (i) implies (ii). Suppose that holds. Let $n\in N^*$. We claim that ${{\mathbf {C}}}_H(n)$ is an $r$-group. Notice that ${{\mathbf {C}}}_H(n)=H\cap H^n$ by Lemma \[lem:general facts\](ii). Seeking a contradiction, suppose that $|{{\mathbf {C}}}_H(n)|$ is divisible by a prime $s \neq r$. Choose $y\in {{\mathbf {C}}}_H(n)$ with $|y|=s$ and let $g:=ny=yn\in G$. We claim that $g\in\Delta_H(G)$, which would be a contradiction since $|g|=|n|s$ is not a power of $r$. Assume that $g \not\in\Delta_H(G)$, so $g\in H^t$ for some $t\in G$. Since $G=HN$, we may assume that $t \in N$. Then $$g^s=(ny)^s=n^sy^s=n^s\in H^t$$ and $n^s\in N{\trianglelefteqslant}G$, so $t(n^s)t^{-1}\in H\cap N=1$ and thus $n^s=1$, which is not possible since $n$ is a nontrivial $r$-element. Therefore, $g=ny\in\Delta_H(G)$ as required. Next we will show that (ii) implies (i). Suppose that ${{\mathbf {C}}}_H(n)$ is an $r$-group for all $n\in N^*$. Let $g \in \Delta_H(G)$, say $g=hn$ with $h\in H$ and $n\in N^*$. We claim that $g$ is an $r$-element. Seeking a contradiction, suppose that $m:=|g|$ is divisible by a prime $s \neq r$. Set $x:=g^{m/s} \in G$ and let $S$ be a Sylow $s$-subgroup of $H$. Then $|x|=s$ and $S$ is also a Sylow $s$-subgroup of $G$ since $|G:H|=|N|$ is coprime to $s$. By Sylow’s theorem, $x^t \in S{\leqslant}H$ for some $t \in G$. Since $g^G \subseteq \Delta_H(G)$, replacing $g$ by $g^{t^{-1}}$ we may assume that $x \in H$. Then $g \in {{\mathbf {C}}}_G(x)={{\mathbf {C}}}_H(x){{\mathbf {C}}}_N(x)$ by Lemma \[lem:general facts\](i). Suppose that ${{\mathbf {C}}}_N(x) \neq 1$, say $1 \neq n \in {{\mathbf {C}}}_N(x)$. Then $x \in {{\mathbf {C}}}_H(n)$, but this is a contradiction since $|x|=s$ and we are assuming that ${{\mathbf {C}}}_H(n)$ is an $r$-group. Therefore, ${{\mathbf {C}}}_N(x)=1$ and thus ${{\mathbf {C}}}_G(x)={{\mathbf {C}}}_H(x)$. Hence $g\in {{\mathbf {C}}}_H(x){\leqslant}H$, which contradicts the fact that $g\in \Delta_H(G)$. This final contradiction shows that $g$ is an $r$-element, so holds. Now let us show that (ii) implies (iii). Suppose that ${{\mathbf {C}}}_H(n)$ is an $r$-group for all $n\in N^*$. Let $x\in H^*$ be an $r'$-element. We claim that ${{\mathbf {C}}}_N(x)=1$. Seeking a contradiction, suppose that $1 \neq n \in {{\mathbf {C}}}_N(x)$. Then $x\in {{\mathbf {C}}}_H(n)$, so $|{{\mathbf {C}}}_H(n)|$ is divisible by $|x|$, which is not an $r$-power. This contradicts the assumption that ${{\mathbf {C}}}_H(n)$ is an $r$-group. To complete the proof, it remains to show that (iii) implies (ii). Suppose that ${{\mathbf {C}}}_N(x)=1$ for every nontrivial $r'$-element $x \in H$. Let $n \in N^*$. If ${{\mathbf {C}}}_H(n)$ is not an $r$-group, then there exists an element $x\in {{\mathbf {C}}}_H(n)$ with $|x|=s$, where $s \neq r$ is a prime. Therefore, $1\neq n \in {{\mathbf {C}}}_N(x)$, which is not possible since ${{\mathbf {C}}}_N(x)=1$. We are now in a position to prove Theorem \[t:main3\]. Let $G = HV {\leqslant}{\rm AGL}(V)$ be a finite affine primitive permutation group with point stabilizer $H=G_0$ and socle $V = ({\mathbb{Z}}_p)^k$, where $p$ is a prime and $k {\geqslant}1$. If property holds, then $r=p$ and Lemma \[lem:equivalence\] implies that no nontrivial $r'$-element of $H$ has fixed points on $V\setminus\{0\}$. Therefore, the pair $(H,V)$ is $r'$-semiregular in the sense of [@FLT]. Conversely, if $r=p$ and $(H,V)$ is $r'$-semiregular, then ${{\mathbf {C}}}_V(x)=0$ for every nontrivial $r^\prime$-element $x\in H$, so Lemma \[lem:equivalence\] implies that $G$ has property . \[r:affine0\] *Note that the equivalence of (i) and (ii) in Lemma \[lem:equivalence\] implies that an affine group $G = HV {\leqslant}{\rm AGL}(V)$ has property if and only if every two-point stabilizer in $G$ is an $r$-group.* If $G = HV {\leqslant}{\rm AGL}(V)$ is an affine group (with $V = ({\mathbb{Z}}_r)^k$) and $r \not\in \pi(H)$, then $G$ is a Frobenius group and property clearly holds. Therefore, we may focus on the case where $r \in \pi(H)$. As noted in the Introduction, detailed information on $r'$-semiregular pairs $(H,V)$ was initially obtained by Guralnick and Wiegand in [@GW Section 4], where this notion arises naturally in their study of the multiplicative structure of field extensions. Similar results were established in later work by Fleischmann et al. [@FLT]. In both papers, the main aim is to determine the structure of $H$. For solvable affine groups, we have the following result (in the statement, ${\mathbf{O}}_{r'}(Y)$ denotes the largest normal $r'$-subgroup of $Y$): \[p:flt1\] Let $G = HV {\leqslant}{\rm AGL}(V)$ be a finite affine primitive permutation group with point stabilizer $H = G_0$ and socle $V = ({\mathbb{Z}}_r)^k$. Assume that $H$ is solvable and $r \in \pi(H)$. Then $G$ has property only if $H \cong X \times Y$ or $(X \times Y){:}2$, where $X \in \{1,{\rm SL}_{2}(3)\}$, $Y = {\mathbf{O}}_{r'}(Y)R$ and $R$ is a Sylow $r$-subgroup of $Y$. This follows from [@FLT Theorem 2.1]. The main result for a perfect group $H$ is Proposition \[p:flt2\] below (see [@FLT Theorem 4.1]; also see [@GW Theorem 4.2]). In part (iv), $\mathcal{S} = \{5,13,37,73, \ldots\}$ is the set of all primes $s$ satisfying the following conditions: - $s = 2^a3^b+1$, where $a {\geqslant}2$ and $b {\geqslant}0$; - $(s+1)/2$ is a prime. It is not known whether or not $\mathcal{S}$ is finite. \[p:flt2\] Let $G= HV {\leqslant}{\rm AGL}(V)$ be a finite affine primitive permutation group with point stabilizer $H=G_0$ and socle $V = ({\mathbb{Z}}_r)^k$. Assume that $H$ is perfect and $r \in \pi(H)$. Then $G$ has property only if one of the following holds: - $H \cong {\rm SL}_{2}(r^a)$, where $a {\geqslant}1$ and $r^a>3$; - $H \cong {}^2{\rm B}_2(2^{2a+1})$, $r=2$ and $a {\geqslant}1$; - $H \cong {}^2{\rm B}_2(2^{2a+1}) \times {\rm SL}_{2}(2^{2b+1})$, $r=2$ and $a,b {\geqslant}1$ such that $(2a+1,2b+1) = 1$; - $H \cong {\rm SL}_{2}(s)$, $r=3$ and $s \in \mathcal{S} \cup \{7,17\}$. For instance, $H = {\rm SL}_{2}(7)$ has a $12$-dimensional faithful, irreducible module $V$ over ${\mathbb{F}}_{3}$, and the corresponding affine group $G = HV$ has property (with ${\mathcal{E}}(G) = \{3,9\}$). In the general case, we refer the reader to [@FLT Theorem 6.1] for a detailed description of the structure of $H$. Finally, let us suppose that $G= HV {\leqslant}{\rm AGL}(V)$ is a finite affine primitive permutation group with property . Set $${\mathcal{E}}(G)={\mathcal{E}}_H(G)=\{|x|\,:\,x\in\Delta_H(G)\}.$$ Can we determine when ${\mathcal{E}}(G) = \{r\}$? In order to address this question, let $P$ be a Sylow $r$-subgroup of $G$. Then $V{\leqslant}P$ since $V$ is a normal $r$-subgroup of $G$, and we have $P=(H\cap P)V=KV$ with $K:=H\cap P$. Note that $P=KV$ is a semidirect product. \[p:prime\] Let $G = HV {\leqslant}{\rm AGL}(V)$ be a finite affine primitive permutation group with point stabilizer $H = G_0$ and socle $V = ({\mathbb{Z}}_r)^k$. Assume that property holds. Let $P$ be a Sylow $r$-subgroup of $G$ and set $K = H \cap P$. Then the following hold: - $P=KV$ is a transitive permutation group on $P/K$. - $\Delta(G)=\bigcup_{g\in G}\Delta_K(P)^g$ and ${\mathcal{E}}(G)={\mathcal{E}}_K(P)$. As above, $P=KV$ is a semidirect product. For part (i), it suffices to show that the core $L$ of $K$ in $P$ is trivial. We have $L{\leqslant}K{\leqslant}H$ and $L{\trianglelefteqslant}P$, so $[L,V]{\leqslant}L\cap V{\leqslant}K\cap V=1$ and thus $L{\leqslant}{{\mathbf {C}}}_K(V){\leqslant}{{\mathbf {C}}}_H(V)=1$ (here we are using the fact that $V$ is a faithful irreducible $H$-module). This proves (i). Now consider part (ii). Clearly, it suffices to show that the first equality holds. By applying Lemma \[lem:general facts\](iii) we have $$\Delta_K(P)=\{tv \, : \, t\in K,v\in V\setminus [V,t]\}.$$ Since $K{\leqslant}H$, a further application of Lemma \[lem:general facts\](iii) (this time for $G=HV$) shows that $\Delta_K(P)\subseteq \Delta(G)$. As $\Delta(G)$ is a normal subset of $G$, it follows that $$\bigcup_{g\in G}\Delta_K(P)^g\subseteq\Delta(G).$$ Since property holds, every $g\in \Delta(G)$ is an $r$-element, so some $G$-conjugate of $g$ is in $P$. Without loss of generality, we may assume that $g\in P=KV$. By Lemma \[lem:general facts\](iii) we have $g=hn$, with $h\in H$ and $n\in V\setminus [V,h]$. Moreover, since $V{\leqslant}P$ and $g\in P$, we have $h=gn^{-1}\in H\cap P=K$. Therefore, by applying Lemma \[lem:general facts\](iii) once again, we conclude that $g=hn \in \Delta_K(P)$, so $\Delta(G)=\bigcup_{g\in G}\Delta_K(P)^g$ and the proof is complete. Now, if we assume that $G=HV$ has property , then part (ii) of Proposition \[p:prime\] implies that ${\mathcal{E}}(G)=\{r\}$ if and only if ${\mathcal{E}}_K(P)=\{r\}$. Clearly, if $P$ has exponent $r$, then ${\mathcal{E}}_K(P)=\{r\}$. Conversely, if ${\mathcal{E}}_K(P)=\{r\}$ with $r=2$ or $3$, then a theorem of Mann and Praeger [@MP Proposition 2] implies that $P$ has exponent $r$. In fact, for this specific transitive group $P$ we can show that the same conclusion holds for *any* prime $r$ (we thank an anonymous referee for pointing this out). \[c:prime\] Let $G = HV {\leqslant}{\rm AGL}(V)$ be a finite affine primitive permutation group with point stabilizer $H = G_0$ and socle $V = (\mathbb{Z}_{p})^k$, where $p$ is a prime and $k {\geqslant}1$. Then every derangement in $G$ has order $r$, for some fixed prime $r$, if and only if $r=p$ and the following two conditions hold: - Every two-point stabilizer in $G$ is an $r$-group; - A Sylow $r$-subgroup of $G$ has exponent $r$. Let $P$ be a Sylow $r$-subgroup of $G$. First assume that $r=p$ and (i) and (ii) hold. By (i), the pair $(H,V)$ is $r'$-semiregular so Theorem \[t:main3\] implies that property holds. Therefore, ${\mathcal{E}}(G)={\mathcal{E}}_K(P)$ by Proposition \[p:prime\](ii) (with $K = H \cap P$) and thus condition (ii) implies that ${\mathcal{E}}(G)=\{r\}$ as required. 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--- abstract: | Analyzing qualitative behaviors of biochemical reactions using its associated network structure has proven useful in diverse branches of biology. As an extension of our previous work, we introduce a graph-based framework to calculate steady state solutions of biochemical reaction networks with synthesis and degradation. Our approach is based on a labeled directed graph $G$ and the associated system of linear non-homogeneous differential equations with first order degradation and zeroth order synthesis. We also present a theorem which provides necessary and sufficient conditions for the dynamics to engender a unique stable steady state. Although the dynamics are linear, one can apply this framework to nonlinear systems by encoding nonlinearity into the edge labels. We answer open question from our previous work concerning the non-positiveness of the elements in the inverse of a perturbed Laplacian matrix. Moreover, we provide a graph theoretical framework for the computation of the inverse of a such matrix. This also completes our previous framework and makes it purely graph theoretical. Lately, we demonstrate the utility of this framework by applying it to a mathematical model of insulin secretion through ion channels and glucose metabolism in pancreatic $\beta$-cells. author: - 'I. Mirzaev[^1]' - 'D. M. Bortz$^{*}$[^2]' bibliography: - 'mathbioCU.bib' title: Analytical Equilibrium Solutions of Biochemical Systems with Synthesis and Degradation --- Introduction ============ In recent years, many researchers have devoted their efforts to developing a systems-level understanding of biochemical reaction networks. In particular, the study of these chemical reaction networks (CRNs) using their associated graph structure has attracted considerable attention. The work led by Craciun and Feinberg on multistationarity [@Craciun2005; @Craciun2010; @Craciun2006; @CraciunFeinberg2006] and the work led by Mincheva and Roussel on stable oscillations [@Mincheva2011a; @Mincheva2007a; @Mincheva2007] are two particularly influential approaches. For a good overview of the various graph theoretic developments, we direct the interested reader to the review provided in Domijan and Kirkilionis [@Domijan2008]. In this work, we focus on applications of graph theory, mainly the Matrix-Tree Theorem (MTT), for deriving equilibrium solutions (ES) for CRNs that fit within a Laplacian dynamics framework. The MTT-based framework was first applied in a biological context by King and Altman [@King1956] to derive steady state rate equations in enzyme kinetics. This framework was then simplified and summarized into rules (known as *Chou’s graphical rules* [@Lin2013]) by Chou and coworkers [@Chou1989; @Chou1990; @Chou1993]. Chou [@Chou1989] has also extended the framework for non-steady state enzyme-catalyzed systems. The main disadvantage of Chou’s graphical rules is that they are only applicable if the underlying digraph structure is *strongly connected*, i.e., every vertex is reachable from every other vertex. This issue was solved and extended for general directed graphs (*digraphs*) by Mirzaev and Gunawardena in 2013 [@MirzaevGunawardena2013bmb] and is applicable to the specific class of linear ordinary differential equations (ODEs) known as *Laplacian dynamics*. Systems described by Laplacian dynamics are created using a weakly connected digraph, $G$, with $n$ vertices, with labelled, directed edges, and without self loops. Note that by *weakly connected* we mean that the graph cannot be expressed as the union of two disjoint digraphs. If there is an edge from vertex $j$ to vertex $i$, we label it with $e_{ij}>0$, and with $e_{ij}=0$ if there is no such edge. [^3] The Laplacian matrix (hereafter, a *Laplacian* $\mathcal{L}$) of given digraph $G$ is then defined as $$\left(\mathcal{L}(G)\right)_{ij}=\begin{cases} e_{ij} & \text{if }i\ne j\\ -\sum_{m\ne j}e_{mj} & \text{if }i=j\,. \end{cases}\label{eq:Laplacian Matrix}$$ The corresponding *Laplacian* *dynamics* are then defined as $$\frac{d\mathbf{x}}{dt}=\mathcal{L}(G)\cdot\mathbf{x}$$ where $\mathbf{x}=\left(x_{1},\cdots,x_{n}\right)^{T}$ is column vector of species concentrations at each vertex, $1,\cdots,n$. In a biochemical context one may think of vertices as different species and edges as rate of transformation from one species to another. However, we note that this framework is symbolic in nature in the sense that the mathematical description of the computed steady states is done without the specification of rate constants, i.e., edge weights $e_{ij}$. In other words, the only information about an individual $e_{ij}$ relevant to our approach is whether or not it is zero. Laplacian matrices were first introduced by Kirchhoff in 1847 in his article about electrical networks [@Kirchhoff1847]. Ever since then Laplacians have been studied and applied in various fields. For an example of studying the applications of Laplacians to spectral theory, we refer the interested reader to Bronski and Deville [@Bronski2014] in which they study the class of *Signed graph Laplacians* (a symmetric matrix, which is special case of above defined Laplacian). In this article we will extend the framework intitially developed in [@MirzaevGunawardena2013bmb] to investigate behaviors of Laplacian dynamics when zero-th order synthesis and first order degradation are added to the system. Specifically, we will examine the following dynamics, $$\frac{d\mathbf{x}}{dt}=\mathcal{L}(G)\cdot\mathbf{x}-D\cdot\mathbf{x}+\mathbf{s}\label{eq:New dynamics}$$ where the degradation matrix $D$ is a diagonal matrix with $\left(D\right)_{ii}=d_{i}\geq0$ and the synthesis vector $\mathbf{s}$ is a column vector with $\left(\mathbf{s}\right)_{i}=s_{i}\geq0$. Hereafter, we refer to this new dynamics as synthesis and degradation dynamics (or simply as SD dynamics). In the biological networks literature this type of dynamics are often referred as *inconsistent* networks[@Marashi2014]. For these dynamics, several questions naturally arise. Under what conditions does this system have non-negative, stable ES solution? Moreover, how can we relate the ES solution to the underlying digraph structure of $G$ as we did for Laplacian dynamics without synthesis and degradation? Our goal is to answer these questions on a theoretical level as well as apply the result to real world CRN examples. The outline of this work is as follows. We will first briefly review the main results of [@Gunawardena2012; @MirzaevGunawardena2013bmb] and present some additional notation (to be used in subsequent sections). In Section \[sec:Theoretical-Development\] we describe our main theoretical results and in Section \[sec: negativity of inverse\] fully discuss the proof of an important result in Section \[sec:Theoretical-Development\]. In Section \[sec:Biochemical-Network-Application\], we illustrate an application of these results to exocytosis cascade of insulin granules and glucose metabolism in pancreatic $\beta$-cells. Lastly, in Section \[sec:Conclusions\], we conclude with a discussion of the implications of these results as well as plans for future work.[^4] \[sec:Preliminary-results\]Preliminaries ======================================== In this section we briefly summarize the important results of Mirzaev and Gunawardena [@MirzaevGunawardena2013bmb] and refer the interested reader to that article for proofs and more extensive discussion and interpretation. For the sake of clarity, we will preserve the original notation while we include some additional definitions that can be found in many introductory graph theory books. Given a digraph $G$, we denote the set of vertices of $G$ with $\mathcal{V}(G)$ and we write $i\Longrightarrow j$ to denote the existence of a path from vertex $i$ to ** vertex $j$. If $i\Longrightarrow j$ and $j\Longrightarrow i$, vertex $i$ is said to be *strongly connected* to vertex $j$, and is denoted $i\Longleftrightarrow j$. A digraph $G$ is *strongly connected* if for each ordered pair $i,j$ of vertices in $G$, we have that $i\Longleftrightarrow j$. The *strongly connected components* (SCCs) of a digraph are the largest strongly connected subgraphs. Let $C[i]$ denote the SCC containing $i$, $i\in\mathcal{V}(C[i])$. Suppose we are given two SCCs, $C[i]$ and $C[j]$, if $i\Longrightarrow j$ we write $C[i]\preceq C[j]$ to denote that $C[i]$ *precedes* $C[j]$. This *precedes* relation is both reflexive and transitive. Moreover, the relation is also antisymmetric as $C[i]\preceq C[j]$ and $C[j]\preceq C[i]$ imply that $i\Longleftrightarrow j$ and $C[i]=C[j]$. From this, we can conclude that the precedes relation allows for a *partial ordering* of the SCCs. Accordingly, this allows us to identify so-called terminal SCCs (tSCC), which are those SCCs $C[i]$ such that, if $C[i]\preceq C[j]$ then $C[i]=C[j]$. These tSCCs are used in many other contexts, for example, they are also known as “attractors” of state transition graphs [@Berenguier2013]. With this terminology, we can devise an insightful relabeling of the vertices of digraph $G$. Such a relabeling will transform the Laplacian matrix into one with a block lower-diagonal structure, which will prove convenient in our theoretical development. Suppose there are $q$ tSCCs out of a total of $p+q$ SCCs. Our goal is to relabel the vertices such that the first $p$ blocks of Laplacian matrix correspond to the $p$ non-terminal SCCs. Since the precedence relation, $\preceq$, is a partial ordering, there exists an ordering of the SCCs, $C_{1},\dots,C_{p+q}$, such that, if $C_{i}\preceq C_{j}$, then $i\le j$. Since a tSCC cannot precede any other SCC, then the tSCCs can be in some arbitrary order $\{C_{i}\}_{i=p+1}^{p+q}$ (which will not impact our results). We denote $a_{i}$ as the number of vertices in $C_{i}$, and $m_{i}=\sum_{k=1}^{i}a_{k}$ as the partial sum of the $a_{i}$’s, (with $m_{0}=0$). Note that the $a_{i}$’s should add up to the number of vertices in digraph $G$, i.e. $\sum_{i=1}^{p+q}a_{i}=n$. Then the vertices of $C_{i}$ are relabeled using only indices $m_{i-1}+1,\dots,m_{i-1}+a_{i}$ for $i=1,\dots,p+q$. Consequently, the new Laplacian matrix, $\mathcal{L}(G),$ is constructed using the relabeled vertices. Since $i<j$ implies $C_{j}\not\preceq C_{i}$, the Laplacian of $G$ can be written in block lower-triangular form $$\mathcal{L}(G)=\left(\begin{array}{ccc|ccc} \boxed{\mathcal{L}_{1}} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0}\\ \vdots & \ddots & \vdots & \vdots & \vdots & \vdots\\ + & + & \boxed{\mathcal{L}_{p}} & \mathbf{0} & \cdots & \mathbf{0}\\ \hline + & \cdots & + & \boxed{\mathcal{L}_{p+1}} & \mathbf{0} & \mathbf{0}\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots\\ + & \cdots & + & \mathbf{0} & \mathbf{0} & \boxed{\mathcal{L}_{p+q}} \end{array}\right)=\left(\begin{array}{c|c} N & \mathbf{0}\\ \hline B & T \end{array}\right)\,,$$ where $+$ stands for some matrix with non-negative real entries, the submatrix $N$ is block lower-triangular with non-negative off-diagonal elements, $B$ is a matrix with non-negative elements, $\mathbf{0}$ is matrix of all zeros, and $T$ is also a block diagonal matrix such that $$N=\left(\begin{array}{ccc} \boxed{\mathcal{L}_{1}} & & \mathbf{0}\\ \vdots & \ddots\\ + & + & \boxed{\mathcal{L}_{p}} \end{array}\right),\, T=\left(\begin{array}{ccc} \boxed{\mathcal{L}_{p+1}} & & \mathbf{0}\\ & \ddots\\ \mathbf{0} & & \boxed{\mathcal{L}_{p+q}} \end{array}\right)\,.\label{eq: Definition of N}$$ By the definition of the Laplacian matrix (see (\[eq:Laplacian Matrix\])) all off-diagonal elements are non-negative real numbers. The blocks in boxes on the main diagonal, denoted by $\mathcal{L}_{1},\dots,\mathcal{L}_{p+q}$, are the submatrices defined by restricting $\mathcal{L}(G)$ to the vertices of the corresponding SCCs, $C_{1},\dots,C_{p+q}$. Note that for $i=p+1,\dots,p+q$ each $\mathcal{L}_{i}$ is Laplacian matrix in its own, $\mathcal{L}_{i}=\mathcal{L}(C_{i})$ . However for the non-terminal SCCs, $\left\{ C_{i}\right\} _{i=1}^{p}$, there is always at least one outgoing edge to some other SCC. This implies that for $i=1,\dots,p$ each matrix $\mathcal{L}_{i}$ is defined as the Laplacian of a corresponding SCC minus some non-zero diagonal matrix corresponding to outgoing edges from this SCC, $\mathcal{L}_{i}=\mathcal{L}(C_{i})-\Delta_{i}$ for some $\Delta_{i}\not\equiv0$. In this case we call $\mathcal{L}_{i}$ a *perturbed Laplacian matrix*, or simply a *perturbed matrix* and note the following property of $\mathcal{L}_{i}$ (proven in [@MirzaevGunawardena2013bmb]). \[lem:perturbed\] The Perturbed Laplacian matrix of strongly connected graph $G$ is non-singular. A *directed spanning subgraph* of digraph $G$ is a connected subgraph of $G$ that includes every vertex of $G$, so that any spanning subgraph which is at the same time is a tree is called *directed spanning tree* (DST) of the digraph $G$. We say that a DST, $\mathscr{T}$, is *rooted* at $i\in G$ if vertex $i$ is the only vertex in $\mathscr{T}$ without any outgoing edges, and denote the set of DSTs of digraph $G$ rooted at vertex $i$ with $\Theta_{i}(G)$. Thus $\Theta_{i}(G)$ is a non-empty set of spanning trees for a strongly connected digraph $G$. However, for an arbitrary digraph there maybe no spanning tree rooted at specific vertex, in which case $\Theta_{i}(G)=\emptyset$. In this case the corresponding element, $\mathcal{L}(G)_{(j)}$, is zero, where $\mathcal{L}(G)_{(ji)}$ denotes the $ji$-th minor of Laplacian matrix $\mathcal{L}(G)$ and is the determinant of the $(n-1)\times(n-1)$ matrix that results from deleting row $j$ and column $i$ of $\mathcal{L}(G)$. Next we review the main theorem from [@MirzaevGunawardena2013bmb] on which the results of this paper are based. The theorem utilizes the digraph structure of digraph $G$ to calculate minors of a Laplacian. The proof of this theorem can be found in several papers, and we direct readers to [@MirzaevGunawardena2013bmb] for a proof with same notations as in this article.[^5] \[thm:(Matrix-Tree-Theorem)\](Matrix-Tree Theorem) If $G$ is digraph with $n$ vertices then the minors of its Laplacian are given by $$\mathcal{L}(G)_{(ij)}=(-1)^{n+i+j-1}\sum_{\mathscr{T}\in\Theta_{j}(G)}P_{\mathcal{\mathscr{T}}}\,,$$ where $P_{\mathcal{\mathscr{T}}}$ is the product of all edge weights in the spanning tree $\mathscr{T}$. An illustration of above theorem is depicted in Figure \[fig:Illustration-of-Algorithm\], where $\mathcal{L}(G)_{(23)}$ and $\mathcal{L}(G)_{32}$ minor of $\mathcal{L}(G)$ are computed using spanning trees of digraph $G$. Consequently, the MTT implies that the $ij$-th minor of the Laplacian (up to sign) is the sum of all $P_{\mathcal{\mathscr{T}}}$ for each spanning tree, $\mathscr{T}$, rooted at vertex $j$. Since all edges of the digraph $G$ are non-negative numbers (zero only if there is no such edge), then the expression $\rho_{i}^{G}=\sum_{\mathscr{T}\in\Theta_{i}(G)}P_{\mathscr{T}}$ will always be non-negative.[^6] If $G$ is strongly connected then $\Theta_{i}(G)\ne\emptyset$, so $\rho_{i}^{G}$ is strictly positive. Uno in his article [@Uno1996] provided an algorithm for enumerating and listing all spanning trees of a general digraph and Ahsendorf et al. [@Ahsendorf2013] utilized Uno’s algorithm to compute minors of Laplacian matrix using the Matrix-Tree theorem. Having an implementation of the MTT available, one can then calculate the kernel elements of the Laplacian, $\mathcal{L}(G)$, using the following two fairly well known Propositions (see [@MirzaevGunawardena2013bmb] for proofs). \[prop:SCC kernel\]If $G$ is strongly connected graph, then $\ker\mathcal{L}(G)=span\left\{ \boldsymbol{\rho}^{G}\right\} $, where $\boldsymbol{\rho}^{G}$ is column vector with $\left(\boldsymbol{\rho}^{G}\right)_{i}=\rho_{i}^{G}>0$. Here, the kernel is defined in the conventional sense, $\ker\mathcal{L}(G)=\left\{ x\in\mathbb{R}^{n\times1}:\,\mathcal{L}(G)\cdot x=\mathbf{0}\right\} $. Moreover, Proposition \[prop:SCC kernel\] guarantees that a kernel element has all positive elements, a fact which is not immediately obvious using standard linear algebraic methods. When $G$ is not a strongly connected digraph, the kernel elements of $\mathcal{L}(G)$ are constructed using kernel elements of its tSCCs. Specifically, since for $i=1,\cdots,q$ each $\mathcal{L}_{p+i}=\mathcal{L}(C_{p+i})$ is a Laplacian matrix on its own, by Proposition \[prop:SCC kernel\] there exists $\boldsymbol{\rho}^{C_{p+i}}\in\mathbb{R}_{>0}^{a_{p+i}\times1}$ such that $\mathcal{L}_{p+i}\cdot\boldsymbol{\rho}^{C_{p+i}}=\mathbf{0}$ and $a_{i}$ is the number of vertices in $C_{i}$. Then we can extend this vector to $\bar{\boldsymbol{\rho}}^{C_{p+i}}\in\mathbb{R}_{>0}^{n\times1}$ by setting all entries with indices outside $C_{p+i}$ to zero: $$\left(\bar{\boldsymbol{\rho}}^{C_{p+i}}\right)_{k}=\begin{cases} \left(\boldsymbol{\rho}^{C_{p+i}}\right)_{k-m_{p+i-1}} & \text{if }m_{p+i-1}\le k\le m_{p+i}\\ 0 & \text{otherwise} \end{cases}\label{eq:Extension of rho}$$ Since $\mathcal{L}(G)$ has lower-block diagonal structure and since $\mathcal{L}_{p+i}\cdot\boldsymbol{\rho}^{C_{p+i}}=\mathbf{0}$ for each $i=1,\cdots,q$ we have $\mathcal{L}(G)\cdot\bar{\boldsymbol{\rho}}^{C_{p+i}}=\mathbf{0}$. This can be summarized in the following Proposition: \[prop:General kernel\]For any graph $G$, $$\ker\mathcal{L}(G)=span\left\{ \bar{\boldsymbol{\rho}}^{C_{p+1}},\dots,\bar{\boldsymbol{\rho}}^{C_{p+q}}\right\} \,,$$ and $\dim\ker\mathcal{L}(G)=q$ To prove stability of the steady states we will use the following theorem and corollary, which provides sufficiency conditions for the solution to a dynamical system $d\mathbf{x}/dt=A\cdot\mathbf{x}$ coupled with initial condition $\mathbf{x}(0)=\mathbf{x}_{0}$ to converge to a steady state (proof in [@MirzaevGunawardena2013bmb]). Typically, the stability of a dynamics depends on the sign of the real parts of the eigenvalues of $A$ as well as the algebraic and geometric multiplicities of the zero eigenvalue. \[thm:Convergence\]Suppose that the real matrix $A$ satisfies following two conditions 1\. If $\lambda$ is an eigenvalue of $A$, then either $\lambda=0$ or $Re(\lambda)<0$ 2\. $alg_{A}(0)=geo_{A}(0)$, where $alg_{A}(0)$ and $geo_{A}(0)$ are the algebraic and geometric multiplicities of zero eigenvalue, respectively. Then the solution of $d\mathbf{x}/dt=A\cdot\mathbf{x}$ converges to a steady state as $t\to\infty$ for any initial condition. \[cor:alg and geo\]The Laplacian of a weakly connected digraph satisfies conditions of Theorem \[thm:Convergence\]. Moreover, $geo_{\mathcal{L}(G)}\left(0\right)=alg_{\mathcal{L}(G)}\left(0\right)=q$, where $q$ is number of tSCCs of $G$. With these preliminary results in hand we provide stability analysis for the SD dynamics as well as a graph theoretical algorithm for the computation of steady states. \[sec:Theoretical-Development\]Theoretical Development ====================================================== In this section we will provide a thorough analysis of the synthesis and degradation dynamics (SD dynamics), (\[prop:General kernel\]), that we defined earlier. Suppose now that we add additional edges to core digraph, $G$, $$\overset{s_{i}}{\longrightarrow}i\:,\, i\overset{d_{i}}{\longrightarrow}$$ corresponding to zeroth-order synthesis and first-order degradation, respectively. Each vertex can have any combination of synthesis and degradation edges and the dynamics can now be described by the following system of linear ordinary differential equations (ODEs): $$\frac{d\mathbf{x}}{dt}=\mathcal{L}(G)\cdot\mathbf{x}-D\cdot\mathbf{x}+\mathbf{s}\,.\label{eq:3}$$ Here $\mathcal{L}(G)$ is the Laplacian matrix of the core digraph $G$, $D$ is a diagonal matrix with $\left(D\right)_{ii}=d_{i}$, and $\mathbf{s}$ is a column vector with $\left(\mathbf{s}\right)_{i}=s_{i}$, using the convention that $d_{i}$ or $s_{i}$ is zero if the corresponding partial edge at vertex $i$ is absent. The presence of synthesis without degradation yields unstable dynamics. Therefore, whenever we have $D\equiv\mathbf{0}$ we assume that $\mathbf{s}\equiv\mathbf{0}$. In this case the system reduces to Laplacian dynamics, for which a thorough analysis was given in [@MirzaevGunawardena2013bmb]. From now on we will assume that at least one element of $D$ is nonzero. The dynamics defined by Equation (\[eq:3\]) have a unique solution for a given initial condition. This can easily be verified as the right hand side of the dynamics, $f(\mathbf{x})=\mathcal{L}(G)\cdot\mathbf{x}-D\cdot\mathbf{x}+\mathbf{s}$ is affine in $\mathbf{x}$ and thus also *Lipschitz* continuous. Thus the existence of unique continuous solution is guaranteed. The next question to be answered is to identify the conditions under which the SD dynamics possess steady state solution(s). In order to derive the necessary and sufficient conditions for existence of steady state solution we define a *complementary digraph*, $G^{\star}$, which is formed by defining new vertex, $\star$, such that $$\star\ce{->[s_{i}]}i\:\mbox{ or }\, i\ce{->[d_{i}]}\star$$ For the sake of simplicity we will divide our results into two cases: when $G^{\star}$ is a strongly connected digraph and when it is not strongly connected. \[sub:SC framework\]Strongly connected case ------------------------------------------- Assume that complementary digraph $G^{\star}$ is strongly connected. Let $F$ denote the matrix of the Laplacian minus the degradation matrix $D$: $$F=\mathcal{L}(G)-D=\left(\begin{array}{ccc|ccc} \boxed{\mathcal{L}_{1}-D_{1}} & \cdots & \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0}\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\* + & \cdots & \boxed{\mathcal{L}_{p}-D_{p}} & \mathbf{0} & \cdots & \mathbf{0}\\ \hline + & \cdots & + & \boxed{\mathcal{L}_{p+1}-D_{p+1}} & \mathbf{0} & \mathbf{0}\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\* + & \cdots & + & \mathbf{0} & \mathbf{0} & \boxed{\mathcal{L}_{p+q}-D_{p+q}} \end{array}\right)\label{eq:partition}$$ Suppose that there is some index $m\in\{p+1,\dots,p+q\}$ for which $D_{m}\equiv0$. Then $\mathcal{L}_{m}-D_{m}=\mathcal{L}_{m}=\mathcal{L}(C_{m})$, which implies that $C_{m}$ is preserved as a tSCC in the complementary digraph $G^{\star}$. In this case, the vertices corresponding to the tSCC $C_{m}$ cannot be reached from any other SCC of $G^{\star}$, which in turn contradicts the fact that graph $G^{\star}$ is strongly connected. Thus each matrix $\mathcal{L}_{i}-D_{i}$ is a perturbed Laplacian matrix of some strongly connected digraph, so that from Remark \[lem:perturbed\] each $\mathcal{L}_{i}-D_{i}$ is a non-singular matrix for $i=p+1,\cdots,p+q$. For $i=1,\cdots,p$ as each of the $\mathcal{L}_{i}$’s is already a *perturbed* matrix, “perturb”-ing them further does not change the fact that they are non-singular. Thus the matrix $F$ is a non-singular matrix, since its diagonal components are all non-singular matrices and the unique steady state solution is given algebraically as $$\mathbf{x}_{{\scriptscriptstyle ES}}=-(\mathcal{L}(G)-D)^{-1}\cdot\mathbf{s}\,.\label{eq:4}$$ However, we are interested in computing the ES by means of graph theory. Towards this end, we define a change of variables $$\mathbf{x}=\mathbf{y}-(\mathcal{L}(G)-D)^{-1}\cdot\mathbf{s}\,.$$ This substitution transforms the original SD dynamics into $$\frac{d\mathbf{y}}{dt}=(\mathcal{L}(G)-D)\cdot\mathbf{y}\,.\label{eq:2}$$ From Proposiotion \[cor:alg and geo\] we know that eigenvalues of the Laplacian*,* $\mathcal{L}(G)$, ** satisfy $Re(\lambda)\le0$, where equality holds if and only if $\lambda=0$. The proof of this result follows from applying the *Gershgorin theorem* to the columns of the matrix $\mathcal{L}(G)$. Then the matrix $D$ in $\mathcal{L}(G)-D$ shifts the centers of the Gershgorin discs further to left on the real line without changing their radii, so we will still have $Re(\lambda)\le0$ for eigenvalues of the matrix $\mathcal{L}(G)-D$. On the other hand, the matrix $\mathcal{L}(G)-D$ is non-singular, so it follows that $Re(\lambda)<0$. This result in turn implies that solution of the system given in (\[eq:2\]) converges to the trivial steady state, $\mathbf{y}_{{\scriptscriptstyle ES}}=0$. Thus we can now state the following theorem. \[thm:Given-a-strongly\]Given a strongly connected digraph $G$, the SD dynamics (\[eq:3\]) have a unique stable steady state solution. The symbolic computation of the ES solution using (\[eq:4\]) can be very expensive even for small number of vertices. Therefore we restate an algorithm, given in [@Gunawardena2012], which uses graphical structure of graph $G^{\star}$ to calculate steady state solution given in (\[eq:4\]). Let $\mathbbm{1}=(1,\cdots,1)^{T}$ be a vector of all ones. At steady state we have $\frac{d\mathbf{x}}{dt}=\mathbf{0}$ and using the fact that $\mathbbm{1}^{T}\cdot\mathcal{L}(G)=\mathbf{0}$ it follows from (\[eq:New dynamics\]) that $$d_{1}x_{1}+\dots+d_{n}x_{n}=s_{1}+\dots+s_{n}\,.\label{eq:balance}$$ In other words at steady state we should have an overall balance in synthesis and degradation. The Laplacian $\mathcal{L}(G^{\star})$ of the digraph $G^{\star}$ can then be related to the Laplacian $\mathcal{L}(G)$ of the digraph $G$ $$\mathcal{L}(G^{\star})=\left(\begin{array}{c|c} \mathcal{L}(G) & \mathbf{0}\\ \hline \mathbf{0} & \mathbf{0} \end{array}\right)+\left(\begin{array}{c|c} -D & \mathbf{s}\\ \hline \mathbbm{1}^{T}\cdot D & -\mathbbm{1}^{T}\cdot\mathbf{s} \end{array}\right)\label{eq:5}$$ Suppose now that we have overall balance in synthesis and degradation then using (\[eq:5\]) it is easy to see that $(x_{1},\cdots,x_{n},1)$ is a steady state of $$\frac{d\mathbf{x}}{dt}=\mathcal{L}(G^{\star})\cdot\mathbf{x}\label{eq:G plus system}$$ if and only if $(x_{1},\cdots,x_{n})$ is a steady state of SD dynamics given in (\[eq:3\]). Since $G^{\star}$ is strongly connected, the MTT provides a basis element for the kernel of the Laplacian matrix $\mathcal{L}(G^{\star})$, $\ker\left\{ \mathcal{L}(G^{\star})\right\} =\text{span}\{\boldsymbol{\rho}^{G^{\star}}\}$ [@Gunawardena2012]. Consequently, the unique steady state $\mathbf{x}_{{\scriptscriptstyle ES}}$ is given by $$\left(\mathbf{x}_{{\scriptscriptstyle ES}}\right)_{i}=\frac{\left(\boldsymbol{\rho}^{G^{\star}}\right)_{i}}{\left(\boldsymbol{\rho}^{G^{\star}}\right)_{\star}}\label{eq: SC steady state MTT}$$ Since any steady state solution of (\[eq:G plus system\]) can be written as scalar multiple of kernel element, $\boldsymbol{\rho}^{G^{\star}}$, that single degree of freedom is used to guarantee that $\left(\mathbf{x}_{{\scriptscriptstyle ES}}\right)_{\star}=1$ (synthesis and degradation vertex, $\star$). This condition also ensures that overall balance in synthesis and degradation (\[eq:balance\]) is satisfied. General Case ------------ In contrast to the strongly-connected case, the steady state solutions do not always exist in the general case. First, we will derive conditions to assure the existence of a steady state solution. Then, we will show that provided we have a steady state solution $\mathbf{x}_{{\scriptscriptstyle ES}}$, the system converges to this $\mathbf{x}_{{\scriptscriptstyle ES}}$ as $t\to\infty$. Third, we provide a framework for construction of $\mathbf{x}_{{\scriptscriptstyle ES}}$ using the underlying graph structure of graph $G$ with illustration of results using a hypothetical example. In the case that the digraph $G^{\star}$ is not strongly connected, in the partition of the matrix $F$ (\[eq:partition\]) there is at least one $i\in\left\{ p+1,\dots,p+q\right\} $ such that $D_{i}\equiv\mathbf{0}$. Let $\left\{ i_{1},\cdots,i_{k}\right\} \subseteq\left\{ p+1,\cdots,p+q\right\} $ be a set for which $D_{i_{1}}=\dots=D_{i_{k}}\equiv\mathbf{0}$, then we can relabel the vertices corresponding to the tSCC ** such that matrices $\mathcal{L}_{i_{1}},\dots,\mathcal{L}_{i_{k}}$ are positioned in the lower right of matrix $F$, $$\begin{aligned} F=\mathcal{L}(G)-D & =\left(\begin{array}{ccc|ccc} \boxed{\mathcal{L}_{1}-D_{1}} & \cdots & \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0}\\ \vdots & \ddots & \vdots\\* + & + & \boxed{\mathcal{L}_{p+q-k}-D_{p+q-k}} & \mathbf{0} & \cdots & \mathbf{0}\\ \hline + & \cdots & + & \boxed{\mathcal{L}_{p+q-k+1}} & \cdots & \mathbf{0}\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\* + & \cdots & + & \mathbf{0} & \cdots & \boxed{\mathcal{L}_{p+q}} \end{array}\right)\end{aligned}$$ $$=\left(\begin{array}{ccc|ccc} \boxed{\mathcal{M}_{1}} & \cdots & \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0}\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\* + & + & \boxed{\mathcal{M}_{r}} & \mathbf{0} & \cdots & \mathbf{0}\\ \hline + & \cdots & + & \boxed{\mathcal{L}_{r+1}} & \mathbf{0} & \mathbf{0}\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\* + & \cdots & + & \mathbf{0} & \mathbf{0} & \boxed{\mathcal{L}_{r+k}} \end{array}\right)=\left(\begin{array}{c|c} N & \mathbf{0}\\ \hline B & T \end{array}\right)\label{eq:General case Decomposition}$$ where *$r=p+q-k$*, each $\mathcal{M}_{i}=\mathcal{L}_{i}-D_{i}=\mathcal{L}(C_{i})-\Delta_{i}-D_{i}$ is a *perturbed* Laplacian matrix of some SCC $C_{i}$ ** and each $\mathcal{L}_{i}$ corresponds to the Laplacian matrix of some tSCC ** in graph $G^{\star}$*.* This relabeling is always possible because the labeling procedure described in Section \[sec:Preliminary-results\] does not provide any restriction on individual labeling of vertices located in the set of tSCCs. Next we present a theorem, which provides the necessary and sufficient conditions in order for an ES to exist. For that we partition the synthesis vector $\mathbf{s}$ such that it matches up with the partition of the matrix $F$, $$F=\left(\begin{array}{c|c} N & \mathbf{0}\\ \hline B & T \end{array}\right),\qquad\mathbf{s}=\left(\begin{array}{c} \mathbf{s}'\\ \hline \mathbf{s}'' \end{array}\right)\,.$$ \[thm:BNS\]When $G^{\star}$ is not strongly connected, the necessary and sufficient conditions for existence of an ES solution are 1. $\mathbf{s}''\equiv\mathbf{0}$ 2. $B\cdot N^{-1}\cdot\mathbf{s}'\equiv\mathbf{0}$ Let us first derive the equivalent statements for the existence of an ES solution. Finding a steady state solution of the system is equivalent to solving the linear system $$\left(\mathcal{L}(G)-D\right)\cdot\mathbf{x}=-\mathbf{s}\,.\label{eq:steady state equivalent system}$$ Thus a steady state solution exists if and only if $-\mathbf{s}\in\text{Range}\left\{ \mathcal{L}(G)-D\right\} $. Let us apply simple row reduction (i.e., Gaussian elimination) to the augmented matrix $(\begin{array}{c:c}\mathcal{L}(G)-D & \mathbf{s}\end{array})$ $$\begin{aligned} &\left(\begin{array}{ccc|ccc:c} \boxed{\mathcal{M}_{1}} & \cdots & \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0}\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & -\mathbf{s}' \\* * & \cdots & \boxed{\mathcal{M}_{r}} & \mathbf{0} & \cdots & \mathbf{0}\\ \cmidrule[0.5pt]{1-6} \cmidrule[1pt]{7-7} & B_{1} & & \boxed{\mathcal{L}_{r+1}} & \mathbf{0} & \mathbf{0} & -\mathbf{s}^{(1)}\\ & \vdots & & \vdots & \ddots & \vdots&\vdots\\ & B_{k} & & \mathbf{0} & \mathbf{0} & \boxed{\mathcal{L}_{r+k}} & -\mathbf{s}^{(k)} \end{array}\right) \\ &\\ \longrightarrow & \left(\begin{array}{ccc|ccc:c} \boxed{\mathbb{I}_{a_{1}}} & \cdots & \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0}\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & -N^{-1}\mathbf{s}'\\ \mathbf{0} & \mathbf{0} & \boxed{\mathbb{I}_{a_{r}}} & \mathbf{0} & \cdots & \mathbf{0}\\ \cmidrule[0.5pt]{1-6} \cmidrule[1pt]{7-7} & B_{1} & & \boxed{\mathcal{L}_{r+1}} & \mathbf{0} & \mathbf{0} & -\mathbf{s}^{(1)}\\ & \vdots & & \vdots & \ddots & \vdots &\vdots\\ & B_{k} & & \mathbf{0} & \mathbf{0} & \boxed{\mathcal{L}_{r+k}} & -\mathbf{s}^{(k)} \end{array}\right)\\ &\\ \longrightarrow &\left(\begin{array}{ccc|ccc:c} \boxed{\mathbb{I}_{a_{1}}} & \cdots & \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0}\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & -N^{-1}\mathbf{s}'\\ \mathbf{0} & \mathbf{0} & \boxed{\mathbb{I}_{a_{r}}} & \mathbf{0} & \cdots & \mathbf{0}\\ \cmidrule[0.5pt]{1-6} \cmidrule[1pt]{7-7} \mathbf{0} & \cdots & \mathbf{0} & \boxed{\mathcal{L}_{r+1}} & \mathbf{0} & \mathbf{0} & -\mathbf{s}^{(1)}+B_{1}N^{-1}\mathbf{s}'\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots\\ \mathbf{0} & \cdots & \mathbf{0} & \mathbf{0} & \mathbf{0} & \boxed{\mathcal{L}_{r+k}} & -\mathbf{s}^{(k)}+B_{k}N^{-1}\mathbf{s}' \end{array}\right)\ . \end{aligned}$$ So the above system (\[eq:steady state equivalent system\]) has a solution if and only if each partial linear system $$\mathcal{L}_{r+i}\cdot\mathbf{z}^{(i)}=-\mathbf{s}^{(i)}+B_{i}\cdot N^{-1}\cdot\mathbf{s}'\label{eq:Eq Condition with L_{r+i}}$$ has a solution. Equation (\[eq:Eq Condition with L\_[r+i]{}\]) provides an equivalent condition for the existence of an ES solution of the SD dynamics. At this point we will prove that (\[eq:Eq Condition with L\_[r+i]{}\]) is satisfied if and only if two conditions of the theorem are satisfied. Let us assume that (\[eq:Eq Condition with L\_[r+i]{}\]) holds true. Then each partial linear system has a solution if the following condition is satisfied: $$\mathbbm{1}^{T}\cdot\mathcal{L}_{r+i}\cdot\mathbf{z}^{(i)}=\mathbbm{1}^{T}\cdot\mathcal{L}(C_{r+i})\cdot\mathbf{z}^{(i)}=\mathbf{0}=-\mathbbm{1}^{T}\cdot\mathbf{s}^{(i)}+\mathbbm{1}^{T}\cdot B_{i}\cdot N^{-1}\cdot\mathbf{s}'\qquad i\in\left\{ 1,\cdots,k\right\} \,.\label{eq:1Lz}$$ To proceed further we need the nontrivial fact that all the entries of the matrix $N^{-1}$ are non-positive real numbers. For that reason, we have devoted all of Section \[sec: negativity of inverse\] for the proof of this fact as well as presenting a graph theoretical algorithm for computation of $N^{-1}$. All entries of the vector $\mathbf{s}^{(i)}$ and matrix $B_{i}$ are non-negative real numbers, because all edge weights are non-negative real numbers by definition. Therefore, each of the products $B_{i}\cdot N^{-1}\cdot\mathbf{s}'$ are the matrices with non-positive entries. This in turn implies that both of the summands in (\[eq:1Lz\]) are equal to zero, $$-\mathbbm{1}^{T}\cdot\mathbf{s}^{(i)}\equiv\mathbf{0}\textrm{ and }\mathbbm{1}^{T}\cdot B_{i}\cdot N^{-1}\cdot\mathbf{s}'\equiv\mathbf{0}\qquad i\in\left\{ 1,\cdots,k\right\}$$ Recall that a sum of non-negative ($\mathbb{R}_{\ge0}$) real numbers is equal to zero if and only if each of the numbers are equal to zero. Hence, $$\mathbf{s}^{(i)}\equiv\mathbf{0}\textrm{ and }B_{i}\cdot N^{-1}\cdot\mathbf{s}'\equiv\mathbf{0}\qquad i\in\left\{ 1,\cdots,k\right\} \label{eq:two conditions partioned}$$ which is equivalent to the two conditions of the theorem, $$\mathbf{s}''=\left(\mathbf{s}_{1},\cdots,\mathbf{s}_{k}\right)^{T}\equiv\mathbf{0}\textrm{ and }B\cdot N^{-1}\cdot\mathbf{s}'\equiv\mathbf{0}\,.\label{eq: 1}$$ Conversely, assume that two conditions of the theorem are satisfied. Then it easy to observe that (\[eq:two conditions partioned\]) also holds true. Consequently, it follows that the linear system (\[eq:Eq Condition with L\_[r+i]{}\]) reduces to $$\mathcal{L}_{r+i}\cdot\mathbf{z}^{(i)}=\mathcal{L}(C_{r+i})\cdot\mathbf{z}^{(i)}=-\mathbf{s}_{i}+B_{i}\cdot N^{-1}\cdot\mathbf{s}'=\mathbf{0}\label{eq:Reduces to LD}$$ which always has a solution. Moreover, the solution of the above linear system (\[eq:Reduces to LD\]) can be constructed graph theoretically by Proposition \[prop:SCC kernel\]. The first condition of the theorem, $\mathbf{s}''\equiv\mathbf{0}$, can be interpreted as follows: a necessary condition for the existence of a steady state is that if a tSCC ** does not have degradation edge, it should also not have a synthesis edge. On the other hand, one can also visualize these conditions in terms of chemical reactions. If there is continuous inflow of substrates into the production part of the reaction and a lack of outflow, then reaction will grow without bound. The second condition of the theorem identifies nodes without degradation, which also contribute directly (or indirectly) to tSCCs. Indeed, this type of nodes also cause the SD dynamics to grow without bound. Next we show that (\[eq:New dynamics\]) can be transformed into homogeneous system of linear differential equations, provided that both conditions of Theorem \[thm:BNS\] are fulfilled. As in the previous section we denote matrix $L(G)-D$ with $F$, and partition $F$ and $\mathbf{s}$ as, $$F=\left(\begin{array}{c|c} N & \mathbf{0}\\ \hline B & T \end{array}\right)\qquad\mathbf{s}=\begin{pmatrix}\mathbf{s}'\\ \hline \mathbf{0} \end{pmatrix}$$ where $F\in\mathbb{R}_{\ge0}^{n\times n},\, N\in\mathbb{R}_{\ge0}^{m\times m},\, B\in\mathbb{R}_{\ge0}^{m\times(n-m)},\, T\in\mathbb{R}_{\ge0}^{(n-m)\times(n-m)}$ and $\mathbf{s}\in\mathbb{R}_{\ge0}^{n\times1},\,\mathbf{s}'\in\mathbb{R}_{\ge0}^{m\times1}$. Let us also define a matrix $Q\in\mathbb{R}_{\le0}^{n\times n}$ $$Q=\left(\begin{array}{c|c} N^{-1} & \mathbf{0}\\ \hline \mathbf{0} & \mathbf{0} \end{array}\right)\,,$$ to be used in the change of variable $\mathbf{x}=\mathbf{y}-Q\cdot\mathbf{s}$. This substitution transforms (\[eq:New dynamics\]) into $$\frac{d\mathbf{x}}{dt}=\frac{d\mathbf{y}}{dt}=F\cdot\mathbf{y}-F\cdot Q\cdot\mathbf{s}+\mathbf{s}=F\cdot\mathbf{y}-\begin{pmatrix}\mathbf{s}'\\ \hline B\cdot N^{-1}\cdot\mathbf{s}' \end{pmatrix}+\begin{pmatrix}\mathbf{s}'\\ \hline \mathbf{0} \end{pmatrix}\,.$$ Assuming that an ES solution of the SD dynamics exists, then by Theorem \[thm:BNS\] we have that $B\cdot N^{-1}\cdot\mathbf{s}'\equiv\mathbf{0}$, from which it follows that $$\frac{d\mathbf{y}}{dt}=F\cdot\mathbf{y}\,.\label{eq:System with matrix F}$$ For any given initial condition \[lem:System–converges\] the dynamics defined in (\[eq:System with matrix F\]) converges to a unique steady state as $t\to\infty$. We will prove this theorem ** by showing that matrix $F$ satisfies both conditions of Theorem \[thm:Convergence\]. First note that by definition, the matrix $F$ is a Laplacian matrix minus a non-negative diagonal matrix, $F=\mathcal{L}(G)-D$. Hence, it follows that $$\sum_{v\ne i}|\left(F\right)_{vi}|=\sum_{v\ne i}\left(\mathcal{L}(G)\right)_{vi}=|\left(\mathcal{L}(G)\right)_{ii}|\le|\left(\mathcal{L}(G)\right)_{ii}|+d_{i}=\left|\left(F\right)_{ii}\right|$$ Therefore, if we apply Gerschgorin’s theorem to the columns of the matrix $F$, we see that each eigenvalue of $F$ is located in the discs of the form $$\left\{ z\in\mathbf{C}\ |\ \left|z+|\left(\mathcal{L}(G)\right)_{ii}|+d_{i}\right|\le|(\mathcal{L}(G))_{ii}|\right\} \,.$$ A disc touches the $y-axis$ from the left hand side if and only if $|\left(\mathcal{L}(G)\right)_{ii}|+d_{i}=|\left(\mathcal{L}(G)\right)_{ii}|$, or $d_{i}=0$. Hence for an eigenvalue, $\lambda$, of the matrix $F$ we conclude that $Re(\lambda)\le0$, where equality holds if and only if $\lambda=0$. Thus, the matrix $F$ satisfies first condition of Theorem \[thm:Convergence\]. On the other hand, from the lower-block diagonal structure of the matrix $F$ and Corollary \[cor:alg and geo\] , it follows that $$\begin{aligned} geo_{F}(0) & =\dim\left\{ \ker F\right\} =\dim\left\{ \ker N\right\} +\dim\left\{ \ker\mathcal{L}_{r+1}\right\} +\cdots+\dim\left\{ \ker\mathcal{L}_{r+k}\right\} \\ & =0+geo_{\mathcal{L}_{r+1}}(0)+\cdots+geo_{\mathcal{L}_{r+k}}(0)=k\,.\end{aligned}$$ Remember that each matrix $\mathcal{L}_{r+i}$ is the Laplacian matrix of the tSCC $C_{r+i}$, ** so the $\dim\ker\left\{ \mathcal{L}_{r+i}\right\} =geo_{\mathcal{L}_{r+i}}(0)=1=alg_{\mathcal{L}_{r+i}}(0)$. ** Again the block diagonal structure of $F$ suggests that $$\begin{aligned} alg_{F}(0) & =alg_{\mathcal{M}_{1}}(0)+\cdots+alg_{\mathcal{M}_{r}}(0)+alg_{\mathcal{L}_{r+1}}(0)+\cdots+alg_{\mathcal{L}_{r+k}}(0)\\ & =0+\cdots+0+1+\cdots+1=k\,.\end{aligned}$$ Since each matrix $\mathcal{M}_{i}$ is non-singular, none of their eigenvalues are zero. Hence, it follows that the dynamics (\[eq:System with matrix F\]) satisfy the second condition of Theorem \[thm:Convergence\], $alg_{F}(0)=geo_{F}(0)=k$. Therefore, the matrix $F$ satisfies both conditions of Theorem ** \[thm:Convergence\], which in turn implies that the dynamics defined in (\[eq:System with matrix F\]) converge to a unique ES for any given initial condition. Now we will provide a framework for finding the ES solution of the SD dynamics. Define $R$ as an $n\times k$ matrix whose columns are a basis elements of the column null space (right kernel) of the matrix $F$.[^7] ** Analogously, define $L$ such that is a $k\times n$ matrix whose rows are basis elements of the row null space (left kernel) of the matrix $F$. Then these matrices satisfy $$F\cdot R=\mathbf{0}\text{ and }L\cdot F=\mathbf{0}$$ Naturally, $L$ and $R$ can be chosen so that the following equation hold $$L\cdot R=\mathbb{I}_{k}\,.\label{eq:LR}$$ Since matrices $L$ and $R$ are not uniquely defined, equation (\[eq:LR\]) serves as a normalization condition. In fact, in the subsequent section we discuss an example of such normalization. The following lemma gives a representation of the ES of the dynamics defined in (\[eq:System with matrix F\]) in terms of matrices $R$ and $L$. \[lem:steady state solution\]Assume that $F$ is a matrix for which (\[eq:LR\]) holds and when coupled with the initial condition $\mathbf{y}(0)=\mathbf{y}_{0}$ the solution of system (\[eq:System with matrix F\]) converges to a steady state $\mathbf{y}_{{\scriptscriptstyle ES}}$ as $t\to\infty$. Then $\mathbf{y}_{{\scriptscriptstyle ES}}=R\cdot L\cdot\mathbf{y}_{0}$. The solution of the dynamics (\[eq:System with matrix F\]) can be given in exponential form as $\mathbf{y}(t)=\exp(Ft)\cdot\mathbf{y}_{0}$, so $$\mathbf{y}(t)=\left(\mathbb{I}_{n}+\left(Ft\right)+\frac{(Ft)^{2}}{2!}+\cdots\right)\cdot\mathbf{y}_{0}=\left(\mathbb{I}_{n}+F\cdot A(t)\right)\cdot\mathbf{y}_{0},\label{eq:exponential}$$ where $A(t)$ is some matrix with time defined functions as entries. From (\[eq:LR\]) and (\[eq:exponential\]) it follows that $L\cdot\mathbf{y}(t)=L\mathbf{y}_{0}+L\cdot F\cdot A(t)\mathbf{y}_{0}=L\mathbf{y}_{0}$. Therefore, asymptotically as $t\to\infty$ we find that $L\cdot\mathbf{y}_{{\scriptscriptstyle ES}}=L\cdot\mathbf{y}(0)$. On the other hand steady state **$\mathbf{y}_{{\scriptscriptstyle ES}}$**, should also satisfy $\frac{d\mathbf{{\scriptstyle \mathbf{y}}}_{{\scriptscriptstyle {\scriptscriptstyle ES}}}}{dt}=\mathbf{0}=F\cdot\mathbf{y}_{{\scriptscriptstyle ES}}$, then vector $\mathbf{y}_{{\scriptscriptstyle ES}}$ is element of the column null space of the matrix $F$. In other words, the vector $\mathbf{y}_{{\scriptscriptstyle ES}}$ can be written as linear combination of the elements of the column null space, so there exist some vector $\mathbf{d}\in\mathbb{R}^{n\times1}$ such that $\mathbf{y}_{{\scriptscriptstyle ES}}=R\cdot\mathbf{d}$ . Therefore, $L\cdot\mathbf{y}(0)=L\cdot\mathbf{y}_{{\scriptscriptstyle ES}}=L\cdot\left(R\cdot\mathbf{d}\right)=\mathbf{d}$, so that $\mathbf{y}_{{\scriptscriptstyle ES}}=R\cdot\mathbf{d}=R\cdot L\cdot\mathbf{y}(0)$, as desired. Once the steady state solution of the dynamics (\[eq:System with matrix F\]) is found, the steady state solution of the SD dynamics (\[eq:3\]) can be found using back substitution: $$\mathbf{x}_{{\scriptscriptstyle ES}}=\mathbf{y}_{{\scriptscriptstyle ES}}-Q\cdot\mathbf{s}=R\cdot L\cdot\mathbf{y}_{0}-Q\cdot\mathbf{s}=R\cdot L\cdot\mathbf{x}_{0}+\left(R\cdot L-\mathbb{I}_{n}\right)\cdot Q\cdot\mathbf{s}\,.\label{eq:SS solution}$$ ### \[sub:Construction-of-matrices\]Construction of the matrices $R$ and $L$ We next discuss the graph theoretical procedure to construct matrices $R$ and $L$ that satisfy (\[eq:LR\]). The general strategy is to calculate matrix $R$ using Proposition \[prop:General kernel\], then construct the uniquely defined matrix $L$ that satisfies (\[eq:LR\]). Consider the block decomposition of matrix $F$ given in (\[eq:General case Decomposition\]), decompose matrices $R$ and $L$ such that $$F= \begin{blockarray}{ccc} \scriptstyle{n-u} &\scriptstyle{u} & \\ \begin{block}{\Left{}{(\;}c|c<{\;})c} N&\mathbf{0}&\scriptstyle{n-u}\\ \BAhhline{--||&} B&T&\scriptstyle{u}\\ \end{block} \end{blockarray},\qquad L=\begin{blockarray}{ccc} \scriptstyle{n-u} & \scriptstyle{u}& \\ \begin{block}{\Left{}{(\;}c|c<{\;})c} X&U&\scriptstyle{k}\\ \end{block} \end{blockarray}, \qquad R=\begin{blockarray}{cc} \scriptstyle{k}& \\ \begin{block}{\Left{}{(\;}c<{\;})c} Y&\scriptstyle{n-u}\\ \BAhhline{-||&} V&\scriptstyle{u}\\ \end{block} \end{blockarray} \label{LR partition}$$where $k$ is the number of tSCCs of complementary digraph $G^{\star}$, $u$ is number of vertices that are in tSCCs $C_{r+1},\dots,C_{r+k}$ , $X$ is $k\times(n-u)$, $U$ is $k\times u$, $N$ is $(n-u)\times(n-u)$, $B$ is $u\times(n-u)$, $T$ is $u\times u$, $Y$ is $(n-u)\times k$ and $V$ is $u\times k$. Consequently, the kernel elements of the matrix $F$ are constructed using Proposition \[prop:General kernel\], $\ker\left\{ F\right\} =span\left\{ \bar{\boldsymbol{\rho}}^{C_{r+1}},\cdots,\bar{\boldsymbol{\rho}}^{C_{r+k}}\right\} $ using the tSCCs, $\left\{ C_{r+1},\cdots,C_{r+k}\right\} $ and thus we have $F\cdot\bar{\boldsymbol{\rho}}^{C_{r+i}}=0,\quad i=1,\dots,k$. Let $\hat{\boldsymbol{\rho}}^{C_{r+i}}$ be normalized version of $\bar{\boldsymbol{\rho}}^{C_{r+i}}$ such that $$\mathbbm{1}^{T}\cdot\hat{\boldsymbol{\rho}}^{C_{r+i}}=1\,.\label{eq:normalize}$$ Then the $i^{th}$ column of $R$ is defined as $R_{i}=\hat{\boldsymbol{\rho}}^{C_{r+i}}$ such that $$F\cdot R_{i}=\mathbf{0}\,.$$ As a result we have matrix $R$ which satisfies $$F\cdot R=\mathbf{0}\,.\label{eq:FR}$$ In other words, for any vertex $j\not\in\mathcal{V}\left(C_{r+i}\right)$, $\left(R_{i}\right)_{j}=0$. This in turn implies that $Y\equiv\mathbf{0}$ and $V$ is non-zero matrix corresponding to the tSCCs of complementary digraph $G^{\star}$. Then we can construct the matrix $L$ using the following process. Let $U$ be the matrix given by first transposing $V$ and then replacing each nonzero element of $V$ by $1$. Since $\mathbbm{1}^{T}\cdot\mathcal{L}(C_{r+i})=\mathbbm{1}^{T}\cdot\mathcal{L}_{r+i}=\mathbf{0}$, we have that $U\cdot T=\mathbf{0}$ and by (\[eq:normalize\]) we have that $U\cdot V=\mathbb{I}_{k}$. Let the matrix $X$ be constructed as $X=-U\cdot B\cdot N^{-1}$. With this definition and (\[LR partition\]), we can see that the matrix $L$ satisfies the criteria $L\cdot F=\mathbf{0}$, $$L\cdot F=\left(\begin{array}{c|c} X & U\end{array}\right)\cdot\left(\begin{array}{c|c} N & \mathbf{0}\\ \hline B & T \end{array}\right)=\left(\begin{array}{c|c} X\cdot N+U\cdot B & U\cdot T\end{array}\right)=\left(\begin{array}{c|c} -U\cdot B\cdot N^{-1}\cdot N+U\cdot B & U\cdot T\end{array}\right)=\mathbf{0}\label{eq:LF}$$ Moreover, by definition, (\[LR partition\]), the matrices $L$ and $R$ also satisfy (\[eq:LR\]) as illustrated by $$L\cdot R=\left(\begin{array}{c|c} X & U\end{array}\right)\cdot\left(\begin{array}{c} Y\\ \hline V \end{array}\right)=X\cdot Y+U\cdot V=0+\mathbb{I}_{k}=\mathbb{I}_{k}\label{eq:LRI}$$ Therefore by (\[eq:FR\]), (\[eq:LF\]) and (\[eq:LRI\]) we can conclude that constructed matrices $R$ and $L$ satisfies (\[eq:LR\]) and so the ES of the SD dynamics (\[eq:3\]) is given by (\[eq:SS solution\]). \[sec:Illustrarion-of-Results\]Illustration of Results ------------------------------------------------------ Consider the directed graph $G$ with $5$ vertices, given in Figure \[fig: 3.3 Digraph G\] with its complementary digraph $G^{\star}$ in Figure \[fig: 3.3 Complementary-digraph\]. We have labeled vertices according to the labeling procedure described in Section \[sec:Preliminary-results\]. We note that the complementary digraph $G^{\star}$ of digraph $G$ is not strongly connected, with non-terminal SCC $C_{1}$ with $\mathcal{V}\left(C_{1}\right)=\{1,2,3,\star\}$ and two tSCCs, $C_{2}$ with $\mathcal{V}\left(C_{2}\right)=\left\{ 4\right\} $ and $C_{3}$ with $\mathcal{V}\left(C_{3}\right)=\left\{ 5\right\} $. The Laplacian matrix for this digraph is $$\mathcal{L}(G)=\left(\begin{array}{ccccc} -(a+c+d) & b & 0 & 0 & 0\\ a & -(b+e+f) & 0 & 0 & 0\\ c & 0 & 0 & 0 & 0\\ d & e & 0 & 0 & 0\\ 0 & f & 0 & 0 & 0 \end{array}\right)\,.\label{eq:Laplacian of illustration}$$ The degradation and synthesis are given by $D=diag\left(0,h,i,0,0\right)$, $\mathbf{s}=\left(g,0,k,0,l\right)^{T},$ respectively. The associated SD dynamics is given by a system of ODEs as in (\[eq:3\]). So the matrix $F=\mathcal{L}(G)-D$ with its corresponding partitioning is given by $$F=\left(\begin{array}{c|c} N & 0\\ \hline B & T \end{array}\right)=\left(\begin{array}{ccc|cc} -(a+c+d) & b & 0 & 0 & 0\\ a & -(b+e+f+h) & 0 & 0 & 0\\ c & 0 & i & 0 & 0\\ \hline d & e & 0 & 0 & 0\\ 0 & f & 0 & 0 & 0 \end{array}\right)\label{eq:illus F}$$ Then we partition the vector $\mathbf{s}$ such that it matches with the partitioning of $F$, so that we have $$\mathbf{s}=\left(\begin{array}{c} \mathbf{s}'\\ \hline \mathbf{s}'' \end{array}\right),\text{ where }\mathbf{s}'=\left(\begin{array}{c} g\\ 0\\ k \end{array}\right)\text{ and }\mathbf{s}''=\left(\begin{array}{c} 0\\ l \end{array}\right)\,.$$ In order for an ES to exist, the vector $\mathbf{s}$ should satisfy the necessary and sufficient conditions given in Theorem \[thm:BNS\]. By the first condition we have that $\mathbf{s}''\equiv\mathbf{0}$, and thus $\boxed{l=0}$ implying that we cannot have synthesis on vertex $5$. Next, we check the second condition of Theorem \[thm:BNS\], $$\mathbf{0}=BN^{-1}\mathbf{s}'=\mathbbm{1}^{T}BN^{-1}\mathbf{s}'=\left(\begin{array}{ccc} -\frac{a(e+f)+d(b+e+f+h)}{b(c+d)+(a+c+d)(e+f+h)} & -\frac{bd+(a+c+d)(e+f)}{b(c+d)+(a+c+d)(e+f+h)} & 0\end{array}\right)\left(\begin{array}{c} g\\ 0\\ k \end{array}\right)\,,$$ and conclude that $\boxed{g=0}$. Hence, the steady state solution exists if and only if $g=l=0$. The digraphs $G$ and $G^{\star}$ which are compatible with having a steady state are depicted in Figure \[fig:ES-compatible-digraph\]. In a comparison with Figure \[fig: 3.3 Digraph G\], we note the absence of synthesis on nodes $1$ and $5$. Every matrix is defined as in (\[eq:illus F\]) except for the vector $\mathbf{s}=(0,0,k,0,0)^{T}.$ As before $G^{\star}$ has two tSCCs, $C_{2}$ with $\mathcal{V}\left(C_{2}\right)=\{4\}$ and $C_{3}$ with $\mathcal{V}\left(C_{3}\right)=\{5\}$. Now with the necessary and sufficient conditions in hand, we will follow the construction process for the matrices $R$ and $L$ described in Section \[sub:Construction-of-matrices\]. Since tSCCs, $C_{2}$ and $C_{3}$, each have only one vertex, we have normalized vectors $\hat{\boldsymbol{\rho}}^{C_{2}}=(0,0,0,1,0)^{T}$ and $\hat{\boldsymbol{\rho}}^{C_{3}}=(0,0,0,0,1)$. So the columns of the matrix $R$ are defined as $R_{1}=\hat{\boldsymbol{\rho}}^{C_{2}},\, R_{2}=\hat{\boldsymbol{\rho}}^{C_{3}}$, $$R=\left(\begin{array}{cc} 0 & 0\\ 0 & 0\\ 0 & 0\\ \hline 1 & 0\\ 0 & 1 \end{array}\right)=\left(\begin{array}{c} Y\\ \hline V \end{array}\right)$$ Therefore, $V=\left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right)$, then transposing $V$ and writing $1$’s instead of its non-zero elements we get $U=\left(\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right)$. And so the matrix $X$ is $$X=-U\cdot B\cdot N^{-1}=\frac{1}{b(c+d)+(a+c+d)(e+f+h)}\left(\begin{array}{ccc} af & (a+c+d)f & 0\\ ae+d(b+e+f+h) & bd+(a+c+d)e & 0 \end{array}\right)$$ So $$L=\left(\begin{array}{c|c} X & U\end{array}\right)=\left(\begin{array}{ccc|cc} \frac{af}{b(c+d)+(a+c+d)(e+f+h)} & \frac{(a+c+d)f}{b(c+d)+(a+c+d)(e+f+h)} & 0 & 0 & 1\\ \frac{ae+d(b+e+f+h)}{b(c+d)+(a+c+d)(e+f+h)} & \frac{bd+(a+c+d)e}{b(c+d)+(a+c+d)(e+f+h)} & 0 & 1 & 0 \end{array}\right)$$ Accordingly, by (\[eq:SS solution\]) the ES, $\mathbf{x}_{{\scriptscriptstyle ES}}$, is given by $$\mathbf{x}_{{\scriptscriptstyle ES}}=R\cdot L\cdot\mathbf{x}_{0}+\left(R\cdot L-\mathbb{I}_{n}\right)\cdot F^{+}\cdot\mathbf{s}=R\cdot L\cdot\mathbf{x}_{0}+\left(\begin{array}{ccccc} 0 & 0 & \frac{k}{i} & 0 & 0\end{array}\right)^{T}\,.$$ \[sec: negativity of inverse\] Inverse of Non-Singular Perturbed ** Matrices ============================================================================ In our previous paper [@MirzaevGunawardena2013bmb], we have proven that the perturbed Laplacian matrix of a strongly connected digraph is non-singular. Nevertheless, we claimed that the inverse of such a matrix has non-positive entries. In this section, we prove our claim and provide a graph theoretic algorithm for the computation of the inverse of perturbed matrices. Once again consider a digraph $G$ with $n$ nodes. As before Laplacian matrix for this digraph is given by matrix $\mathcal{L}(G)\in\mathbb{R}^{n\times n}$ and perturbed matrix is defined as $P=\mathcal{L}(G)-\Delta$, where $\Delta$ is diagonal matrix with non-negative entries, $$\left(\Delta\right)_{ij}=\begin{cases} \delta_{i} & i=j\\ 0 & i\not=j \end{cases}\,.$$ Remember from Remark \[lem:perturbed\] that a perturbed matrix of a strongly connected digraph is a non-singular matrix. However, a perturbed matrix of an arbitrary digraph is not necessarily non-singular. Here, we will prove that the inverse of any non-singular perturbed matrix is a non-positive matrix. By that we mean all the elements of the inverse matrix, $P^{-1}$, are non-positive real numbers (henceforth, $P$ represents a non-singular perturbed matrix). To accomplish this we first prove the statement for the case when the digraph $G$ is strongly connected and then prove it for an arbitrary digraph. \[sub:Strongly-connected-case inverse\]Strongly connected case -------------------------------------------------------------- When the digraph $G$ is strongly connected we will use the explicit formulation of the inverse of a non-singular matrix $P$, derived from Laplace expansion of the determinant, $$P^{-1}=\frac{1}{\det(P)}adj(P)\label{eq:adjugate}$$ where $\left(adj(P)\right)_{ij}=(-1)^{i+j}P_{(ji)}$, the $ij$-th entry of the adjugate being the $(ji)$-th minor of $P$ (up to the sign). At this point we can reconstruct the $i$-th row and $j$-th column of the matrix $P$ such that the constructed matrix is the Laplacian matrix of a strongly connected digraph denoted as $G^{ij}$. Then it follows that $\mathcal{L}(G^{ij})_{(ij)}=P_{(ij)}$. An example illustration of such a construction is given in Figure \[fig:Reconstruction-of-the\]. Thus the $ij$-th minor of the matrix $P$ can be calculated as the $ij$-th minor of the matrix $\mathcal{L}(G^{ij})$, which in turn can be calculated using the MTT, $$\begin{aligned} \left(adj(P)\right)_{ji} & =(-1)^{i+j}P_{(ij)}=(-1)^{i+j}\mathcal{L}(G^{ij})_{(ij)}\nonumber \\ & =(-1)^{i+j}(-1)^{n+i+j-1}\left(\boldsymbol{\rho}^{G^{ij}}\right)_{i}=(-1)^{n-1}\left(\boldsymbol{\rho}^{G^{ij}}\right)_{i}\,.\label{eq:constructed G}\end{aligned}$$ +=\[remember picture\] (n1) at (-10,0) [$\displaystyle P=\left(\begin{array}{ccc} -a-\delta_{1} & 0 & c\\ \tikz[baseline]{\node[rectangle,draw=black!90,thick, anchor=base] (t1){$a$};} & -b-\delta_{2} & 0\\ 0 & b & -c \end{array}\right)$]{}; (n3) at (-10,-5) [$\displaystyle \mathcal{L}(G^{21})=\left(\begin{array}{ccc} -a-\delta_{1} & 0 & c\\ \tikz[baseline]{\node[rectangle,draw=black!90,thick, anchor=base] (t2){$a+\delta_1$};} & -b & 0\\ 0 & b & -c \end{array}\right)$]{}; (n4) at (-4,-2.5) [$\displaystyle P_{(21)}=\mathcal{L}(G^{21})_{(21)}=\left|\begin{array}{cc} 0 & c\\ b & -c \end{array}\right|$]{}; \(1) at (1,0) [1]{}; (2) at (4,0) [2]{}; (3) at (2.5,2) [3]{}; (55) at (5.5,1.5) [$G$]{}; (11) at (1,-6) [1]{}; (22) at (4,-6) [2]{}; (33) at (2.5,-4) [3]{}; (44) at (5.5,-4.5) [$G^{21}$]{}; \(1) edge node \[below\][$a$]{} (2); (2) edge node \[right\][$b$]{} (3); (3) edge node \[left\]\[left\][$c$]{} (1); (1) edge node \[left\][$\delta_1$]{} (1,-2); (2) edge node \[left\][$\delta_2$]{} (4,-2); (11) edge node \[below\][$a+\delta_1$]{} (22); (22) edge node \[right\][$b$]{} (33); (33) edge node \[left\]\[left\][$c$]{} (11); (t1) edge \[out=220, in=180\] (n4); (t2) edge \[out=150, in=180\] (n4); (n1.west) edge \[bend right\] (n3.west); Since the digraph $G^{ij}$ is strongly connected, from Proposition \[prop:SCC kernel\] we find that $(\boldsymbol{\rho}^{G^{ij}})_{i}>0$ for each index $i$. As for the determinant of the matrix $P$, we can again add one row and one column to the matrix $P$ such that the constructed $(n+1)\times(n+1)$ matrix is the Laplacian matrix of strongly connected digraph $G^{\, n+1\, n+1}$. For the convenience of the notation, hereafter, we refer to the digraph $G^{\, n+1\, n+1}$ simply as $G^{\, n+1}$. Then by ** the MTT it is implied that $$\det(P)=\mathcal{L}(G^{\, n+1})_{(n+1\, n+1)}=(-1)^{3(n+1)-1}\left(\boldsymbol{\rho}^{G^{\, n+1}}\right)_{n+1}=(-1)^{n}\left(\boldsymbol{\rho}^{G^{\, n+1}}\right)_{n+1}\,.\label{eq:determinant}$$ Note that the construction of such a row and a column is independent of the existing rows, so we can always reconstruct a strongly connected digraph $G^{\, n+1}$. Consequently, from Proposition \[prop:SCC kernel\] it follows that $(\boldsymbol{\rho}^{G^{n+1}})_{n+1}>0$. Therefore, ** (\[eq:adjugate\])*,* (\[eq:constructed G\]) and (\[eq:determinant\]) together imply that $$(P^{-1})_{ij}=\frac{1}{det(P)}\left(adj(P)\right)_{ij}=\frac{(-1)^{n-1}\left(\boldsymbol{\rho}^{G^{ij}}\right)_{i}}{(-1)^{n}\left(\boldsymbol{\rho}^{G^{n+1}}\right)_{n+1}}=-\frac{\left(\boldsymbol{\rho}^{G^{ij}}\right)_{i}}{\left(\boldsymbol{\rho}^{G^{n+1}}\right)_{n+1}}<0\,,$$ and thus all the entries of the matrix $P^{-1}$ are strictly less than zero. General Case ------------ In the general case, the perturbed matrix of an arbitrary digraph $G$ may not be a non-singular matrix. In Section \[sec:Theoretical-Development\] we have seen that for a perturbed matrix to be non-singular, the diagonal blocks associated to each SCC should be a perturbed matrix on their own. Consider an arbitrary digraph $G$ with $q$ SCCs, $C_{1},\cdots,C_{q}$, we assume that the matrix $P$ can be partitioned analogous to (\[eq:partition\]), $$P=\left(\begin{array}{ccc} \boxed{\mathcal{P}_{1}} & \cdots & \mathbf{0}\\ \vdots & \ddots & \vdots\\ + & + & \boxed{\mathcal{P}_{q}} \end{array}\right)\,,$$ where each $\mathcal{P}_{i}=\mathcal{L}(C_{i})-\Delta_{i}$ is an $a_{i}\times a_{i}$ perturbed matrix of SCC $C_{i}$. Having this decomposition in hand, we are ready to prove that the inverse of the matrix $P$ has all non-positive entries. To do that we will use the results of Section \[sub:Strongly-connected-case inverse\] and follow the standard path for finding the inverse using the method of Gaussian elimination,$$\begin{aligned} (\begin{array}{c:c}P & \mathbb{I}_{n}\end{array})= &\left(\begin{array}{ccc:ccc}\boxed{\mathcal{P}_{1}} & \cdots & \mathbf{0} & \boxed{\mathbb{I}_{a_{1}}} & \cdots & \mathbf{0}\\\vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\+ & + & \boxed{\mathcal{P}_{q}} & \mathbf{0} & \mathbf{0} & \boxed{\mathbb{I}_{a_{q}}}\end{array}\right)\\ \longrightarrow&\left(\begin{array}{ccc:ccc}\boxed{\mathbb{I}_{a_{1}}} & \cdots & \mathbf{0} & \boxed{\mathcal{P}_{1}^{-1}} & \cdots & \mathbf{0}\\\vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\- & - & \boxed{\mathbb{I}_{a_{q}}} & \mathbf{0} & \mathbf{0} & \boxed{\mathcal{P}_{q}^{-1}}\end{array}\right)\end{aligned}$$ where minus sign, $-$, stands for some matrix with non-positive real entries. From Section \[sub:Strongly-connected-case inverse\] each $\mathcal{P}_{i}^{-1}$ has negative entries. Hence multiplying a block of rows having non-negative real numbers by $\mathcal{P}_{i}^{-1}$ transforms elements of those rows into non-positive real numbers. Consider the $ij$-the entry, $-a_{ij}$, of the LHS. Note that $jj$-th entry of LHS is equal to $1$, e.g. $a_{jj}=1$. Then in order to eliminate this negative element on the LHS of the augmented matrix (i.e., $-a_{ij}$ ) we have to multiply the $j$-th row by a positive real number, $a_{ij}$, and add resulting row to the $i$-th row. Since all the elements of RHS are non-positive, this operation places non-positive real numbers on $i$-th row of RHS. Performing operations consecutively, on columns $1,2,\dots,n$ lead to the following matrix $$\longrightarrow\left(\begin{array}{ccc:ccc}\boxed{\mathbb{I}_{a_{1}}} & \cdots & \mathbf{0} & \boxed{\mathcal{P}_{1}^{-1}} & \cdots & \mathbf{0}\\\vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\\mathbf{0} & \mathbf{0} & \boxed{\mathbb{I}_{a_{q}}} & - & - & \boxed{\mathcal{P}_{q}^{-1}}\end{array}\right)=\left(\begin{array}{c:c}\mathbb{I}_{n} & P^{-1}\end{array}\right) \label{eq:general inverse}$$ As it can be observed from the above equation, (\[eq:general inverse\]), all the entries of the matrix $P^{-1}$ are non-positive real numbers, as desired. Note that the inverse matrix $P^{-1}$ preserves lower-block diagonal structure of the perturbed matrix $P$. Moreover, recall that the matrix $N$ defined in (\[eq: Definition of N\]) is also a non-singular perturbed matrix. Hence, all the entries of the inverse matrix $N^{-1}$ are non-positive real numbers. \[sec:Computation-of inverse of N\]Symbolic computation of $P^{-1}$ based on the Matrix-Tree Theorem ---------------------------------------------------------------------------------------------------- For a given set of constant edge weights numerical computation of the inverse matrix $P^{-1}$ is challenging, and even more challenging is symbolic computation of the inverse. Therefore, here we provide a graph theoretic algorithm for the symbolic computation of $P^{-1}$. The algorithm is again based on the MTT, and utilizes the theory developed in Section \[sub:SC framework\]. We start by introducing strictly positive synthesis edges at each vertex, i.e. $\mathbf{s}=(s_{1},\dots,s_{n})^{T}\in\mathbb{R}_{>0}^{n\times1}$. This makes the complementary digraph $G^{\star}$ strongly connected, since the vertex $\star$ can be reached from any other vertex and any vertex can be reached from $\star$. **** After applying the framework of the strongly connected case (Section \[sub:SC framework\], (\[eq: SC steady state MTT\])) for the symbolic synthesis vector, $\mathbf{s}\in\mathbb{R}_{>0}^{n\times1}$, we get the graph theoretic representation of the ES, $\mathbf{p}$, i.e. $$\left(\mathbf{p}\right)_{i}=\frac{\left(\boldsymbol{\rho}^{G^{\star}}\right)_{i}}{\left(\boldsymbol{\rho}^{G^{\star}}\right)_{\star}}\,.\label{eq:ES for P}$$ Consider the standard basis of $\mathbb{R}^{n}$ $$\left\{ \mathbf{e}^{(i)}=(0,\dots,0,1,0,\dots,0)\right\} _{i=1}^{n}\,.$$ Then, multiplying $P^{-1}$ by the vector $\mathbf{e}^{(i)}$ yields the $i$-th column of the inverse matrix $P^{-1}$. Thus we construct $P^{-1}$ by constructing its one column at a time. For that consider the following combinations of specific synthesis edges, $$\mathbf{s}^{(0)}=(1,\dots,1)^{T},\ ,\mathbf{s}^{(i)}=(1,\dots,1,2,1,\dots,1)^{T}\quad i=1,\dots,n\label{eq:Specific synthesis}$$ where only $2$ is the $i$-th entry of the vector $\mathbf{s}^{(i)}$. One can then easily observe that $$\left\{ \mathbf{s}^{(i)}-\mathbf{s}^{(0)}=\mathbf{e}^{(i)}=(0,\dots,0,1,0,\dots,0)^{T}\ i=1,\dots,n\right\}$$ is standard basis for **$\mathbb{R}^{n}$**. On the other hand substituting the vectors $\left\{ \mathbf{s}^{(0)},\mathbf{s}^{(1)},\dots,\mathbf{s}^{(n)}\right\} $ into (\[eq:ES for P\]) gives rise to the steady states $\left\{ \mathbf{p}^{(0)},\mathbf{p}^{(1)},\dots,\mathbf{p}^{(n)}\right\} $, respectively.[^8] Then again these steady states can be computed algebraically using ** (\[eq:4\]), $$-P^{-1}\cdot\mathbf{s}^{(i)}=\mathbf{p}^{(i)}\qquad\forall i=0,1,\dots,n.$$ This in turn can be simplified to $$P^{-1}\cdot\left(\mathbf{s}^{(i)}-\mathbf{s}^{(0)}\right)=P^{-1}\cdot\mathbf{e}^{(i)}=\mathbf{p}^{(0)}-\mathbf{p}^{(i)}\,.$$ Since $\left\{ \mathbf{e}^{(1)},\dots,\mathbf{e}^{(n)}\right\} $ is a standard basis for $\mathbb{R}^{n\times1}$, $i$-th column of the matrix $P^{-1}$ is given by the vector $\mathbf{p}^{(0)}-\mathbf{p}^{(i)}$ or simply as $$P^{-1}=\left(\begin{array}{c|c|c|c} \mathbf{p}^{(0)} & \mathbf{p}^{(1)} & \mathbf{\cdots} & \mathbf{p}^{(n)}\end{array}\right)\cdot\left(\begin{array}{cccc} 1 & 1 & \cdots & 1\\ -1 & 0 & \cdots & 0\\ 0 & -1 & \ddots & \vdots\\ \vdots & 0 & \ddots & 0\\ 0 & \vdots & \cdots & -1 \end{array}\right)\label{eq:Symbolic inverse}$$ Note that since spanning trees rooted at vertex $\star$ cannot contain any outgoing edge from vertex $\star$, the synthesis edges $\left\{ s_{1},\dots,s_{n}\right\} $ will not contribute to $\left(\boldsymbol{\rho}^{G^{\star}}\right)_{\star}$. Hence, $\left(\boldsymbol{\rho}^{G^{\star}}\right)_{\star}$ remains same for each substitution of $\mathbf{s}^{(i)}$. Consequently, we don’t have to calculate $\left(\boldsymbol{\rho}^{G^{\star}}\right)_{\star}$ each time and divide the other terms by it, we can just factor out and perform the division at the end. We will illustrate the algorithm presented in this subsection with a simple example. Consider the perturbed matrix $$P=\left(\begin{array}{cc} -a & 0\\ a & -b \end{array}\right)$$ which is simple a $2\times2$ matrix whose inverse is $$P^{-1}=\frac{1}{ab}\left(\begin{array}{cc} -b & 0\\ -a & -a \end{array}\right)\,.$$ On the other hand, by (\[eq:ES for P\]) the symbolic ES is given by $$\mathbf{p}=\frac{1}{ab}\left(\begin{array}{c} s_{1}b\\ s_{2}a+s_{1}a \end{array}\right)\,.\label{eq:illus P inverse}$$ Then, substituting the synthesis vectors defined in (\[eq:Specific synthesis\]) into (\[eq:illus P inverse\]) we find that $$\mathbf{p}^{(0)}=\frac{1}{ab}\left(\begin{array}{c} b\\ 2a \end{array}\right),\ \mathbf{p}^{(1)}=\frac{1}{ab}\left(\begin{array}{c} 2b\\ 3a \end{array}\right),\ \mathbf{p}^{(2)}=\frac{1}{ab}\left(\begin{array}{c} b\\ 3a \end{array}\right)\,.$$ Thus the inverse matrix $P^{-1}$ is given by (\[eq:Symbolic inverse\]), $$P^{-1}=\left(\begin{array}{c|c|c} \mathbf{p}^{(0)} & \mathbf{p}^{(1)} & \mathbf{p}^{(3)}\end{array}\right)\cdot\left(\begin{array}{cc} 1 & 1\\ -1 & 0\\ 0 & -1 \end{array}\right)=\frac{1}{ab}\left(\begin{array}{ccc} b & 2b & b\\ 2a & 3a & 3a \end{array}\right)\cdot\left(\begin{array}{cc} 1 & 1\\ -1 & 0\\ 0 & -1 \end{array}\right)=\frac{1}{ab}\left(\begin{array}{cc} -b & 0\\ -a & -a \end{array}\right)\,.$$ \[sec:Biochemical-Network-Application\]Biochemical Network Application ====================================================================== In this section we will describe how the above developed framework is useful for symbolic computation of the steady state solutions of biochemical reaction networks. Secretion of insulin granules in $\beta$-cells ---------------------------------------------- One of the most prevalent diseases, diabetes mellitus (or simply diabetes) is characterized by high level of blood glucose. Diabetes results from either pancreas does not release enough insulin, or cells do not respond to insulin produced with increased consumption of sugar, or combination of both [@Barg2003]. Insulin is blood glucose-lowering hormone produced, processed and stored in secretory granules by pancreatic $\beta$-cells in Langerhans islets [@Olofsson2002]. Consequently, secretory granules are released to extracellular space, which is regulated by Ca^2+^- dependent exocytosis [@Wollheim1981]. Since diabetes is related to secretional malfunctions [@Rorsman2003], studying mechanism of both normal and pathological insulin release in molecular level is crucial for understanding of disease process. ![Schematic drawing of exocytosis cascade in **$\beta$**-cells. *On the left,* insulin granules produced in Golgi network is transported into extracellular space through exocytosis. *On the right,* particular steps involved in the exocytosis of the insulin granules. The numbers stand for: 1) Re-supply 2) Priming 3) Domain Binding 4) Ca Triggering 5) Fusion 6) Pore Expansion 7) Insulin Release \[fig:scheme\]](Insulin) Chen et al. [@Chen2008] developed mathematical model of $\beta$-cell to calculate both rate of granule fusion and the rate of insulin secretion in $\beta$-cells stimulated with electrical potential. The model is based on five-state kinetic model of granule fusion proposed by Voets et al. [@Voets1999]. Figure \[fig:scheme\] illustrates kinetic scheme proposed for exocytosis in pancreatic $\beta$-cells. As it is shown in the figure the model accounts for steps involved in exocytosis cascade such as re-suply, priming, domain binding, Ca^2+^ triggering, fusion, pore expansion and insulin release. It is assumed that L-type (not the R-type) voltage-sensitive Ca^2+^-channels are used for secretion of primed granules through cell membrane. During this process “microdomains” with high Ca^2+^ concentration are formed at the inner mouth of L-type channels (illustrated as circles in Figure \[fig:scheme\]). Concentration of Ca^2+^ in cytosol and microdomain at time $t$ are denoted by $C_{i}(t)$ and $C_{md}(t)$, respectively. Since the number of granules are far less than number of Ca^2+^, it is also assumed that dynamics of Ca^2+^ is independent of exocytosis cascade. For further details we refer the reader to the original paper [@Chen2008]. Figure \[fig:Digraph Exocytosis\] illustrates the dynamics associated with exocytosis cascade as a digraph $G$. Since complementary digraph of $G$ (Figure \[fig:Complementary-digraph Exocytosis\]) is strongly connected, the steady state solutions of dynamics can be calculated using the algorithm described in Section \[sub:SC framework\]. The ES is given as $$N_{{\scriptscriptstyle ES}}=\Delta\left(\begin{array}{c} 6k_{-1}^{3}+2u_{1}k_{-1}^{2}+C_{\text{md}}k_{1}u_{1}k_{-1}+2C_{\text{md}}^{2}k_{1}^{2}u_{1}\\ 18C_{\text{md}}k_{1}k_{-1}^{2}+6C_{\text{md}}k_{1}u_{1}k_{-1}+3C_{\text{md}}^{2}k_{1}^{2}u_{1}\\ 18C_{\text{md}}^{2}k_{-1}k_{1}^{2}+6C_{\text{md}}^{2}u_{1}k_{1}^{2}\\ 6C_{\text{md}}^{3}k_{1}^{3}\\ \frac{6r_{-1}k_{-1}^{3}}{r_{1}}+\frac{2r_{-1}u_{1}k_{-1}^{2}}{r_{1}}+\frac{C_{\text{md}}k_{1}r_{-1}u_{1}k_{-1}}{r_{1}}+\frac{6C_{\text{md}}^{3}k_{1}^{3}u_{1}}{r_{1}}+\frac{2C_{\text{md}}^{2}k_{1}^{2}r_{-1}u_{1}}{r_{1}}\\ \frac{6r_{-2}r_{-1}k_{-1}^{3}}{r_{1}r_{2}}+\frac{2r_{-2}r_{-1}u_{1}k_{-1}^{2}}{r_{1}r_{2}}+\frac{C_{\text{md}}k_{1}r_{-2}r_{-1}u_{1}k_{-1}}{r_{1}r_{2}}+\frac{6C_{\text{md}}^{3}k_{1}^{3}u_{1}}{r_{2}}+\frac{6C_{\text{md}}^{3}k_{1}^{3}r_{-2}u_{1}}{r_{1}r_{2}}+\frac{2C_{\text{md}}^{2}k_{1}^{2}r_{-2}r_{-1}u_{1}}{r_{1}r_{2}}\\ \frac{6C_{\text{md}}^{3}k_{1}^{3}u_{1}}{u_{2}}\\ \frac{6C_{\text{md}}^{3}k_{1}^{3}u_{1}}{u_{3}} \end{array}\right)$$ where $\Delta$ is given as follows $$\begin{array}{c} \Delta=\frac{r_{1}r_{2}r_{3}}{r_{-1}r_{-2}r_{-3}\left(\frac{6k_{1}^{3}r_{1}u_{1}C_{\text{md}}^{3}}{r_{-2}r_{-1}}+\frac{6k_{1}^{3}r_{1}r_{2}u_{1}C_{\text{md}}^{3}}{r_{-3}r_{-2}r_{-1}}+\frac{6k_{1}^{3}u_{1}C_{\text{md}}^{3}}{r_{-1}}+k_{1}k_{-1}u_{1}C_{\text{md}}+2k_{1}^{2}u_{1}C_{\text{md}}^{2}+2k_{-1}^{2}u_{1}+6k_{-1}^{3}\right)}\end{array}$$ As we can see the ES gets complicated for the large graphs. However, our framework provides steady state value of any given substrate (see (\[eq: SC steady state MTT\])), which is not easily found by numerical simulations. At the resting state (electric potential set to $V=-70\, mV$ ), concentration of Ca^2+^ in the microdomain is very low, so it is assumed that $C_{md}=C_{md}(t)\approx0$. In this case, the complementary digraph of $G$ is no longer strongly connected, which is given in Figure \[fig:Resting state\]. Then the ES solution have to be computed by the process described in Section \[sub:Construction-of-matrices\], and is given as $$N_{r,{\scriptscriptstyle ES}}=\left(\begin{array}{c} \frac{r_{1}r_{2}r_{3}}{r_{-3}r_{-2}r_{-1}}\\ 0\\ 0\\ 0\\ \frac{r_{2}r_{3}}{r_{-3}r_{-2}}\\ \frac{r_{3}}{r_{-3}}\\ 0\\ 0 \end{array}\right)$$ In the above example the dynamics were essentially linear. Although the framework in this paper is linear in nature, it can be applied to nonlinear systems as well. This can be done by incorporating nonlinearity into the framework through the edge labels. So far we treated edge weights as uninterpreted symbols. In fact, edge weights can be an arbitrary positive rational expressions. For example, the Michaelis-Menten formula used in enzyme kinetics is a legitimate edge weight $$a=\frac{V_{max}[S]}{K_{m}+[S]}\,,$$ where $[S]$ stands for the concentration of the substrate $S$, $K_{m}$ and $V_{max}$ are reaction specific constants. However, in most cases a chemical reaction network modeled with mass action kinetics, which gives rise to a nonlinear system of ODEs. The steady states of this type of dynamics can also be algorithmically computed using our framework. For instance, a chemical reaction of type $$A+B\overset{k}{\longrightarrow}C$$ can be represented in our way as $$A\overset{kB}{\longrightarrow}C$$ One can then use above formalism to transform chemical reactions into a digraph with time dependent edge weights. Consequently, this digraph can be used to calculate the steady states using our framework. One should keep in mind that in a such transformation only the equilibrium solutions coincide not the transient dynamics [@Gunawardena2012]. For more extensive discussion of the topic we refer the reader to [@Gunawardena2012] and [@Gunawardena2014]. Next we illustrate such incorporation by applying it to a nonlinear biochemical network. Glucose metabolism in $\beta$-cells ----------------------------------- In their paper Sweet and Matschinsky [@Sweet1995] setup a mathematical model of pancreatic $\beta$-cell glucose metabolism to investigate the relation between glucose and the rate of glycolysis (see Figure \[fig:Schematic-glycolysis\]). Since glucose metabolism in $\beta$-cells indirectly affects the rate of insulin secretion [@Pedersen2009], this type of models have implications for the diabetes treatment. All reactions together make dynamics overwhelmingly complex. To avoid this authors assumed that reactions inside dashed rectangles (pools) are operating sufficiently fast, and have reached thermodynamic equilibrium. Then ordinary differential equations are written for the rates of transfers between these pools. Consequently, equilibrium metabolites in a pool are calculated algebraically using equilibrium assumptions. Although the model is minimalistic, it still includes parameters describing overall behavior glycolysis in $\beta$-cell. Nonlinear dynamics associated with the model can be given as in figure \[fig:Glycolysis-in-pancreatic\]A. In this case nonlinearity is hidden in label $f$, $$f=\frac{G_{PK}D\sqrt{b^{2}-b-\frac{8K_{TPI}}{V_{c}}{\scriptstyle GIP(t)}}}{{\scriptstyle GIP(t)}}$$ where ${\scriptstyle GIP(t)}$ stands for concentration of ${\scriptstyle GIP}$ at time $t$, and every other letter are reaction rate constants. Since complementary digraph of $G$ (Figure \[fig:Glycolysis-in-pancreatic\]b) is strongly connected, the ES solutions to the system can be obtained by the procedure described in Section \[sub:SC framework\]: $$\left(\begin{array}{c} Glu^{*}\\ H-6-P^{*}\\ \frac{1}{2}GIP^{*}\\ Pyr^{*} \end{array}\right)=\frac{1}{bdh+beh+ceh}\left(\begin{array}{c} adh+aeh\\ ach\\ \frac{aceh}{f}\\ ace+bdg+beg+ceg \end{array}\right)$$ ![Schematic model of glycolysis in pancreatic $\beta$-cells \[fig:Schematic-glycolysis\]](glucose) \[sec:Conclusions\]Conclusions and future work ============================================== In our previous work, we have developed a “linear framework” for symbolical computation of equilibrium solutions of Laplacian dynamics, which has applications in many diverse fields of biology such as enzyme kinetics, pharmacology and receptor theory, gene regulation, protein post-translation modification [@Gunawardena2012; @Gunawardena2014; @Ahsendorf2013]. Our effort here was to extend existing framework for the case when zeroth order synthesis and first order degradation is added to Laplacian dynamics. The main motivation came from [@Gunawardena2012], where the author discusses the addition of synthesis and degradation to Laplacian dynamics of strongly connected digraph. Here we extended the proposed framework for arbitrary digraph with synthesis and degradation, and showed that synthesis and degradation dynamics possesses unique stable steady state solution under certain necessary and sufficient conditions. These conditions can be also used to identify whether given synthesis and degradation dynamics reaches a steady state. Moreover, as before, we have developed a mathematical framework to compute that unique ES. Our algorithm uses underlying digraph structure of dynamics and computer implementation of previous framework [@Ahsendorf2013] can be revised for automatic computations. This type of dynamics are frequently encountered in biological literature. In fact, to illustrate utility of our framework we have applied it to several examples in biochemistry such as exocytosis cascade of insulin granules and glucose metabolism in pancreatic $\beta$-cells. Since computed steady states are exact (not an approximation), they can be used to check correctness of numerical solutions. On the other hand, one of the greatest challenges in mathematical modeling is finding required parameters using given set of experimental data. Yet another feature of framework is that it can prove useful in parameter estimation problems. Particularly, one can calibrate computed symbolic ES solutions to experimental results. Although the latter example, glucose metabolism in $\beta$-cells, demonstrates application of framework to nonlinear system of differential equations, the scope of application of our framework to such nonlinear systems is limited. Therefore, as our future plan we intend to further extend framework such that it can be applied to broader range of nonlinear dynamics. Acknowledgements ================ Funding for this research was supported in part by grants NIH-NIGMS 2R01GM069438-06A2 and NSF-DMS 1225878. The authors would also like to thank Clay Thompson (Systems Biology Group, Pfizer, Inc.) for his suggestion of the insulin synthesis example used in Section \[sec:Biochemical-Network-Application\]. \[5cm\] [^1]: Applied Mathematics, Univeristy of Colorado, Boulder, CO 80309-0526 [^2]: Corresponding author (dmbortz@colorado.edu) [^3]: If a negative edge weight is encountered in applications, one can reverse orientation of that edge, hence preserving positivity of edge labels. [^4]: For the convenience of the reader and to promote clarity, we include at the end of this document a list of nomenclature used throughout this work. [^5]: For more generalized versions of MTT such as all-minors Matrix-Tree theorem and Matrix Forest Theorem we refer reader to [@Agaev2000; @Chebotarev2002]. [^6]: Later, we will define $\rho_{i}^{G}$ as an entry of the kernel element of Laplacian. [^7]: Recall that dimensions of row and column null spaces of a matrix are same. In fact, from Corollary \[cor:alg and geo\] we have this dimension equal to number of tSCC of digraph $G^{\star}$. [^8]: Note that we cannot substitute $\mathbf{e}^{(i)}$ directly to (\[eq:ES for P\]), because this would make $G^{\star}$ not strongly connected.

--- author: - 'Andrew Cotter[^1]' - 'Heinrich Jiang[^2]' - 'Karthik Sridharan[^3]' bibliography: - 'main.bib' title: 'Two-Player Games for Efficient Non-Convex Constrained Optimization' --- [^1]: acotter@google.com [^2]: heinrichj@google.com [^3]: sridharan@cs.cornell.edu

--- abstract: 'In this paper we analyse the double vector meson production in photon – hadron ($\gamma h$) interactions at $pp/pA/AA$ collisions and present predictions for the $\rho\rho$, $J/\Psi J/\Psi$ and $\rho J/\Psi$ production considering the double scattering mechanism. We estimate the total cross sections and rapidity distributions at LHC energies and compare our results with the predictions for the double vector meson production in $\gamma \gamma$ interactions at hadronic colliders. We present predictions for the different rapidity ranges probed by the ALICE, ATLAS, CMS and LHCb Collaborations. Our results demonstrate that the $\rho\rho$ and $J/\Psi J/\Psi$ production in $PbPb$ collisions is dominated by the double scattering mechanism, while the two - photon mechanism dominates in $pp$ collisions. Moreover, our results indicate that the analysis of the $\rho J/\Psi$ production at LHC can be useful to constrain the double scattering mechanism.' author: - 'V.P. Gonçalves $^{1,2}$, B.D. Moreira$^{3}$ and F.S. Navarra$^3$' title: 'Double vector meson production in photon - hadron interactions at hadronic colliders' --- LU TP 16-XX\ April 2016 Recent theoretical and experimental studies has demonstrated that hadronic colliders can also be considered photon – hadron and photon – photon colliders [@upc] which allow us to study the photon – induced interactions in a new kinematical range and probe e.g. the nuclear gluon distribution [@gluon; @gluon2; @gluon3; @Guzey; @vicwerluiz], the dynamics of the strong interactions [@vicmag_mesons1; @outros_vicmag_mesons; @vicmag_update; @motyka_watt; @Lappi; @griep; @bruno1; @bruno2], the Odderon [@vicodd1; @vicodd2], the mechanism of quarkonium production [@Schafer; @mairon1; @mairon2; @cisek; @bruno1; @bruno2] and the photon flux of the proton [@vicgus1; @vicgus2]. In particular, the installation of forward detectors in the LHC [@ctpps; @marek] should allows to separate more easily the exclusive processes, where the incident hadrons remain intact, allowing a detailed study of more complex final states as e.g. the exclusive production of two vector mesons explore other final states. Recent results from the LHCb Collaboration for the exclusive double $J/\Psi$ production [@lhcb_dif] has demonstrate that the experimental analysis of this process is feasible, motivating the improvement of the theoretical description of this process [@kmr_duplo; @vic_cris_dif; @antonirhorho; @antonipsipsi; @bruno_doublegama]. In particular, in Ref. [@bruno_doublegama] we have revisited the double vector production in $\gamma \gamma$ interactions, proposed originally in Refs. [@vicmagvv1; @vicmagvv2; @vicmagvv3], taking into account recent improvements in the description of the $\gamma \gamma \rightarrow VV$ ($V = \rho, J/\Psi$) cross section at low [@antonirhorho; @antonipsipsi] and high [@brunodouble] energies. A typical diagram for this process is represented in Fig. \[dia1\]. The results presented in Ref. [@bruno_doublegama] has demonstrated that the analysis of this process is feasible in hadronic collisions, mainly in $pp$ collisions, and that its study may be useful to constrain the QCD dynamics at high energies, as proposed originally in Ref. [@vicmagvv1]. However, double vector mesons can also be produced in photon – hadron ($\gamma h$) interactions if a double scattering occurs in a same event, as represented in Fig. \[dia2\]. The treatment of this double scattering mechanism (DSM) for $\gamma h$ interactions in heavy ion collisions was proposed originally in Ref. [@klein] and the double $\rho$ production was recently discussed in detail in Ref. [@mariola]. Such results demonstrated that the contribution of the double scattering mechanism is important for high energies, which motivates a more detailed analysis of this process. In this paper we extend these previous studies for the double $J/\Psi$ and $\rho J/\Psi$ production in $AA$ collisions and present, by the first time, predictions for the double vector meson in $pp$ and $pA$ collisions. Additionally, we compare our results for double vector meson production in $\gamma h$ interactions with those obtained in Ref. [@bruno_doublegama] for $\gamma \gamma$ interactions. As we will demonstrate below, the $\rho\rho$ and $J/\Psi J/\Psi$ production in $PbPb$ collisions is dominated by the double scattering mechanism, while the two - photon mechanism dominates in $pp$ collisions. Moreover, our results indicate that the analysis of the $\rho J/\Psi$ production at LHC can be useful to constrain the double scattering mechanism. -- -- -- -- -- -- Lets start our analysis presenting a brief review of the main concepts and formulas to describe the single and double vector meson production in $\gamma h$ interactions at hadronic colliders. The basic idea in the photon-induced processes is that a ultra relativistic charged hadron (proton or nucleus) give rise to strong electromagnetic fields, such that the photon stemming from the electromagnetic field of one of the two colliding hadrons can interact with one photon of the other hadron (photon - photon process) or can interact directly with the other hadron (photon - hadron process) [@upc; @epa]. In these processes the total cross section can be factorized in terms of the equivalent flux of photons into the hadron projectile and the photon-photon or photon-target production cross section. In this paper our main focus will be diffractive vector meson production in photon – hadron interactions in hadronic collisions. The differential cross sections for the production of a single vector meson $V$ at rapidity $y$ at fixed impact parameter $b$ of the hadronic collision can be expressed as follows: $$\begin{aligned} \frac{d\sigma \,\left[h_1 + h_2 \rightarrow h_1 \otimes V \otimes h_2\right]}{d^2b dy} = \left[\omega N_{h_1}(\omega,b)\,\sigma_{\gamma h_2 \rightarrow V \otimes h_2}\left(\omega \right)\right]_{\omega_L} + \left[\omega N_{h_2}(\omega,b)\,\sigma_{\gamma h_1 \rightarrow V \otimes h_1}\left(\omega \right)\right]_{\omega_R}\, \label{dsigdy}\end{aligned}$$ where the rapidity ($y$) of the vector meson in the final state is determined by the photon energy $\omega$ in the collider frame and by mass $M_{V}$ of the vector meson \[$y\propto \ln \, ( \omega/M_{V})$\]. Moreover, $\sigma_{\gamma h_i \rightarrow V \otimes h_i}$ is the total cross section for the diffractive vector meson photoproduction, with the symbol $\otimes$ representing the presence of a rapidity gap in the final state and $\omega_L \, (\propto e^{-y})$ and $\omega_R \, (\propto e^{y})$ denoting photons from the $h_1$ and $h_2$ hadrons, respectively. One have that Eq. (\[dsigdy\]) takes into account that both incident hadrons can be source of photon which will interact with the other hadron. The equivalent photon spectrum $N(\omega,b)$ of a relativistic hadron for photons of energy $\omega$ at the distance ${\mathbf b}$ to the hadron trajectory, defined in the plane transverse to the trajectory, can be expressed in terms of the charge form factor $F$ as follows $$\begin{aligned} N(\omega,b) = \frac{Z^{2}\alpha_{em}}{\pi^2}\frac{1}{b^{2}\omega} \cdot \left[ \int u^{2} J_{1}(u) F\left( \sqrt{\frac{\left( \frac{b\omega}{\gamma_L}\right)^{2} + u^{2}}{b^{2}}} \right ) \frac{1}{\left(\frac{b\omega}{\gamma_L}\right)^{2} + u^{2}} \mbox{d}u \right]^{2} \,\,, \label{fluxo}\end{aligned}$$ where $\gamma_L$ is the Lorentz factor. The double vector meson production can occur if two $\gamma h$ interactions are present in the same event, as represented in Fig. \[dia2\]. In order to treat this double - scattering mechanism we will follow the approach from Refs. [@klein; @mariola] that proposed that the double differential cross section for the production of a vector meson $V_1$ at rapidity $y_1$ and a second vector meson $V_2$ at rapidity $y_2$ will be given by $$\begin{aligned} \frac{d^2\sigma_{h_1 h_2 \rightarrow h_1 V_1 V_2 h_2}}{dy_1 dy_2} = {\cal{C}} \int_{b_{min}} \frac{d\sigma \,\left[h_1 + h_2 \rightarrow h_1 V_1 h_2\right]}{d^2b dy_1} \times \frac{d\sigma \,\left[h_1 + h_2 \rightarrow h_1 V_2 h_2\right]}{d^2b dy_2} \,\, d^2b \,\,, \label{Eq:double}\end{aligned}$$ where ${\cal{C}}$ is equal to 1 (1/2) for $V_1 \neq V_2$ ($V_1 = V_2$) and $b_{min} = R_{h_1} + R_{h_2}$ excludes the overlap between the colliding hadrons and allows to take into account only ultra peripheral collisions. Consequently, the double vector meson production can be easily estimated in terms of the cross sections for the single vector meson production, which is determined by the photon flux and the $\gamma h \rightarrow V h$ cross section. In what follows we will consider the color dipole formalism to describe the diffractive vector meson photoproduction, which successfully describe the HERA data and recent LHC data [@amir; @bruno1; @bruno2]. In this approach the description of the single vector meson production can be factorized as follows: i) a photon is emitted by one of the incident hadrons, ii) the photon fluctuates into a quark-antiquark pair (the dipole), iii) this color dipole interact with the other hadron by the exchange of a color single state, denoted Pomeron ($I\!\!P$) and, iv) the pair converts into the vector meson final state. The $\gamma h \rightarrow V h$ cross section is given by $$\begin{aligned} \sigma (\gamma h \rightarrow V h) = \int_{-\infty}^0 \frac{d\sigma}{d{t}}\, d{t} = \frac{1}{16\pi} \int_{-\infty}^0 |{\cal{A}}_T^{\gamma h \rightarrow V h }(x,\Delta)|^2 \, d{t}\,\,, \label{sctotal_intt}\end{aligned}$$ with the scattering amplitude is given by $$\begin{aligned} {\cal A}_{T}^{\gamma h \rightarrow V h}({x},\Delta) = i \int dz \, d^2{\mbox{\boldmath $r$}}\, d^2{\mbox{\boldmath $b$}}_h e^{-i[{\mbox{\boldmath $b$}}_h-(1-z){\mbox{\boldmath $r$}}].{\mbox{\boldmath $\Delta$}}} \,\, (\Psi^{V*}\Psi)_{T} \,\,2 {\cal{N}}_h({x},{\mbox{\boldmath $r$}},{\mbox{\boldmath $b$}}_h) \,\,, \label{sigmatot2}\end{aligned}$$ where $(\Psi^{V*}\Psi)_{T}$ denotes the overlap of the transverse photon and vector meson wave functions. The variable $z$ $(1-z)$ is the longitudinal momentum fractions of the quark (antiquark) and $\Delta$ denotes the transverse momentum lost by the outgoing pion (${t} = - \Delta^2$). The variable ${\mbox{\boldmath $b$}}_h$ is the transverse distance from the center of the target $h$ to the center of mass of the $q \bar{q}$ dipole and the factor in the exponential arises when one takes into account non-forward corrections to the wave functions [@non]. As in our previous studies [@bruno1; @bruno2] in what follows we will assume that the vector meson is predominantly a quark-antiquark state and that the spin and polarization structure is the same as in the photon [@dgkp; @nnpz; @sandapen; @KT] (for other approaches see, for example, Ref. [@pacheco]). As a consequence, the overlap between the photon and the vector meson wave function, for the transversely polarized case, is given by (For details see Ref. [@KMW]) $$\begin{aligned} (\Psi^*_V\Psi)_T &=& \frac{\hat e_fe}{4\pi}\frac{N_c}{\pi z(1-z)} \left\{m_f^2K_0(\epsilon r)\phi_T(r,z)-\left[z^2+(1-z)^2\right]\epsilon K_1(\epsilon r)\partial_r\phi_T(r,z)\right\}, \end{aligned}$$ where $ \hat{e}_f $ is the effective charge of the vector meson, $m_f$ is the quark mass, $N_c = 3$, $\epsilon^2 = z(1-z)Q^2 + m_f^2$ and $\phi_i(r,z)$ define the scalar parts of the vector meson wave functions. In the Gauss-LC model one have that $$\begin{aligned} \phi_T(r,z) &=& N_T\left[z(1-z)\right]^2\exp\left(-r^2/2R_T^2\right)\,.\end{aligned}$$ with the parameters $N_T$ and $R_T$ being determined by the normalization condition of the wave function and by the meson decay width (For details see Table 1 in Ref. [@bruno2]). It is important to emphasize that predictions based on this model for the wave functions have been tested with success in $ep$ and ultra peripheral hadronic collisions (See, e. g. Refs. [@amir; @anelise; @bruno1; @bruno2]). Moreover, ${\cal{N}}_h (x,{\mbox{\boldmath $r$}},{\mbox{\boldmath $b$}}_h)$ denotes the non-forward scattering amplitude of a dipole of size ${\mbox{\boldmath $r$}}$ on the hadron $h$, which is directly related to the QCD dynamics. In what follows we will assume that for the proton case ${\cal{N}}_p (x,{\mbox{\boldmath $r$}},{\mbox{\boldmath $b$}}_p)$ is given by the bCGC model proposed in Ref. [@KMW], which improves the Iancu - Itakura - Munier (IIM) model [@iim] with the inclusion of the impact parameter dependence in the dipole - proton scattering amplitude. Following [@KMW] we have: $$\begin{aligned} \mathcal{N}_p(x,{\mbox{\boldmath $r$}},{{\mbox{\boldmath $b$}}_p}) = \left\{ \begin{array}{ll} {\mathcal N}_0\, \left(\frac{ r \, Q_{s,p}}{2}\right)^{2\left(\gamma_s + \frac{\ln (2/r Q_{s,p})}{\kappa \,\lambda \,Y}\right)} & \mbox{$r Q_{s,p} \le 2$} \\ 1 - \exp \left[-A\,\ln^2\,(B \, r \, Q_{s,p})\right] & \mbox{$r Q_{s,p} > 2$} \end{array} \right. \label{eq:bcgc}\end{aligned}$$ with $Y=\ln(1/x)$ and $\kappa = \chi''(\gamma_s)/\chi'(\gamma_s)$, where $\chi$ is the LO BFKL characteristic function [@bfkl]. The coefficients $A$ and $B$ are determined uniquely from the condition that $\mathcal{N}_p(x,{\mbox{\boldmath $r$}},{\mbox{\boldmath $b$}}_p)$, and its derivative with respect to $rQ_s$, are continuous at $rQ_s=2$. In this model, the proton saturation scale $Q_{s,p}$ depends on the impact parameter: $$Q_{s,p}\equiv Q_{s,p}(x,{{\mbox{\boldmath $b$}}_p})=\left(\frac{x_0}{x}\right)^{\frac{\lambda}{2}}\; \left[\exp\left(-\frac{{b_p}^2}{2B_{\rm CGC}}\right)\right]^{\frac{1}{2\gamma_s}}. \label{newqs}$$ The parameter $B_{\rm CGC}$ was adjusted to give a good description of the $t$-dependence of exclusive $J/\psi$ photoproduction. The factors $\mathcal{N}_0$, $x_0$, $\lambda$ and $\gamma_s$ were taken to be free. Recently the parameters of this model have been updated in Ref. [@amir] (considering the recently released high precision combined HERA data), giving $\gamma_s = 0.6599$, $B_{CGC} = 5.5$ GeV$^{-2}$, $\mathcal{N}_0 = 0.3358$, $x_0 = 0.00105 \times 10^{-5}$ and $\lambda = 0.2063$. As demonstrate in Ref. [@armesto_amir], this phenomenological dipole describes quite well the HERA data for the exclusive $\rho$ and $J/\Psi$ production. Moreover, the results from Refs. [@bruno1; @bruno2] demonstrated that this model allows to describe the recent LHC data for the exclusive vector meson photoproduction in $pp$ and $pPb$ collisions. Another motivation to use the bCGC model, is that this model is based on the CGC physics, which was used in Ref. [@bruno_doublegama] to estimate the double vector meson production in $\gamma \gamma$ interactions. A common approach for the QCD dynamics in $\gamma \gamma$ and $\gamma h$ interactions is important to minimize the theoretical uncertainty and to perform a realistic comparison between the predictions of the two different mechanisms for the double vector production. In order to describe the vector meson production in $\gamma A$ interactions we need to specify the forward dipole - nucleus scattering amplitude, $\mathcal{N}_A(x,{\mbox{\boldmath $r$}},{\mbox{\boldmath $b$}}_A)$. Following [@bruno1] we will use in our calculations the model proposed in Ref. [@armesto], which describes the current experimental data on the nuclear structure function as well as includes the impact parameter dependence in the dipole nucleus cross section. In this model the forward dipole-nucleus amplitude is given by $$\begin{aligned} {\cal{N}}_A(x,{\mbox{\boldmath $r$}},{\mbox{\boldmath $b$}}_A) = 1 - \exp \left[-\frac{1}{2} \, \sigma_{dp}(x,{\mbox{\boldmath $r$}}^2) \,T_A({\mbox{\boldmath $b$}}_A)\right] \,\,, \label{enenuc}\end{aligned}$$ where $\sigma_{dp}$ is the dipole-proton cross section given by $$\begin{aligned} \sigma_{dp} (x,{\mbox{\boldmath $r$}}^2) = 2 \int d^2{\mbox{\boldmath $b$}}_p \,\,\mathcal{N}_p(x,{\mbox{\boldmath $r$}},{{\mbox{\boldmath $b$}}_p}) \end{aligned}$$ and $T_A({\mbox{\boldmath $b$}}_A)$ is the nuclear profile function, which is obtained from a 3-parameter Fermi distribution for the nuclear density normalized to $A$. The above equation sums up all the multiple elastic rescattering diagrams of the $q \overline{q}$ pair and is justified for large coherence length, where the transverse separation ${\mbox{\boldmath $r$}}$ of partons in the multiparton Fock state of the photon becomes a conserved quantity, [*i.e.*]{} the size of the pair ${\mbox{\boldmath $r$}}$ becomes eigenvalue of the scattering matrix. In the case of the double vector meson production in $\gamma \gamma$ interactions at hadronic colliders, represented in Fig. \[dia1\], we have that the total cross section is given by (For details see Ref. [@bruno_doublegama]) $$\begin{aligned} \sigma \left( h_1 h_2 \rightarrow h_1 \otimes V_1V_2 \otimes h_2 ;s \right) &=& \int \hat{\sigma}\left(\gamma \gamma \rightarrow V_1V_2 ; W_{\gamma \gamma} \right ) N\left(\omega_{1},{\mathbf b_{1}} \right ) N\left(\omega_{2},{\mathbf b_{2}} \right ) S^2_{abs}({\mathbf b}) \frac{W_{\gamma \gamma}}{2} \mbox{d}^{2} {\mathbf b_{1}} \mbox{d}^{2} {\mathbf b_{2}} \mbox{d}W_{\gamma \gamma} \mbox{d}Y \,\,\, . \label{cross-sec-2}\end{aligned}$$ where $\omega_1$ and $\omega_2$ are the photon energies, $W_{\gamma \gamma} = \sqrt{4 \omega_1 \omega_2}$ is the invariant mass of the $\gamma \gamma$ system and $Y$ is the rapidity of the outgoing double meson system. Moreover, $S^2_{abs}({\mathbf b})$ is the absorption factor, given in what follows by $$\begin{aligned} S^2_{abs}({\mathbf b}) = \Theta\left( \left|{\mathbf b}\right| - R_{h_1} - R_{h_2} \right ) = \Theta\left( \left|{\mathbf b_{1}} - {\mathbf b_{2}} \right| - R_{h_1} - R_{h_2} \right ) \,\,, \label{abs}\end{aligned}$$ where $R_{h_i}$ is the radius of the hadron $h_i$ ($i = 1,2$). In the dipole picture, the $\gamma \gamma \rightarrow V_1 V_2$ cross section can be expressed as follows $$\begin{aligned} \sigma\, (\gamma \gamma \rightarrow V_1 \, V_2) = \frac{[{\cal I}m \, {\cal A}(W_{\gamma \gamma}^2,\,t=0)]^2}{16\pi\,B_{V_1 \,V_2}} \;, \label{totalcs}\end{aligned}$$ where we have approximated the $t$-dependence of the differential cross section by an exponential with $B_{V_1 \, V_2}$ being the slope parameter. The imaginary part of the amplitude at zero momentum transfer ${\cal A}(W_{\gamma \gamma}^2,\,t=0)$ reads as $$\begin{aligned} {\cal I}m \, {\cal A}\, (\gamma \gamma \rightarrow V_1 \, V_2) & = & \int dz_1\, d^2{\mbox{\boldmath $r$}}_1 \,[\Psi^\gamma(z_1,\,{\mbox{\boldmath $r$}}_1)\,\, \Psi^{V_1*}(z_1,\,{\mbox{\boldmath $r$}}_1)]_T \nonumber \\ &\times & \int dz_2\, d^2{\mbox{\boldmath $r$}}_2 \,[\Psi^\gamma(z_2,\,{\mbox{\boldmath $r$}}_2)\,\, \Psi^{V_2 *}(z_2,\,{\mbox{\boldmath $r$}}_2)]_T \, \sigma_{d d}({\mbox{\boldmath $r$}}_1, {\mbox{\boldmath $r$}}_2,Y) \, , \label{sigmatot}\end{aligned}$$ where $\Psi^{\gamma}$ and $\Psi^{V_i}$ are the light-cone wave functions of the photon and vector meson, respectively, and $T$ the transverse polarization. The variable ${\mbox{\boldmath $r$}}_1$ defines the relative transverse separation of the pair (dipole) and $z_1$ $(1-z_1)$ is the longitudinal momentum fraction of the quark (antiquark). Similar definitions are valid for ${\mbox{\boldmath $r$}}_2$ and $z_2$. Moreover, $\sigma_{d d}$ is the dipole - dipole cross section, which can be estimated taking into account the nonlinear effects in the QCD dynamics. In what follows, we assume the Gauss-LC model for the vector meson wave functions and estimate $\sigma_{d d}$ using the approach proposed in Refs. [@nosfofo; @bruno_doublegama], which is based on the CGC physics. We refer the reader to the Ref. [@bruno_doublegama] for more details about the double vector meson production in $\gamma \gamma$ interactions. -- -- -- -- -- -- -- -- ----------------- ----------------- ----------------------------- ---------------------------- -------------------------- ---------------------------- ----------------------------- Final state Mechanism $PbPb$ $PbPb$ $pPb$ $pp$ $pp$ $\sqrt{s}=2.76\,\mbox{TeV}$ $\sqrt{s}=5.5\,\mbox{TeV}$ $\sqrt{s}=5\,\mbox{TeV}$ $\sqrt{s}=7\,\mbox{TeV}$ $\sqrt{s}=14\,\mbox{TeV}$ $J/\Psi J/\Psi$ DSM 402.301 nb 1054.951 nb 28.473 pb 3.223 $\times$10$^{-4}$ pb 7.256$\times$10$^{-4}$ pb $\gamma \gamma$ 235.565 nb 658.589 nb 310.194 pb 0.2412 pb 0.4793 pb $\rho \rho$ DSM 21.150 mb 29.421 mb 702.595 nb 4.354 pb 7.083 pb $\gamma \gamma$ 1.389 mb 1.973 mb 536.432 nb 182.442 pb 237.006 pb $\rho J/\Psi$ DSM 0.18 mb 0.35 mb 8.929 nb 7.469 $\times$10$^{-2}$ pb 14.288 $\times$10$^{-2}$ pb ----------------- ----------------- ----------------------------- ---------------------------- -------------------------- ---------------------------- ----------------------------- : Total cross sections for the double vector meson production considering the double scattering and two - photon mechanisms and different center - of - mass energies considering the full kinematical range covered by the LHC.[]{data-label="tab1"} In what follows we present our predictions for the rapidity distributions and total cross sections for the $\rho \rho$, $\rho J/\Psi$ and $J/\Psi J/\Psi$ production in $\gamma h$ interactions at $pp/pPb/PbPb$ collisions. We will denote the predictions associated to the double scattering mechanics by DSM hereafter. Following Ref. [@mariola] we will estimate the equivalent photon spectra for $A = Pb$ assuming the nucleus as a point - like object, i.e. $F(q^2) = 1$. In the proton case, we will take $F(q^2) = 1/[1 + q^2/(0.71 \mbox{GeV}^2)]^2$ and $R_p = 0.7$ fm as in Ref. [@bruno_doublegama]. Moreover, we will compare our predictions for the $J/\Psi J/\Psi$ and $\rho \rho$ production with the results obtained in Ref. [@bruno_doublegama] for the production of these final states in $\gamma \gamma$ interactions. In Fig. \[fig2\] we present our predictions for the energy dependence of the total cross sections for the double vector meson production in $\gamma h$ and $\gamma \gamma$ interactions. For the double $J/\Psi$ production (upper panels), the double scattering mechanism becomes competitive with the two - photon one only in $PbPb$ collisions, being a factor 10 (100) smaller in $pPb$ ($pp$) collisions. In particular, for $pp$ collisions, the DSM contribution is negligible. On the other hand, our results demonstrate that the associated production of a $J/\Psi$ and a $\rho$ meson by the double scattering mechanism is important the LHC range. It is important to emphasize that this final state also can be produced by $\gamma \gamma$ interactions. However, as its contribution in hadronic collisions still is an open question due to the current large uncertainty on the normalization of the $\gamma \gamma \rightarrow \rho J\Psi$ cross section (For a detailed discussion see Ref. [@brunodouble]), we do not present the associated predictions. In the case of the double $\rho$ production (lower panels), the double scattering mechanism is dominant in $PbPb$ collisions, in agreement with the results presented in Ref. [@mariola]. On the other hand, the contribution of the double scattering and two - photon mechanisms are similar in $pPb$ collisions, while the $\gamma \gamma$ dominates in the $pp$ collisions. These results demonstrate that the analysis of this final state in $PbPb / pPb / pp$ can be useful to disentangle the different mechanisms for the $\rho \rho$ production. The corresponding total cross sections at different values of the center - of - mass energy are presented in Table \[tab1\] considering the full kinematical range covered by the LHC. In Figs. \[fig3\] and \[fig4\] we present our predictions for the rapidity distributions for the double vector meson production by the double scattering mechanism in $PbPb$ and $pPb$ collisions, respectively. For $PbPb$ collisions, as expected, one have symmetric distributions for the $J/\Psi J/\Psi$ and $\rho \rho$ production. On the other hand, in the case of the $\rho J/\Psi$ production, the distribution is asymmetric, being wider for the rapidity associated to the $\rho$ meson. In the case of $pPb$ collisions, one have that the photon flux of the nucleus is amplified by a factor $Z^2$ in comparison to the photon flux associated to the proton. As a consequence, the double scattering mechanism is dominated by $\gamma h$ interactions where the photons are emitted by the nucleus. The contribution associated to one photon emitted by the nucleus and the other by the proton is suppressed by a factor $Z^2$, while the contribution associated to $\gamma h$ interactions with photons emitted by the proton is suppressed by a factor $Z^4$. It implies that the rapidity distributions are asymmetric for all final states considered (See Fig. \[fig4\]). Similarly as observed in $PbPb$ collisions, the rapidity distribution associated to the $\rho$ meson is wider in comparison to the $J/\Psi$ one. -- -- -- -- -- -- -- -- -- -- -- -- Finally, in Table \[tab2\] we present our predictions for the total cross sections for the double vector production by the double scattering mechanism in $PbPb$ and $pPb$ collisions considering the rapidity ranges covered by the ATLAS, CMS, ALICE and LHCb Collaborations. In the particular case of the ALICE Collaboration we estimate the cross sections considering: (a) that both mesons are produced in the range $-1 < y_{1,2} < 1$ (denoted ALICE1 in the Table) and (b) that one meson is produced in the range $-1 < y_{1} < 1$ and the other in the range $-3.6 < y_{2} < -2.6$ (denoted ALICE2). For the $\rho J/\Psi$ production in the ALICE2 range, we present our results for the two possible configurations: $(y_1, y_2) = (y_{\rho}, y_{J/\Psi})$ and $(y_1, y_2) = (y_{J/\Psi},y_{\rho})$, with the results associated to the latter one being presented in parenthesis in Table \[tab2\]. We predict large values for the total cross sections, in particular, for the $\rho \rho$ and $\rho J/\Psi$ production in $PbPb$ collisions, in the phase space covered by the different collaborations. Consequently, we believe that the analysis of these different final states is feasible in the future, which will allow to probe the double scattering mechanism at the LHC. ----------------- ----------------------------------------- --------------------- -------------------- -------------------- -------------------------------------------- Final state LHCb ATLAS/CMS ALICE1 ALICE2 $2 < y_{1,2} < 4.5$ $-2 < y_{1,2} < 2$ $-1 < y_{1,2} < 1$ $-1 < y_{1} < 1$ and $-3.6 < y_{2} < -2.6$ $J/\Psi J/\Psi$ $PbPb \,\, (\sqrt{s}=2.76\,\mbox{TeV})$ 5.51 nb 234.94 nb 69.91 nb 6.94 nb $PbPb \,\, (\sqrt{s}=5.5\,\mbox{TeV})$ 30.85 nb 446.11 nb 118.03 nb 25.45 nb $pPb \,\, (\sqrt{s}=5\,\mbox{TeV})$ 3.25 pb 8.87 pb 2.16 pb 0.37 pb $\rho \rho$ $PbPb \,\, (\sqrt{s}=2.76\,\mbox{TeV})$ 0.93 mb 6.08 mb 1.58 mb 0.54 mb $PbPb \,\, (\sqrt{s}=5.5\,\mbox{TeV})$ 1.50 mb 7.06 mb 1.79 mb 0.73 mb $pPb \,\, (\sqrt{s}=5\,\mbox{TeV})$ 84.09 nb 122.03 nb 30.11 nb 8.53 nb $\rho J/\Psi$ $PbPb \,\, (\sqrt{s}=2.76\,\mbox{TeV})$ 4.48 $\mu$b 75.17 $\mu$b 20.94 $\mu$b 2.06 (7.25) $\mu$b $PbPb \,\, (\sqrt{s}=5.5\,\mbox{TeV})$ 13.42 $\mu$b 112.00 $\mu$b 29.06 $\mu$b 6.21 (11.86) $\mu$b $pPb \,\, (\sqrt{s}=5\,\mbox{TeV})$ 1.02 nb 2.08 nb 0.51 nb 87.31 (144.56) pb ----------------- ----------------------------------------- --------------------- -------------------- -------------------- -------------------------------------------- : Total cross sections for the double vector meson production by the double scattering mechanism (DSM) for different center - of - mass energies considering the distinct phase space in rapidity covered by the ALICE, ATLAS, CMS and LHCb Collaborations.[]{data-label="tab2"} Let us summarize our main conclusions. In recent years, a series of studies have discussed in detail the treatment of the total cross section and the exclusive production of different final states in $\gamma \gamma$ and $\gamma h$ interactions considering very distinct theoretical approaches. In particular, recent results for the double vector meson production in $\gamma \gamma$ interactions at hadronic colliders has demonstrated that this process can be used to constrain the QCD dynamics at high energies. However, this final state can also be generated if double $\gamma h$ interactions are present in the same event. In this paper we have estimated the magnitude of this contribution for the $J/\Psi J/\Psi$, $\rho \rho$ and $\rho J/\Psi$ production in $PbPb/pPb/pp$ collisions. We have treated the double scattering and two - photon mechanisms using the dipole formalism and a same approach for the QCD dynamics and the vector meson wave function. Our results indicated that the DSM contribution is dominant for the $J/\Psi J/\Psi$ and $\rho \rho$ production in $PbPb$ collisions. On the other hand, the two - photon production dominates the double $J/\Psi$ production in $pPb$ and $pp$ collisions. In the case of the double $\rho$ production, the DSM and two - photon contributions are similar in $pPb$ collisions, with the two - photon being dominant in $pp$ collisions. Therefore, the analysis of double vector production considering different projectiles can be useful to disentangle the different mechanisms of production. In particular, the analysis of the DPS production in heavy ion collisions can be used to complement our understanding of the description of the diffractive vector meson photoproduction. Moreover, our results demonstrated that the DPS $\rho J/\Psi$ production is large in the LHC kinematical range. Finally, our predictions for the double vector meson production in the phase space covered by the different experimental collaborations at the LHC indicate that the study of the double vector meson production is feasible in the future. [*Note added in the proof:*]{} One month after the submission of this paper, a report has appeared [@LNS] where it has been estimated the exclusive double $\rho$ production in $pp$ collisions. The total cross section was estimated in [@LNS] taking into account pomeron and reggeon exchanges and considering the tensor pomeron model proposed in Ref. [@N1] and discussed in detail in Ref. [@N2]. The cross sections found in [@LNS] are more than three orders of magnitude larger than our predictions. Therefore, the double $\rho$ production in $pp$ collisions is predicted to be dominated by pomeron - pomeron interactions, which implies that the analysis of this process can be useful to probe the tensor pomeron model. An alternative to study the photon - induced $\rho \rho$ production in $pp$ collisions analysed here is the reconstruction of the entire event with a cut on the summed transverse momentum of the event. As the typical photon virtualities are very small, the hadron scattering angles are very low. Consequently, we expect a different transverse momentum distribution of the scattered hadrons, with pomeron - pomeron interactions predicting larger $p_T$ values. Surely this subject deserve a more detailed analysis in the future. Finally, it is important to emphasize that in contrast to $pp$ collisions, the photon - induced interactions are expected to be dominant in $pA (AA)$ collisions due to the $Z^2 \, (Z^4)$ enhancement associated to the presence of nuclear photon flux. VPG thanks G. Contreras, S. Klein, R. McNulty and D. Tapia Takaki by useful discussions. This work was partially financed by the Brazilian funding agencies CNPq, CAPES, FAPERGS and FAPESP. [99]{} C. A. Bertulani and G. 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--- abstract: 'We propose a boundary value correction approach for cases when curved boundaries are approximated by straight lines (planes) and Lagrange multipliers are used to enforce Dirichlet boundary conditions. The approach allows for optimal order convergence for polynomial order up to 3. We show the relation to the Taylor series expansion approach used by Bramble, Dupont and Tomée [@BrDuTh72] in the context of Nitsche’s method and, in the case of *inf–sup* stable multiplier methods, prove a priori error estimates with explicit dependence on the meshsize and distance between the exact and approximate boundary.' author: - Erik Burman - Peter Hansbo - 'Mats G. Larson' date: - - title: Dirichlet Boundary Value Correction using Lagrange Multipliers --- Introduction ============ In this contribution we develop a modified Lagrange multiplier method based on the idea of boundary value correction originally proposed for standard finite element methods on an approximate domain in [@BrDuTh72] and further developed in [@Du74]. More recently boundary value correction have been developed for cut and immersed finite element methods [@BuHaLa18; @BuHaLa18b; @BBCL18; @MaSc18; @MaSc18b]. Using the closest point mapping to the exact boundary, or an approximation thereof, the boundary condition on the exact boundary may be weakly enforced using multipliers on the boundary of the approximate domain. Of particular practical importance in this context is the fact that we may use a piecewise linear approximation of the boundary, which is very convenient from a computational point of view since the geometric computations are simple in this case and a piecewise linear distance function may be used to construct the discrete domain. We prove optimal order a priori error estimates, in the energy and $L^2$ norms, in terms of the error in the boundary approximation and the meshsize. The proof utilizes the a priori error estimates derived in [@BuHaLa18] for the cut boundary value corrected Nitsche method together with a bound, which shows that the solution to the boundary value corrected Lagrange method is close to the corresponding Nitsche solution for which optimal bounds are available. We obtain optimal order convergence for polynomial approximation up to order 3 of the solution. Note that without boundary correction one typically requires $O(h^{p+1})$ accuracy in the $L^\infty$ norm for the approximation of the domain, which leads to significantly more involved computations on the cut elements for higher order elements, see [@JoLa13]. We present numerical results illustrating our theoretical findings. The outline of the paper is as follows: In Section 2 we formulate the model problem and our method, in Section 3 we present our theoretical analysis, in Section 4 we discuss the choice of finite element spaces in cut finite element methods, in Section 5 we present the numerical results, and finally in Section 6 we include some concluding remarks. Model problem and method ======================== The domain ---------- Let $\Omega$ be a domain in $\mathbb{R}^d$ with smooth boundary $\partial \Omega$ and exterior unit normal ${\boldsymbol n}$. We let $\varrho$ be the signed distance function, negative on the inside and positive on the outside, to $\partial \Omega$ and we let $U_\delta(\partial \Omega)$ be the tubular neighborhood $\{{\boldsymbol x}\in {\mathbb{R}}^d : |\varrho({\boldsymbol x})| < \delta\}$ of $\partial \Omega$. Then there is a constant $\delta_0>0$ such that the closest point mapping ${\boldsymbol p}({\boldsymbol x}):U_{\delta_0}(\partial \Omega) \rightarrow \partial \Omega$ is well defined and we have the identity ${\boldsymbol p}({\boldsymbol x}) = {\boldsymbol x}- \varrho({\boldsymbol x}){\boldsymbol n}({\boldsymbol p}({\boldsymbol x}))$. We assume that $\delta_0$ is chosen small enough that ${\boldsymbol p}({\boldsymbol x})$ is well defined. See [@GilTru01], Section 14.6 for further details on distance functions. The model problem ----------------- We consider the problem: find $u:\Omega \rightarrow {\mathbb{R}}$ such that $$\begin{aligned} {2}\label{eq:poissoninterior_strong} -\Delta u &= f \qquad && \text{in $\Omega$} \\ \label{eq:poissonbc_strong} u &= g \qquad && \text{on $\partial\Omega$}\end{aligned}$$ where $f\in H^{-1}(\Omega)$ and $g\in H^{1/2}(\partial \Omega)$ are given data. It follows from the Lax-Milgram Lemma that there exists a unique solution to this problem and we also have the elliptic regularity estimate $$\label{eq:ellipticregularity} \|u\|_{H^{s+2}(\Omega)} \lesssim \|f\|_{H^s(\Omega)}, \qquad s \geq -1.$$ Here and below we use the notation $\lesssim$ to denote less or equal up to a constant. Using a Lagrange multiplier to enforce the boundary condition we can write the weak form of – as: find $(u,\lambda) \in H^1(\Omega) \times H^{-1/2}(\partial\Omega)$ such that $$\begin{aligned} {2}\label{eq:poissoninterior} \int_{\Omega}\nabla u \cdot\nabla v \,\text{d}\Omega +\int_{\partial\Omega}\lambda\, v\, \text{d}s &= \int_{\Omega}f v\, \text{d}\Omega\qquad \forall v\in H^1(\Omega) \\ \label{eq:poissonbc} \int_{\partial\Omega}u\, \mu\, \text{d}s &= \int_{\partial\Omega}g\, \mu\, \text{d}s\qquad \forall \mu\in H^{-1/2}(\partial\Omega)\end{aligned}$$ The mesh and the discrete domain -------------------------------- Let ${\mathcal{K}}_{h}, h \in (0,h_0]$, be a family of quasiuniform partitions, with mesh parameter $h$, of $\Omega$ into shape regular triangles or tetrahedra $K$. The partitions induce discrete polygonal approximations $\Omega_h = \cup_{K \in {\mathcal{K}}_h}K$, $h \in (0,h_0]$, of $\Omega$. We assume neither $\Omega_h \subset \Omega$ nor $\Omega \subset \Omega_h$, instead the accuracy with which $\Omega_h$ approximates $\Omega$ will be crucial. To each $ \Omega_h$ is associated a discrete unit normal ${\boldsymbol n}_h$ and a discrete signed distance $\varrho_h:\partial \Omega_h \rightarrow \mathbb{R}$, such that if ${\boldsymbol p}_h({\boldsymbol x},\varsigma):={\boldsymbol x}+ \varsigma {\boldsymbol n}_h({\boldsymbol x})$ then ${\boldsymbol p}_h({\boldsymbol x},\varrho_h({\boldsymbol x})) \in \partial \Omega$ for all ${\boldsymbol x}\in \partial \Omega_h$. We will also assume that ${\boldsymbol p}_h({\boldsymbol x},\varsigma) \in U_{\delta_0}(\Omega):=U_{\delta_0}(\partial\Omega)\cup\Omega$ for all ${\boldsymbol x}\in \partial \Omega_h$ and all $\varsigma$ between $0$ and $\varrho_h({\boldsymbol x})$. For conciseness we will drop the second argument of ${\boldsymbol p}_h$ below whenever it takes the value $\varrho_h({\boldsymbol x})$, and thus we have the map $\partial \Omega_h \ni {\boldsymbol x}\mapsto {\boldsymbol p}_h({\boldsymbol x}) \in \partial \Omega$. We assume that the following assumptions are satisfied $$\label{eq:geomassum-a} \delta_h := \| \varrho_h \|_{L^\infty(\partial \Omega_h)} = o(h), \qquad h \in (0,h_0]$$ and $$\label{eq:geomassum-c} \| {\boldsymbol n}_h - {\boldsymbol n}\circ {\boldsymbol p}_h \|_{L^\infty(\partial \Omega_h)} = o(1), \qquad h \in (0,h_0]$$ where $o(\cdot)$ denotes the little ordo. We also assume that $h_0$ is small enough to guarantee that $$\label{eq:geomassum-b} \partial \Omega_h \subset U_{\delta_0}(\partial \Omega), \qquad h\in(0,h_0]$$ and that there exists $M>0$ such for any ${\boldsymbol y}\in U_{\delta_0}(\partial \Omega)$ the equation, find ${\boldsymbol x}\in \partial \Omega_h$ and $ |\varsigma| \leq \delta_h$ such that $$\label{eq:assump_olap} {{\boldsymbol p}}_h({\boldsymbol x},\varsigma) = {\boldsymbol y}$$ has a solution set $\mathcal{P}_h$ with $$\label{eq:card_hyp} \mbox{card}(\mathcal{P}_h) \leq M$$ uniformly in $h$. The rationale of this assumption is to ensure that the image of ${\boldsymbol p}_h$ can not degenerate for vanishing $h$; for more information cf. [@BuHaLa18]. We note that it follows from (\[eq:geomassum-a\]) that $$\label{eq:geomassum-exact-normal} \|\varrho \|_{L^\infty(\partial \Omega_h)} \lesssim \|\varrho_h \|_{L^\infty(\partial \Omega_h)} = o(h)$$ since $|\varrho_h({\boldsymbol x})| \geq |\varrho({\boldsymbol x})|$, ${\boldsymbol x}\in U_{\delta_0}(\partial \Omega)$. We also assume the additional regularity $$\label{eq:residualregularity} f+ \Delta u \in H^{l\textcolor{black}{+\frac12+\epsilon}}(U_{\delta_0}(\Omega))$$ The finite element method ------------------------- ### Boundary value correction The basic idea of the boundary value correction of [@BrDuTh72] is to use a Taylor series at ${\boldsymbol x}\in {\partial\Omega_h}$ in the direction ${\boldsymbol n}_h$, and let this series represent $u_h \vert_{\partial\Omega}$. In the present work we will restrict ourselves to $$\label{def:Taylor} u_h\circ {\boldsymbol p}_h({\boldsymbol x}) \approx u_h({\boldsymbol x}) + \varrho_h({\boldsymbol x}){\boldsymbol n}_h({\boldsymbol x})\cdot\nabla u_h({\boldsymbol x})$$ which is the case of most practical interest. Choosing appropriate discrete spaces $V_h$ and $\Lambda_h$ for the approximation of $u$ and $\lambda$, respectively (particular choices are considered in Section \[sec:numex\]), we thus seek $(u_h,\lambda_h)\in V_h\times\Lambda_h$ such that $$\begin{aligned} {2}\label{eq:poissoninterior_FEM} \int_{\Omega_h}\nabla u_h \cdot\nabla v \,\text{d}\Omega_h +\int_{\partial\Omega_h}\lambda_h\, v\, \text{d}s &= \int_{\Omega_h}f v\, \text{d}\Omega_h\qquad \forall v\in V_h \\ \label{eq:poissonbc_FEM} \int_{\partial\Omega_h}(u_h+\varrho_h{\boldsymbol n}_h\cdot\nabla u_h)\, \mu\, \text{d}s &= \int_{\partial\Omega_h}\tilde{g}\, \mu\, \text{d}s\qquad \forall \mu\in \Lambda_h\end{aligned}$$ where we introduced the notation $\tilde{g}:= g\circ {\boldsymbol p}_h$ for the pullback of $g$ from $\partial \Omega$ to $\partial \Omega_h$. Using Green’s formula we note that the first equation implies that $\lambda_h = -{\boldsymbol n}_h\cdot\nabla u_h$, and therefore we now propose the following modified method: Find $(u_h,\lambda_h)\in V_h\times\Lambda_h$ such that $$\begin{aligned} {2}\label{eq:multinterior} \int_{\Omega_h}\nabla u_h \cdot\nabla v \,\text{d}\Omega_h +\int_{\partial\Omega_h}\lambda_h\, v\, \text{d}s &= \int_{\Omega_h}f v\, \text{d}\Omega_h\qquad \forall v\in V_h \\ \label{eq:multbc} \int_{\partial\Omega_h}u_h\, \mu\, \text{d}s-\int_{\partial\Omega_h} \varrho_h\lambda_h\,\mu\, \text{d}s &= \int_{\partial\Omega}\tilde{g}\, \mu\, \text{d}s\qquad \forall \mu\in \Lambda_h\end{aligned}$$ or $$A(u_h,\lambda_h;v,\mu) = (f,v)_{\Omega_h} + (\tilde{g},\mu)_{\partial\Omega_h}\quad \forall (u_h,\lambda_h)\in V_h\times\Lambda_h\label{eq:mainproblem}$$ where $(\cdot,\cdot)_{M}$ denotes the $L_2$ scalar product over $M$, with $\| \cdot\|_{M}$ the corresponding $L_2$ norm, and $$A(u,\lambda;v,\mu) := (\nabla u ,\nabla v )_{\Omega_h} +(\lambda , v)_{\partial\Omega_h} +( u,\mu)_{\partial\Omega_h} -(\varrho_h\lambda ,\mu)_{\partial\Omega_h}.$$ Relation to Nitsche’s method with boundary value correction ----------------------------------------------------------- Problem (\[eq:mainproblem\]) can equivalently be formulated as finding the stationary points of the Lagrangian $$\mathcal{L}(u,\lambda) := \frac12\|\nabla u\|^2_{\Omega_h} + (\lambda,u)_{\partial\Omega_h}-\|\varrho^{1/2}_h\lambda\|^2_{\partial\Omega_h} -(f,u)_{\Omega_h} - (\tilde{g},\lambda)_{\partial\Omega_h}$$ We now follow [@BuHa17] and add a consistent penalty term and seek stationary points of the augmented Lagrangian $$\mathcal{L}_\text{aug}(u,\lambda) := \mathcal{L}(u,\lambda) + \frac{1}{2} \|\gamma^{1/2}(u-\varrho_h\lambda-\tilde{g})\|^2_{\partial\Omega_h}$$ where $\gamma > 0$ remains to be chosen. The corresponding optimality system is $$\begin{aligned} \nonumber (f,v)_{\Omega_h} + (\tilde{g},\mu)_{\partial\Omega_h} = {}& A(u_h,\lambda_h;v,\mu) +(\gamma (u_h-\varrho_h\lambda_h-\tilde{g}),v)_{\partial\Omega_h}\\ & -(\gamma\varrho_h (u_h-\varrho_h\lambda_h-\tilde{g}),\mu)_{\partial\Omega_h} \end{aligned}$$ Now, formally replacing $\lambda_h$ by $-{\boldsymbol n}_h\cdot\nabla u_h$ and $\mu$ by $-{\boldsymbol n}_h\cdot\nabla v$ we obtain $$\begin{aligned} \nonumber (f,v)_{\Omega_h} - (\tilde{g},{\boldsymbol n}_h\cdot\nabla v)_{\partial\Omega_h} = {}& (\nabla u_h ,\nabla v )_{\Omega_h} -({\boldsymbol n}_h\cdot\nabla u_h,v)_{\partial\Omega_h} \\ \nonumber & -(u_h,{\boldsymbol n}_h\cdot\nabla v)_{\partial\Omega_h} -(\varrho_h {\boldsymbol n}_h\cdot\nabla u_h,{\boldsymbol n}_h\cdot\nabla v)_{\partial\Omega_h}\\ & +(\gamma (u_h+\varrho_h{\boldsymbol n}_h\cdot\nabla u_h-\tilde{g}),v+\varrho_h{\boldsymbol n}_h\cdot\nabla v)_{\partial\Omega_h}\end{aligned}$$ Setting now $\gamma = \gamma_0/h$, with $\gamma_0$ sufficiently large to ensure coercivity, we obtain the symmetrized version of the boundary value corrected Nitsche method proposed in [@BrDuTh72] with optimal convergence up to order $p=3$ assuming $\varrho_h\ge - C h$, for some sufficiently small constant. This means that $\partial \Omega_h$ either has to be a good approximation of $\partial \Omega$, or where it approximates poorly, $\Omega_h$ must approximation $\Omega$ from the inside. For future reference we define this method as: Find $u_h \in V_h$ such that $$\label{eq:Nitform} A_{Nit}(u_h,v_h) = (f,v_h)_{\partial \Omega_h} + (\tilde g, {\boldsymbol n}_h \cdot \nabla v_h)_{\partial \Omega_h}+ (\gamma \tilde g, v_h + \varrho_h {\boldsymbol n}_h \cdot \nabla v_h)_{\partial \Omega_h}$$ for all $v_h \in V_h$. Here the bilinear form is defined by $$\begin{aligned} \nonumber A_{Nit}(w_h,v_h) &:= (\nabla w_h ,\nabla v_h )_{\Omega_h} -({\boldsymbol n}_h\cdot\nabla w_h,v_h+\varrho_h {\boldsymbol n}_h\cdot\nabla v_h)_{\partial\Omega_h} -(w_h+\varrho_h {\boldsymbol n}_h\cdot\nabla w_h,{\boldsymbol n}_h\cdot\nabla v)_{\partial\Omega_h} \\ &\qquad +(\varrho_h {\boldsymbol n}_h\cdot\nabla w_h,{\boldsymbol n}_h\cdot\nabla v)_{\partial\Omega_h} + (\gamma (w_h+\varrho_h{\boldsymbol n}_h\cdot\nabla w_h,v+\varrho_h{\boldsymbol n}_h\cdot\nabla v_h)_{\partial\Omega_h}.\end{aligned}$$ Elements of analysis ==================== In this section we will prove some basic results on the stability and the accuracy of the method (\[eq:mainproblem\]). We will restrict ourselves to a discussion of the case $- C h \leq \varrho_h$, for some $C$ small enough. We assume that $\Lambda_h$ is the space of piecewise polynomial functions of order $k-1$ and $V_h$ is the space of continuous piecewise polynomial functions of order $k$, that we will denote $V_h^k$, enriched with higher order bubbles on the faces in $\partial \Omega_h$ so that inf-sup stability holds. The precise condition is given in equation (\[eq:infsup\]) below. For details on stable choices of the multiplier space we refer to [@BM97; @BD98; @KLPV01]. We introduce the triple norm defined on $ H^1(\Omega_h) \times L^2(\partial \Omega_h)$: $${|\mspace{-1mu}|\mspace{-1mu}|}(v,\mu) {|\mspace{-1mu}|\mspace{-1mu}|}:= \|\nabla v\|_{\Omega_h} + \|h^{-\frac12} v\|_{\partial \Omega_h} + \|h^{\frac12} \mu\|_{\partial \Omega_h}.$$ We let $\pi_h:L^2(\partial \Omega_h) \to \Lambda_h$ denote the $L^2$-orthogonal projection and we assume that the bound $$\|v - \pi_h v\|_{\partial \Omega_h} \lesssim h \|\nabla_\partial v\|_{\partial \Omega_h}$$ for all $v \in H^1(\partial \Omega_h)$ and where $\nabla_\partial$ denotes the gradient on the boundary. The formulation (\[eq:mainproblem\]) satisfies the following stability result \[prop:infsup\] Assume that $\varrho_h \ge - C_{\partial \Omega} h$ and that $V_h \times \Lambda_h$ satisfies the inf-sup condition. Then for $C_{\partial \Omega}$ sufficiently small, for all $(y_h,\eta_h) \in V_h \times \Lambda_h$, there exists $(v_h,\mu_h) \in V_h \times \Lambda_h$ such that $${|\mspace{-1mu}|\mspace{-1mu}|}(y_h,\eta_h) {|\mspace{-1mu}|\mspace{-1mu}|}\lesssim A(y_h,\eta_h;v_h,\mu_h)$$ and $${|\mspace{-1mu}|\mspace{-1mu}|}(v_h,\mu_h) {|\mspace{-1mu}|\mspace{-1mu}|}\lesssim {|\mspace{-1mu}|\mspace{-1mu}|}(y_h,\eta_h){|\mspace{-1mu}|\mspace{-1mu}|}.$$ First observe that $$A(y_h,\eta_h;y_h,-\eta_h) = \|\nabla y_h\|^2_{\Omega_h} + (\varrho_h \eta_h, \eta_h)_{\partial \Omega_h}.$$ Then recall that since the space satisfies the inf-sup condition there exists $v_\eta \in V_h$ such that $$\label{eq:infsup} \|h^{\frac12} \eta_h\|_{\partial \Omega_h} \lesssim (\eta_h,v_\eta)_{\partial \Omega_h} \quad \mbox{and} \quad \|\nabla v_\eta\|_{\Omega_h}+\|h^{-\frac12} v_\eta\|_{\partial \Omega_h} \lesssim \|h^{\frac12} \eta_h\|_{\partial \Omega_h}.$$ It follows that for some $c_\eta,\, C_{\partial \Omega_h}$ sufficiently small $$\begin{aligned} \|\nabla y_h\|^2_{\Omega_h}+\|h^{\frac12} \eta_h\|^2_{\partial \Omega} &\lesssim \|\nabla y_h\|^2_{\Omega_h} + (\varrho_h \eta_h, \eta_h)_{\partial \Omega_h} + \|h^{\frac12} \eta_h\|^2_{\partial \Omega} \\ &\lesssim A(y_h,\eta_h;y_h + c_\eta v_\eta,-\eta_h ).\end{aligned}$$ Here we used equation , $$(\varrho_h \eta_h, \eta_h)_{\partial \Omega_h} + \|h^{\frac12} \eta_h\|^2_{\partial \Omega} \ge (1 - C_{\partial \Omega_h}) \|h^{\frac12} \eta_h\|^2_{\partial \Omega}$$ and $$(\nabla y_h, y_h + c_\eta v_\eta)_{\Omega_h} \ge -\frac12 \|\nabla y_h\|^2_{\Omega_h} - 2 c_\eta^2 \|v_\eta\|^2_{\partial \Omega}.$$ Finally let $\mu_y =\pi_h y_h$ and observe that $$\begin{aligned} \nonumber &(y_h, h^{-1} \mu_y)_{\partial \Omega} - (\rho_h \eta_h, h^{-1} \mu_y) _{\partial \Omega} \\ &\qquad \ge \|h^{-\frac12} y_h\|_{\partial \Omega_h}^2 - \|h^{-\frac12} (y_h - \mu_y)\|_{\partial \Omega_h}^2 - \frac12 C_{\partial \Omega_h}^2 \|h^{\frac12} \eta_h\|_{\partial \Omega_h}^2 - \frac12 \|h^{-\frac12} \mu_y\|_{\partial \Omega_h}^2 \\ &\qquad \ge \frac12 \|h^{-\frac12} y_h\|_{\partial \Omega_h}^2 - C^2_t \|\nabla y_h\|^2_{\Omega_h} - \frac12 C_{\partial \Omega}^2 \|h^{\frac12} \eta_h\|_{\partial \Omega_h}^2.\end{aligned}$$ Where we used the approximation property of $\pi_h$ and a trace inequality $$\label{eq:trace} h_K^{\frac12}\|v_h\|_{\partial K} + h_K \|\nabla v_h\|_{K} \lesssim \|v_h\|_K.$$ for all elements $K \in {{\mathcal{K}}_h}$ and polynomials $v_h \in \mathbb{P}(K)$, to show that $$\|h^{-\frac12} (y_h - \mu_y)\|_{\partial \Omega_h} \leq C_t \|\nabla y_h\|_{\Omega_h}.$$ The first claim follows by taking $v_h = y_h + c_ \eta v_\eta$ and $\mu_h = - \eta_h + c_y h^{-1} \mu_y$ with $c_\eta$ and $c_y$ both $O(1)$, sufficiently small and assuming that $C_{\partial \Omega_h}$ is small enough. To conclude the proof we need to show that $${|\mspace{-1mu}|\mspace{-1mu}|}(v_h,\mu_h) {|\mspace{-1mu}|\mspace{-1mu}|}\lesssim {|\mspace{-1mu}|\mspace{-1mu}|}(y_h,\eta_h) {|\mspace{-1mu}|\mspace{-1mu}|}.$$ By the triangle inequality we have $${|\mspace{-1mu}|\mspace{-1mu}|}(v_h,\mu_h) {|\mspace{-1mu}|\mspace{-1mu}|}\leq {|\mspace{-1mu}|\mspace{-1mu}|}(y_h,\eta_h) {|\mspace{-1mu}|\mspace{-1mu}|}+{|\mspace{-1mu}|\mspace{-1mu}|}( c_ \eta v_\eta,c_y h^{-1} \mu_y) {|\mspace{-1mu}|\mspace{-1mu}|}.$$ By definition $${|\mspace{-1mu}|\mspace{-1mu}|}( c_ \eta v_\eta,c_y h^{-1} \mu_y) {|\mspace{-1mu}|\mspace{-1mu}|}=c_ \eta \|\nabla v_\eta\|_{\Omega_h} + c_ \eta\|h^{-\frac12} v_\eta\|_{\partial \Omega_h} + c_y \|h^{-\frac12} \mu_y\|_{\partial \Omega_h}$$ and the proof follows from (\[eq:infsup\]) together with the stability of $\pi_h$ in $L^2$. We will now use this stability result to prove an error estimate. For simplicity we here assume that $\varrho_h>0$, i.e. $\Omega_h \subset \Omega$. Let $u \in H^{k+1}(\Omega)$ denote the solution to (\[eq:poissoninterior\])–(\[eq:poissonbc\]). Let $u_h, \lambda_h \in V_h \times \Lambda_h$ denote the solution of (\[eq:mainproblem\]). Assume that the polynomial order of $V_h$ is $k \in \{1,2,3\}$, with enrichment on the boundary and $\Lambda_h \equiv X_h^{k-1}$. Assume that $V_h \times \Lambda_h$ satisfies (\[eq:infsup\]). Then there holds, with $\tilde \lambda = {\boldsymbol n}_h \cdot \nabla u\vert_{\partial \Omega_h}$, $$\begin{aligned} \label{eq:errorestenergy} {|\mspace{-1mu}|\mspace{-1mu}|}(u - u_h, \tilde \lambda- \lambda_h) {|\mspace{-1mu}|\mspace{-1mu}|}&\lesssim h^{k} \|u \|_{H^{k+1}(\Omega)} + h^{-1/2} \delta_h^{2} \sup_{0\leq t \leq \delta_0} \| D^{k+1} u\|_{L^2(\partial \Omega_t)} \\ \nonumber &\qquad + \textcolor{black}{h^{1/2}} \delta_h^{l+1} \sup_{-\delta_0\leq t < 0} \| D_n^{l} (f + \Delta u)\|_{L^2(\partial \Omega_t)}.\end{aligned}$$ Let $\tilde u_h \in V_h$ denote the solution to (\[eq:Nitform\]). We recall from [@BrDuTh72; @BuHaLa18] that the following error bound holds $$\begin{aligned} \nonumber {|\mspace{-1mu}|\mspace{-1mu}|}(u - \tilde u_h,0) {|\mspace{-1mu}|\mspace{-1mu}|}&+\|h^{\frac12} {\boldsymbol n}_h \cdot \nabla (u - \tilde u_h)\|_{\partial \Omega_h}+ \|h^{-\frac12} (\tilde u_h + \varrho {\boldsymbol n}_h \cdot \nabla \tilde u_h - \tilde g)\|_{\partial \Omega_h} \\ \label{eq:errorNit} &\lesssim h^{k} \|u \|_{H^{k+1}(\Omega)} + h^{-1/2} \varrho_h^{2} \sup_{0\leq t \leq \delta_0} \| D^{k+1} u\|_{L^2(\partial \Omega_t)} \\ \nonumber &\qquad + \textcolor{black}{h^{1/2}} \varrho_h^{l+1} \sup_{-\delta_0\leq t < 0} \| D_n^{l} (f + \Delta u)\|_{L^2(\partial \Omega_t)}.\end{aligned}$$ Let $i_h u$ denote the nodal interpolant of $u$. We then form the discrete errors $e_h = u_h - \tilde u_h$ and $\varsigma_h = \lambda_h - \zeta_h$ for some $\zeta_h \in \Lambda_h$. Using the triangle inequality and we have $${|\mspace{-1mu}|\mspace{-1mu}|}(u - u_h, \tilde \lambda- \lambda_h) {|\mspace{-1mu}|\mspace{-1mu}|}\leq {|\mspace{-1mu}|\mspace{-1mu}|}(u - \tilde u_h, \tilde \lambda- \zeta_h) {|\mspace{-1mu}|\mspace{-1mu}|}+ {|\mspace{-1mu}|\mspace{-1mu}|}(e_h, \varsigma_h) {|\mspace{-1mu}|\mspace{-1mu}|}.$$ Since the first term on the left hand side is bounded by standard interpolation and (\[eq:errorNit\]). We only need to consider the second term. By the stability estimate of Proposition \[prop:infsup\] we have $${|\mspace{-1mu}|\mspace{-1mu}|}(e_h, \varsigma_h) {|\mspace{-1mu}|\mspace{-1mu}|}\lesssim A(e_h,\varsigma_h; v_h,\mu_h).$$ Using the method (\[eq:mainproblem\]) and the definition of $\tilde u_h$ we find that $$\begin{aligned} \label{eq:gal_ortho} A(e_h,\varsigma_h; v_h,\mu_h) & = (f,v_h)_{\Omega_h} + (\tilde g, \mu_h)_{\partial \Omega_h} \\ \nonumber &\qquad - (\nabla \tilde u_h,\nabla v_h) _{\Omega_h}+(\zeta_h,v_h)_{\partial \Omega_h} \\ \nonumber &\qquad -(\tilde u_h, \mu_h)_{\partial \Omega_h} - (\varrho_h\zeta_h, \mu_h) _{\partial \Omega_h}.\end{aligned}$$ The definition of Nitsche’s method (\[eq:Nitform\]) implies the equality $$\begin{aligned} (f,v_h)_{\Omega_h} - (\nabla \tilde u_h ,\nabla v_h )_{\Omega_h} = {}& (\tilde{g},{\boldsymbol n}_h\cdot\nabla v_h)_{\partial\Omega_h} -({\boldsymbol n}_h\cdot\nabla \tilde u_h,v_h)_{\partial\Omega_h} \\ \nonumber & -(\tilde u_h,{\boldsymbol n}_h\cdot\nabla v_h)_{\partial\Omega_h} -(\varrho_h {\boldsymbol n}_h\cdot\nabla \tilde u_h,{\boldsymbol n}_h\cdot\nabla v_h)_{\partial\Omega_h} \\ \nonumber & +(\gamma (\tilde u_h+\varrho_h{\boldsymbol n}_h\cdot\nabla \tilde u_h-\tilde{g}),v_h+\varrho_h{\boldsymbol n}_h\cdot\nabla v_h)_{\partial\Omega_h} \\ = {} & -({\boldsymbol n}_h\cdot\nabla \tilde u_h,v_h)_{\partial\Omega_h} + (\tilde{g} - \tilde u_h - \varrho_h {\boldsymbol n}_h\cdot\nabla \tilde u_h,{\boldsymbol n}_h\cdot\nabla v_h)_{\partial\Omega_h} \\ \nonumber & +(\gamma (\tilde u_h+\varrho_h{\boldsymbol n}_h\cdot\nabla \tilde u_h-\tilde{g}),v_h+\varrho_h{\boldsymbol n}_h\cdot\nabla v_h)_{\partial\Omega_h}.\end{aligned}$$ Combining then (\[eq:gal\_ortho\]) with (\[eq:Nitform\]) we have $$\begin{aligned} A(e_h,\varsigma_h; v_h,\mu_h) = &(\tilde g-\tilde u_h-\varrho_h \zeta_h, \mu_h)_{\partial \Omega_h} \\ \nonumber &+(\zeta_h-{\boldsymbol n}_h \cdot \nabla \tilde u_h,v_h)_{\partial \Omega_h}\\ \nonumber & + (\tilde{g} - \tilde u_h - \varrho_h {\boldsymbol n}_h\cdot\nabla \tilde u_h,{\boldsymbol n}_h\cdot\nabla v_h)_{\partial\Omega_h} \\ \nonumber & +(\gamma (\tilde u_h+\varrho_h{\boldsymbol n}_h\cdot\nabla \tilde u_h-\tilde{g}),v_h+\varrho_h{\boldsymbol n}_h\cdot\nabla v_h)_{\partial \Omega_h}\\ = & I+II+III+IV.\end{aligned}$$ We will now bound the terms $I-IV$. First note that, $$\begin{aligned} I+III+IV \leq & (\|h^{-\frac12}(\tilde g-\tilde u_h-\varrho_h {\boldsymbol n}_h \nabla\cdot \tilde u_h)\|_{\partial \Omega_h}+\|h^{-\frac12}\varrho_h (\zeta_h- {\boldsymbol n}_h \cdot \nabla \tilde u_h) \|_{\partial \Omega_h}) \\ \nonumber & \times (\| \nabla v_h\|_{\Omega_h} + \|h^{-\frac12} v_h\|_{\partial \Omega_h} + \|h^{\frac12} \mu_h\|_{\partial \Omega_h}).\end{aligned}$$ For term $II$ there holds using Cauchy-Schwarz inequality $$\begin{aligned} II=(\zeta_h-{\boldsymbol n}_h \cdot \nabla \tilde u_h,v_h)_{\partial \Omega_h} \lesssim \|h^{\frac12}(\zeta_h - {\boldsymbol n}_h \cdot\nabla \tilde u_h)\|_{\partial \Omega_h} \|h^{-\frac12} v_h\|_{\partial \Omega_h}.\end{aligned}$$ Summing up we have using the assumption that $\|\rho_h\|_{L^\infty(\partial \Omega_h)} \leq O(h)$, $$\begin{aligned} I+II+III+IV &\leq (\|h^{-\frac12}(\tilde g-\tilde u_h-\varrho_h {\boldsymbol n}_h \cdot \nabla \tilde u_h)\|_{\partial \Omega_h} \\ \nonumber &\qquad +\|h^{\frac12} (\zeta_h- {\boldsymbol n}_h \cdot \nabla \tilde u_h) \|_{\partial \Omega_h}) {|\mspace{-1mu}|\mspace{-1mu}|}(v_h,\mu_h){|\mspace{-1mu}|\mspace{-1mu}|}.\end{aligned}$$ For the term $\|h^{-\frac12}(\tilde g-\tilde u_h-\varrho_h {\boldsymbol n}_h \cdot \nabla \tilde u_h)\|_{\partial \Omega_h}$ we may use the bound (\[eq:errorNit\]). It only remains to bound $\|h^{\frac12} (\zeta_h- {\boldsymbol n}_h \nabla \tilde u_h) \|_{\partial \Omega_h} $. To this end consider $\pi_{k-1} \nabla \tilde u_h \in [X_h]^d$ and let $\zeta_h = {\boldsymbol n}_h \cdot \pi_{k-1} \nabla \tilde u_h\vert_{\partial \Omega_h}$. For this choice we have using a trace inequality $$\begin{aligned} \|h^{\frac12} (\zeta_h- {\boldsymbol n}_h \nabla \tilde u_h) \|_{\partial \Omega_h} = {}&\|h^{\frac12} {\boldsymbol n}_h \cdot (\pi_{k-1} \nabla \tilde u_h- \nabla \tilde u_h) \|_{\partial \Omega_h} \\ \leq & \|\pi_{k-1} \nabla \tilde u_h- \nabla \tilde u_h\|_{\Omega_h}.\end{aligned}$$ To bound the term in the right hand side we add and subtract $\nabla u - \pi_{k-1} \nabla u$ and use the triangle inequality and the stability of the $L^2$-projection $\pi_{k-1}$ to obtain $$\begin{aligned} \|\pi_{k-1} \nabla \tilde u_h- \nabla \tilde u_h\|_{\Omega_h} &\leq \|\pi_{k-1} (\nabla \tilde u_h- \nabla u)\|_{\Omega_h}+ \|\pi_{k-1} \nabla u- \nabla u\|_{\Omega_h}+\|\nabla u- \nabla \tilde u_h\|_{\Omega_h} \\ & \leq \|\pi_{k-1} \nabla u- \nabla u\|_{\Omega_h}+2 \|\nabla u- \nabla \tilde u_h\|_{\Omega_h}.\end{aligned}$$ For the first term in the right hand side we have the approximation bound $$\|\pi_{k-1} \nabla u- \nabla u\|_{\Omega_h} \lesssim h^k \|D^{k+1} u\|_{\Omega_h}.$$ The second term is bounded by (\[eq:errorNit\]). We conclude by applying the second inequality of Proposition \[prop:infsup\]. Remarks on cut finite element methods ===================================== In the context of cut finite element methods the discontinuous multiplier spaces used above can no longer be expected to be stable. It is possible to stabilise the multiplier using Barbosa-Hughes stabilisation. However, fluctuation based multipliers are unlikely to be suitable in this context since the weak consistency of the fluctuations of the multiplier between elements depends on the geometry approximation through the interface normal. Since the method is of interest when the geometry approximation is of relatively low order, this limits the possibility to use fluctuation based stabilisation. For closed smooth boundaries, one may prove inf-sup stability and optimal convergence, without stabilisation, when using continuous approximation of polynomial order less than or equal to $2$, for both the bulk variable and the multiplier provided $\rho_h = O(h^2)$. The approximation order of the interface normal, which is $O(h)$ prohibits higher order convergence if the interface approximation is piecewise affine. For instance, piecewise cubic continuous approximation will not necessarily achieve higher order convergence that the piecewise quadratic approximation. Numerical examples {#sec:numex} ================== We show examples of higher order triangular elements with linearly interpolated boundary and low order rectangular elements with staircase boundary, using discontinuous multiplier spaces. In all examples we define the meshsize $h=1/\sqrt{\text{NNO}}$, where NNO corresponds to the number of nodes of the lowest order FEM on the mesh in question (bilinear or affine). Triangular elements ------------------- We first consider the case of affine triangulations of a ring $1/4\leq r\leq 3/4$, $r=\sqrt{x^2+y^2}$. We use the manufactured solution $u=(r-1/4)(3/4-r)$ and compute the corresponding right–hand side analytically. An elevation of the a typical discrete solution is given in Fig.  \[fig:trisol\]. We use continuous piecewise $P^k$ polynomials, $k=2,3$ for the approximation of $u$, and for the approximation of $\lambda$ we use piecewise $P^{k-1}$ polynomials, discontinous on each element edge on $\Gamma_h$. To ensure [*inf–sup*]{} stability, we add hierarchical $P^{k+1}$ bubbles on each edge in the approximation of $u$. #### Second order elements. In Fig.  \[fig:errtri\] we show the convergence in $L_2(\Omega_h)$ and $H^1(\Omega_h)$ with and without boundary modification. In Fig.  \[fig:errlam\] we show the error in multiplier computed as $\| (-{\boldsymbol n}\cdot\nabla u)\vert_{\partial\Omega_h} - \lambda_h\|_{\partial\Omega_h}$. Optimal order convergence is observed for the modified method, convergence $O(h^3)$ in $L_2(\Omega_h)$ and $O(h^2)$ in $H^1(\Omega_h)$; the multiplier error is approximately $O(h^2)$. #### Third order elements. Next we use continuous piecewise third order polynomials for the approximation of $u$, and for the approximation of $\lambda$ we use piecewise quadratic polynomials, discontinous on each element edge on $\Gamma_h$. In Fig. \[fig:errtri2\] we show the convergence in $L_2(\Omega_h)$ and $H^1(\Omega_h)$ with and without boundary modification. In Fig.  \[fig:errlam2\] we show the error in multiplier computed as above. Optimal order convergence is again observed for the modified method, convergence $O(h^4)$ in $L_2(\Omega_h)$ and $O(h^3)$ in $H^1(\Omega_h)$; the multiplier error is approximately $O(h^3)$. Note that no improvement over $P^2$ approximations can be seen in the unmodified method due to the geometry error being dominant. #### An unstable pairing of spaces. We finally make the observation that our modification has a stabilising influence on the approximation. We try continuous $P^2$ approximations of $u$ and discontinuous $P^2$ approximations of $\lambda$. In this case we get no convergence without the modification due to the violation of the [*inf–sup*]{} condition, whereas with modification we obtain the optimal convergence pattern in $u$ and a stable multiplier convergence given in Fig \[fig:conunstab\]. Rectangular elements -------------------- This example shows that it is possible to achieve optimal convergence even on a staircase boundary. We use a continuous piecewise $Q_1$ approximation on the (affine) rectangles, again enhanced for [*inf–sup*]{}, now by hierarchical $P^2$ bubble function on the boundary edges, together with edgewise constant multipliers on $\Gamma_h$. We use the manufactured solution $u=\sin(x^3)\cos(8y^3)$ on the domain inside the ellipse $x^2/4+y^2 = 1$. Our computational grids consist of elements completely inside this ellipse; a typical coarse grid is shown if Fig. \[fig:coarse\] where we note the staircase boundary. In Fig. \[fig:elevq\] we show elevations of the numerical solutions on a finer grid without and with boundary correction. In Fig. \[fig:errquad\] we show the errors of the unmodified and modified methods. Again we observe optimal order convergence for the modified method, $O(h^2)$ in $L_2(\Omega_h)$ and $O(h)$ in $H^1(\Omega_h)$. Concluding remarks ================== We have introduced a symmetric modification of the Lagrange multiplier approach to satisfying Dirichlet boundary conditions for Poisson’s equation. This novel approach allows for affine approximations of the boundary, and thus affine elements, up to polynomial approximation order 3 without loss of convergence rate as compared to higher order boundary fitted meshes. The modification is easy to implement and only requires that the distance to the exact boundary in the direction of the discrete normal can be easily computed. In fact, the modification stabilises the multiplier method so that unstable pairs of spaces can be used, as long as there is a uniform distance to the boundary. #### Acknowledgement. This research was supported in part by EPSRC, UK, Grant No. EP/P01576X/1, the Swedish Foundation for Strategic Research Grant No. AM13-0029, the Swedish Research Council Grants No. 2013-4708, 2017-03911, 2018-05262, and Swedish strategic research programme eSSENCE. [10]{} F. Ben Belgacem and Y. Maday. The mortar element method for three-dimensional finite elements. , 31(2):289–302, 1997. T. Boiveau, E. Burman, S. Claus, and M. Larson. Fictitious domain method with boundary value correction using penalty-free [N]{}itsche method. , 26(2):77–95, 2018. D. Braess and W. Dahmen. Stability estimates of the mortar finite element method for [$3$]{}-dimensional problems. , 6(4):249–263, 1998. J. H. Bramble, T. Dupont, and V. Thom[é]{}e. Projection methods for [D]{}irichlet’s problem in approximating polygonal domains with boundary-value corrections. , 26:869–879, 1972. E. Burman and P. Hansbo. Deriving robust unfitted finite element methods from augmented [L]{}agrangian formulations. In S. P. A. Bordas, E. Burman, M. G. Larson, and M. A. Olshanskii, editors, [*Geometrically Unfitted Finite Element Methods and Applications*]{}, pages 1–24. Springer, Cham, 2017. E. Burman, P. Hansbo, and M. G. Larson. A cut finite element method with boundary value correction. , 87(310):633–657, 2018. E. Burman, P. Hansbo, and M. G. Larson. A cut finite element method with boundary value correction for the incompressible [S]{}tokes’ equations. In F. A. Radu, K. Kumar, I. Berre, J. M. Nordbotten, and I. S. Pop, editors, [*Numerical Mathematics and Advanced Applications ENUMATH 2017*]{}, pages 183–192. Springer, Cham, 2019. T. Dupont. ${L}_2$ error estimates for projection methods for parabolic equations in approximating domains. In C. de Boor, editor, [*Mathematical Aspects of Finite Elements in Partial Differential Equations*]{}, pages 313–352. Academic Press, New York, 1974. D. Gilbarg and N. S. Trudinger. . Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. A. Johansson and M. G. Larson. A high order discontinuous [G]{}alerkin [N]{}itsche method for elliptic problems with fictitious boundary. , 123(4):607–628, 2013. C. Kim, R. D. Lazarov, J. E. Pasciak, and P. S. Vassilevski. Multiplier spaces for the mortar finite element method in three dimensions. , 39(2):519–538, 2001. A. Main and G. Scovazzi. The shifted boundary method for embedded domain computations. [Part I: Poisson and Stokes]{} problems. , 372:972–995, 2018. A. Main and G. Scovazzi. The shifted boundary method for embedded domain computations. [Part II: Linear advection–diffusion and incompressible Navier–Stokes equations]{}. , 372:996–1026, 2018. ![Elevation of the discrete solution on triangles.[]{data-label="fig:trisol"}](elevation.eps) ![Errors with and without boundary modification, $P^2$ case.[]{data-label="fig:errtri"}](convp2.eps) ![Errors in the multiplier with and without boundary modification, $P^2$ case.[]{data-label="fig:errlam"}](convp2lam.eps) ![Errors with and without boundary modification, $P^3$ case.[]{data-label="fig:errtri2"}](convp3.eps) ![Errors in the multiplier with and without boundary modification, $P^3$ case.[]{data-label="fig:errlam2"}](convp3lam.eps) ![Error plots for the unstable triangular element example.[]{data-label="fig:conunstab"}](conunstab.eps) ![A coarse mesh inside the elliptical domain.[]{data-label="fig:coarse"}](ellipse.eps) ![Elevation of the discrete solution on rectangles for the unmodified (top) and for the modified (bottom) schemes.[]{data-label="fig:elevq"}](elevationunstab.eps) ![Elevation of the discrete solution on rectangles for the unmodified (top) and for the modified (bottom) schemes.[]{data-label="fig:elevq"}](elevationstab.eps) ![Error plots for the rectangular element example.[]{data-label="fig:errquad"}](errorplotquad.eps)

--- author: - Paritosh Garg - Sagar Kale - Lars Rohwedder - Ola Svensson bibliography: - 'ref.bib' title: 'Robust Algorithms under Adversarial Injections[^1]' --- [^1]: Research supported in part by the Swiss National Science Foundation project 200021-184656 “Randomness in Problem Instances and Randomized Algorithms.”

--- abstract: 'Dipolar interactions are ubiquitous in nature and rule the behavior of a broad range of systems spanning from energy transfer in biological systems to quantum magnetism. Here, we study magnetization-conserving dipolar induced spin-exchange dynamics in dense arrays of fermionic erbium atoms confined in a deep three-dimensional lattice. Harnessing the special atomic properties of erbium, we demonstrate control over the spin dynamics by tuning the dipole orientation and changing the initial spin state within the large 20 spin hyperfine manifold. Furthermore, we demonstrate the capability to quickly turn on and off the dipolar exchange dynamics via optical control. The experimental observations are in excellent quantitative agreement with numerical calculations based on discrete phase-space methods, which capture entanglement and beyond-mean field effects. Our experiment sets the stage for future explorations of rich magnetic behaviors in long-range interacting dipoles, including exotic phases of matter and applications for quantum information processing.' author: - 'A.Patscheider' - 'B.Zhu' - 'L.Chomaz' - 'D.Petter' - 'S.Baier' - 'A-M.Rey' - 'F.Ferlaino' - 'M.J.Mark' bibliography: - 'Spindynamics.bib' date: April 2019 title: Controlling Dipolar Exchange Interactions in a Dense 3D Array of Large Spin Fermions --- Spin lattice models of localized magnetic moments (spins), which interact with one another via exchange interactions, are paradigmatic examples of strongly correlated many-body quantum systems. Their implementation in clean, isolated, and fully controllable lattice confined ultra-cold atoms opens a path for a new generation of synthetic quantum magnets, featuring highly entangled states, especially when driven out-of-equilibrium, with broad applications ranging from precision sensing and navigation, to quantum simulation and quantum information processing [@Bloch2008; @Gross2017]. However, the extremely small energy scales associated with the nearest-neighbor spin interactions in lattice-confined atoms with dominant contact interactions, have made the observation of quantum magnetic behaviors extremely challenging [@Bloch2008r; @Greif2013]. On the contrary, even under frozen motional conditions, dipolar gases, featuring long-range and anisotropic interactions, offer the opportunity to bring ultra-cold systems several steps ahead towards the ambitious attempt of modeling and understanding quantum magnetism. While great progress in studying quantum magnetism has been achieved using arrays of Rydberg atoms [@Zeiher2017; @Bernien2017; @Barredo2018; @Guardado2018], trapped ions [@Neyenhuise2017; @Blatt2012; @Britton2012], polar molecules [@Yan2013ood; @Hazzard2014], and spin-$3$ bosonic chromium atoms [@dePaz2013; @dePaz2016; @Lepoutre2018], most of the studies so far have been limited to spin-$1/2$ mesoscopic arrays of at the most few hundred particles or to macroscopic but dilute ($<0.1$ filling fractions) samples of molecules in lattices. In this work, we report the first investigations of non-equilibrium quantum magnetism in a dense array of fermionic magnetic atoms confined in a deep three-dimensional optical lattice. Our platform realizes a quantum simulator of the long-range XXZ Heisenberg model. The simulator roots on the special atomic properties of ${}^{167}$Er, whose ground state bears large angular momentum quantum numbers with $I=7/2$ for the nuclear spin and $J=6$ for the electronic angular momentum, resulting in a $F=19/2$ hyperfine manifold, as depicted in Fig.\[fig:1\]A. Such a complexity enables new control knobs for quantum simulations. First, it is responsible for the large magnetic moment in Er. Second, it gives access to a fully controllable landscape of $20$ internal levels, all coupled by strong magnetic dipolar interactions, up to $49$ times larger than the ones felt by $F=1/2$ alkali atoms in the same lattice potential [@Stamper-Kurn2013]. Finally, it allows fast optical control of the energy splitting between the internal states which can be tuned on and off resonance using the large tensorial light shift [@Becher2017pol], which adds to the usual quadratic Zeeman shift. {width="95.00000%"} Using all these control knobs, we explore the dipolar exchange dynamics and benchmark our simulator with an advanced theoretical model, which takes entanglement and beyond mean-field effects into account [@suppmat]. In particular, we initialize the system into a desired spin state and activate the spin dynamics, while the motional degree of freedom mainly remains frozen. Here, we study the spreading of the spin population in the different magnetic sublevels as a function of both the specific initial spin state and the dipole orientation. We demonstrate that the spin dynamics at short evolution time follows a scaling that is invariant under internal state initialization (choice of macroscopically populated initial Zeeman level) and that is set by the effective strength of the dipolar coupling. On the contrary, at longer times, the many-body dynamics is affected by the accessible spin space and the long-range character of dipolar interactions beyond nearest neighbors. Finally, we show temporal control of the exchange dynamics using off resonant laser light. The XXZ Heisenberg model that rules the magnetization-conserving spin dynamics of our system can be conveniently written using spin-$19/2$ dimensionless angular momentum operators $\hat{\mathbf{F}}_i=\{\hat{F}^x_i,\hat{F}^y_i,\hat{F}^z_i\}$, acting on site $i$ and satisfying the commutation relation $[\hat F_i^x,\hat F_i^y]=i\hat F_i^z$. We use the eigenbasis of $\hat{F}^z$ denoted as $|m_F\rangle$ with $0\leq |m_F|\leq F$ [@Auerbach1994iea; @Dutta2015; @suppmat]: $$\begin{aligned} \hat{H} &=& \frac{1}{2}\sum_{i,j\neq i}V_{i,j} \left(\hat{F}_ {i}^z\hat{F}_{j}^z-\frac{1}{4}(\hat{F}^+_i\hat{F}^-_{j}+\hat{F}^-_{i}\hat{F}^+_{j}) \right) \nonumber \\ &&+ \sum_{i} \delta_{i}(\hat{F}^z_{i} )^2\end{aligned}$$ The coupling constants $V_{i, j}\,=\,V_{dd}d_y^3\frac{1-3\cos^2(\theta_{i,j})}{r_{ij}^3}$, describe the direct dipole-dipole interactions (DDI), which have long-range character and thus couple beyond nearest neighbors. The dipolar coupling strength between two dipoles located at $\vec{r}_{i}$ and $\vec{r}_{j}$ depends on their relative distance $r_{ij}=|\vec{r}_{i}-\vec{r}_{j}|$ and on their orientation, described by the angle $\theta_{i,j}$ between the dipolar axis, set by the external magnetic field, and the interparticle axis; see Fig.\[fig:1\]B. Here, $V_{dd}\,\approx\,\frac{\mu_0g_F^2\mu_B^2}{4\pi d_y^3}$ denotes the dipolar coupling strength, with $g_F\approx 0.735$ for ${}^{167}$Er, $\mu_0$ the magnetic permeability of vacuum, $\mu_B$ the Bohr magneton, and $d_y$ the shortest lattice constant. The $\hat{F}_ {i}^z\hat{F}_{j}^z$ terms in the Hamiltonian account for the diagonal part of the interactions while the $\hat{F}^+_i\hat{F}^-_{j}+\hat{F}^-_{i}\hat{F}^+_{j}$ terms describe dipolar exchange processes. The second sum denotes the single particle quadratic term $\delta_{i}(\hat{F}^z_{i})^2$ with $\delta_{i}=\delta_{i}^Z+\delta_{i}^T$, accounting for the quadratic Zeeman effect $\propto \delta_{i}^Z$ and tensorial light shifts $\propto \delta_{i}^T$. These two contributions can be independently controlled in our experiment. The quadratic Zeeman shift allows us to selectively prepare all atoms in one target state of the spin manifold [@suppmat]. The tensorial light shift can compete or cooperate with the quadratic Zeeman shift and can be used as an additional control knob to activate/deactivate the exchange processes. Note that, for all measurements, a large linear Zeeman shift is always present, but since it does not influence the spin-conserving dynamics, it is omitted in Eq.1. In the experiment, we first load a spin-polarized quantum degenerate Fermi gas of $\approx 10^4$ Er atoms into a deep 3D optical lattice, following the scheme of Ref. [@Baier2017sif]. The cuboid lattice geometry with lattice constants $(d_x,d_y,d_z) = (272,266,544)\,$nm results in weakly coupled 2D planes, with typical tunneling rates of $\sim10\,$Hz inside the planes and $\sim\,$mHz between them [@suppmat]. The external magnetic field orientation, setting the quantization axis as well as the dipolar coupling strengths, is defined by the polar angles $\Theta$ and $\phi$ in the laboratory frame; see Fig.\[fig:1\]B. The fermionic statistics of the atoms enables us to prepare a dense band insulator with at most one atom per lattice site, as required for a clean implementation of the XXZ Heisenberg model. This is an advantage of fermionic atoms as compared to bosonic systems, which typically require filtering protocols to remove doublons [@Lepoutre2018]. Our experimental sequence to study the spin dynamics is illustrated in Fig.\[fig:1\]C. In particular, we prepare the system into the targeted $m_F^0$ state by using the lattice-protection protocol demonstrated in Ref.[@Baier2017sif]; see also Ref.[@suppmat]. At the end of the preparation, the majority of atoms are in the desired $m_F^0$ ($>80\%$) at $B\approx 4\,$G. We note that atom losses during the spin preparation stage reduces the filling factor to about 60% of the initial one [@suppmat]. We then activate the spin dynamics by quenching the magnetic field to a value for which $\bar{\delta}=\sum_{i} \delta_{i} =0$, providing a resonance condition for the magnetization-conserving spin-exchange processes; see Fig.\[fig:2\]A. After a desired time of evolution, we stop the dynamics by rapidly increasing the magnetic field, leaving the resonance condition. We finally extract the atom number in each spin state via a spin-resolved band-mapping technique [@suppmat] and derive the relative state populations by normalization to the initial total atom number. {width="0.95\linewidth"} We now probe the evolution of the spin-state population as a function of the hold time on resonance. We observe a redistribution of the population from the initial state to multiple neighboring states in $m_F$ space, as exemplary shown for an initial state of ${|\text{--}13/2\rangle}$ in Fig.\[fig:2\]B-C. The dynamics preserves the total magnetization; see inset of Fig.\[fig:2\]B. We observe similar behavior independently of the initialized $m_F^0$ states. The spin transfer happens sequentially. At short times it is dominated by the transfer to states directly coupled by the dipolar exchange Hamiltonian, i.e.those ones which differ by plus/minus one unit of angular momentum ($\Delta m_F = \pm 1$). At longer times, subsequent processes transfer atoms to states with $|\Delta m_F| \geq 2$; see Fig.\[fig:2\]C-D. To benchmark our quantum simulator, we use a semiclassical phase-space sampling method, the so called generalized discrete truncated Wigner approximation (GDTWA) [@Lepoutre2018; @Schachenmayer2015a; @Schachenmayer2015b; @Polkovnikov2010; @suppmat]. The method accounts for quantum correlation in the many-body dynamics and is adapted to tackle the complex dynamics of a large-spin system in a regime where exact diagonalization techniques are impossible to implement with current computers. The GDTWA calculations take into account actual experimental parameters such as spatial inhomogeneites, density distribution after the lattice loading, initial spin distribution, and effective lattice filling, including the loss during the spin preparation protocol [@suppmat]. Figure \[fig:2\]B shows the experimental dynamics together with the GDTWA simulations. Although the model does not include corrections due to losses and tunneling during the dynamics, it successfully captures the behavior of our dense system not only at short time, but also at long time, where the population dynamics slows down and starts to reach an equilibrium. Similar level of agreement between experiment and theory is shown in Fig.\[fig:2\]C-D where we directly compare the spreading of the spin population as a function of time. The important role of quantum effects in the observed spin dynamics can be illustrated by contrasting the GDTWA simulation with a mean-field calculation. Indeed, the mean-field calculation fails in capturing the system behavior. It predicts a too slow population dynamics for non-perfect spin-state initialization, as in the experiments shown in Fig.\[fig:2\]B, and no dynamics for the ideal case where all atoms are prepared in the same internal state [@suppmat]. To emphasize the beyond nearest-neighbor effects, we also compare the experiment with a numerical simulation that only includes nearest-neighbor interactions (NNI-GDTWA). Here, we again observe a very slow spin evolution, which largely deviates from the measurements. The agreement of the full GDTWA predictions with our experimental observations points to the long-range many-body nature of the underlying time evolution. Our theory calculations also support the built up of entanglement during the observed time evolution. Different spin configurations feature distinct effective interaction strengths, which also depend on the orientation of the dipoles with respect to the lattice. We demonstrate our ability to control this interaction, which governs the rate of population exchange, by the choice of the initial $m_F^0$ state and the orientation of the external magnetic field. This capability provides us with two tuning knobs to manipulate dipolar exchange interactions in our quantum simulator. Figure \[fig:3\]A-F plots the dynamics of the populations for three neighboring spin states after the quench, starting from different initial spin states. Solid lines show the results of the full-GDTWA calculations. For each initial $m_F^0$, we find a remarkable agreement between theory and experiment. We observe a strong speedup for states with large spin projections perpendicular to the quantization axis, as it is expected from the expectation value of $\hat{F}^+_i\hat{F}^-_j$, which gives a prefactor $\gamma(m_F^0)=\sqrt{F(F+1)-m_F^0(m_F^0+1)}\sqrt{F(F+1)-m_F^0(m_F^0-1)}$. The initial dynamics can be well described by a perturbative expansion up to the second order [@suppmat], resulting in the analytic expression for the normalized population $n_{m_F}(t)$ of the initial state: $$\begin{aligned} n_{m_F^0}(t)&=&n_{m_F^0}(0)\Big(1-n_{m_F^0}(0)\frac{V_{\rm eff}^2}{\hbar^2}t^2\Big)\end{aligned}$$ {width="0.95\linewidth"} Here, $V^2_\text{eff}\equiv \frac{\gamma^2(m_F^0)}{8 N}\sum_{i,j\neq i} V_{ij}^2$ is the overall effective interaction strength summed over $N$ atoms and $n_{m_F^0}(0)$ denotes the purity of the initial state preparation. For a quantitative analysis of the early-time spin evolution, we compare the theoretically calculated $V_\text{eff}$ from the initial atomic distribution used in the GDTWA model with the one extracted from a fit of Eq.2 to the experimental data. Here we consider the data up to $t<0.5\frac{\hbar}{V_\text{eff}}$ estimated using the theoretically calculated $V_\text{eff}$ [@Note1]. Figure \[fig:3\]G plots both, the theoretical and experimental $V_\text{eff}$ as a function of the initial $m_F^0$ and highlights once more their quantitative agreement. The interaction parameter $V_\text{eff}$, can also be used to rescale the time axis. As shown in Fig.\[fig:3\]H, all data sets now collapse onto each other for $\frac{t V_\text{eff}}{\hbar}<0.5$, revealing the invariant character of the short-time dynamics under the internal state initialization. At longer timescales, the theory shows that the timetraces start to deviate from each others and saturate to different values, indicating that thermal-like states are on reach. In the experiment, we observe a similar behavior but here the saturation value might also be affected by losses and residual tunneling. Because of the anisotropic character of the DDI, the strength of the dipolar exchange can be controlled by changing the angle $\Theta$; see Fig.\[fig:1\]B. As exemplary shown in Fig.\[fig:4\]A for ${|\text{--}17/2\rangle}$, the observed evolution speed of the spin populations strongly depends on $\Theta$, changing by about a factor of $2$ between $\Theta=40^\circ$ and $80^\circ$. The GDTWA results show a very good quantitative agreement with the experiment. We repeat the above measurements for different values of $\Theta$ and we extract $V_\text{eff}$; Fig.\[fig:4\]B. It is worth to notice that, while the dipolar interactions can be completely switched off at a given angle in a 1D chain, in a 3D system the situation is more complicated. However, as expected by geometrical arguments, we observe that the overall exchange strength becomes minimal for a specific dipole orientation $(\Theta_c\approx 35^\circ,\phi_c=45^\circ)$. We compare our measured $V_\text{eff}$ with the ones calculated from the initial spin distribution, which is a good quantity to describe the early time dynamics. Theory and experiment show a similar trend, in particular reaching a minimum at about $\Theta_c$. Note that the simple analytic formula (Eq.2), used for fitting the data, deviates from the actual evolution at longer times. This leads to a small down-shift of the experimental values [@suppmat]. ![Angle dependence of the spin exchange dynamics and dynamical control. (A) Exemplary measurements of the time evolution for the starting spin state ${|\text{--}17/2\rangle}$ for $\Theta\,=\,40^\circ,80^\circ$. Solid lines show the full-GDTWA results. (B) Extracted $V_\text{eff}$ as a function of $\Theta$ from a fit to the experimental data (orange circles) and numerically computed from the initial spin distribution (black circles). Errorbars denote the $68\%$ confidence interval of the fits. (C) Time evolution of the initial state ${|\text{--}9/2\rangle}$ at $\bar \delta = 0$ and $\Theta = 0^\circ$ without (filled circles) and with (open circles) switching on an additional light field after $20\, \rm ms$ of evolution. Solid (dashed) lines are the corresponding full-GDTWA calculations. The inset shows the population of the initial spin state after $50\, \rm ms$ evolution time as a function of the quadratic Zeeman shift without (filled circles) and with (open circles) the additional light field. Determining the centers of the resonances via a fit yields an absolute shift of the resonance condition by $h \times 27(1)\, \rm Hz$ between both conditions.[]{data-label="fig:4"}](Fig4){width="0.95\linewidth"} Finally, we demonstrate fast optical control of the spin dynamics relying on the remarkably large tensorial light shift of erbium compared to alkali atoms. As shown in Fig.\[fig:4\]C, we can almost fully suppress the spin exchange dynamics by suddenly switching on a homogeneous light field after an initial evolution time on resonance. Therefore the tensorial light shift, inducing a detuning from the resonance condition (see inset), allows a full spatial and temporal control over the exchange processes as the light power can be changed orders of magnitude faster than the typical interaction times and can address even single lattice sites in quantum gas microscopes. This capability can be an excellent resource for quantum information processing, i.e.we could use dipolar exchange processes to efficiently prepare highly entangled states between different parts of a quantum system and then store the quantum information at longer times by turning the interactions off. The excellent agreement between the experiment and the theory, not only verifies our quantum simulator but sets the stage towards its use for the study of new regimes intractable to theory. For example by operating at weaker lattice when motion is involved, the dynamics is no longer described by a spin model but by a high spin Fermi-Hubbard model with long-range interactions. Alternatively by treating the internal hyperfine levels as a synthetic dimension [@Mancini1510] while adding Raman transitions to couple them, one could engineer non-trivial synthetic gauge field models even when tunneling is only allowed in one direction. Moreover, the demonstrated control over the different hyperfine level structure, our capability to tune the strength of the direct dipolar exchange coupling via the magnetic field angle, and the possibility of the dynamical and spatial control of the resonance condition via tensorial light shifts make our system a potential resource for quantum information processing with a qudit-type multi-level encoding using the 20 different interconnected hyperfine levels [@Lanyon2008; @Brion2007; @Kues2017]. We thank J. Schachenmayer for fruitful discussions and Arghavan Safavi-Naini for helping us understanding the loading into the lattice. We thank J.H.Becher and G.Natale for their help in the experimental measurements and for fruitful discussions. We also thank Rahul Nandkishore and Itamar Kimchi for reviewing the manuscript. The Innsbruck group is supported through an ERC Consolidator Grant (RARE, no.681432) and a FET Proactive project (RySQ, no.640378) of the EU H2020, and by a Forschergruppe (FOR 2247/PI2790) of the DFG and the FWF. LC is supported within a Marie Curie Project (DipPhase, no.706809) of the EU H2020. A.M.R is supported by the AFOSR grant FA9550-18-1-0319 and its MURI Initiative, by the DARPA and ARO grant W911NF-16-1-0576, DARPA-DRINQs, the ARO single investigator award W911NF-19-1-0210, the NSF PHY1820885, JILA-NSF PFC-173400 grants, and by NIST. B.Z. is supported by the NSF through a grant to ITAMP. \* Correspondence and requests for materials should be addressed to M.J.M. (email: manfred.mark@uibk.ac.at). Supplementary Materials {#supplementary-materials .unnumbered} ======================= Experimental setup and lattice loading ====================================== In our experiment we start with a degenerate Fermi gas of about $2.4\times 10^4$ ${}^{167}$Er atoms in the lowest spin state $|F=19/2, m_F=\text{--}19/2\rangle={|\text{--}19/2\rangle}$ and a temperature of $T \approx 0.3\,T_\text{F}$ [@Aikawa2014rfd; @Baier2017sif]. The atoms are confined in a crossed optical dipole trap (ODT) and the trap frequencies are $(\nu_{\perp},\nu_{\parallel},\nu_{z})=(63(1),36(2),137(1))\,\rm Hz$, where $\nu_{\perp}$ ($\nu_\parallel$) are the trap frequencies perpendicular to (along) the horizontal ODT and $\nu_{z}$ is measured along the vertical direction defined by gravity. We load the atomic sample adiabatically into a 3D lattice that is created by two retro-reflected laser beams at $532\,$nm in the x-y plane and one retro-reflected laser beam at $1064\,$nm nearly along the z direction, defined by gravity and orthogonal to the x-y plane. Note, that due to a small angle of about $10^\circ$ between the vertical lattice beam and the z axis we obtain a 3D-lattice, slightly deviating from an ideal situation of a rectangular unit cell and our parallelepipedic cell has the unit lattice distances of $d_{x}\,=\,272\,{\rm nm}$, $d_{y}\,=\,266\,{\rm nm}$, and $d_{z}\,=\,544\,{\rm nm}$. The lattice geometry is similar to the one used in our previous works [@Baier2016ebh; @Baier2017sif]. We ramp up the lattice beams in $150\,$ms to their final power and switch off the ODT subsequently in $10\,$ms and wait for $500\,$ms to eliminate residual atoms in higher bands [@Baier2017sif]. For our typical lattice depths used in the measurements reported here of $(s_{x},s_{y},s_{z})\,=\,(20,20,80)$, where $s_i$ with $i\in {x,y,z}$ is given in the respective recoil energy $\mathrm{E}_{\mathrm{R},i}$ with $E_{\text{R};x,y} = h \times 4.2\, \rm kHz$ and $E_{\text{R};z} = h \times 1.05\, \rm kHz$, the atoms can be considered pinned on single lattice sites with low tunneling rates $J_{x,y} = h \times 10.5\, \rm Hz$ and $J_{z} = h \times 1\, \rm mHz$. State preparation and detection efficiency ========================================== To prepare the atoms in the desired spin state, after loading them into the lattice, we use a technique based on a rapid-adiabatic passage (RAP). During the full preparation procedure, the optical lattice operates as a protection shield to avoid atom loss and heating due to the large density of Feshbach resonances [@Baier2017sif]. To reach a reliable preparation with high fidelity it is necessary that the change in the energy difference between subsequent neighboring spin states is sufficiently large. Therefore, we ramp the magnetic field in $40\,\rm ms$ to $40.6\,\rm G$ to get a large enough differential quadratic Zeeman shift, which is on the order of $\approx h \times 40\,$kHz. After the magnetic field ramp we wait for $80\,\rm ms$ to allow the latter to stabilize before performing the RAP procedure. The follow up RAP protocol depends on the target state. For example, to transfer the atoms from ${|\text{--}19/2\rangle}$ into the ${|\text{--}7/2\rangle}$ state, we apply a radio-frequency (RF) pulse at $41.31\,\rm MHz$ and reduce the magnetic field with a linear ramp, e.g.by $500\,\rm mG$ in $42\,\rm ms$. The variation of the magnetic field is analogous to the more conventional scheme where the frequency of the RF is varied (see Fig.\[fig:S1\]A). For the preparation of higher (lower) spin states we perform a larger (smaller) reduction of the magnetic field on a longer (shorter) timescale. After the RAP ramp we switch off the RF field and ramp the magnetic field in $10\,\rm ms$ to $B = 3.99\,\rm G$. Here we wait again for $100\,$ms to allow the magnetic field to stabilize. During the ramp up and down to $40\,$G of the magnetic field we loose $25(2)\,\%$ of the atoms. We attribute this loss mainly to the dense Feshbach spectra that we are crossing when ramping the magnetic field. The exact loss mechanism has not been yet identified, constituting a topic of interest for latter investigation. At $3.99\,$G, before switching on the spin dynamics, about $1.7\times 10^4$ atoms remain in the lattice. The losses affect the lattice filling at which the spin dynamics occur. Our simulations account for this initially reduced filling; see Sec.S8. ![State preparation and detection efficiency. (A) Energy levels of the ground state hyperfine manifold in the dressed-state picture in dependence of the detuning between the applied RF-frequency and the atomic resonance condition for the ${|\text{--}1/2\rangle}$ to the ${|1/2\rangle}$ hyperfine levels. The solid red arrow exemplary shows the RAP for preparation of atoms into the $|m_F^0 \rangle = {|\text{--}7/2\rangle}$ state. The insets show a zoom of one avoided crossing. (B-C) Spin-preparation and atom-counting efficiency measured for ${|\text{--}17/2\rangle}$, ${|\text{--}15/2\rangle}$, ${|\text{--}9/2\rangle}$, and ${|9/2\rangle}$. The obtained values are interpolated linearly assuming a linear dependence on the $m_F^0$ state.[]{data-label="fig:S1"}](FigS1){width="1\linewidth"} Additionally to the losses due to the magnetic field ramps, we also observe losses caused by the RAP itself. To quantify the preparation efficiency, i.e. the loss of atoms due to the spin preparation via RAP as a function of the target $m_F$ state, we perform additional measurements where we either do not perform a RAP or where we add an inverse RAP to transfer all atoms back into the ${|\text{--}19/2\rangle}$ state. By comparing the atom numbers from measurements without RAP and with double-RAP and assuming that the loss process is symmetric, we derive the preparation efficiency as plotted in Fig.\[fig:S1\]B. We also account for the difference in the counting efficiency of the individual spin states, which arises from different effective scattering cross sections for the imaging light. Here we compare the measured atom number in a target $m_F$ state to the expected atom number taking into account the previously determined preparation efficiency as discussed above and the atom number without RAP; see Fig.\[fig:S1\]C. The counting and preparation efficiencies are determined for the ${|\text{--}17/2\rangle}$, ${|\text{--}15/2\rangle}$, ${|\text{--}9/2\rangle}$, and ${|9/2\rangle}$ states and interpolated assuming a linear dependency of these efficiencies on $m_F$ (see Fig.\[fig:S1\]B,C). We estimate the preparation efficiency of the respective m$_F$ state to be $0.86(1) - 0.008(1)~\times~\text{m}_F$. We attribute the lower preparation efficiency for higher spin states to the larger number of avoided crossing between spin states that come into play during the RAP procedure. Overall we expect that the lattice filling over the whole sample, taking into account the losses due to magnetic field ramping and spin state preparation, reduces from close to unity down to a value between $0.6$ and $0.7$; see also Tab.S1 – S2. Quench protocol and detection sequence ====================================== In our experiment we exploit both, the light and the magnetic shifts of the energies of each spin state to reach a resonant condition where the energy difference between neighboring spin states is cancelled and therefore spin changing dynamics preserving the total magnetization become energetically allowed. In particular we exploit the tensorial light shift of the spin states energies [@Becher2017pol] $$U_t = \frac{3m_F^2 - F(F+1)}{F(2F-1)}\frac{3 \cos ^2 \theta _p -1}{2} \alpha _t (\omega),$$ present in atomic erbium to initialize the dynamic evolution of the spin population. The tensorial light shift depends quadratically on the $m_F$ state as well as on the angle $\theta _p$ between the magnetic field axis and the axis of polarization of the light. Here, $\alpha _t$ refers to the tensorial polarizability coefficient and $\omega$ to the light frequency. After the preparation of the respective spin state we start all our measurements at $B=3.99\, \rm G$, pointing in the $z$ direction. However, to reach the resonance condition we use two slightly different quench protocols for the measurement sets with fixed $\Theta = 0^\circ$ for the different initial spin state and for the sets of measurements where $|m_F \rangle ={|\text{--}17/2\rangle}$ and $\Theta \in (0^\circ, 80^\circ)$. The measurement sequences differ on the one hand by the way we jump on resonance to initialize the spin dynamics and on the other hand by shining in an additional laser beam of wavelength $1064\, \rm nm$ and power of $7\, \rm W$. This additional light is necessary because changing $\Theta$ reduces simultaneously $\theta _p$ resulting in a smaller tensorial light shift and therefore in a shift of the resonance position to lower magnetic field values. For large $\Theta$ the light shift of the lattice beams is smaller and therefore the resonance is very close to $0\,$G which we want to avoid to prevent spin relaxation processes. For the sets of measurements where we keep $\Theta = 0^\circ$ but vary the initial $m_F^0$ state we quench the magnetic field directly after the preparation, from $3.99\, \rm G$ to resonance. In contrast we use a different approach for the measurements where $\Theta$ is varied. After the preparation we ramp in $10\,\rm ms$ the additional laser beam to $7\,\rm W$. Due to the reduced speed of our magnetic field coils in $x$ and $y$ direction we first rotate the magnetic field such that the transverse components $B_{\rm{x}}$ and $B_{\rm{y}}$ are already at their target values while keeping an additional offset of $2\,$G in the $z$ direction. The quench to resonance is then done using only the coils for the magnetic field in the $z$ direction. The additional offset field of $2\,$G is large enough to suppress dynamics. We measure the evolution of the magnetic field by performing RF spectroscopy and find that for both quench procedures the magnetic field evolves exponentially towards its quench value with $1/e$ decay times of $1.4\,\rm ms$ and $1.2\,\rm ms$, respectively. After holding on resonance for a certain time we quench the magnetic field back to $3.99\, \rm G$ and we rotate the latter back to $\Theta = 0^\circ$. After a waiting time of $50 \, \rm ms$ we perform a band-mapping measurement combined with a Stern-Gerlach technique, i.e. we ramp the lattice down in $1\, \rm ms$ and apply a magnetic field gradient that is large enough to separate the individual spin states after a time of flight (TOF) of $15\,\rm ms$. This allows us to image the first Brillouin-zone for the different spin states. During TOF the magnetic field is rotated towards the imaging axis. We typically record the population of the initially prepared ${|m_F \rangle}$, of its four neighbors, and of ${|\text{--}19/2\rangle}$ by summing the 2D atomic density over a region of interest. Figure \[fig:S2\] shows examples of the imaging of different spin states for the cases of a non adjusted RAP as well as for the preparation of the atoms in ${|\text{--}9/2\rangle}$, ${|3/2\rangle}$, and ${|5/2\rangle}$. In the case of ${|3/2\rangle}$ residual atoms in ${|\text{--}19/2\rangle}$, ${|\text{--}17/2\rangle}$, and ${|5/2\rangle}$ are visible due to a non perfect preparation. ![Spin resolved imaging. Absorption images for a non adjusted RAP and for the preparation of ${|\text{--}9/2\rangle}$, ${|3/2\rangle}$, and ${|5/2\rangle}$. Whereas for the ${|\text{--}9/2\rangle}$ and ${|5/2\rangle}$ case no residual atoms in other spin states are visible, for the ${|3/2\rangle}$ case we observe a small amount of residual atoms in other spin states due to a non perfect preparation. []{data-label="fig:S2"}](FigS2.pdf){width="0.95\linewidth"} Lifetime and losses in the lattice ================================== Off-resonance, i.e. at a magnetic field of $B=3.99\,$G, we measure the lifetime of the prepared spin state to be on the order of a few seconds, being slightly shorter for higher spin states. Note that, here, we do not observe any population growing in the neighboring spin states. Differently, for the measurements on resonance we observe a faster loss happening on the timescale of the first $20$–$30\,$ms followed by loss at lower speed for the remaining atoms. We fit an exponential decay to extract the atoms loss and change in filling over the timescale that we use to extract $V_{\rm{eff}}$, $t_{\rm{fit}}$, (see S8) as well as over the full $100\,$ms of the dynamics reported in the main text (Fig.\[fig:2\]-\[fig:3\]). Table S1 gives the corresponding numbers for the sets of data for the different initial $m_F^0$ states. During the fitting timescale we observe atoms loss on the order of $5$–$10$%. This atom loss can be converted into a change of the effective filling of the lattice compared to the state obtained after the lattice loading giving a minimum filling of $\nu=0.58$ for the $|m_F^0 \rangle ={|1/2\rangle}$ case. For longer timescales larger losses in the range between $10-35$% are observed. In general, we note that the amount of loss depends on the initial $m_F^0$ state, resulting larger for the central ${|m_F \rangle}^0$ states. Similar numbers are obtained for the sets of measurement where we vary $\Theta$ (see Tab.S2). The exact mechanism leading to these losses is not yet understood and will be the topic of future studies. Thanks to their limited importance over the early time dynamics, we here compare our results to theoretical prediction without losses; see S 8. A proper description of the long time dynamics will certainly require to account and understand such effects. $m_F^0$ $t_{\rm{fit}}$ (ms) $N_{\rm{loss}}(t_{\rm{fit}})$ (%) $\nu$ $(0)$ $\nu$ $(t_{\rm{fit}})$ $N_{\rm{loss}}(100\,$ms) (%) $\nu$ $(100\,$ms) ----------------- --------------------- ----------------------------------- ------------- ------------------------ ------------------------------ ------------------- $-\frac{17}{2}$ $34.2$ $1.8$ $0.7$ $0.68$ $5.3$ $0.66$ $-\frac{13}{2}$ $15.7$ $7.2$ $0.69$ $0.64$ $19.6$ $0.55$ $-\frac{9}{2}$ $11.3$ $8.7$ $0.67$ $0.62$ $25.1$ $0.51$ $-\frac{5}{2}$ $9.6$ $13.7$ $0.66$ $0.58$ $27.7$ $0.48$ $-\frac{1}{2}$ $9.0$ $12.2$ $0.65$ $0.57$ $36.1$ $0.42$ $\frac{1}{2}$ $9.0$ $13.5$ $0.65$ $0.56$ $36.7$ $0.41$ $\frac{3}{2}$ $9.2$ $9.2$ $0.64$ $0.58$ $34.0$ $0.43$ $\frac{9}{2}$ $11.3$ $6.7$ $0.63$ $0.59$ $23.1$ $0.49$ $\frac{13}{2}$ $15.7$ $4.9$ $0.62$ $0.59$ $21.9$ $0.48$ $\Theta$ ($^\circ$) $t_{\rm{fit}}$ (ms) $N_{\rm{loss}}(t_{\rm{fit}})$ (%) $\nu$ $(0)$ $\nu$ $(t_{\rm{fit}})$ $N_{\rm{loss}}(100\,$ms) (%) $\nu$ $(100\,$ms) --------------------- --------------------- ----------------------------------- ------------- ------------------------ ------------------------------ ------------------- $0$ $26.8$ $13.8$ $0.7$ $0.60$ $30.1$ $0.49$ $10$ $30.1$ $11.9$ $0.7$ $0.61$ $25.6$ $0.52$ $20$ $36.7$ $6.9$ $0.7$ $0.65$ $18.2$ $0.57$ $30$ $47.0$ $8.6$ $0.7$ $0.64$ $17.8$ $0.57$ $35$ $52.0$ $6.7$ $0.7$ $0.65$ $11.4$ $0.62$ $40$ $53.2$ $7.2$ $0.7$ $0.65$ $13.2$ $0.60$ $50$ $46.1$ $12$ $0.7$ $0.61$ $17.2$ $0.58$ $60$ $37.0$ $8.9$ $0.7$ $0.63$ $19.2$ $0.56$ $80$ $30.0$ $10.4$ $0.7$ $0.62$ $19.3$ $0.56$ Experimental uncertainties and inhomogeneities ============================================== Ideally, all atoms in the sample experience the same linear and quadratic Zeeman shift and the same quadratic light shift. However, in the experiment inhomogeneities from the magnetic field and light intensities lead to a spatial dependence of those shifts. An upper bound of the variation of Zeeman shifts can be deduced from RF-spectroscopy measurements done with bosonic erbium. From the width of the RF-resonance ($\approx 500\,$Hz) and the size of the cloud ($\approx 15\, \rm{\mu}$m) we estimate a maximum magnetic field gradient of $\leq 230\,$mG/cm, assuming the gradient as main broadening mechanism for the resonance width, neglecting magnetic field noise and Fourier broadening. This translates into a differential linear Zeeman shift of $\leq h \times 6\,$Hz between adjacent lattice sites in the horizontal x-y plane and $\leq h \times 12\,$Hz between adjacent planes along the z-direction. Together with the magnetic field values used in the spin dynamic experiments, the variation of the quadratic Zeeman shift is negligible compared to other inhomogeneities ($\leq h \times 0.1\,$Hz). The inhomogeneity of the quadratic light shifts can be estimated by considering the shape of the lattice light beams (Gaussian beams with waists of about $(w_x,w_y,w_z) = (160,160,300)\, \rm{\mu}$m) and the resonance condition of the magnetic field, translated into a quadratic Zeeman shift of $h \times 71(1)\, \rm Hz$. This considerations can be used to obtain an estimation for a site dependent light shift compared to the center of the atomic sample. If we take a possible displacement of the atoms by $\leq 10\rm \mu m$ in all directions, from the center of the lattice to the center of the beams, into account, we can estimate an upper bound for the light shift of $\delta_i^T \leq h \times 2\,$Hz at $20$ lattice sites away from the center along the $y$ direction. Spin Hamiltonian {#genh} ================ The experiment operates in a deep lattice regime, where tunneling is suppressed. At the achieved initial conditions, the $^{167}$Er atoms are restricted to occupy the lowest lattice band, and Fermi statistics prevents more than one atom per lattice site. In the presence of a magnetic field strong enough to generate Zeeman splittings larger than nearest-neighbor dipolar interactions, only those processes that conserve the total magnetization are energetically allowed [@dePaz2016]. Under these considerations, the dynamics is described by the following secular Hamiltonian: $$\begin{aligned} \hat H &=&\sum_i\delta_{ i} (\hat F_{ i} ^z)^2+\sum_{ i} B_{ i} \hat F_{ i} ^z \nonumber \\ && +\frac{1}{2}\sum_{i,j\neq i}V_{{ i} ,{ j}}[\hat F_{ i}^z\hat F_{ j}^z-\frac{1}{4}(\hat F_{ i}^+\hat F_{ j}^-+h.c)].\label{eq:H}\end{aligned}$$ Here the operators $F_{ i}^{z,\pm}$ are spin $19/2$ angular momentum operators acting on lattice site $i$. The first two terms account for the site-dependent quadratic and linear shifts respectively, where $\delta_{ i}$ includes both Zeeman terms and tensorial light shifts as discussed in the main text. $B_{i}=B+\Delta B_i$ denotes the linear Zeeman shift at site $i$. While the constant and uniform contribution, $B$, commutes with all other terms, thus can be rotated out, the spatially varying contribution, $\Delta B_i$, is relatively small in the experiment but still is accounted for in the theory calculations. The last term is the long-range dipolar interaction between atoms in different sites, with $$\begin{aligned} V_{{\bf i},{\bf j}}&\equiv&V_{dd} d_y^3 \frac{1-3\cos^2\theta_{{ i},{ j}}}{|{\bf r}_{i}-{\bf r}_{ j}|^3},\end{aligned}$$ where $\theta_{ij}$ is the angle between the dipolar orientation set by an external magnetic field and the inter-particle spacing ${\bf r}_{ i}-{\bf r}_{ j} $. $V_{dd}\,\approx\,\frac{\mu_0g_F^2\mu_B^2}{4\pi d_y^3}$ corresponds to the interaction strength between two atoms, $i$ and $j$, separated by the smallest lattice constant $|{\bf r}_{ i}-{\bf r}_{ j}|= d_y=266$nm and forming an angle $\theta_{{ i},{ j}}=\pi/2$ with the quantization axis. Here $g_F \approx 0.735$ is the Lande g-factor for Er atoms, $\mu_0$ is the magnetic permeability of vacuum and $\mu_B$ is the Bohr magneton. We compute $V_{dd}$ from $$\begin{aligned} V_{dd}&=&\frac{\mu_0(\mu_Bg_F)^2}{4\pi}\nonumber \\ && \times \int d^3{\bf r}d^3{\bf r}'\frac{1-3\cos^2\theta_{rr'}}{|{\bf r}-{\bf r}'|^3}|\phi_{ i}({\bf r})|^2|\phi_{ j}({\bf r}')|^2, \label{eq:Vdip}\end{aligned}$$ where $\phi_{i}({ r})$ denotes the lowest band Wannier function centered at lattice site ${ i}$. For the experimental lattice depths $(s_x,s_y,s_z)=(20,20,80)$ in units of the corresponding recoil energies, $V_{dd}$ is estimated to be $h \times 0.336\,$Hz. The GDTWA method ================ To account for quantum many-body effects during the dynamics generated by long-range dipolar interactions in these complex macroscopic spin $F=19/2$ 3D lattice array, we apply the so called Generalize Discrete Truncated Wigner Approximation (GDTWA) first introduced in Ref. [@Lepoutre2018]. The underlying idea of the method is to supplement the mean field dynamics of a spin $F$ system with appropriate sampling over the initial conditions in order to quantitatively account for the build up of quantum correlations. For a spin-F atom $i$ with $\mathcal{N}=2F+1$ spin states, its density matrix $\hat \rho_i$ consists of $\mathcal{D}=\mathcal{N}\times\mathcal{N}$ elements. Correspondingly, we can define $\mathcal{D}$ Hermitian operators, $\Lambda^i_{\mu}$, with $\mu=1,... D$, using the generalized Gell-Mann matrices (GGM) and the identity matrix [@gmm2008]: $$\begin{aligned} \Lambda^i_{\mu=1,...\mathcal{N}(\mathcal{N}-1)/2}&=&\frac{1}{\sqrt{2}}(\ket{\beta}\bra{\alpha}+h.c.), \end{aligned}$$ for $\alpha>\beta$, $1\le \alpha$,$\beta\le \mathcal{N}$, $$\begin{aligned} \Lambda^i_{\mu=\mathcal{N}(\mathcal{N}-1)/2+1,...\mathcal{N}(\mathcal{N}-1)}&=&\frac{1}{\sqrt{2}i}(\ket{\beta}\bra{\alpha}-h.c.),\end{aligned}$$ for $\alpha>\beta$, $1\le \alpha$,$\beta\le \mathcal{N}$, $$\begin{aligned} \Lambda^i_{\mu=\mathcal{N}(\mathcal{N}-1)+1,...\mathcal{N}^2-1}&=& \frac{1}{\sqrt{\alpha(\alpha+1)}} \nonumber \\ &&\times (\sum_{\beta=1}^\alpha\ket{\beta}\bra{\beta}\nonumber \\&&-\alpha\ket{\alpha+1}\bra{\alpha+1}),\end{aligned}$$ for $1\leq\alpha<\mathcal{N}$ $$\begin{aligned} \Lambda^i_{\mathcal{D}}&=&\sqrt{\frac{1}{\mathcal{D}}} \mathbb{I}.\end{aligned}$$ With these operators, the local density matrix $\hat\rho_i$, as well as any operator $\hat O^i$ of local observables can be represented as $$\begin{aligned} \hat O^i&=&\sum_\mu c^i_\mu \Lambda^i_\mu, ~~~\text{with}\\ c^i_\mu&=&{\rm Tr}[\Lambda^i_\mu\hat O^i], \end{aligned}$$ and $\mu=1,2,...\mathcal{D}$. This allows expressing both one-body and two-body Hamiltonians in the form $ \hat H_i=\sum_\mu c^i_\mu\Lambda^i_\mu$, and $\hat H_{ij}=\sum_{\mu,\nu}c^{ij}_{\mu\nu}\Lambda^i_\mu\Lambda^j_\nu$. The Heisenberg equations of motion for $\Lambda^i_\mu$ can be written as $$\begin{aligned} i\hbar\frac{d \Lambda^i_\mu}{dt}&=&[\Lambda^i_\mu,\hat H]\nonumber\\ &=&\sum_\mu c^i_\nu[\Lambda^i_\mu,\Lambda^i_\nu]+\sum_{\sigma,j,\nu}c^{ij}_{\sigma,\nu}[\Lambda^i_\mu,\Lambda^i_\sigma]\Lambda^j_\nu.\label{eq:dLam} \end{aligned}$$ In the experiment, the initial state is a product state of single atom density matrices, $\hat\rho(t=0)=\prod\hat\rho^i(t=0)$. If we adopt a factorization $\langle \Lambda^i_\mu\Lambda^j_\nu...\Lambda^k_\sigma\rangle=\langle\Lambda^i_\mu\rangle\langle\Lambda^j_\nu\rangle...\langle\Lambda^k_\sigma\rangle$ for any non-equal $i,j,...k$ (i.e.each operator acts on a different atom) and arbitrary $\mu,\nu,\sigma$, Eq.S10 becomes a closed set of nonlinear equations for $\lambda^i_\mu=\langle\Lambda^i_\mu\rangle$. Within a mean-field treatment, the initial condition is fixed by $\lambda^i_\mu(t=0)={\rm Tr}[\Lambda^i_\mu\hat\rho(t=0)]$, which determines the ensuing dynamics from Eq.S10. This treatment neglects any correlations between atoms. In the GDTWA method, the initial value of $ \lambda^i_\mu$ is instead sampled from a probability distribution in phase space, with statistical average $\overline{\lambda^i_\mu(0)}={\rm Tr}[\Lambda^i_\mu\hat\rho(t=0)]$. Specifically, each $\Lambda^i_\mu$ can be decomposed via its eigenvalues and eigenvectors as $\Lambda^i_\mu=\sum_{a^i_\mu}a^i_\mu\ket{a^i_\mu}\bra{a^i_\mu}$. We take $a^i_\mu$ as the allowed values of $\Lambda^i_\mu$ in phase space, then for an initial state $\hat \rho^i(t=0)$, the probability distribution is $p(a^i_\mu)={\rm Tr}[\hat \rho^i(t=0)\ket{a^i_\mu}\bra{a^i_\mu}]$. From Eq.S10, each sampled initial configuration for the $N$ atom array, $\{a_\mu\}=\{a^{i1}_{\mu_1},a^{i2}_{\mu_2},...a^{iN}_{\mu_N}\}$ leads to a trajectory of $\Lambda^i_{\mu}$, which we denote as $\lambda^i_{\mu,\{a_\mu\}}(t)$. The quantum dynamics can be obtained by averaging over sufficient number of trajectories $$\begin{aligned} \lambda^i_\mu(t)&\approx&\overline{\lambda^i_\mu(t)}=\sum_{\{a_\mu\}}p(\{a_\mu\})\lambda^i_{\mu,\{a_\mu\}}(t).\end{aligned}$$ This approach has been shown capable of capturing the buildup of quantum correlations [@Schachenmayer2015b; @Lepoutre2018]. Incorporating experimental conditions in numerical simulation ============================================================= In our experiment, the lattice filling fraction is not unity when the spin dynamics takes place. The reduced filling fraction is due to two effects: the finite temperature and atom loss during the initial state preparation. To account for the effect of a finite temperature, we first obtain the density distribution before ramping up the lattice from a Fermi-Dirac distribution $n^0({\bf r}_i)=\frac{1}{1+{\rm exp}(\beta(\epsilon({\bf r}_i)-\mu))}$, with parameters $\beta=1/k_BT$ and $\mu$ matching the inferred experiment temperature $T$, and the total atom number $N_0=2.4\times 10^4$. The function $\epsilon({\bf r}_i)$ accounts for the weak external harmonic confinement. We compute the density distribution function after loading the atoms in the lattice, $n^F({\bf r}_i)$ by simulating the lattice ramp which is possible since to an excellent approximation we can treat the system as non-interacting. Indeed, we neglect the dipolar interaction in the loading given that their magnitude is much lower than the Fermi Energy of the gas. In the numerical simulation, we then sample the position of atoms ${{\bf r}_i}$ in the lattice according to a distribution $p({\bf r}_i)=n^F({\bf r}_i)/N_0$. In practice, to reduce computation cost we need to reduce the total atom number in our calculations and use a smaller lattice with fewer populated lattice sites. In this case, we reduce the number of lattice sites by a factor $\xi=(N_{sim}/N_{exp})^{1/3}$, where $N_{sim(exp)}$ are the number of atoms in the simulation (experiment), while keeping the lattice spacings the same as in experiment, $(d_x,d_y,d_z)=(272,266,544)\,$nm. That is, for an initial lattice with $L_x$ sites along $x$ direction, in our simulations there are $\xi L_x$ sites while the separation between two adjacent lattice sites is still $d_x$. We then sample the initial distribution of atoms in the lattice with $\tilde{p}(\tilde{\bf r}_i)=\xi^3p(\xi\tilde{\bf r}_i)$, which preserves the local density and is similar to sampling in a coarse-grained lattice. In our simulations, we chose $N_{sim}\gtrsim 350$ and checked that the convergence in $N_{sim}$ has been reached. As discussed in Sec.S2, a fraction of atoms is lost during the ramp up and down of the magnetic field before initializing the spin dynamics over the sample. While a rigorous treatment on how these losses modify the distribution is not currently accessible with our current experimental setup, we try to account for it in the simulation by preferentially removing those atoms with a probablity $\propto p({\bf r}_i) N_{\rm nn}$, where $N_{\rm nn}$ is the number of nearest neighbors (separation $\le d_y$), until $N=\nu(0) N_0$ atoms are left. According to experiment estimates, the filling fractions before the initialization of the spin dynamics are $\nu(0) = 0.6 \sim 0.7$ (see Tab.S1 and S2). Figure \[fig:S3\] shows the histogram of neighbors in the resulting atom distribution. Such distribution effectively reduces the nearest-neighbor interactions and is found to give a better agreement with experiment. ![Atom distribution Histogram showing the average number of atoms in distances normalized to the lattice direction along $y$ for random removal of atoms and for removal depending on the number of nearest neighbors (NN-dependent removal).[]{data-label="fig:S3"}](FigS3){width="0.95\linewidth"} Both the quadratic and linear shifts in the experiment are inhomogeneous across the lattice as discussed in Sec.S5, and we include them in our numerical simulation as site-dependent terms $\delta_{i} (\hat F_{i}^z)^2$ and $B_{ i}\hat F_{ i}^z$, with $\delta_{ i}=a |{\bf r}_i|^2$ and $B_{ i}=b(x_i+y_i+z_i)$. Based on experimental estimation, we have chosen the values of $a$ and $b$ such that $\delta_{i}=h\times 1.6\,$Hz ($h\times 0.7\,$Hz) at 20 sites along $y$ away from the lattice center, and $B_{i}$ differs by $h\times 6\,$Hz ($h\times 1.8\,$Hz) between adjacent sites, for Fig.\[fig:2\] and\[fig:3\] (Fig.\[fig:4\]) in the simulation. Short-time population dynamics ============================== Considering a fixed initial atomic distribution over the lattice, the population dynamics at early times can be derived via a perturbative short time expansion $$\begin{aligned} n_{m_F}(t)&\equiv& \langle\hat n_{m_F}(t)\rangle=\langle\hat n_{m_F}\rangle+i\langle[\hat H, \hat n_{m_F}]\rangle t/\hbar\nonumber\\ &&-\langle[\hat H,[\hat H,\hat n_{m_F}]]\rangle t^2/2\hbar^2\nonumber\\ &&-i\langle[\hat H,[\hat H,[\hat H, \hat n_{m_F}]]]\rangle t^3/3!\hbar^3 \nonumber\\ &&+\langle[\hat H,[\hat H,\allowbreak[\hat H,[\hat H,\hat n_{m_F}]]]\rangle t^4/4!\hbar^4\nonumber\\ &&+\mathcal{O}(t^5) \end{aligned}$$ Here the average $\langle\cdot\rangle$ is over the initial state, which is assumed to be a pure state, $\hat n_{m_F}=(\sum_{i} \mathcal{P}^{m_F}_i)/N$, where $\mathcal{P}^{m_F}_i=|m_F\rangle_i{}_i\langle m_F| $ is the onsite projector for an atom at site $i$ in state $|m_F\rangle$ and $N$ denotes the total number of atoms. Note that here the sums are always carried out over the populated lattice sites in the initial lattice configuration. We obtain $$\begin{aligned} n_{m_F^0}(t)&=&n_{m_F^0}(0)\Big(1-n_{m_F^0}(0)\frac{V_{\rm eff}^2}{\hbar^2}t^2\nonumber\\ &&+\mathcal{O}(t^4)\Big),\end{aligned}$$ with $$\begin{aligned} V_{\rm eff}^2&=&\frac{\gamma^2(m_F^0)}{8N}\sum_{{ i},{ j}\neq { i}} V_{{ i},{ j}}^2,\end{aligned}$$ $$\begin{aligned} \gamma(m_F^0)&=&\sqrt{F(F+1)-m_F^0(m_F^0+1)} \nonumber \\ && \times \sqrt{F(F+1)-m_F^0(m_F^0-1)},\end{aligned}$$ where $n_{m_F^0}$ denotes the population on the selected target state. To obtain Eq.S13, we have assumed that initially most of the population is in this target state, i.e. $n_{m_F^0}(0)\sim 1$. In the experiment, this assumption is always satisfied and therefore Eq.S15 is expected to reproduce well the short time dynamics. The dependence of $\gamma(m_F^0)$ on the initial state $m_F^0$ is a consequence of the dependence of dipolar exchange processes on the spin coherences, i.e. $|\langle i:m_{F}^0+1,j:m_{F}^0-1| \hat F_i^+\hat F_j^-|i:m_{F}^0,j:m_{F }^0\rangle|$. Therefore the smaller the value $|m_F^0|$ of the initial populated states, the faster the early time dynamics. Notably, up to order $t^2$ the initial dynamics is independent of quadratic shifts and external magnetic field gradients. This is because both of their corresponding Hamiltonians commute with the spin population operator $\hat n_{m_F}$. From this simple perturbative treatment one learns that by preparing different initial states with different $m_F^0$, the decay rates of the short time population dynamics provide information of $V_{\rm eff}$ and thus of the underlying dipolar couplings. As discussed in Sec.S2 and S8, the lattice filling fraction is not unity and the initial atomic density distribution in the lattice may vary from shot to shot. To account for this effect, we perform a statistical average of Eq.S14 calculated for each lattice configuration generated with the procedure in Sec.S8 to obtain the theoretical values in Fig.\[fig:3\]G and Fig.\[fig:4\]B. It is important here to compare the predictions obtained from a simple mean-field analysis. In contrast to Eq.S15, neglecting quantum correlations yields $$\begin{aligned} n_{m_F^0}^{\rm Mean-Field}(t)&=&n_{m_F^0}(0)\Big(1-n_{m_F^0}(0)[1-n_{m_F^0}(0)]\frac{V_{\rm eff}^2}{\hbar^2}t^2\nonumber\\ &&+\mathcal{O}(t^4)\Big ). \label{eq:shortT:mf}\end{aligned}$$ At the mean field level therefore if initially the atoms are prepared such that $n_{m_F^0}(0)=1$, then there is no population dynamics. This is in stark contrast to the quantum systems where dynamics is enabled by quantum fluctuations. To extract $V_{\rm{eff}}$ from our experimental data and to compare it to the theoretical simulations we fit the initial dynamics with Eq.S15. We define the time scale for the fitting via $t_{\rm{fit}} < 0.5\frac{\hbar}{V_\text{eff}}$, which corresponds to the timescale on which each atom did on average half a spin flip. We note that on this timescale the time evolution starts already to deviate from the short time expansion (Eq.2), leading to a systematic downshift of the experimentally fitted $V_\text{eff}$; see Fig.\[fig:4\]B. However, a minimum timescale has to be chosen to ensure that the fit is performed using a large enough number of datapoints. Figure \[fig:S4\] shows exemplary the fit to the experimental data for $|m_F^0 \rangle= {|\text{--}9/2\rangle}$. ![Fitting example to extract $V_{\rm{eff}}$. The dotted-dashed line exemplary shows the fit of Eq.S13 to the experimental data to extract $V_{\rm{eff}}$ for $|m_F^0 \rangle = {|\text{--}9/2\rangle}$. The solid green line indicates the time $t_{\rm{fit}}$ up to which the fit is performed.[]{data-label="fig:S4"}](FigS4){width="0.95\linewidth"}

--- abstract: | There is increasing evidence that supermassive black holes in active galactic nuclei (AGN) are scaled-up versions of Galactic black holes. We show that the amplitude of high-frequency X-ray variability in the hard spectral state is inversely proportional to the black hole mass over eight orders of magnitude. We have analyzed all available hard-state data from [*RXTE*]{} of seven Galactic black holes. Their power density spectra change dramatically from observation to observation, except for the high-frequency ($\ga$ 10 Hz) tail, which seems to have a universal shape, roughly represented by a power law of index -2. The amplitude of the tail, $C_M$ (extrapolated to 1 Hz), remains approximately constant for a given source, regardless of the luminosity, unlike the break or QPO frequencies, which are usually strongly correlated with luminosity. Comparison with a moderate-luminosity sample of AGN shows that the amplitude of the tail is a simple function of black hole mass, $C_M = C/M$, where $C \approx 1.25$ M$_\odot$ Hz$^{-1}$. This makes $C_M$ a robust estimator of the black hole mass which is easy to apply to low- to moderate-luminosity supermassive black holes. The high-frequency tail with its universal shape is an invariant feature of a black hole and, possibly, an imprint of the last stable orbit. author: - | Marek Gierli[ń]{}ski$^{1,2}\thanks{E-mail:Marek.Gierlinski@durham.ac.uk}$, Marek Niko[ł]{}ajuk$^{3}$ and Bo[ż]{}ena Czerny$^{4}$\ $^1$Department of Physics, University of Durham, South Road, Durham DH1 3LE, UK\ $^2$Astronomical Observatory, Jagiellonian University, Orla 171, 30-244 Krak[ó]{}w, Poland\ $^3$Department of Physics, University of Bia[ł]{}ystok, Lipowa 41, 15-424 Bia[ł]{}ystok, Poland\ $^3$Copernicus Astronomical Centre, Bartycka 18, 00-716 Warszawa, Poland\ date: Submitted to MNRAS title: 'High-frequency X-ray variability as a mass estimator of stellar and supermassive black holes' --- = -0.5cm \[firstpage\] X-rays: binaries – galaxies: active – accretion, accretion discs Introduction {#sec:introduction} ============ Astrophysical black holes are very simple objects, completely characterized by their mass and spin. Hence, the gravitational potential around a black hole simply scales with its mass. An important question in high-energy astrophysics is whether the accretion flow properties scale with the black hole mass in a simple manner, or, more specifically, whether active galactic nuclei (AGN) are scaled-up versions of Galactic black hole binaries (BHB). One of the ways of tackling this problem is to study X-ray variability, which is observed in accreting black holes of all masses. Recent advances in mass estimates of AGN central black holes lead to discovery of dependence of the observed variability properties on mass. Long X-ray monitoring campaigns allowed to construct power density spectra (PDS) of accreting supermassive black holes which turned out to have a roughly of (broken) power-law shape. The variability amplitude (the excess variance; e.g. Lu & Yu 2001; Markowitz & Edelson 2001) and the frequency of the break (e.g. M$^c$Hardy et al. 2004, 2006) can depend on the black hole mass. In order to use the X-ray variability for mass measurement we need a property which scales only with the black hole mass, and does not change with accretion rate. The break frequency does not satisfy this condition, as it changes significantly with the accretion rate, in X-ray binaries (e.g. Done & Gierli[ń]{}ski 2005). M$^c$Hardy et al. (2006) showed that a more general relation holds between the break frequency, $\nu_b$, and the black hole mass, $M$: $\nu_b = A L_{\rm bol}^B / M^C$, where $A$, $B$ and $C$ are constants. This relation includes a significant dependence on the source bolometric luminosity, $L_{\rm bol}$. It was already suggested by Hayashida et al. (1998) that measuring the normalization of the high-frequency tail of the power spectrum, well above the high-frequency break, is an interesting possibility for black hole mass measurement. Equivalently, one can use the excess variance, $\sigma^2_{\rm NXS}$, measured for short data sets. This general line was followed by Czerny et al. (2001), Papadakis (2004), Niko[ł]{}ajuk, Papadakis & Czerny (2004, hereafter N04) and Niko[ł]{}ajuk et al. (2006, hereafter N06). However, the method was not reliably checked against the dependence on the source accretion rate. In this paper we put the idea of $\sigma^2_{NXS} \propto M^{-1}$ correlation to the test. We use an extensive set of BHB observations to see if and when $\sigma^2_{\rm NXS}$ is constant for a given mass and whether it anticorrelates with the black hole mass. High-frequency power {#sec:power} ==================== Power density spectra of many AGN can be approximated by a broken power law, with power $P_\nu \propto \nu^{-1}$ below and $P_\nu \propto \nu^{-2}$ above the break frequency, $\nu_b$ (e.g. Markowitz et al. 2003b), where $P_\nu$ is the power spectral density normalized to the mean and squared. A second break at lower frequencies, below which the power is roughly $P_\nu \propto \nu^{0}$, has been also observed (e.g. Pounds et al. 2001; Markowitz, Edelson & Vauhgan 2003a). At the zeroth order of approximation this is consistent with the PDS observed in stellar-mass BHB in the hard spectral state. Fig. \[fig:pds\] shows a sample of PDS from Galactic BHB in the hard state (details of the data reduction are described in Section \[sec:data\]). Plainly, these spectra are much more complex that a doubly-broken power law, with multiple broad and narrow noise components, usually well described by a series of Lorentzians (e.g. Pottschmidt et al. 2003). However, despite this complexity, the entire spectral shape roughly resembles a (doubly) broken power law. N04 assumed that the break frequency is inversely proportional to the black hole mass, while the $P_\nu \propto \nu^{-1}$ part of the PDS below the break (the ‘flat top’ in $\nu P_\nu$ diagrams) has constant normalization, independent of the black hole mass. Yet inspection of several BHB power spectra clearly shows that neither of these is constant for a given source. The break frequency is known to change with accretion rate (e.g. Done & Gierli[ń]{}ski 2005). The ‘flat top’ normalization can change as well, as one can see in GX 339–4 spectra in Fig. \[fig:pds\]. There is, however, one feature of these power spectra that remains remarkably invariant: the high-frequency spectral shape, above $\nu_b$. For a given source it can be roughly described by a single power law with constant index of 1.5–2.0, and constant normalization for various observations differing in luminosity by more than one order of magnitude. In this paper we test the idea that the high-frequency part of the PDS remains fairly constant for a given source and scales with the black hole mass. This is a simple refinement of the idea proposed by N04 and later developed by N06. Here we do not make any assumptions about how the characteristic frequencies (e.g. break frequency) depend on black hole mass. Instead, we assume that the the PDS [*above*]{} the break frequency (the high-frequency tail) has a universal spectral shape (roughly $\propto \nu^{-2}$) with normalization depending on the black hole mass. This can be written as $$P_\nu = C_M (\nu/\nu_0)^{-2},\label{eq:tail}$$ where $\nu_0$ is an arbitrary frequency which we chose to be $\nu_0$ = 1 Hz. Thus, $C_M$ (in units of Hz$^{-1}$) is the normalization of the (extrapolated) high-frequency tail at 1 Hz. The assumption that $C_M$ is unique function of the black hole mass would directly correspond to the original assumption of N04 about constancy of $P(\nu_b)\nu_b$ if $\nu_b$ were constant for a given black hole mass. Due to limited statistics it is often difficult to study details of the high-frequency shape of the PDS. Therefore, we simplify the situation by calculating the amplitude, or the excess variance, of variability in a given frequency band significantly above the break. This can be done directly from a light curve or by integrating the PDS. The excess variance calculated between frequencies $\nu_1$ and $\nu_2$ (both greater than $\nu_b$) is: $$\sigma_{\rm NXS}^2 = \int_{\nu_1}^{\nu_2} P_\nu d\nu = C_M \nu_0 \left( {\nu_0 \over \nu_1} - {\nu_0 \over \nu_2} \right).\label{eq:sigma_nxs}$$ The key assumption, which we want to test in this paper, is that $C_M$ is inversely proportional to the black hole mass, $C_M = C/M$, where $C$ is a constant. We note that our $C$ is the same constant as constant $C$ defined by N04 in eq. 4, divided by $\nu_0^2$. Data reduction and selection {#sec:data} ============================ Source Name Start End --------------- ------------ ------------ XTE J1118+480 2000-03-29 2000-08-08 2005-01-13 2005-02-26 4U 1543–47 2002-06-17 2002-10-11 XTE J1550–564 1998-09-07 1999-05-20 2000-04-10 2000-07-16 2001-01-28 2001-04-29 2002-01-10 2002-03-05 2003-03-27 2003-05-16 XTE J1650–500 2001-09-06 2002-04-21 GRO J1655–40 2005-02-20 2005-11-11 GX 339–4 2002-04-02 2003-05-06 2003-12-28 2005-08-12 Cyg X-1 1996-02-12 2006-01-12 : Log of [*RXTE*]{} observations. Each set of data corresponds to one transient outburst. For Cyg X-1 we used all data publicly available in February 2007.[]{data-label="tab:obslog"} We used publicly available [*Rossi X-ray Timing Explorer*]{} ([*RXTE*]{}) Proportional Counter Array (PCA) data of seven black hole binaries, listed in Table \[tab:obslog\]. First, we extracted background-corrected energy spectra from Standard-2 data (top layer, detector 2 only) for each pointed observation. These were used to create hardness-intensity diagrams (HID). Intensity is defined as the total 2–60 keV count rate and the hardness ratio is the ratio of count rates in energy bands 6.3–10.5 and 3.8–6.3 keV. We also calculated power density spectra (PDS) for each observation from full-band (2–60 keV) data in 0.0039–128 Hz frequency band. We subtracted the Poissonian noise from the PDS, corrected them for dead-time effects (Revnivtsev, Gilfanov & Churazov 2000) and background (Berger & van der Klis 1994) We used HIDs to identify hard state spectra. In transients these spectra lie on the vertical branch of the diagram at hardness ratio $\ga$0.8, corresponding to the rise or decay of the outburst. While in GX 339–4 the transition from the vertical hard branch into intermediate horizontal branch is abrupt and well defined, in XTE J1550–564, XTE J1650–500 and GRO J1655–40 we also included a few points from the part of the HID where the hard branch turns into intermediate, as they have sufficiently large hardness ratio to be (potentially) classified as the hard state. In Cyg X-1 we arbitrarily selected observations with hardness ratio greater than 0.9. We also arbitrarily rejected all data with average count rate less than 10 counts per s per PCU as they did not have enough statistics to robustly calculate the high-frequency power. All the selections are shown in Figs. \[fig:cm1\] and \[fig:cm2\]. For a given source (and separately for each outburst in case of GX 339–4 and XTE J1550–564) we measured the high-frequency power, $\sigma^2_{{\rm NXS}}$, with their statistical errors, $\alpha$, for each pointed observation in the hard state. The power (or excess variance) was measured by integrating the PDS over the frequency band 10–128 Hz (eq. \[eq:sigma\_nxs\]). Then, we computed the mean amplitude, $\langle C_M \rangle$ for a given source (or a particular outburst of the source) weighted by errors of $C_{M}$. The error of $\langle C_M \rangle$ was estimated from $\chi^2$ statistics for $\Delta\chi^2 = 2.7$, i.e. corresponding to 90 per cent confidence limits. Despite the initial count rate selection, some data sets (in particular from short observations) gave high-frequency power with large errors. We discarded all these, setting an arbitrary upper limit on error, $\alpha < 0.3 \sigma^2_{{\rm NXS}}$. Results {#sec:results} ======= =8.5cm =8.5cm =8.5cm =8.5cm =8.5cm =8.5cm =8.5cm Figs. \[fig:cm1\] and \[fig:cm2\] show selection of hard-state data (hardness-intensity diagrams on the left) and measured amplitude of the high-frequency tail, $C_M$ in the right-hand panels. It is obvious from these diagrams that $C_M$ is not constant for a given source and varies from one pointed observation to another. These variations are not very significant, though. With very few exceptions (2 observations of GRO J1655–40 and 3 of GX 339–4) individual $C_M$ measurements are within 3$\sigma$ of the mean. The dispersion is higher in Cyg X-1 where statistics is better and errors on $C_M$ smaller. Generally, we find that $C_M$ does not significantly depend on the source brightness (count rate). Source Name Outburst $\langle C_M \rangle$ (Hz$^{-1}$) $T_{\rm hard}$ (days) ---------------- ---------- ----------------------------------- ----------------------- XTE J1118+480 2000 $0.098\pm0.005$ 130\* 2005 $0.14\pm0.02$ 26\* 4U 1543–47 2002 $0.14\pm0.01$ – XTE J1550–564 1998 $\sim$0.05 3 2000 $0.137\pm0.002$ 14 2001 $0.095\pm0.001$ 90\* 2002 $0.104\pm0.004$ 50\* 2003 $0.101\pm0.002$ 50\* XTE J1650–500 2001 $0.113\pm0.001$ 8 GRO J1655–40 2005 $0.179\pm0.003$ 12 GX 339–4 2002 $0.088\pm0.001$ 20 2004 $0.121\pm0.003$ 180 Cyg X-1 (hard) – $0.0868\pm0.0003$ – Cyg X-1 (soft) – $0.139\pm0.004$ – : Mean amplitude of the high-frequency power, $\langle C_M \rangle$ (eq. \[eq:tail\]). We show data for each outburst separately. $T_{\rm hard}$ is approximate duration of the initial hard state during the rise of the outburst. Asterisks denote hard-state only outbursts. 4U 1543–47 was not observed in the initial hard state and Cyg X-1 is a persistent source.[]{data-label="tab:results"} More significant differences can be found for different outbursts of a given source. We have analyzed five outbursts of XTE J1550–564. The three hard-state outbursts between 2001 and 2003 (see Table \[tab:results\]) were similar in all properties, so we analyzed them together. They yield mean $\langle C_M \rangle = 0.101\pm0.002$ Hz$^{-1}$. The hard state in 2000 outburst gave higher $\langle C_M\rangle = 0.137\pm0.002$ Hz$^{-1}$. However, the 1998 outburst was very different, with much lower and quickly changing high-frequency power, $C_M \sim 0.05$ Hz$^{-1}$. We have excluded the onset of 1998 outburst from further analysis, as it might have represented a different accretion state (we discuss this in detail in Sec. \[sec:discussion\]). Similarly, GX 339–4 and XTE J1118+480 showed different $C_M$ during different outbursts, though at least in the case of the latter one this could have been due to systematic effects at low count rates, which we discuss later in this section. We looked into the dependence of $C_M$ on the hardness ratio, which can be regarded as a crude indicator of the spectral state. We did not find any clear general trend, as illustrated in Fig. \[fig:hrc\]. Additionally, we found variability amplitude for three other black hole candidates with no black hole mass estimates. The relatively high value of $\langle C_M \rangle = 0.15_{-0.04}^{+0.03}$ Hz$^{-1}$ obtained for XTE J1720–318 seems to be consistent with a rather low black hole mass of 5 M$_\odot$, as also suggested by Cadolle Bel (2004) from disc spectral fitting. For H1743–322 (=XTE J1746–322) and XTE J1748-288 we found $\langle C_M \rangle = 0.090\pm0.015$ and $0.056\pm0.008$ Hz$^{-1}$, respectively. =8.5cm =8.5cm =8.5cm The hard X-ray spectral state, which we study in this paper, is dominated by Comptonized emission. It would be very interesting to see if similar behaviour can be seen in the other spectral state dominated by Comptonization, the very high (or steep power law) state. We have selected the very high state data from hardness-intensity diagrams of XTE J1550–564 and GRO J1655–40, as shown in Fig. \[fig:vhs\]. These observations have significantly less variability at frequencies $>$10 Hz then the hard state data, so the resulting $C_M$ is small (black squares in Fig. \[fig:vhs\]). But we must remember that the soft X-ray part of the spectrum (where PCA has highest sensitivity) has strong contribution from the cold accretion disc, which can dilute the variability coming from Comptonization (e.g. Done & Gierli[ń]{}ski 2005) hence suppressing the observed power. Therefore, we extracted additional power spectra at higher energies (above PCA channel 36, roughly corresponding to energy of 14 keV), where the disc influence is negligible. The high-frequency variability amplitude from these data is much higher, as shown in blue triangles in Fig. \[fig:vhs\]. Though there is a large scatter in $C_M$ from individual observations, the mean $\langle C_M \rangle$ is of order of 0.1 Hz$^{-1}$ in both sources, consistent with the hard state results. Therefore, the high-frequency variability from Comptonization appears to be very similar in both hard and very high states (but see discussion in Sec. \[sec:discussion\]). The soft-state data from our BHB sample are strongly dominated by the disc emission. The only source with reasonable count rates at higher energies in the soft state is Cyg X-1. We applied the same approach to Cyg X-1 soft-state data (arbitrarily chosen hardness ratio in the range 0.45–0.55), as to the very high state data in the previous paragraph. The result can be seen in Fig. \[fig:soft\]. The scatter in individual data points in significant, with mean $\langle C_M \rangle = 0.138\pm0.004$ Hz$^{-1}$. Since the fit of the constant to the data is rather poor, $\chi^2_\nu$ = 156/73, the error on $\langle C_M \rangle$ is indicative only. Caveats ======= We tested our results for possible systematic effects. Many of our observations have low count rate, which can affect the resulting power spectra. At low count rates, PCA channels 0–7 can create artificial power due to problems with the on-board computer (Revnivtsev, private communication). To test the effect of this we have removed PCA channels 0–7, where PCA configuration allowed for that, and calculated new values of $C_M$. We found that this had a negligible effect on our results. The white noise level was subtracted from the power spectra during data reduction. If the white noise level was not estimated correctly, this could have influenced the amplitude of variability and the resulting $C_M$. To test this we have selected observations where high timing resolution was available and calculated power spectra up to 1024 Hz. As we do not expect any significant power above a few hundred Hz from black holes (Sunyaev & Revnivtsev 2000), we assumed that the 512–1024 Hz power could be used as a good white noise estimator. We reanalyzed these data and calculated new values of $C_M$. Again, the effect on our results turned out to be negligible. Background effects can be potentially important for estimating the amplitude of variability, which is defined as $({\rm rms} / {\rm mean})^2$. Our power spectra were calculated from light curves not corrected for background, so their power is $[{\rm rms} / (R_s + R_b)]^2$, where $R_s$ and $R_b$ are source and background mean count rates, respectively. These PDS were then multiplied by $[(R_s + R_b) / R_s]^2$, where source and background count rates were estimated from light curves. In the dimmest observations $R_s$ and $R_b$ are comparable. To estimate possible background inaccuracy (which is modelled in the PCA rather then measured) we extracted one PCA and HEXTE spectrum (using standard HEASARC reduction techniques, adding 1 per cent systematic errors in the PCA) of XTE J1118+480, corresponding to a high-$C_M$ and low count rate point circled in Fig. \[fig:cm1\] (ID 90111-01-02-07, observed on 2005-01-24). We have fitted the joined PCA/HEXTE spectrum with a simple Comptonization model (see Done & Gierli[ń]{}ski 2003 for details), using PCA in 3–40 and HEXTE in 20–200 keV band. The fit was good ($\chi^2 = 137/145$) with no strong residuals. Then we changed the level of PCA background by $\pm$10 per cent. The fit with 90 per cent of background was only marginally worse ($\chi^2 = 142/145$), while the 110 per cent background resulted in a rather poor fit ($\chi^2 = 171/145$). In both cases there were strong residuals in the PCA and disagreement with HEXTE data above $\sim$25 keV. This shows that the background in the low count rate data is estimated with accuracy much better then 10 per cent. Hence, the uncertainty on $C_M$ due to background estimation is no more than a few per cent. The increase of $C_M$ by factor 2–5 at low count rates seen in Fig. \[fig:cm1\] cannot be caused by incorrect background. Since we see this effect in most sources below the same count rate of $\sim$20 s$^{-1}$ (regardless of the distance, hence at different luminosities) it must be of (unknown) instrumental origin. We would like to stress, however, that the increase is not statistically significant, typically less then 2$\sigma$. The high-frequency variability is known to depend on energy in some sources (e.g. Nowak et al. 1999). This is important when comparing BHB with AGN, as we look at different parts of the Comptonized spectrum, AGN data showing higher scattering orders than BHB. We have tested our data for energy dependence. This was possible only in bright observations from XTE J1550-564, GX 339–4 and Cyg X-1, where statistics at higher energies was good enough. The high-energy (above $\sim$14 keV) data give $\langle C_M \rangle = 0.13\pm0.01$ Hz$^{-1}$ for the 2000 outburst of XTE J1550–564 and $\langle C_M \rangle = 0.09\pm0.01$ Hz$^{-1}$ for the 2002 outburst of GX 339–4. These values are consistent with the broad-band data, which in the PCA is dominated by soft X-rays (see Table \[tab:results\]), which suggests that the high-frequency amplitude is not energy-dependent in these sources. On the other hand, similar approach to Cyg X-1 gave $\langle C_M \rangle = 0.109\pm0.001$ Hz$^{-1}$, higher by about 25 per cent higher then the broad-band data. Source Name $M$ (M$_\odot$) Reference --------------- ----------------- ------------------------------- XTE J1118+480 8.5 (7.9–9.1) Gelino et al. (2006) 4U 1543–47 9.4 (7.4–11.4) Park et al. (2004) XTE J1550–564 10 (9.7–11.6) Orosz et al. (2002) XTE J1650–500 5 (2.7–7.3) Orosz et al. (2004) GRO J1655–40 6.3 (5.8–6.8) Greene, Bailyn & Orosz (2001) GX 339–4 6 (2.5–10) Cowley et al. (2002) Cyg X-1 20 (13.5–29) Zi[ó]{}[ł]{}kowski (2005) : Black hole masses in X-ray binaries, used in this paper.[]{data-label="tab:sources"} Dependence on mass ================== Fig. \[fig:mcm\]($a$) shows the dependence of $C_M$ on black holes mass. We used best currently available mass estimates of black holes in X-ray binaries, as summarized in Table \[tab:sources\]. XTE J1650–500 does not have good mass estimate, though Orosz et al. (2004) found an upper limit of 7.4 M$_\odot$. We assumed the mass function, 2.7 M$_\odot$, as the lower limit and we adopted the actual mass in the middle of this interval, at 5 M$_\odot$. We show data from different outbursts of XTE 1118+480, XTE J1550–564 and GX 339–4 separately. Clearly, there is no apparent correlation between black hole mass and $C_M$, though we must bear in mind that mass estimates in X-ray binaries are not very accurate. Moreover, different outbursts giving slightly different $\langle C_M \rangle$ create additional dispersion. Hence, the overall uncertainties are rather large, so the expected relation $C_M = C/M$ cannot be robustly confirmed from X-ray binaries. To do this, we need to extend our studies to supermassive black holes. N06 compared masses of a sample of Seyfert 1 galaxies measured by reverberation method with masses from high-frequency variability, using the value of constant $C$ derived from Cyg X-1 observations. Here we use the same sample of AGN in order to compare them with our much larger sample of BHB, constrain the $C_M = C/M$ relation better and get a ‘big picture’ overview of variability properties for all masses of black holes. In Fig. \[fig:mcm\]($c$) we plotted the sample of of N06; panel $b$ shows the overview of stellar-mass and supermassive black holes. The red diagonal line in the diagrams represents the best-fitting function $C_M = C/M$, with $C$ = 1.25$\pm0.06$ M$_\odot$ Hz$^{-1}$. We would like to point out that this particular value depends on the selection of X-ray binary data, as different outbursts can give different $\langle C_M \rangle$. Also, errors on mass are non-Gaussian in many cases, so the error on $C$, given here, is indicative only. It is interesting to notice that the constant $C$ is 1.24$\pm$0.06 M$_\odot$ Hz$^{-1}$ for X-ray binaries only and 1.51$_{-0.29}^{+0.37}$ M$_\odot$ Hz$^{-1}$ for AGN only; both values are consistent within error limits. We also plotted a line (green dashed) corresponding to the soft state of Cyg X-1, for comparison (with $C$ = 2.77 M$_\odot$ Hz$^{-1}$). Some of the AGN are consistent with the soft-state line. We also tested whether the relation between variability and mass is really linear. We fitted a more general form of a power-law dependence, $C_M = C_f (M/$M$_\odot)^{-\alpha}$ to all BHB and AGN data and found $C_f = 1.19\pm0.05$ Hz$^{-1}$ and index $\alpha = 0.98\pm0.01$ very close to linear relation. Hence, we opt for the simpler solution and regard the linear relation as well established. The $C_M = C/M$ relation is in excellent agreement with the data spanning over eight orders of magnitude in mass. It gives us robust confirmation of our hypothesis that the high-frequency power is inversely correlated with black hole mass. Even more firm confirmation would come from objects with intermediate black holes mass, filling the big gap in Fig. \[fig:mcm\]($b$). One category of sources potentially useful for testing this is the ultra-luminous X-ray sources (ULX) with possible masses of hundreds of M$_\odot$. Alas, they are most likely in the very high spectral state, with soft X-ray emission dominated by the disc. Besides, there are no reliable high-frequency power spectra available from ULX. An interesting source for comparison turns out to be a dwarf galaxy NGC 4395, with black hole mass estimates between 0.13 and 3.6$\times10^5$ M$_\odot$ (Kraemer et al. 1999; Ho 2002; Filippenko & Ho 2003; Vaughan et al. 2005; Greene & Ho 2006). The luminosity is low, $L/L_{\rm Edd} \la 0.1$ (Vaughan et al. 2005). We analyzed its [*ASCA*]{} observations of 2000-05-24 and 2000-05-26 and found the excess variance for this source and $C_M = 3.2_{-1.1}^{+1.4}\times10^{-5}$ Hz$^{-1}$. We show the range of masses and $C_M$ for NGC 4395 in Fig. \[fig:mcm\]($b$). They are in excellent agreement with our mass-variability relation! We can also give an independent estimate of the black hole mass in NGC 4395 based on the best-fitting value of $C$: $M \approx 3.9\times10^4$ M$_\odot$. Discussion {#sec:discussion} ========== We have analyzed all available hard-state data from seven Galactic black hole systems and found that the amplitude of the high-frequency tail, $C_M$, is roughly constant for a given source, changing by no more than a factor two. There is no apparent dependence of $C_M$ on luminosity or hardness ratio. This contrasts QPO or break frequency behaviour, which typically show a strong correlation with luminosity. The scale of change of $C_M$ is also much less than the observed span of QPO/break frequency, which can be one order of magnitude within the hard state. This makes $C_M$ much more invariant feature of a given black hole and a robust estimator of its mass. There are, certainly, some departures from this rule. Different outbursts of the same transient can have slightly different high-frequency tail amplitudes. Most of the sources show an increase in $C_M$ below count rate of $\sim$20 s$^{-1}$ per PCU. This might be attributed to systematic instrumental effects at low count rates. Due to large errors this has a negligible effect on our results. But, if this effect extends to higher count rates, than the difference between 2000 and 2005 outbursts of XTE J1118+480 (increase of $C_M$ below 40 s$^{-1}$, see Fig. \[fig:cm1\]) could be of instrumental origin. XTE J1550–564 gives a much clearer exception to this rule. The first outburst in 1998 had a very short initial hard state with the high-frequency tail amplitude much lower than in the next outbursts. In contrast, the three hard-state only outbursts in 2001, 2002 and 2003 produced very consistent results. The explanation of this phenomenon might lie in the stability of the accretion flow. The spectral state during the onset of the 1998 outburst was changing faster than in any other outburst analyzed here and took a different track on the colour-colour diagram (Done & Gierli[ń]{}ski 2003). It is possible that the accretion flow during such a rapid transition was in a different state than in slower transitions, possibly due to higher ionization state of the reflector (Wilson & Done 2001; Done & Gierli[ń]{}ski 2003). Clearly, this also affected its variability properties, decreasing the high-frequency power. This particular hard state was different in many aspects, so we rejected it from our sample. The more stable hard-state only outbursts give probably a better estimate of $C_M$. Similarly, the persistent source, Cyg X-1, gave a very stable and robust high-frequency power. Thus, fast transitions, characterized by an unstable and quickly changing accretion flow, can apparently break the $C_M = C/M$ law. In Table \[tab:results\] we show the approximate duration of the initial hard state after the onset of the outburst. XTE J1650–500 showed a rather quick transition, so might have suffered from a similar instability, as XTE J1550–564 in 1998. On the other hand, the tail amplitude was very stable and returned to the same level in the hard state at the end of the outburst (see Fig. \[fig:cm1\]). The 2004/2005 outburst of GX 339–4 had a much longer hard state than the 2002/2003 outburst, so perhaps it was a better estimate of the high-frequency power. In this work we assumed the constant power-law high-frequency tail with the spectral index of -2. This is not necessarily true and detailed fits to PDS of X-ray binaries, where statistics is sufficient, show indices between 1.5 and 2.0. We note that the particular choice of the spectral index does not affect the overall result of our paper, as different index would only introduced a constant offset in $C_M$. Scatter in spectral indices from one observation to another might introduce some additional uncertainty, though this would be very difficult to estimate for AGN. The luminosities of the Seyfert 1 sample used in this paper are generally low, $\la$5 per cent of $L_{\rm Edd}$, except for 3C120 and NGC 7469, which are significantly brighter. The 3–10 keV photon power-law indices are $\Gamma \sim$ 1.5–1.8 (Nandra et al. 1997). This suggests that these sources are in the hard X-ray spectral state. On the other hand, the spectral index of Comptonized component alone is not enough to establish the X-ray spectral state. Many BHB show soft-state disc-dominated spectra with a flat ($\Gamma \approx 2$) power-law Comptonized tail, while measuring the [*intrinsic*]{} spectral index in AGN is not straightforward due to presence of complex absorption and/or reflection (Gierli[ń]{}ski & Done 2004). We cannot rule out (in particular for brighter AGN) that some of the sources in the sample are actually in the soft X-ray spectral state. Despite that the mass-variability amplitude correlation seems to hold well for all of them. Some of the objects are more consistent with the soft-state Cyg X-1 line in Fig. \[fig:mcm\]($c$). On the other hand, the brightest 3C120 and NGC 7469 lay [*below*]{} the hard-state line. Clearly, the dependence of high-frequency amplitude on luminosity and spectral state in AGN requires further studies. Our analysis of the very high (steep power law) state shows that after a simple bandwidth correction the tail power is consistent with the hard-state results. This is very encouraging, showing that perhaps the universal shape of the high-frequency tail is the inherent property of Comptonization, regardless of the spectral state. Narrow-line Seyfert 1 galaxies (NLS1) are most likely the supermassive counterparts of Galactic sources in the bright very high state (e.g. Pounds, Done & Osborne 1995). N04, and later Niko[ł]{}ajuk, Gurynowicz & Czerny (2007) showed that the excess variance from the NLS1 sample is by a factor $\sim$20 larger than expected from $C_M = C/M$ correlation established for moderate-luminosity AGN. NLS1 have also systematically higher break frequencies for a given mass, so the luminosity dependence of the break frequency (M$^c$Hardy et al. 2006) can perhaps be related to the increase in $C_M$. One explanation could be a bandpass effect. The seed photons in X-ray binaries in the very high state are at $\sim$1 keV, while in AGN they are at much lower energies, $\la$10 eV. Therefore, we see a different part of the Comptonized spectrum in X-rays. Gierli[ń]{}ski & Zdziarski (2005) showed that the variability amplitude strongly increases with energy in the very high state, while it is almost constant in the hard state. As we see higher orders of scattering in NLS1 than in the very-high-state stellar mass black holes, we expect higher variability in the former ones, as observed. Another possible explanation is contribution from complex absorption to the observed variability, which can significantly increase the rms at energies $\sim$1–2 keV (Markowitz et al. 2003a; Gierli[ń]{}ski & Done 2006). We expect more complex absorption from bright NLS1 sources with strong outflows. The low- and moderate-luminosity AGN are less likely to be affected by this additional variability. Fig. \[fig:mcm\] shows that the amplitude of the high-frequency tail in the PDS scales very well with the black hole mass. The correlation holds for over seven orders of magnitude. Thus, the high-frequency power in accreting black holes in the hard spectral states seems to be universal. This part of the PDS corresponds to the shortest timescales producing power in the accretion flow. Features (e.g. QPOs) occasionally observed at higher frequencies (e.g. Strohmayer 2001) have much less power. We do not understand the origin of rapid X-ray variability from accretion flows very well. It might be created by fluctuations propagating in the accretion flow (Lyubarskii 1997). The closer to the centre, the shorter characteristic timescales of fluctuations. Inevitable, there is a final barrier for propagation, the last stable orbit, which acts as a low-pass filter, removing all frequencies higher than a certain limit. The limiting frequency is inversely proportional to the radius of the last stable orbit and hence to the black hole mass. With a certain shape of the filter any initial spectrum of fluctuations will be truncated to a similar shape at higher frequencies. This might (at least qualitatively) explain the observed universal shape of the high-frequency tail (Done, Gierli[ń]{}ski & Kubota 2007). Obviously, the size of the last stable orbit depends not only on the black hole mass, but also on its spin. Higher spin would shrink the last stable orbit as if the black hole mass was smaller. This would increase $C_M$ by factor 4.8 for a maximally spinning Kerr black hole with respect to a Schwarzschild one. Alas, the black hole spin is notoriously difficult to measure (compare, e.g., McClintock et al. 2006 and Middleton, Done & Gierli[ń]{}ski 2006). The potentially highly spinning GRS 1915+105 has never been observed in the low-luminosity hard spectral state. Another culprit is XTE J1650–500, in which a broad iron line, suggesting high black hole spin, has been reported (Miller et al. 2002, but see also Done & Gierli[ń]{}ski 2006). However, as one can see in Fig. \[fig:mcm\] $C_M$ of this source is well below the $C/M$ line, so its high spin does not seem to be supported by our data. On the other hand the mass of black hole in this source is not very well established, so we cannot make any strong statements about it. Fig. \[fig:mcm\] shows that the scatter in $C_M$ for all sources is only factor two, so we do not expect large scatter in black hole spins in the sample. Yet another factor that might influence the results is the inclination of the disc with respect to the observer. If rapid X-ray variability is produced in flares or fluctuations corotating with the disc, then it is affected by Doppler effects for highly inclined discs. [Ż]{}ycki & Nied[ź]{}wiecki (2005) calculated these effects and predicted that high inclinations would give rise to strong increase in the high-frequency power. However, the additional signal appears above $\sim$100 Hz and would not be easily detected in PDS. We calculated $C_M$ from 10–128 Hz band, so our results should not be affected by the inclination. Comparison with break-frequency scaling {#sec:comparison} ======================================= An alternative approach of linking variability with black hole mass is the mass-luminosity-break frequency scaling, $\nu_b = A L_{\rm bol}^B / M^C$ (M$^c$Hardy et al. 2006; K[ö]{}rding et al. 2007), where the three observables: mass, luminosity and break frequency form a ‘fundamental plane’ along which the BHB and AGN data correlate. We would like to point out few advantages of the method proposed in this paper over the break-frequency scaling. Generally, it is easier to find variability amplitude than the break frequency, as the latter requires modelling of the power spectrum (but see Pessah 2007). The amplitude method does not involve luminosity or accretion rate, so it doesn’t require distance to the source, which in case of many Galactic black holes is poorly constrained. Another advantage of this approach is its simplicity, as it relates amplitude of variability with black hole mass directly, with just one scaling constant. The break-frequency method requires three independent constants. This suggests that high-frequency variability amplitude is more fundamental in nature. One of the drawbacks of the amplitude method is its limitation to the hard X-ray spectral state in BHB. Another state dominated by Comptonization, the very high (steep power law) state is potentially useful, but its application to bright AGN requires better understanding of energy dependence of rms. Soft-state spectra of BHB are strongly diluted by the (stable) disc and good-statistics high-energy data, required to establish $C_M$ reliably, is not available (except for Cyg X-1 and, perhaps, GRS 1915+105). But most of the AGN spectra in the 2–10 keV band are dominated by Comptonization, so this method might be valid in the soft state. The same problem seems to affect the break-frequency scaling. K[ö]{}rding et al. (2007) point out that their method is mostly limited to the hard state, as measuring and defining the break frequency in soft and very high states is very difficult. Both break-frequency and amplitude correlations require a shift in the relation when applied to Cyg X-1 in the soft spectral state (K[ö]{}rding et al. 2007), though this is rather difficult to extend to soft states of other BHB. The high-frequency amplitude depends, to some extend, on energy, while the break-frequency is energy-independent. Another potential disadvantage of the amplitude method is additional variability introduced by ionized smeared absorption or reflection in some sources, which is most pronounced around 1–2 keV (Markowitz et al. 2003b; Gierli[ń]{}ski & Done 2006; Crummy et al. 2006). Conclusions {#sec:conclusions} =========== Black hole X-ray binaries have a universal shape of the high-frequency tail (above the break frequency) in their PDS, as illustrated in Fig. \[fig:pds\]. Though the exact shape of the tail is not easy to establish, it can be approximated by a power law, $P_\nu = C_M(\nu/\nu_0)^{-2}$. The amplitude, $C_M$ of the tail is remarkably constant for any given BHB in the hard state, regardless of the luminosity. When extended to supermassive black holes in moderate luminosity AGN, the tail amplitude scales very well with black hole mass, $C_M = C/M$. The best-fitting value of the scaling constant from our sample of BHB and AGN is $C$ = 1.25$\pm0.06$ M$_\odot$ Hz$^{-1}$. This method can be applied to estimate black hole masses in many AGN. If the universal shape of the high-frequency tail is an inherent property of Comptonization, it might be applied to other spectral states. We speculate that the constancy of the tail is an imprint of the last stable orbit around the black hole. Acknowledgements {#acknowledgements .unnumbered} ================ We thank the anonymous referee for their valuable comments. MG acknowledges support through a PPARC PDRF and Polish Ministry of Science and Higher Education grant 1P03D08127. MN and BC acknowledges support through Polish Ministry of Science and Higher Education grant 1P03D00829. 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--- author: - Liming Jiang - Changxu Zhang - Mingyang Huang - Chunxiao Liu - | \ Jianping Shi - 'Chen Change Loy$^{\textrm{\Letter}}$\' bibliography: - 'sections\_arxiv/egbib.bib' title: 'TSIT: A Simple and Versatile Framework for Image-to-Image Translation\' ---

--- abstract: 'Building on the work of Nogin [@Nogin], we prove that the braid group $B_4$ acts transitively on full exceptional collections of vector bundles on Fano threefolds with $b_2=1$ and $b_3=0$. Equivalently, this group acts transitively on the set of simple helices (considered up to a shift in the derived category) on such a Fano threefold. We also prove that on threefolds with $b_2=1$ and very ample anticanonical class, every exceptional coherent sheaf is locally free.' address: 'Department of Mathematics, University of Oregon, Eugene, OR 97405' author: - 'A. Polishchuk' title: Simple helices on Fano threefolds --- \[section\] \[thm\][Proposition]{} \[thm\][Lemma]{} \[thm\][Corollary]{} [^1] Background and the main results =============================== We refer to the paper [@GK] for the review of the theory of exceptional bundles and exceptional collections (see also section 3.1 of [@Bridge] for a short account of the basic definitions and some results). Let $X$ be a (smooth) Fano threefold over ${{\Bbb C}}$ with $b_2=1$ and $b_3=0$. By the classification of Fano threefolds (see [@IP]), it is known that $X$ is either ${{\Bbb P}}^3$, or the $3$-dimensional quadric, or $V_5$, or $V_{22}$ (in the latter case there are moduli for Fano threefolds of this type). It is known that these Fano threefolds can be characterized by the condition ${\operatorname{rk}}K_0(X)=4$. Furhermore, in all of these cases the derived category $D^b(X)$ of coherent sheaves on $X$ admits a [*full exceptional collection of vector bundles*]{} $(E_1,E_2,E_3,E_4)$. By definition, this means that ${\operatorname{Ext}}^n(E_i,E_j)=0$ for $i>j$ and $n\ge 0$, ${\operatorname{Ext}}^n(E_i,E_i)=0$ for $n>0$, ${\operatorname{End}}(E_i)={{\Bbb C}}$, and the collection $(E_1,\ldots,E_4)$ generates $D^b(X)$. The constructions of full exceptional collections in the above four cases are due to Beilinson [@Be], Kapranov [@Ka], Orlov [@Orlov], and Kuznetsov [@Ku], respectively. There is a natural action of the braid group $B_n$ on the set of full exceptional collections of objects in $D^b(X)$, where $X$ is a smooth projective variety with ${\operatorname{rk}}K_0(X)=n$ given by [*left and right mutations*]{}. Bondal proved that in the case when ${\operatorname{rk}}K_0(X)=\dim X+1$, the property of a collection to consist of pure sheaves (as opposed to complexes) is preserved under mutations (see [@Bondal]). Furthermore, Positselski showed in [@Posic] that in this case all full exceptional collections of sheaves actually consist of vector bundles. Thus, in the case when $n=rk K_0(X)=\dim X+1$ there is an action of the braid group $B_n$ on the set of full exceptional collections of vector bundles on $X$. In this paper we will prove transitivity of this action for the case of Fano threefolds of the above type, by reducing it to the similar transitivity result on the level of $K_0(X)$ established by Nogin [@Nogin]. \[main-thm\] Let $X$ be a Fano threefold with $b_2=1$ and $b_3=0$. Then the action of the braid group on the set of complete exceptional collections of bundles on $X$ is transitive. Note that this result does not establish the conjecture on transitivity of the braid group action on the set of all full exceptional collections (up to a shift of each object) proposed in [@BP], since we do not know whether every full exceptional collection consists of shifts of sheaves only. Neither does it lead to the classification of exceptional bundles on $X$ since we do not know whether one can include every exceptional bundle in a full exceptional collection. One can restate Theorem \[main-thm\] using the notion of a [*simple helix*]{}[^2]. By definition, a [*simple helix of period $n$*]{} in $D^b(X)$ for a smooth projective variety $X$ is a collection of objects $(E_i)$ numbered by $i\in{{\Bbb Z}}$, such that for every $m\in{{\Bbb Z}}$ the sequence $(E_{m+1},\ldots,E_{m+n})$ is full and exceptional, and ${\operatorname{Hom}}^p(E_i,E_j)=0$ for $p\neq 0$ and $i\le j$. Simple helices can exist only if $X$ is a Fano variety with ${\operatorname{rk}}K_0(X)=\dim X+1$ (see [@BP]) and necessarily consist of shifts of vector bundles (see [@Posic]). One can show that in this case $E_{i-n}\simeq E_i(K)$, where $K$ is the canonical class on $X$ (see [@Bondal]). Conversely, starting with any full exceptional collection of vector bundles $(E_1,\ldots,E_n)$, one gets a simple helix by considering $$\label{helix-eq} (\ldots,E_1,\ldots,E_n,E_1(-K),\ldots, E_n(-K),E_1(-2K),\ldots).$$ Similarly to the case of exceptional collections one defines an action of the braid group on the set of simple helices. Our theorem can be restated as follows: [*the action of the braid group on simple helices in $D^b(X)$, where $X$ is a Fano threefold, is transitive.*]{} The difficult part of the proof of Theorem \[main-thm\] was done by Nogin in [@Nogin] where he proved the transitivity of the action of the braid group on semiorthogonal bases in $K_0(X)$ in the above situation. In the case when $X$ is not of type $V_{22}$, this easily implies our result, as was observed by A. Bondal. Indeed, if $X$ is either ${{\Bbb P}}^3$, or a quadric, or $V_5$ then there exists a full exceptional collection of vector bundles on $X$, two of which are line bundles. Studying exceptional objects in the triangulated subcategory generated by the remaining two bundles, one finds that such an object is determined by its class in $K_0$ up to a shift, which concludes the proof in this case. So, in our proof of Theorem \[main-thm\] the reader may assume (but does not have to) that $X$ is of type $V_{22}$. Our argument is based on the following result, perhaps of independent interest. \[exc-K0-thm\] Let $X$ be a Fano threefold with very ample anticanonical class and $b_2=1$. Let $E_1$ and $E_2$ be exceptional bundles on $X$ with the same class in $K_0(X)$. Assume that ${\operatorname{Ext}}^1(E_1,E_1(-K))=0$. Then $E_1\simeq E_2$. The proof of this theorem will be given in the next section. It is based on the trick of considering restrictions to a generic anticanonical K3 surface in $X$, which was exploited by S. Zube in [@Zube] to prove the stability of an exceptional bundle on ${{\Bbb P}}^3$. Using the same trick we will prove that in the situation of Theorem \[exc-K0-thm\] every exceptional sheaf on $X$ is locally free and stable (see Theorem \[stab-thm\] below). Now let us show how Theorem \[exc-K0-thm\] implies our main result. [*Proof of Theorem \[main-thm\].*]{} Given a pair of complete exceptional collections of bundles on $X$, we can mutate one of them to obtain the situation when the two collections will give identical classes in $K_0$ (by the transitivity of the braid group action on the set of semiorthogonal bases in $K_0$ proved by Nogin [@Nogin]). It remains to note that every exceptional collection $(E_1,\ldots, E_n)$ of vector bundles on $X$ extends to a simple helix . Hence, ${\operatorname{Ext}}^1(E_i,E_i(-K))=0$ for $i=1,\ldots,n$, and we can apply Theorem \[exc-K0-thm\]. It would be nice to get rid of the assumption on the vanishing of ${\operatorname{Ext}}^1$ in Theorem \[exc-K0-thm\]. So far, we were able to do this only in the case of rank $2$ bundles assuming that the index of $X$ is $\ge 2$. \[rk-2-thm\] Let $X$ be a Fano threefold with very ample anticanonical class, $b_2=1$, and index $\ge 2$. Then every exceptional bundle $E$ of rank $2$ on $X$ satisfies ${\operatorname{Ext}}^1(E_1,E_1(-K))=0$. Hence, such a bundle is uniquely determined by its class in $K_0(X)$. Proofs via Zube’s trick ======================= In this section when talking about stability of a vector bundle we always mean Mumford’s stability with respect to the slope function corresponding to an ample generator of the Neron-Severi group (we will only need this for varieties with Picard number $1$ and for curves). Let $S$ be an algebraic K3 surface. Recall that a [*spherical object*]{} $F\in D^b(S)$ is an object satisfying ${\operatorname{Hom}}^i(F,F)=0$ for $i\neq 0,2$, and ${\operatorname{Hom}}^0(F,F)={\operatorname{Hom}}^2(F,F)={{\Bbb C}}$ (see [@ST]). If $F$ is a coherent sheaf then $F$ is spherical if and only if it is simple and rigid. Let us recall some well-known properties of spherical and rigid sheaves. \[sph-stab-lem\] Let $S$ be a K3 surface, $F$ a spherical sheaf on $S$. \(i) Let $TF{\subset}F$ be the torsion subsheaf. Then $TF$ is rigid. \(ii) If $F$ is torsion free then it is locally free. \(iii) If $F_1$ and $F_2$ are stable spherical bundles with the same class in $K_0(S)$ then $F_1\simeq F_2$. \(iv) A nonzero rigid sheaf on $S$ cannot have zero-dimensional support. \(v) Assume $S$ has Picard number $1$. Then every spherical sheaf on $S$ is locally free and stable. [[*Proof*]{}]{}. Parts (i), (ii) and (iii) follow from Corollary 2.8, Proposition 3.3 and Corollary 3.5 of [@Mukai], respectively. \(iv) This follows immediately from the observation that for a sheaf $F$ with zero-dimensional support one has $\chi(F,F)=0$. Indeed, if $F$ is also rigid then we have $0=\chi(F,F)\ge\dim{\operatorname{Hom}}(F,F)$, so $F=0$. \(v) Let $TF{\subset}F$ be the torsion subsheaf of a spherical sheaf $F$. By part (i) we know that $TF$ is rigid. But then $-c_1(TF)^2=\chi(TF,TF)>0$ which is impossible since the Picard number of $S$ is $1$. Hence, $F$ is torsion free, which implies that it is locally free by part (ii). Finally, the stability of $F$ follows from Proposition 3.14 of [@Mukai]. The proof of the next result is based on the idea of Zube in [@Zube]: to check the stability of a bundle on a Fano threefold $X$ we restrict it to a smooth anticanonical divisor in $X$. The same trick allows to check that an exceptional coherent sheaf on $X$ is locally free. \[stab-thm\] Let $X$ be a Fano threefold with very ample anticanonical class and $b_2=1$. Then every exceptional coherent sheaf on $X$ is locally free and stable. The proof is based on the following well-known observation (see [@Kul]; [@ST], Ex. 3.14). \[sph-lem\] Let $E$ be an exceptional object in $D^b(X)$, where $X$ is a Fano threefold $X$, $i:S{\hookrightarrow}X$ a smooth anticanonical surface. Then $Li^*E$ is a spherical object in $D^b(S)$. [[*Proof*]{}]{}. This is derived immediately by applying the functor ${\operatorname{Hom}}(E,?)$ to the exact triangle $$E(-S)\to E\to i_*Li^*E\to \ldots$$ and using the Serre duality on $X$. [*Proof of Theorem \[stab-thm\].*]{} Let $E$ be an exceptional sheaf on $X$, and let $i:S{\hookrightarrow}X$ be a generic K3 surface in the anticanonical linear system. Since we can assume that $S$ does not contained the associated points of $E$, we have $E|_S\simeq Li^*E$. Therefore, by Lemma \[sph-lem\], the sheaf $E|_S$ is spherical. Note also that by Moishezon’s theorem (see [@Mo]), the Picard number of $S$ equals $1$. Hence, by Lemma \[sph-stab-lem\](v), $E|_S$ is locally free. Since $-K$ is ample, this immediately implies that away from a finite number of points $E$ has constant rank, and hence is locally free. Now let us consider an arbitrary smooth anticanonical divisor $i:S{\hookrightarrow}X$ and the corresponding spherical object $Li^*E\in D^b(S)$. It is easy to see that the cohomology sheaves of a spherical object in $D^b(S)$ are rigid (see Proposition 3.5 of [@IU]). Hence, the sheaf $L^1i^*E$ is rigid. But it also has zero-dimensional support, which implies that $L^1i^*E=0$ by Lemma \[sph-stab-lem\](iv). Therefore, $E|_S\simeq Li^*E$ is a spherical sheaf on $S$, locally free outside a finite number of points. It follows that the torsion subsheaf of $E|_S$ is at the same time rigid (by Lemma \[sph-stab-lem\](i)) and has zero-dimensional support, hence, it is zero. By Lemma \[sph-stab-lem\](ii), this implies that $E|_S$ is locally free. Since there exists a smooth anticanonical divisor passing through every point of $X$, we derive that the rank of $E$ is constant on $X$, therefore, $E$ is locally free. Assume $E$ is not stable. Then there exists an exact sequence $$0\to F\to E\to Q\to 0$$ with torsion-free sheaf $Q$, $0<{\operatorname{rk}}F<{\operatorname{rk}}E$, $\mu(F)\ge\mu(E)$. Let $i:S{\hookrightarrow}X$ be a generic anticanonical surface $S$. Since we can choose $S$ not containing the associated points of $F$, $E$ and $Q$, $Li^*F\simeq F|_S$ will be a subsheaf of $Li^*E\simeq E|_S$. Since by Moishezon’s theorem, $S$ has the Picard number $1$, applying Lemma \[sph-stab-lem\](v), we derive that the bundle $E|_S$ is stable. But this contradicts to the inequality $$\mu(F|_S)=\mu(F)\ge \mu(E)=\mu(E|_S),$$ where we use $-K$ and $-K|_S$ to define the slope functions on $X$ and on $S$, respectively. [*Proof of Theorem \[exc-K0-thm\].*]{} By Theorem \[stab-thm\], $E_1$ and $E_2$ are stable bundles of the same slope, so it is enough to construct a nonzero map between them. Consider a generic anticanonical K3 surface $S{\subset}X$. By Lemma \[sph-lem\], the restrictions $E_1|_S$ and $E_2|_S$ are spherical. Also, by Moishezon’s theorem, $S$ has Picard number $1$. Hence, by Lemma \[sph-stab-lem\](iii), we get an isomorphism $E_1|_S\simeq E_2|_S$. The long exact sequence $$\ldots\to{\operatorname{Hom}}(E_1,E_2)\to{\operatorname{Hom}}(E_1,E_2|_S)\to{\operatorname{Ext}}^1(E_1,E_2(K))\to\ldots$$ shows that it is enough to prove the vanishing of ${\operatorname{Ext}}^1(E_1,E_2(K))$ (then one can lift the nonzero element of ${\operatorname{Hom}}(E_1|_S,E_2|_S)$ to a nonzero map $E_1\to E_2$). The long exact sequences for $n\ge 1$ $$\ldots\to {\operatorname{Ext}}^1(E_1,E_2((n+1)K))\to{\operatorname{Ext}}^1(E_1,E_2(nK))\to{\operatorname{Ext}}^1(E_1,E_2(nK)|_S)\to\ldots$$ together with the vanishing of ${\operatorname{Ext}}^1(E_1,E_2((n+1)K))\simeq{\operatorname{Ext}}^2(E_2,E_1(-nK))^*$ for $n\gg 0$, reduce the problem to showing the vanishing of $${\operatorname{Ext}}^1(E_1,E_2(nK)|_S)\simeq{\operatorname{Ext}}^1(E_1,E_1(nK)|_S)$$ for $n\ge 1$. Now we can use the exact sequences $$\ldots\to{\operatorname{Ext}}^1(E_1,E_1(nK))\to{\operatorname{Ext}}^1(E_1,E_1(nK)|_S)\to{\operatorname{Ext}}^2(E_1,E_1((n+1)K))\to\ldots$$ that show that it would be enough to know that ${\operatorname{Ext}}^1(E_1,E_1(nK))={\operatorname{Ext}}^2(E_1,E_1(nK))=0$ for $n\ge 1$. Set $F=\underline{{\operatorname{End}}}_0(E_1)$, the bundle of traceless endomorphisms of $E_1$. Since $E_1$ is exceptional and ${\operatorname{Ext}}^1(E_1,E_1(-K))=0$ by our assumption, we get $$\begin{array}{l} H^3(F(K))\simeq H^0(F)^*=0,\nonumber\\ H^2(F)=0,\\ H^1(F(-K))=0 \end{array}$$ Hence, the sheaf $F$ is $2$-regular in the sense of Castelnuovo-Mumford with respect to $-K$ (see [@Mum]). It follows that $H^2(F(-nK))=0$ for $n\ge 0$, and $H^1(F(-nK))=0$ for $n\ge 1$. Using Serre duality we derive that $H^1(F(nK))=H^2(F(nK))=0$ for all $n\in{{\Bbb Z}}$. On the other hand, Kodaira vanishing theorem together with Serre duality imply that $H^1({{\cal O}}_X(nK))=H^2({{\cal O}}_X(nK))=0$ for $n\in{{\Bbb Z}}$. Hence, ${\operatorname{Ext}}^1(E_1,E_1(nK))={\operatorname{Ext}}^2(E_1,E_1(nK))=0$ for $n\in{{\Bbb Z}}$. [*Proof of Theorem \[rk-2-thm\].*]{} Let $S{\subset}X$ be a generic anticanonical K3 surface. Set $F=E|_S$. The exact sequence $$0={\operatorname{Ext}}^1(E,E)\to{\operatorname{Ext}}^1(E,E(-K))\to{\operatorname{Ext}}^1(E,E(-K)|_S)\to\ldots$$ shows that it suffices to check the vanishing of ${\operatorname{Ext}}^1_S(F,F(L))$, where $L=-K|_S$. By Serre duality on $S$, this is equivalent to the vanishing of ${\operatorname{Ext}}^1_S(F,F(-L))$. Let $C{\subset}S$ be a smooth curve in the linear system $|L|$. Then the exact sequence $$\ldots{\operatorname{Hom}}(F,F)\to{\operatorname{Hom}}(F|_C,F|_C)\to{\operatorname{Ext}}^1_S(F,F(-L))\to{\operatorname{Ext}}^1_S(F,F)=0$$ shows that ${\operatorname{Ext}}^1_S(F,F(-L))=0$ if and only if the restriction $F|_C$ is a simple vector bundle on $C$. Therefore, it is enough to check that $F|_C$ is stable. To this end we will use the effective version of Bogomolov’s theorem on restriction of stable bundles from surfaces to curves (see [@Bog]). Recall that since $F$ is a spherical bundle and $S$ has Picard number $1$, $F$ is stable (by Lemma \[sph-stab-lem\](v)). Using the condition $\chi(F,F)=2$ one easily computes that ${\Delta}(F)=4c_2(F)-c_1^2(F)=2$. Let $H$ be the fundamental ample divisor class on $X$, so that $-K=kH$, where $k$ is the index of $X$. Then $L=kH|_S$, where $k\ge 2={\Delta}(F)/2+1$. Hence, the restriction $F|_C$ is stable (cf. [@HL] Thm. 7.3.5 or [@La] Thm. 1.4.1). 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A. Ishii, H. Uehara, [*Autoequivalences of derived categories on the minimal resolutions of $A_n$-singularities on surfaces*]{}, J. Diff. Geom. 71 (2005), 385–435. D. Huybrechts, M. Lehn, [*The Geometry of Moduli Spaces of Sheaves*]{}, Friedr. Vieweg & Sohn, Braunschweig, 1997. M. Kapranov, [*Derived category of coherent bundles on a quadric*]{}, Functional Anal. Appl. 20 (1986), no. 2, 141–142. S. Kuleshov, [*Exceptional bundles on $K3$ surfaces*]{}, in [*Helices and vector bundles*]{}, 105–114, Cambridge Univ. Press, Cambridge, 1990. A. Kuznetsov, [*An exceptional set of vector bundles on the varieties $V_{22}$*]{}, Univ. Math. Bull. 51 (1996), no. 3, 35–37. A. Langer, [*Lectures on torsion-free sheaves and their moduli*]{}, in [*Algebraic Cycles, Sheaves, Shtukas, and Moduli; IMPANGA Lecture Notes*]{}, Trends in Mathematics, Birkhauser, to appear. B. Moishezon, [*Algebraic homology classes on algebraic varieties*]{}, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 225–268. S. Mukai, [*On the moduli space of bundles on $K3$ surfaces. I*]{}, in [*Vector bundles on algebraic varieties (Bombay, 1984)*]{}, 341–413, Tata Inst. Fund. Res., Bombay, 1987. D. Mumford, [*Lectures on curves on an algebraic surface*]{}, Princeton University Press, Princeton, N.J. 1966. D. Nogin, [*Helices on some Fano threefolds: constructivity of semiorthogonal bases of $K\sb 0$*]{}, Ann. Sci. ƒcole Norm. Sup. (4) 27 (1994), no. 2, 129–172. D. Orlov, [*Exceptional set of vector bundles on the variety $V_5$*]{}, Moscow Univ. Math. Bull. 46 (1991), no. 5, p.48–50 L. Positselski, [*All strictly exceptional collections in ${{\cal D}}^b_{coh}({{\Bbb P}}^n)$ consist of vector bundles*]{}, preprint alg-geom/9507014. P. Seidel, R. Thomas, [*Braid group actions on derived categories of coherent sheaves*]{}, Duke Math. J. 108 (2001), 37–108. S. Zube, [*The stability of exceptional bundles on three-dimensional projective space*]{}, in [Helices and Vector Bundles]{}, 115–117, Cambridge Univ. Press, Cambride,1990. [^1]: Supported in part by NSF grant [^2]: this terminology is due to Bridgeland [@Bridge]; the original term from [@BP] is a [*geometric helix*]{}

--- address: | Department of Physics and Astronomy\ Michigan State University\ East Lansing, MI 48824 USA\ E-mail: shri@pa.msu.edu, yuan@pa.msu.edu author: - 'Shrihari Gopalakrishna and C.–P. Yuan' title: | B-physics Signature of a\ Supersymmetric U(2) Flavor Model[^1] --- Introduction ============ The Standard Model (SM) of high energy physics suffers from the gauge hierarchy problem and the flavor problem. Supersymmetry (SUSY) eliminates the gauge hierarchy problem, and a (horizontal) flavor symmetry in generation space could explain the flavor problem. A SUSY theory with a flavor symmetry might relate the quark/lepton flavor structure with that of the scalar quark/lepton sector. Such a theory would imply certain predictions for flavor changing neutral current (FCNC) processes that we wish to investigate in this work, along with the constraints from experimental FCNC data. We do not assume an alignment of the quark/lepton flavor structure with that of the scalar quark/lepton sector, leading to a non-minimal flavor violation (NMFV) scenario. We consider a spontaneously broken U(2) flavor symmetry [@Pomarol:1995xc; @Barbieri:1995uv] in the framework of “effective supersymmetry” [@Cohen:1996vb], in which the first two generation scalars are relatively heavy (a few TeV mass), thereby satisfying neutron electric dipole moment constraint, etc., while still allowing large CP violating phases in the scalar sector. We analyze the implications of such a framework to B-physics observables. We will present details in a forthcoming paper [@shricp]. Consider that the first and second generation superfields ($\psi_a$, a=1,2) transform as a U(2) doublet while the third generation superfield ($\psi$) is a singlet [@Barbieri:1995uv]. The most general U(2) symmetric superpotential can be written as = \_1 H + \_2 H \_a + \_a \_3 H \_b + \_a \_4 H \_b\ + \_a \_5 H \_b + H\_u H\_d  , where $M$ is the cutoff scale below which such an effective description is valid, the $\alpha_i$ are O(1) constants, $\phi^a$ is a U(2) doublet, $\phi^{ab}$ and $S^{ab}$ are second rank antisymmetric and symmetric U(2) tensors respectively. If U(2) is broken spontaneously by the Vacuum Expectation Values (VEV) = 0\ V ;    = v \^[ab]{};    = 0, = V , with $\frac{V}{M} \equiv \epsilon \sim 0.02$ and $\frac{v}{M} \equiv \epsilon ' \sim 0.004$, and if U(2) is broken below the SUSY breaking scale, the SUSY breaking masses would also have a structure dictated by U(2). The resulting quark and scalar down-type masses are $$\begin{aligned} {\cal M}_d = v_d \begin{pmatrix} O & -\lambda_1\epsilon ' & O \\ \lambda_1\epsilon ' & \lambda_2\epsilon & \lambda_4\epsilon \\ O & \lambda_4'\epsilon & \lambda_3 \end{pmatrix} \ &,& \quad {\cal M}^2_{RL} = v_d \begin{pmatrix}O & -A_1 \epsilon ' & O \\ A_1 \epsilon ' & A_2\epsilon & A_4 \epsilon \\ O & A_4'\epsilon & A_3 \end{pmatrix} \ , \\ {\cal M}^2_{LL} = \begin{pmatrix}m_1^2 & 0 & 0 \\ 0 & m_1^2+\epsilon^2 m_2^2 & \epsilon m_4^{2*} \\ 0 & \epsilon m_4^2 & m_3^2 \end{pmatrix}_{LL}&,& \quad {\cal M}^2_{RR} = \begin{pmatrix} m_1^2 & 0 & 0 \\ 0 & m_1^2+\epsilon^2 m_2^2 & \epsilon m_4^{2*} \\ 0 & \epsilon m_4^2 & m_3^2\end{pmatrix}_{RR}, \nonumber \label{MSQ.EQ}\end{aligned}$$ where $v_d = \left< h_d \right>$ is the VEV of the Higgs field, the $\lambda_i$’s are O(1) coefficients, and, $m_i$ and $A_i$ (complex in general) are determined by the SUSY breaking mechanism. It has been shown [@Barbieri:1995uv] that such a pattern of the quark mass matrix explains the quark masses and CKM elements. For our study, we consider the following values for the various SUSY parameters: ${m_{\tilde b_R,\tilde t_R}}=100$GeV, the other squark masses given by $m_0=1000$GeV, $A=1000$GeV, $\tan{\beta}=5$, $|\mu|=150$GeV, $M_2=250$GeV, $M_{\tilde g}=250$GeV and $m_{H^\pm}=250$GeV. ($m_0$ and $A$ denote generic SUSY breaking mass scales.) Here, we consider processes that go through the $b\rightarrow s$ quark level transition, and in our framework the dominant SUSY contributions are due to $\delta_{32,23}^{RL,RR,LL} \equiv \frac{({\cal M}^2_{RL,RR,LL})_{32,23}}{m_0^2}$. For the chosen values of the parameters, we find $|\delta_{32,23}^{RL}| \sim \frac{v_d A\epsilon}{\tilde{m}_0^2}=6.8\times 10^{-4}$, and, $|\delta_{32}^{LL,RR}| \sim \epsilon \frac{m_4^2}{m_0^2} = 0.02$. B-physics probes ================ Given such an effective SUSY theory we estimate the sizes of various B-physics observables that we expect are modified from their SM predictions. In addition to the SM contribution, we include the charged Higgs, chargino and gluino contributions. We analyze the $\Delta B=1$ FCNC processes, , , , ; and the $\Delta B=2$ processes  mixing and the dilepton asymmetry in $B_s$. We find regions in U(2) SUSY parameter space that are consistent with current experimental data and obtain expectations for measurements that are forthcoming. To illustrate the effects, we present here expectations for CP asymmetries in  and , and,  mixing. A more exhaustive analysis will be presented elsewhere [@shricp]. The CP asymmetry in  is given by A\_[CP]{}\^ &&  , \[ACPBSG.EQ\] and the expectation in the U(2) SUSY theory is shown in Fig. (\[BSGACPLR.FIG\]). We see that significant CP asymmetry is possible in the scenario we are considering, while satisfying experimental constraints. The CP asymmetry in  is defined by A\_[CP]{}\^ &&\ &=& -C\_[K]{} + S\_[K]{}, and Fig. (\[BPHIKCPBS.FIG\] left) shows the CP asymmetry in  for a scan on $\delta_{32}^{RL}$ while satisfying all experimental constraints. The  mixing parameter $\Delta m_{B_s}$ depends quite sensitively on $\delta_{32}^{RR}$ and can be significantly altered from the SM prediction as shown in Fig. (\[BPHIKCPBS.FIG\] right). In conclusion, we note that similar results hold for flavor models that have the same order of magnitude for the 23 element in the squark mass matrix. In such cases, the prospects look exciting for discovering SUSY in B meson processes at current and upcoming colliders. [0]{} A. Pomarol and D. Tommasini, Nucl. Phys. B [**466**]{}, 3 (1996). R. Barbieri, G. R. Dvali and L. J. Hall, Phys. Lett. B [**377**]{}, 76 (1996); R. Barbieri, L. J. Hall and A. Romanino, Phys. Lett. B [**401**]{}, 47 (1997). A. G. Cohen, D. B. Kaplan and A. E. Nelson, Phys. Lett. B [**388**]{}, 588 (1996). S. Gopalakrishna and C.–P. Yuan, In preparation. [^1]: alk presented by at [ *2003: upersymmetry in the esert*]{}, held at the niversity of rizona, ucson, , une 5-10, 2003. o appear in the roceedings.

--- abstract: 'Let $K$ be a number field, and let $C$ be a hyperelliptic curve over $K$ with Jacobian $J$. Suppose that $C$ is defined by an equation of the form $y^{2} = f(x)(x - \lambda)$ for some irreducible monic polynomial $f \in \mathcal{O}_{K}$ of discriminant $\Delta$ and some element $\lambda \in \mathcal{O}_{K}$. Our first main result says that if there is a prime $\mathfrak{p}$ of $K$ dividing $(f(\lambda))$ but not $(2\Delta)$, then the image of the natural $2$-adic Galois representation is open in ${\mathrm{GSp}}(T_{2}(J))$ and contains a certain congruence subgroup of ${\mathrm{Sp}}(T_{2}(J))$ depending on the maximal power of $\mathfrak{p}$ dividing $(f(\lambda))$. We also present and prove a variant of this result that applies when $C$ is defined by an equation of the form $y^{2} = f(x)(x - \lambda)(x - \lambda'')$ for distinct elements $\lambda, \lambda'' \in K$. We then show that the hypothesis in the former statement holds for almost all $\lambda \in \mathcal{O}_{K}$ and prove a quantitative form of a uniform boundedness result of Cadoret and Tamagawa.' author: - Jeffrey Yelton bibliography: - 'bibfile.bib' title: 'Boundedness results for $2$-adic Galois images associated to hyperelliptic Jacobians' --- Introduction {#S1} ============ Let $K$ be a number field with absolute Galois group $G_{K}$, and let $C$ be a hyperelliptic curve defined over $K$; i.e. $C$ is a smooth projective curve defined by an equation of the form $y^{2} = f(x)$ for some squarefree polynomial $f$ of degree $d \geq 3$. (Note that in the case of $d = 3$, $C$ is an elliptic curve.) It is well known that the genus of $C$ is given by $g = \lfloor (d + 1) / 2 \rfloor$. We denote the Jacobian variety of $C$ by $J$; it is an abelian variety of dimension $g$. For each prime $\ell$, we let $T_{\ell}(J)$ denote the $\ell$-adic Tate module of $J$, which is a free ${\mathbb{Z}}_{\ell}$-module of rank $2g$. We write $\rho_{\ell} : G_{K} \to {\mathrm{Aut}}(T_{\ell}(J))$ for the natural $\ell$-adic Galois action on this Tate module. The Tate module $T_{\ell}(J)$ is endowed with the Weil pairing defined with respect to the canonical principal polarization on $J$, which we write as $e_{\ell} : T_{\ell}(J) \times T_{\ell}(J) \to {\mathbb{Z}}_{\ell}$; it is a ${\mathbb{Z}}_{\ell}$-bilinear skew-symmetric pairing. Let ${\mathrm{Sp}}(T_{\ell}(J))$ denote the group of symplectic automorphisms of $T_{\ell}(J)$ with respect to the pairing $e_{\ell}$, and let $${\mathrm{GSp}}(T_{\ell}(J)) := \{\sigma \in \mathrm{Aut}_{{\mathbb{Z}}_{\ell}}(T_{\ell}(J))\ |\ e_{\ell}(P^{\sigma}, Q^{\sigma}) = e_{\ell}(P, Q)^{\chi_{\ell}(\sigma)}\ \forall P, Q \in T_{2}(J)\}$$ denote the group of symplectic similitudes, where $\displaystyle \chi_{\ell} : G_{K} \to {\mathbb{Z}}_{\ell}^{\times}$ is the $\ell$-adic cyclotomic character. It is well known that the image $G_{\ell}$ of $\rho_{\ell}$ is always a closed subgroup of ${\mathrm{GSp}}(T_{\ell}(J))$ and that in fact there is some hyperelliptic Jacobian $J$ of a given dimension $g$ such that the inclusion $G_{\ell} \subseteq {\mathrm{GSp}}(T_{\ell}(J))$ has finite index (or equivalently, that $G_{\ell}$ is an open subgroup of the $\ell$-adic Lie group ${\mathrm{GSp}}(T_{\ell}(J))$); see for instance [@yelton2015images Theorem 1.1]. Note that the subgroup $G_{\ell} \cap {\mathrm{Sp}}(T_{\ell}(J)) \subset G_{\ell}$ coincides with the image of the Galois subgroup which fixes the extension $K(\mu_{\ell}) / K$ obtained by adjoining all $\ell$-power roots of unity to $K$. Since $K$ is a number field, the extension $K(\mu_{\ell}) / K$ is infinite; it follows that $G_{\ell} \not\subset {\mathrm{Sp}}(T_{\ell}(J))$ and that $G_{\ell}$ has finite index in ${\mathrm{GSp}}(T_{\ell}(J))$ if and only if $G_{\ell} \cap {\mathrm{Sp}}(T_{\ell}(J))$ has finite index in ${\mathrm{Sp}}(T_{\ell}(J))$. There have been many results stating that $G_{\ell}$ has finite index in ${\mathrm{GSp}}(T_{\ell}(J))$ under various hypotheses for the polynomial defining the hyperelliptic curve. For instance, Y. Zarhin has proven this for large enough genus in the case of hyperelliptic curves defined by equations of the form $y^{2} = f(x)$ or $y^{2} = f(x)(x - \lambda)$ with $\lambda \in K$, where the Galois group of $f$ is the full symmetric or alternating group ([@zarhin2002very Theorem 2.5] and [@zarhin2010families Theorem 8.3]; see also [@zarhin2013two Theorem 1.3] for a variant of this where the curve is defined using two parameters). A. Cadoret and A. Tamagawa have also proven ([@cadoret2012uniform Theorems 1.1 and 5.1]) that for any family of hyperelliptic Jacobians over a smooth, geometrically connected, separated curve over $K$, this openness condition will be satisfied for the $\ell$-adic Galois action associated to all but finitely many fibers, and that in fact the indices of the $\ell$-adic Galois images corresponding to these fibers are uniformly bounded. However, there have been very few results which give explicit bounds for the index of $G_{\ell}$ in ${\mathrm{GSp}}(T_{\ell}(J))$ in such cases. Our aim in this paper is to give some similar results on the openness of the $2$-adic Galois images in the group of symplectic similitudes associated to Jacobians of hyperelliptic curves whose defining polynomials satisfy certain hypotheses, and to provide formulas giving explicit bounds for the indices of the $2$-adic Galois images in these cases. (Unfortunately, our method currently cannot tell us anything about the $\ell$-adic Galois images for odd primes $\ell$. However, we are hopeful that it can be strengthened to show the openness of the $\ell$-adic Galois images as well under the same or similar hypotheses as is implied by the Mumford-Tate conjecture, and to show that the $\ell$-adic Galois images contain the full symplectic group for almost all $\ell$.) We state our main results below. In these statements as well as in the rest of the paper, we use the following notation. For any integer $N \geq 1$, we denote the level-$N$ congruence subgroup of ${\mathrm{Sp}}(T_{2}(J))$ by $\Gamma(N) := \{\sigma \in {\mathrm{Sp}}(T_{2}(J)) \ | \ \sigma \equiv 1 \ (\mathrm{mod} \ N)\}$. We denote the ring of integers of a number field $K$ by $\mathcal{O}_{K}$. Finally, we write $v_{2} : {\mathbb{Q}}^{\times} \to {\mathbb{Z}}$ for the (normalized) $2$-adic valuation on ${\mathbb{Q}}$. \[thm main1\] Let $K$ be a number field, and let $f \in \mathcal{O}_{K}[x]$ be an irreducible monic polynomial of degree $d \geq 2$ with discriminant $\Delta$. Let $J$ be the Jacobian of the hyperelliptic curve with defining equation $y^{2} = f(x)(x - \lambda)$ for some $\lambda \in \mathcal{O}_{K}$, and define the $2$-adic Galois image $G_{2}$ as above. Then if there is a prime $\mathfrak{p}$ of $\mathcal{O}_{K}$ which divides $(f(\lambda))$ but not $(2\Delta)$, the Lie subgroup $G_{2} \subset {\mathrm{GSp}}(T_{2}(J))$ is open. In fact, we have $G_{2} \cap {\mathrm{Sp}}(T_{2}(J)) \supsetneq \Gamma(2^{2v_{2}(m) + 2})$, where $m \geq 1$ is the greatest integer such that $\mathfrak{p}^{m} \mid (f(\lambda))$. If in addition $d = 3$, then $G_{2} \cap {\mathrm{Sp}}(T_{2}(J)) \supsetneq \Gamma(2^{v_{2}(m) + 1})$. \[thm main2\] Assume the same notation as in the statement of Theorem \[thm main1\], except that the defining equation of the hyperelliptic curve is $y^{2} = f(x)(x - \lambda)(x - \lambda')$ for distinct elements $\lambda, \lambda' \in \mathcal{O}_{K}$. Then if there is a prime $\mathfrak{p}$ of $\mathcal{O}_{K}$ which divides $(f(\lambda))$ but not $(\lambda - \lambda')$ or $(2\Delta)$ and a prime $\mathfrak{p}'$ which divides $(\lambda - \lambda')$ but not $(f(\lambda))$ or $(2\Delta)$, the Lie subgroup $G_{2} \subset {\mathrm{GSp}}(T_{2}(J))$ is open. In fact, we have $G_{2} \cap {\mathrm{Sp}}(T_{2}(J)) \supsetneq \Gamma(2^{2v_{2}(m) + 2})$ if $v_{2}(m') \leq v_{2}(m)$ or $d = 2g$ and $G_{2} \cap {\mathrm{Sp}}(T_{2}(J)) \supsetneq \Gamma(2^{v_{2}(m) + v_{2}(m') + 2})$ otherwise, where $m \geq 1$ is the greatest integer such that $\mathfrak{p}^{m} \mid (f(\lambda))$ and $m' \geq 1$ is the greatest integer such that $\mathfrak{p}'^{m'} \mid (\lambda - \lambda')$. If in addition $d = 2$, then $G_{2} \cap {\mathrm{Sp}}(T_{2}(J)) \supsetneq \Gamma(2^{\max \{v_{2}(m), v_{2}(m')\} + 1})$. \[rmk almost all fibers\] For a fixed irreducible monic polynomial $f \in \mathcal{O}_{K}[x]$ of degree $d \geq 2$ with discriminant $\Delta$, it is not hard to show using Faltings’ Theorem that the hypotheses in Theorems \[thm main1\] is satisfied for almost all $\lambda \in \mathcal{O}_{K}$, as we will see in §\[S5\] (Remark \[rmk not PID\] below). It follows immediately that there are also infinitely many choices of $(\lambda, \lambda') \in K \times K$ satisfying the hypotheses in Theorem \[thm main2\] for this polynomial $f$. \[rmk end\] It is known that there is a finite algebraic extension $K'$ of $K$ over which every endomorphism of an abelian variety over a field $K$ is defined (see [@silverberg1992fields Theorem 2.4]), so that each endomorphism commutes with the action of ${\mathrm{Gal}}(\bar{K} / K')$ on torsion points. Note that the only endomorphisms in ${\mathrm{End}}(T_{\ell}(J))$ which commute with everything in an open subgroup of ${\mathrm{GSp}}(T_{\ell}(J))$ are scalars. It therefore follows from the above theorems that the endomorphism ring of any hyperelliptic Jacobian $J$ satisfying the hypotheses of Theorem \[thm main1\] or of Theorem \[thm main2\] coincides with ${\mathbb{Z}}$ and that such a $J$ is absolutely simple. The key ingredient used in proving the above theorems is a method of describing Galois actions on $\ell$-adic Tate modules of hyperelliptic Jacobians defined over strictly Henselian local fields of residue characteristic $p \neq 2, \ell$ by looking at the valuations of the differences between the roots of the defining polynomial, which is derived from results shown in joint work with H. Hasson ([@hasson2017prime]). This way of looking at $\ell$-adic Galois actions associated to hyperelliptic Jacobians over local fields is very similar to the “method of clusters" used by S. Anni and V. Dokchitser in [@anni2017constructing]. Our approach seems quite powerful and should lead to many similar boundedness results in a number of situations where one can compute valuations of the differences between the roots of the defining polynomial with respect to various primes of the ground field. Unfortunately, in practice, these valuations (or even the roots themselves) may be difficult to calculate, and so our main focus here is on obtaining results such as the ones stated above where the hypotheses are very easy to verify. The rest of this paper is organized as follows. In §\[S2\], we use the main results of [@hasson2017prime], which show that over a strictly Henselian local field of characteristic $p \neq 2$, for primes $\ell \neq p$, the $\ell$-adic Galois action factors through the tame quotient of the absolute Galois group and can be described in terms of Dehn twists with respect to certain loops on a complex hyperelliptic curve. In particular cases such as when exactly two roots of the defining polynomial coalesce in the reduction over the residue field, we will show (Proposition \[prop Galois action local\]) that such a Dehn twist induces a transvection in the symplectic group. We will later put local data together to show that over a number field $K$, the $2$-adic Galois image contains certain powers of several sufficiently “independent" transvections. In §\[S3\], we will demonstrate using elementary matrix algebra that the group generated by these powers of transvections contains a certain congruence subgroup. In §\[S4\], we will use what we have shown in §\[S2\] and §\[S3\] to prove Theorems \[thm main1\] and \[thm main2\] as well as to prove an auxiliary result that applies to a more general situation (Theorem \[thm several primes\]). Finally, in §\[S5\], we will assume that $K$ has class number $1$ and show using Theorem \[thm main1\] that for a given $f \in \mathcal{O}_{K}[x]$ of degree $d \geq 3$, the $2$-adic Galois image associated to the hyperelliptic curves defined by $y^{2} = f(x)(x - \lambda)$ for all but finitely many $\lambda \in \mathcal{O}_{K}$ contains a principal congruence subgroup which depends only on $d$ (Theorem \[thm uniform bounds\]). In fact, for $d \geq 4$ even, in Theorem \[thm uniform bounds\](c) we will provide a uniform bound for indices of the $2$-adic Galois images associated to almost all fibers of such a one-parameter family over the $K$-line, as is guaranteed by [@cadoret2012uniform Theorems 1.1 and 5.1]. The author is grateful to a MathOverflow user whose comment on question 264281 helped to inspire the arguments for the results in §\[S5\]. Hyperelliptic Jacobians over local fields and tame Galois actions {#S2} ================================================================= We retain all notation introduced in the previous section. In this section, we write $\widehat{{\mathbb{Z}}}$ for the profinite completion of ${\mathbb{Z}}$ and use the symbol $\widehat{\pi}_{1}$ to denote the profinite completion of the fundamental group of a topological space. For any profinite group $G$, we write $G^{(p')}$ for its maximal prime-to-$p$ quotient. For any profinite group $G$, let $G^{(p')}$ denote its prime-to-$p$ quotient. Note that since $G^{(p')}$ is a characteristic quotient of $G$, any action on $G$ induces an action on $G^{(p')}$. Now we choose a prime $\mathfrak{p}$ of $K$ of residue characteristic $p \neq 2$. Fix a strict Henselization of the localization of $K$ at the prime $\mathfrak{p}$ and denote it by $\mathcal{R}_{\mathfrak{p}}$ and its fraction field by $\mathcal{K}_{\mathfrak{p}}$; this comes with an embedding $\mathcal{K}_{\mathfrak{p}} \hookrightarrow \bar{K}$. Let $\pi \in K$ be a uniformizer of the discrete valuation ring $\mathcal{R}_{\mathfrak{p}}$. We fix a compatible system of $N$th roots of unity $\zeta_{N} \in \bar{K}$ for $N = 1, 2, 3, ...$; that is, we require that $\zeta_{N'N}^{N'} = \zeta_{N}$ for any integers $N, N' \geq 1$. Note that since $R$ is strictly Henselian, $\zeta_{N} \in R \subset K$ for any $N$ not divisible by $p$. Let $G_{K, \mathfrak{p}}$ denote the absolute Galois group of $\mathcal{K}_{\mathfrak{p}}$, and let $G_{K, \mathfrak{p}}^{{\mathrm{tame}}}$ denote its tame quotient. It follows from a special case of Abhyankar’s Lemma that the maximal tamely ramified extension $\mathcal{K}_{\mathfrak{p}}^{{\mathrm{tame}}}$ is given by $\mathcal{K}_{\mathfrak{p}}(\{\pi^{1/N}\}_{(N, p) = 1})$, where $\pi^{1/N}$ denotes an $N$th root of $\pi$, and that $G_{K, \mathfrak{p}}^{{\mathrm{tame}}} \cong \widehat{{\mathbb{Z}}}^{(p')}$ is topologically generated by the automorphism which acts on $K^{{\mathrm{tame}}}$ by fixing $K$ and sending each $\pi^{1/N}$ to $\zeta_{N}\pi^{1/N}$. We fix, once and for all, an embedding $\bar{K} \hookrightarrow {\mathbb{C}}$ where $\zeta_{N}$ is sent to $e^{2 \pi \sqrt{-1} / N}$ for $N \geq 1$, so that we have an inclusion $\mathcal{K}_{\mathfrak{p}} \subset {\mathbb{C}}$. Let $d \geq 2$ be an integer and choose distinct integral elements $\alpha_{1}, ... , \alpha_{d} \in K$. Choose polynomials $\tilde{\alpha}_{1}, ... , \tilde{\alpha}_{d} \in {\mathbb{C}}[x]$ satisfying $\tilde{\alpha}_{i}(\pi) = \alpha_{i}$ for $1 \leq i \leq d$ and such that the $x$-adic valuation of $\tilde{\alpha}_{i}$ and $\tilde{\alpha}_{j}$ and the $\pi$-adic valuation of $\alpha_{i}$ and $\alpha_{j}$ are equal (such polynomials exist as is shown in the discussion in [@hasson2017prime §3.3]). Let $\varepsilon > 0$ be a real number small enough that $\tilde{\alpha}_{i}(z) \neq \tilde{\alpha}_{j}(z)$ for all $i \neq j$ and for all $z \in B_{\varepsilon}^{*} := \{z \in {\mathbb{C}}\ | \ |z| < \varepsilon\} \smallsetminus \{0\}$. We define a family $\mathcal{F} \to B_{\varepsilon}^{*}$ of $d$-times-punctured Riemann spheres by letting $$\mathcal{F} = {\mathbb{P}}_{{\mathbb{C}}}^{1} \times B_{\varepsilon}^{*} \smallsetminus \bigcup_{i = 1}^{d} \{(\tilde{\alpha}_{i}(z), z) \ | \ z \in B_{\varepsilon}^{*}\}.$$ Choose a basepoint $z_{0} \in B_{\varepsilon}^{*}$. The fundamental group $\pi_{1}(B_{\varepsilon}^{*}, z_{0})$ acts by monodromy on the fundamental group $\pi_{1}(\mathcal{F}_{z_{0}}, \infty)$ of the fiber over $z_{0}$ with basepoint $\infty$. We write $\rho_{{\mathrm{top}}} : \pi_{1}(B_{\varepsilon}^{*}, z_{0}) \to {\mathrm{Aut}}(\pi_{1}(\mathcal{F}_{z_{0}}, \infty))$ for this action. The action $\rho_{{\mathrm{top}}}$ extends uniquely to a continuous action of the profinite completion $\widehat{\pi}_{1}(B_{\varepsilon}^{*}, z_{0})$ on the profinite completion $\widehat{\pi}_{1}(\mathcal{F}_{z_{0}}, \infty)$ (see the discussion in [@hasson2017prime §1.1]), which we also denote by $\rho_{{\mathrm{top}}}$. We note that $\widehat{\pi}_{1}(B_{\varepsilon}^{*}, z_{0})$ is isomorphic to $\widehat{{\mathbb{Z}}}$ and is topologically generated by the element $\delta \in \pi_{1}(B_{\varepsilon}^{*}, z_{0})$ represented by the loop given by $t \mapsto e^{2 \pi \sqrt{-1} t}z_{0}$ for $t \in [0, 1]$. Meanwhile, the absolute Galois group $G_{K, \mathfrak{p}}$ acts naturally on the étale fundamental group $\pi_{1}^{{\mathrm{\acute{e}t}}}({\mathbb{P}}_{\bar{K}_{\mathfrak{p}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}, \infty)$ via the $K_{\mathfrak{p}}$-point lying under the geometric point $\infty : {\mathrm{Spec}}({\mathbb{C}}) \to {\mathbb{P}}_{K_{\mathfrak{p}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}$. After identifying $\pi_{1}^{{\mathrm{\acute{e}t}}}({\mathbb{P}}_{\bar{K}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}, \infty)^{(p')}$ with $\widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}, \infty)^{(p')}$ via Riemann’s Existence Theorem and the inclusion of algebraically closed fields $\bar{\mathcal{K}}_{\mathfrak{p}} \subset {\mathbb{C}}$, we write $\rho_{{\mathrm{alg}}} : G_{K, \mathfrak{p}} \to {\mathrm{Aut}}(\widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}, \infty)^{(p')})$ for this action. We denote the actions on prime-to-$p$ quotients of étale fundamental groups induced by $\rho_{{\mathrm{top}}}$ and $\rho_{{\mathrm{alg}}}$ by $\rho_{{\mathrm{top}}}^{(p')}$ and $\rho_{{\mathrm{alg}}}^{(p')}$ respectively. For the statement of Theorem \[thm comparison punctured projective line\](a) below, we require the terminology of Dehn twists. Let $\gamma : [0, 1] \to M$ be a simple loop on any complex manifold $M$; we will often identify $\gamma$ with its image in $M$. We define the *Dehn twist* on $M$ with respect to the loop $\gamma$. It is an element of the mapping class group of $M$ represented by a self-homeomorphism of $M$ which can be visualized in terms of a small tubular neighborhood of $\gamma \subset M$, in the following way: the Dehn twist keeps the outer edge of the tubular neighborhood fixed while twisting the inner edge one full rotation counterclockwise and acts as the identity everywhere else on $M$. Since this Dehn twist depends only on the homology class $[\gamma] \in H_{1}(M, {\mathbb{Z}})$ of any loop $\gamma$, we will denote it by $D_{[\gamma]}$. (See [@farb2011primer Chapter 3] for more details.) The following theorem is a compilation of all the necessary results describing and comparing $\rho_{{\mathrm{top}}}$ and $\rho_{{\mathrm{alg}}}$ that are proven in [@hasson2017prime] (Theorems 1.2 and 2.3 and Remark 3.10 of that paper). \[thm comparison punctured projective line\] In the above situation, we have the following. a\) Let $\mathcal{I}$ be the set of all pairs $(I, n)$ where $I \subseteq \{1, ... , d\}$ is a subset and $n \geq 1$ is an integer such that $x^{n} \mid \tilde{\alpha}_{i} - \tilde{\alpha}_{j} \in {\mathbb{C}}[x]$ for all $i, j \in I$ and such that $I$ is maximal among intervals with this property. If $\varepsilon$ is small enough, there exist pairwise nonintersecting loops $\gamma_{I, n} : [0, 1] \to \mathcal{F}_{z_{0}} \smallsetminus \{\infty\}$ for each $(I, n) \in \mathcal{I}$ such that $\delta \in \pi_{1}(B_{\varepsilon}^{*}, z_{0})$ acts on $\pi_{1}(\mathcal{F}_{z_{0}}, \infty)$ in the same way that the product $\prod_{(I, d) \in \mathcal{I}} D_{[\gamma_{I, n}]}$ of Dehn twists on $\mathcal{F}_{z_{0}} \smallsetminus \{\infty\}$ does. These loops $\gamma_{I, n}$ each have the property of separating the subset $\{\tilde{\alpha}_{i}(z_{0})\}_{i \in I}$ from its complement in $\{\tilde{\alpha}_{j}(z_{0})\}_{j = 1}^{d} \cup \{\infty\}$, and two such loops $\gamma_{I, n}$ and $\gamma_{I', n'}$ are homologous if and only if $I = I'$. b\) The actions $\rho_{{\mathrm{top}}}^{(p')}$ and $\rho_{{\mathrm{alg}}}^{(p')}$ factor through $\pi_{1}(B_{\varepsilon}^{*}, z_{0})^{(p')}$ and $G_{K, \mathfrak{p}}^{{\mathrm{tame}}}$ respectively. c\) We have isomorphisms $\widehat{\pi}_{1}(B_{\varepsilon}^{*}, z_{0})^{(p')} \stackrel{\sim}{\to} G_{K, \mathfrak{p}}^{{\mathrm{tame}}}$ and $\phi : \widehat{\pi}_{1}(\mathcal{F}_{z_{0}}, \infty)^{(p')} \stackrel{\sim}{\to} \widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}, \infty)^{(p')}$ inducing an isomorphism of the actions $\rho_{{\mathrm{top}}}^{(p')}$ and $\rho_{{\mathrm{alg}}}^{(p')}$. Moreover, we can choose the isomorphism $\phi$ so that it takes any element represented by a loop on $\mathcal{F}_{z_{0}}$ whose image in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ separates some singleton $\{\tilde{\alpha}_{i}(z_{0})\}$ from its complement in $\{\tilde{\alpha}_{j}(z_{0})\}_{j = 1}^{d} \cup \{\infty\}$ to an element represented by a loop on ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}$ whose image in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ separates the singleton $\{\alpha_{i}\}$ from its complement in $\{\alpha_{j}\}_{j = 1}^{d} \cup \{\infty\}$. In other words, the prime-to-$p$ monodromy action $\rho_{{\mathrm{top}}}^{(p')}$ can be described in terms of Dehn twists with respect to loops surrounding certain subsets of the removed points which coalesce at a certain rate as one approaches the center of $B_{\varepsilon}^{*}$; moreover, this is isomorphic to the algebraic action $\rho_{{\mathrm{alg}}}^{(p')}$ via an isomorphism of prime-to-$p$ étale fundamental groups which takes the image of a loop wrapping around a given $a_{i}(z_{0})$ to the image of a loop wrapping around $\alpha_{i}$. We now want to relate this to the action of $G_{K, \mathfrak{p}}$ on the prime-to-$p$ étale fundamental group of a smooth hyperelliptic curve over $K_{\mathfrak{p}}$. Let $\mathfrak{C}$ be a smooth, projective hyperelliptic curve over ${\mathbb{C}}$ of degree $d$ and genus $g$ defined by an equation of the form $y^{2} = \prod_{i = 1}^{d} (x - z_{i})$ for distinct roots $z_{i} \in \mathcal{K}_{\mathfrak{p}}$; if $d$ is odd (resp. even), then $d = 2g + 1$ (resp. $d = 2g + 2$). The hyperelliptic curve $\mathfrak{C}$ comes with a surjective degree-$2$ morphism $\mathfrak{C} \to {\mathbb{P}}_{{\mathbb{C}}}^{1}$ defined by projecting onto the $x$-coordinate. It is well known that this projection ramifies at $\infty$ if and only if $d = 2g + 1$; in this case, we write $z_{2g + 2} = \infty$. Then the projection is ramified at exactly the $(2g + 2)$-element set of $z_{i}$’s. Write $\mathfrak{B} \subset \mathfrak{C}({\mathbb{C}})$ for the set of inverse images of these ramification points in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ ($\mathfrak{B}$ is the set of *branch points* of $\mathfrak{C}$). Clearly the restriction to $\mathfrak{C} \smallsetminus \mathfrak{B}$ of the above projection map yields a finite degree-$2$ étale morphism $\mathfrak{C} \smallsetminus \mathfrak{B} \to {\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}$. Choose a basepoint $P$ of the topological space ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{d}\}$ and a basepoint $Q$ of the topological space $\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}$ such that $Q$ lies in the inverse image of $P$. Then after making identifications via Riemann’s Existence Theorem, we get an inclusion and surjections $$\label{eq maps of fundamental groups} \widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}, P) \rhd \widehat{\pi}_{1}(\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}, Q) \twoheadrightarrow \widehat{\pi}_{1}(\mathfrak{C}({\mathbb{C}}), Q) \twoheadrightarrow H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$$ induced by the maps ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\} \leftarrow \mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B} \hookrightarrow \mathfrak{C}({\mathbb{C}})$ and by identifying the first singular homology group of $\mathfrak{C}({\mathbb{C}})$ with the abelianization of its fundamental group. The inclusion in (\[eq maps of fundamental groups\]) is an inclusion of a characteristic subgroup, so any automorphism of $\widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}, P)$ induces an automorphism of $\widehat{\pi}_{1}(\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}, P)$. We write $\mathfrak{J}$ for the Jacobian of $\mathfrak{C}$. There is a well-known identification of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}) \otimes {\mathbb{Z}}_{\ell}$ with $H_{1}(\mathfrak{J}({\mathbb{C}}), {\mathbb{Z}}) \otimes {\mathbb{Z}}_{\ell}$ and in turn with $T_{\ell}(\mathfrak{J})$ for any prime $\ell$. Moreover, the intersection pairing on $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ defined above carries over to the canonical Riemann form on the complex abelian variety $\mathfrak{J}({\mathbb{C}})$ and in turn to the Weil pairing $e_{\ell}$ on $T_{\ell}(\mathfrak{J})$ (see the results in [@lang2012introduction §IV.4 and §VIII.1] and in [@mumford1974abelian §24]). Given an element $c \in H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$, we also write $c$ for the element $c \otimes 1 \in H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}) \otimes {\mathbb{Z}}_{\ell} = T_{\ell}(\mathfrak{J})$. It is not difficult to show that the action of $G_{K, \mathfrak{p}}$ on $T_{\ell}(\mathfrak{J})$ induced by $\rho_{{\mathrm{alg}}}^{(p')}$ via these identifications is the natural $\ell$-adic Galois action $\rho_{\ell}$: see, for instance, step 5 of the proof of [@yelton2015images Proposition 2.2]. \[dfn symplectic basis\] Given any complex hyperelliptic curve $\mathfrak{C}$ as above, we define the following objects. a\) For any $c \in H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$, the *transvection* with respect to $c$, denoted $T_{c} \in {\mathrm{Aut}}(H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}))$, is the automorphism given by $v \mapsto v + \langle v, c \rangle c$. As above, we may identify $c$ with its image in $T_{\ell}(\mathfrak{J})$, and then $T_{c}$ is identified with the automorphism of $T_{\ell}(\mathfrak{J})$ given by $v \mapsto e_{\ell}(v, c) c$. b\) A *sympectic basis* of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ is an ordered basis $\{a_{1}', ... , a_{g}', b_{1}', ... , b_{g}'\}$ satisfying the following properties:    (i) each $a_{i}'$ (resp. each $b_{i}'$) is represented by a loop on $\mathfrak{C}({\mathbb{C}})$ whose image in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ separates $\{z_{2i - 1}, z_{2i}\}$ (resp. $\{z_{2i}, ... , z_{2g + 1}\}$) from its complement in $\{z_{j}\}_{j = 1}^{2g + 2}$; and    (ii) the (skew-symmetric) intersection pairing $\langle \cdot, \cdot \rangle$ on $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ is determined by $\langle a_{i}', b_{i}' \rangle = -1$ for $1 \leq i \leq g$ and $\langle a_{i}', a_{j}' \rangle = \langle b_{i}', b_{j}' \rangle = \langle a_{i}', b_{j}' \rangle = 0$ for $1 \leq i < j \leq g$. (See, for instance, the first figure in [@arnold1968remark], or [@farb2011primer Figure 6.1].) We note that a transvection in ${\mathrm{Aut}}(H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}))$ respects the intersection pairing $\langle \cdot, \cdot \rangle$; similarly, a transvection in ${\mathrm{Aut}}(T_{\ell}(\mathfrak{J}))$ respects the Weil pairing $e_{\ell}$ and thus lies in ${\mathrm{Sp}}(T_{\ell}(\mathfrak{J}))$. From now on, given a complex hyperelliptic curve $\mathfrak{C}$, we fix a symplectic basis $\{a_{1}', ... , a_{g}', b_{1}', ... , b_{g}'\}$ of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$. In order to prove Proposition \[prop Galois action local\] below, we require a couple of lemmas, the first of which is purely topological. \[lemma lifts to Dehn twists\] Assume all of the above notation. a\) Let $\gamma \subset {\mathbb{P}}_{{\mathbb{C}}}^{1}$ be the image of a simple closed loop on ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{2g + 2}, P\}$ which separates the set of $\alpha_{i}$’s into two even-cardinality subsets. Then the inverse image of $\gamma$ under the ramified degree-$2$ covering map $\mathfrak{C}({\mathbb{C}}) \to {\mathbb{P}}_{{\mathbb{C}}}$ consists of two connected components, each of which are simple closed loops whose homology classes $\pm c \in H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ differ by sign. As a particular case, if $\gamma_{i} \subset {\mathbb{P}}_{{\mathbb{C}}}^{1}$ is the image of a loop which separates $\{z_{i}, z_{2g + 1}\}$ from its complement in $\{z_{j}\}_{j = 1}^{2g + 2}$ for some $i$, then the homology classes of these simple closed loops on $\mathfrak{C}({\mathbb{C}})$ lying above $\gamma_{i}$ are given by $\pm c_{i}'$ for some element $c_{i}' \in H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ which is equivalent modulo $2$ to $$\label {eq c_i} \begin{cases} a_{(i + 1)/2}' + ... + a_{g}' + b_{(i + 1)/2}' & i \ \mathrm{odd}\\ a_{i/2 + 1}' + ... + a_{g}' + b_{i/2 + 1}' & i \ \mathrm{even} \end{cases}$$ b\) With $\gamma$ and $c$ as above, the Dehn twist $D_{[\gamma]}$ induces an automorphism of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ via the inclusion and quotient maps in (\[eq maps of fundamental groups\]), which is given by $T_{c}^{2} \in {\mathrm{Aut}}(H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}))$. The essential ideas of this argument are contained in the proof of [@mumford1984tata Lemma 8.12]. The maximal abelian exponent-$2$ quotient of $\pi_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}, P)$ is isomorphic to $({\mathbb{Z}}/ 2{\mathbb{Z}})^{2g + 1}$ and is identified with the group of partitions of $\{z_{i}\}_{i = 1}^{2g + 2}$ into two subsets (where the addition law is given by symmetric differences), by sending the homology class of any loop $\gamma \subset {\mathbb{P}}_{{\mathbb{C}}}^{1}$ to the subsets of $z_{i}$’s lying in each connected component of ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \gamma$. We have a homomorphism of homology groups (composed with reduction modulo $2$) $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}) \to H_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}, {\mathbb{Z}}/ 2{\mathbb{Z}})$ coming from the inclusion of fundamental groups in (\[eq maps of fundamental groups\]). By [@arnold1968remark Lemma 1], this homomorphism factors through $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}/ 2{\mathbb{Z}})$. It is clear from this that we have an inclusion of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}/ 2{\mathbb{Z}})$ as the subgroup of $H_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}, {\mathbb{Z}}/ 2{\mathbb{Z}})$ which is generated by the partitions $\{z_{2i - 1}, z_{2i}\} \cup \{z_{1}, ... , z_{2i - 2}, z_{2i + 1}, ... , z_{2g + 2}\}$ and $\{z_{2i}, ... , z_{2g + 1}\} \cup \{z_{1}, ... , z_{2i - 1}, z_{2g + 2}\}$ for $1 \leq i \leq g$. It follows from an easy combinatorial argument that this subgroup consists of the partitions of $\{z_{i}\}_{i = 1}^{2g + 2}$ into even-cardinality subsets. Thus, any loop $\gamma \in \pi_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}, P)$ whose image modulo $2$ is such a partition lifts to a loop $c \in \pi_{1}(\mathfrak{C}({\mathbb{C}}), Q)$; the only other choice of loop on $\mathfrak{C}({\mathbb{C}})$ whose image is $\gamma$ must be based at $\iota(Q)$ and equal to the composition of the path $\gamma$ with $\iota$, where $\iota : \mathfrak{C}({\mathbb{C}}) \to \mathfrak{C}({\mathbb{C}})$ is the only nontrivial deck transformation. Since $\iota$ acts on the homology group by sign change (see [@farb2011primer §7.4]), this loop must be $-c$. If the image of $\gamma$ modulo $2$ is the partition of $\{z_{i}, z_{2g + 1}\}$ and its complement, then the desired statement in (a) follows by the straightforward verification that this partition is equal to the symmetric sum of the partitions corresponding to the basis elements $a_{i}'$ and $b_{i}'$ appearing in the formulas given in (\[eq c\_i\]). Thus, (a) is proved. Every self-homeomorphism of ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}$ fixing $P$ lifts uniquely to a self-homeomorphism of $\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}$ fixing $Q$ under which the image of a loop wrapping around a single point in $\mathfrak{B}$ is also a loop wrapping around a single point in $\mathfrak{B}$. Since the kernel of the quotient map $\pi_{1}(\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}, Q) \twoheadrightarrow \pi_{1}(\mathfrak{C}({\mathbb{C}}), Q)$ is generated by squares of homotopy classes of such loops, it follows that $D_{[\gamma]}$ induces an automorphism of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ via the maps in (\[eq maps of fundamental groups\]). It is clear that the unique self-homeomorphism of $\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}$ fixing $Q$ lifting a representative of $D_{[\gamma]}$ must represent the composition of Dehn twists on $\mathfrak{C}({\mathbb{C}}) \smallsetminus (\mathfrak{B} \cup \{Q\})$ with respect to the two lifts of $\gamma$ on $\mathfrak{C}({\mathbb{C}}) \smallsetminus (\mathfrak{B} \cup \{Q\})$, which are $\pm c$. Therefore, the induced automorphism of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}) = H_{1}(\mathfrak{C}({\mathbb{C}}) \smallsetminus \{Q\}, {\mathbb{Z}})$ is determined by the product of $D_{c}$ and $D_{-c}$. Since Dehn twists do not depend on the orientation of loops, this product is $D_{c}^{2}$. Now (b) follows from the well known fact (see [@farb2011primer Proposition 6.3]) that the Dehn twist $D_{c}$ acts on $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ as the transvection $T_{c}$ since $\mathfrak{C}({\mathbb{C}})$ is a compact smooth manifold. \[lemma branch point infty\] Let $\alpha_{1}, ... , \alpha_{d}$ be distinct elements in $\mathcal{R}_{\mathfrak{p}}$. There exist elements $\alpha_{1}', ... , \alpha_{d + 1}' \in \mathcal{R}_{\mathfrak{p}}$ satisfying the following: \(i) The elements $\alpha_{i}' - \alpha_{j}'$ and $\alpha_{i} - \alpha_{j}$ have equal valuation for $1 \leq i < j \leq d$, and $\alpha_{d + 1}' - \alpha_{i}' \in \mathcal{R}_{\mathfrak{p}}^{\times}$ for $1 \leq i \leq d$. \(ii) Let $\alpha_{d + 1} = \infty \in {\mathbb{P}}_{\mathcal{K}_{\mathfrak{p}}}^{1}$. There is a $\mathcal{K}_{\mathfrak{p}}$-isomorphism $$\psi : {\mathbb{P}}_{\bar{\mathcal{K}}_{\mathfrak{p}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d + 1}\} \stackrel{\sim}{\to} {\mathbb{P}}_{\bar{\mathcal{K}}_{\mathfrak{p}}}^{1} \smallsetminus \{\alpha_{1}', ... , \alpha_{d + 1}'\}$$ such that the induced isomorphism $\psi_{*} : \widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d + 1}\}, \psi^{-1}(\infty)) \stackrel{\sim}{\to} \widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}', ... , \alpha_{d + 1}'\}, \infty)$ yields an isomorphism of $\rho_{{\mathrm{alg}}}$ with the analogously defined representation $\rho_{{\mathrm{alg}}}'$. The isomorphism $\psi_{*}$ takes any element represented by a loop on ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d + 1}\}$ whose image in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ separates some singleton $\{\alpha_{i}\}$ from its complement in $\{\alpha_{j}\}_{j = 1}^{d + 1}$ to an element represented by a loop on ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}', ... , \alpha_{d + 1}'\}$ whose image in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ separates the singleton $\{\alpha_{i}'\}$ from its complement in $\{\alpha_{j}'\}_{j = 1}^{d + 1}$. Choose $\beta \in \mathcal{R}_{\mathfrak{p}}^{\times}$ satisfying $\beta \not\equiv \alpha_{j}$ (mod $\pi$) for $1 \leq j \leq d$ (this is always possible because the residue field $\mathcal{R}_{\mathfrak{p}} / (\pi)$ is infinite). Let $\alpha_{j}' = \alpha _{j}\beta / (\beta - \alpha_{j}) \in \mathcal{R}_{\mathfrak{p}}$ for $1 \leq j \leq d$ and $\alpha_{d + 1}' = \beta \in \mathcal{R}_{\mathfrak{p}}$, and let $\psi$ be the $\mathcal{K}_{\mathfrak{p}}$-morphism given by $x \mapsto x\beta / (\beta - x)$. Then property (i) follows from straightforward computation. Since $\psi$ is defined over $\mathcal{K}_{\mathfrak{p}}$, the isomorphism $\psi_{*}$ is equivariant with respect to the action of $G_{K, \mathfrak{p}} = {\mathrm{Gal}}(\bar{\mathcal{K}}_{\mathfrak{p}} / \mathcal{K}_{\mathfrak{p}})$. Moreover, after base change to ${\mathbb{C}}$, $\psi$ is a homeomorphism of punctured Riemann spheres, and the property given in (ii) follows. We are finally ready to state and prove the main result of this section, which is essentially a more concrete version of a particular case of Grothendieck’s criterion for semistable reduction ([@grothendieck1972modeles Proposition 3.5(iv)]). For the statement of the below proposition, we fix a symplectic basis $\{a_{1}, ... , a_{g}, b_{1}, ... , b_{g}\}$ of $H_{1}(C({\mathbb{C}}), {\mathbb{Z}})$; the image $\{a_{1}, ... , a_{g}, b_{1}, ... , b_{g}\} \subset T_{\ell}(J)$ forms a symplectic basis of $T_{\ell}(J)$ with respect to the Weil pairing. \[prop Galois action local\] Let $C$ be a hyperelliptic curve of genus $g$ over $\mathcal{K}_{\mathfrak{p}}$ given by an equation of the form $y^{2} = h(x)$ for some squarefree polynomial $h$ of degree $d = 2g + 1$ or $d = 2g + 2$ with distinct roots $\alpha_{1}, ... , \alpha_{d} \in \mathcal{R}_{\mathfrak{p}}$, and let $J$ be its Jacobian. Suppose that exactly $2$ of the roots, $\alpha_{i}$ and $\alpha_{j}$, are equivalent modulo $\pi$, and let $m \geq 1$ be the maximal integer such that $\pi^{m} \mid (\alpha_{i} - \alpha_{j})$. Then for any prime $\ell \neq 2, p$, the image of the natural action of $G_{K, \mathfrak{p}}$ on $T_{\ell}(J)$ is topologically generated by the element $T_{c}^{2m} \in {\mathrm{Sp}}(T_{\ell}(J))$ for some $c \in T_{\ell}(J)$ determined by a loop on ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ whose image separates $\{\alpha_{i}, \alpha_{j}\}$ from the rest of the roots. As a particular case, if $i \leq 2g$ and $j = 2g + 1$, then this $\ell$-adic Galois image is topologically generated by $T_{c_{i}}^{2m}$ for some $c_{i} \in H_{1}(C({\mathbb{C}}), {\mathbb{Z}})$ equivalent modulo $2$ to $a_{(i + 1)/2} + ... + a_{g} + b_{(i + 1)/2}$ (resp. $a_{i/2 + 1} + ... + a_{g} + b_{i/2 + 1}$) if $i$ is odd (resp. even). We first assume that $d = 2g + 2$. Let $\tilde{\alpha}_{1}, ... , \tilde{\alpha}_{2g + 2} \in {\mathbb{C}}[x]$ be the elements constructed from the $\alpha_{i}$’s as above, and define the family $\mathcal{F} \to B_{\varepsilon}^{*}$ as above. It is clear from the hypothesis on the roots $\alpha_{j}$ that the set $\mathcal{I}$ in the statement of Theorem \[thm comparison punctured projective line\] consists of only the elements $(\{i, 2g + 1\}, n)$ for $1 \leq n \leq m$. Theorem \[thm comparison punctured projective line\](a) then implies that a topological generator of $\widehat{\pi}_{1}(B_{\varepsilon}^{*}, z_{0})$ acts on $\widehat{\pi}_{1}(\mathcal{F}_{z_{0}}, \infty)$ via the monodromy action $\rho_{{\mathrm{top}}}$ as $D_{[\gamma_{i}]}^{m}$, where $\gamma_{i}$ is a loop on $\mathcal{F}_{z_{0}} \smallsetminus \{\infty\}$ whose image in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ separates the subset $\{\tilde{\alpha}_{i}(z_{0}), \tilde{\alpha}_{2g + 1}(z_{0})\}$ from its complement in $\{\tilde{\alpha}_{j}(z_{0})\}_{j = 1}^{2g + 2}$. Let $\mathfrak{C}$ be the complex hyperelliptic curve of degree $d = 2g + 2$ ramified over ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ at the points $\tilde{\alpha}_{1}(z_{0}), ... , \tilde{\alpha}_{d}(z_{0}) \in {\mathbb{C}}$, which has a symplectic basis $\{a_{1}', ... , a_{g}', b_{1}', ... , b_{g}'\}$ of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$. By Lemma \[lemma lifts to Dehn twists\], the automorphism of $\widehat{\pi}_{1}(\mathcal{F}_{z_{0}}, \infty)$ determined by $D_{[\gamma_{i}]}^{m}$ induces the automorphism of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}$ given by $T_{c_{i}}^{2m} : v \mapsto v + 2m \langle v, c_{i}' \rangle c_{i}'$, where $c_{i}' \in H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ is equivalent modulo $2$ to the formula in (\[eq c\_i\]). Now parts (b) and (c) of Theorem \[thm comparison punctured projective line\] say that there is an isomorphism $\phi : \widehat{\pi}_{1}(\mathcal{F}_{z_{0}}, \infty)^{(p')} \stackrel{\sim}{\to} \widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}, \infty)^{(p')}$ making the action $\rho_{{\mathrm{top}}}$ isomorphic to the Galois action $\rho_{{\mathrm{alg}}}$. Since $p \neq 2$ and $\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}$ and $C({\mathbb{C}}) \smallsetminus \{(\alpha_{i}, 0)\}_{i = 1}^{2g + 2}$ are the only degree-$2$ covers of $\mathcal{F}_{z_{0}}$ and ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{2g + 2}\}$ respectively, we see that the isomorphism $\phi$ induces an isomorphism $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}^{(p')} \stackrel{\sim}{\to} H_{1}(C({\mathbb{C}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}^{(p')}$ which we also denote by $\phi$. It is clear from the property of $\phi$ given in Theorem \[thm comparison punctured projective line\](c) and from our characterization of homology groups with coefficients in ${\mathbb{Z}}/ 2{\mathbb{Z}}$ in the proof of Lemma \[lemma lifts to Dehn twists\] that $\phi(a_{j}') \equiv a_{j}$ and $\phi(b_{j}') \equiv b_{j}$ (mod $2$) for $1 \leq j \leq g$. In the case that $d = 2g + 1$, we get the same results by applying Lemma \[lemma branch point infty\], which allows us to replace $\alpha_{1}, ... , \alpha_{2g + 1}, \alpha_{2g + 2} := \infty$ with elements $\alpha_{1}', ... , \alpha_{2g + 2}' \in \mathcal{R}_{\mathfrak{p}}$ whose differences have the same valuations with respect to $\pi$. Putting this all together, we see that the tame Galois action on $H_{1}(C({\mathbb{C}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}^{(p')}$ induced by $\rho_{{\mathrm{alg}}}$ sends a generator of $G_{K, \mathfrak{p}}^{{\mathrm{tame}}}$ to the automorphism of $H_{1}(C({\mathbb{C}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}^{(p')}$ given by $v \mapsto v + 2m \langle v, c_{i} \rangle_{\phi} c_{i}$ for some $c_{i} \in H_{1}(C({\mathbb{C}}), \infty) \otimes \widehat{{\mathbb{Z}}}^{(p')}$ which is equivalent modulo $2$ to the formula given in the statement. Here $\langle \cdot, \cdot \rangle_{\phi}$ is the skew-symmetric pairing on $H_{1}(C({\mathbb{Z}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}^{(p')}$ induced by the interection pairing on $H_{1}(\mathfrak{C}, {\mathbb{Z}})$ via $\phi$; note that $\langle \cdot, \cdot \rangle_{\phi}$ is normalized so that $\langle H_{1}(C({\mathbb{C}}), {\mathbb{Z}}), H_{1}(C({\mathbb{C}}), {\mathbb{Z}}) \rangle_{\phi} = {\mathbb{Z}}$. As above, we identify the maximal pro-$\ell$ quotient of $H_{1}(C({\mathbb{C}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}^{(p')}$ with $T_{\ell}(J)$ and see that the natural $\ell$-adic Galois action $\rho_{\ell}$ factors through $G_{K, \mathfrak{p}}^{{\mathrm{tame}}}$ and takes a generator to the automorphism of $T_{\ell}(J)$ given by $v \mapsto v + 2m\langle v, c_{i} \rangle_{\phi} c_{i}$. But this automorphism must lie in ${\mathrm{Sp}}(T_{\ell}(J))$ by the Galois equivarience of the Weil pairing $e_{\ell}$ and the fact that $\mathcal{K}_{\mathfrak{p}}$ contains all $\ell$-power roots of unity. It is now an easy exercise to verify that this implies that $\langle v, c_{i} \rangle_{\phi} = \pm e_{\ell}(v, c_{i})$ for all $v \in T_{\ell}(J)$, and so the image of $\rho_{\ell}$ is generated by $T_{c_{i}}^{2m} : v \mapsto v + e_{\ell}(v, c_{i}) c_{i}$, as desired. In order to prove Theorems \[thm main1\] and \[thm main2\], we will put some local data together to show using Proposition \[prop Galois action local\] that the $\ell$-adic Galois image contains $\{T_{c_{1}}^{2m}, ... , T_{c_{2g}}^{2m}\}$ for homology classes $c_{i}$ as above and a certain integer $m$. Therefore, it is of interest to investigate the subgroup that this set generates. Unfortunately, since the $c_{i}$’s are only known modulo $2$, not much can be deduced except if $\ell = 2$. In this case, the subgroup of ${\mathrm{Sp}}(T_{2}(J))$ generated by the above set can be determined, which is the goal of the next section. Subgroups generated by powers of transvections {#S3} ============================================== For this section, we let $M$ be a free ${\mathbb{Z}}$-module of rank $2g$, equipped with a nondegenerate skew-symmetric ${\mathbb{Z}}$-bilinear pairing $\langle \cdot, \cdot \rangle : M \times M \to {\mathbb{Z}}$. For any ring $A$, this pairing induces in an obvious way a nondegenerate skew-symmetric $A$-bilinear pairing on the free $A$-module $M \otimes A$ which we also denote by $\langle \cdot, \cdot \rangle$. We write ${\mathrm{Sp}}(M \otimes A)$ for the symplectic group of $A$-automorphisms of $M \otimes A$ which respect this pairing; for any integer $N \geq 1$, we write $\Gamma(N)$ for the level-$N$ congruence subgroup of ${\mathrm{Sp}}(M \otimes A)$ as defined in §\[S1\]. For any element $c \in M \otimes A$, we write $T_{c} \in {\mathrm{Sp}}(M \otimes A)$ for the transvection with respect to $c$ and the pairing $\langle \cdot, \cdot \rangle$, as in Definition \[dfn symplectic basis\](a). We fix a basis $\{a_{1}, ... , a_{g}, b_{1}, ... , b_{g}\}$ of $M$ and assume that the pairing is determined by $\langle a_{i}, b_{i} \rangle = -1$ for $1 \leq i \leq g$ and $\langle a_{i}, a_{j} \rangle = \langle b_{i}, b_{j} \rangle = \langle a_{i}, b_{j} \rangle = 0$ for $1 \leq i < j \leq g$. We also write $a_{i}, b_{i} \in M \otimes A$ for the elements $a_{i} \otimes 1$ and $b_{i} \otimes 1$ respectively for $1 \leq i \leq g$. Our main goal is to prove the following purely algebraic result. \[prop open subgroup\] Let $c_{1}, ... , c_{2g} \in M$ be elements such that $c_{i}$ is equivalent modulo $2$ to $$\begin{cases} a_{(i + 1)/2} + ... + a_{g} + b_{(i + 1)/2} & i \ \mathrm{odd}\\ a_{i/2 + 1} + ... + a_{g} + b_{i/2 + 1} & i \ \mathrm{even} \end{cases}$$ for $1 \leq i \leq 2g$. Then given integers $n, n' \geq 1$, the subgroup $G \subset \Gamma(2^{n})$ generated by the elements $T_{c_{1}}^{2^{n}}, ... , T_{c_{2g - 1}}^{2^{n}}, T_{c_{2g}}^{2^{n'}} \in \Gamma(2^{n})$ contains $\Gamma(2^{2n})$ (resp. $\Gamma(2^{n + n'})$) if $n' \leq n$ (resp. if $n' > n$). In particular, as a topological subgroup of ${\mathrm{Sp}}(M \otimes {\mathbb{Z}}_{2})$, $G$ is open, and its associated Lie algebra $\mathfrak{g}$ coincides with the $2$-adic symplectic Lie algebra $\mathfrak{sp}(M \otimes {\mathbb{Q}}_{2})$. Note that the image of any transvection $T \in {\mathrm{Sp}}(M \otimes {\mathbb{Z}}_{2})$ under the logarithm map is $T - 1 \in \mathfrak{sp}(M \otimes {\mathbb{Q}}_{2})$ and that $\mathfrak{g}$ is generated as a Lie algebra by the logarithms of the transvections $T_{c_{i}}$. Therefore, in order to prove the second statement of this proposition, it suffices to show that $\{T_{c_{s}} - 1\}_{1 \leq s \leq 2g}$ generates the full Lie algebra $\mathfrak{sp}(M \otimes {\mathbb{Q}}_{2})$. However, in order to get both statements, we will prove something slightly stronger which is given by the following lemma. \[lemma transvection commutator basis\] Let $t_{i} = T_{c_{i}} - 1 \in \mathfrak{sp}(M \otimes {\mathbb{Q}}_{2})$ and let $\bar{t}_{i}$ denote the reduction of $t_{i}$ modulo $2$ for $1 \leq i \leq 2g$. Then an ${\mathbb{F}}_{2}$-basis for $\mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$ is given by the set $\{\bar{t}_{i}\}_{1 \leq i \leq 2g} \cup \{[\bar{t}_{i}, \bar{t}_{j}]\}_{1 \leq i < j \leq 2g}$. Let $\bar{c}_{i}$ denote the reductions modulo $2$ of the basis element $c_{i}$ for $1 \leq i \leq 2g$. We know from the characterization of the $c_{i}$’s modulo $2$ given in the statement that $\{\bar{c}_{1}, ... , \bar{c}_{2g}\}$ is an ordered basis of $M \otimes {\mathbb{F}}_{2}$ with $\langle \bar{c}_{i}, \bar{c}_{j} \rangle = \langle \bar{c}_{j}, \bar{c}_{i} \rangle = 1 \in {\mathbb{F}}_{2}$ for $1 \leq i < j \leq 2g$. With respect to this basis, we may view $\mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$ as the Lie algebra $\mathfrak{sp}_{2g}({\mathbb{F}}_{2})$. Now it is well known that $\mathfrak{sp}_{2g}({\mathbb{F}}_{2})$ is a $(2g^{2} + g)$-dimensional vector space over ${\mathbb{F}}_{2}$. Since there are $2g^{2} + g$ elements listed in the set of $\bar{t}_{s}$’s and their commutators, it suffices to prove that this set is linearly independent over ${\mathbb{F}}_{2}$, using only the fact that $\langle \bar{c}_{i}, \bar{c}_{j} \rangle = \langle \bar{c}_{j}, \bar{c}_{i} \rangle = 1$ for $1 \leq i < j \leq 2g$. In order to do this, we first note that for any elements $a, b \in M \otimes {\mathbb{F}}_{2}$, the commutator of the logarithms of the transvections with respect to $a$ and to $b$ is the ${\mathbb{F}}_{2}$-linear operator in $\mathfrak{sp}_{2g}({\mathbb{F}}_{2})$ given by $v \mapsto \langle v, a \rangle \langle a, b \rangle b + \langle v, b \rangle \langle b, a \rangle a$ for all $v \in M \otimes {\mathbb{F}}_{2}$. Therefore, for $1 \leq i < j \leq 2g$, the linear operator $[\bar{t}_{i}, \bar{t}_{j}] = [\bar{t}_{j}, \bar{t}_{i}]$ is given by $$\label{eq commutator} v \mapsto \langle v, \bar{c}_{i} \rangle \bar{c}_{j} + \langle v, \bar{c}_{j} \rangle \bar{c}_{i}.$$ We compute using this formula that the upper left-hand $2 \times 2$ submatrices of $\bar{t}_{1}$, $\bar{t}_{2}$, and $[\bar{t}_{1}, \bar{t}_{2}]$ respectively are $$\label{eq base case} \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix},$$ and therefore these elements of $\mathfrak{sp}_{2g}({\mathbb{F}}_{2})$ are linearly independent over ${\mathbb{F}}_{2}$. This in particular proves the statement for $g = 1$. Now assume inductively that $g \geq 2$ and that the statement is true for $g - 1$. Clearly, $\{\bar{c}_{3}, ... , \bar{c}_{2g}\}$ generates a $2(g - 1)$-dimensional subspace $\mathfrak{g}'$ of $M \otimes {\mathbb{F}}_{2}$ which it generates, and the intersection pairing of any two distinct elements of this set is $1$. Therefore, the inductive assumption implies that the $(2(g - 1)^{2} + (g - 1))$-element set $\{\bar{t}_{i}\}_{3 \leq i \leq g} \cup \{[\bar{t}_{i}, \bar{t}_{j}]\}_{3 \leq i < j \leq 2g}$ is linearly independent; in fact, $\mathfrak{g}'$ is a copy of $\mathfrak{sp}_{2g - 2}({\mathbb{F}}_{2})$ lying inside $\mathfrak{sp}_{2g}({\mathbb{F}}_{2})$. We first claim that the subset $S_{1} := \{\bar{t}_{i}\}_{1 \leq i \leq 2g} \cup \{[\bar{t}_{1}, \bar{t}_{2}]\} \cup \{[\bar{t}_{i}, \bar{t}_{j}]\}_{3 \leq i < j \leq 2g}$ is linearly independent. To see this, note that we have already shown that $\{\bar{t}_{1}, \bar{t}_{2}, [\bar{t}_{1}, \bar{t}_{2}]\}$ is linearly independent, and from the fact that the matrices in $\mathfrak{g}'$ have all $0$’s in their first and second rows, it is clear that $\bar{t}_{1}, \bar{t}_{2}, [\bar{t}_{1}, \bar{t}_{2}] \notin \mathfrak{g}'$. The elements of $S_{1}$ generating $\mathfrak{g}'$ are linearly independent by the inductive assumption, and so the claim follows. We next claim that the subset $S_{2} := \{[\bar{t}_{1}, \bar{t}_{i}]\}_{3 \leq i \leq 2g} \cup \{[\bar{t}_{2}, \bar{t}_{i}]\}_{3 \leq i \leq 2g}$ is linearly independent. In order to show this, consider a linear combination of elements of the set $S_{2}$ written as $\sum_{i = 3}^{2g} \beta_{i} [\bar{t}_{1}, \bar{t}_{i}] + \sum_{i = 3}^{2g} \gamma_{i} [\bar{t}_{2}, \bar{t}_{i}]$ with $\beta_{i}, \gamma_{i} \in {\mathbb{F}}_{2}$. Using the formula in (\[eq commutator\]), we see that this is the linear operator $u \in \mathfrak{sp}_{2g}({\mathbb{F}}_{2})$ given by $$u : v \mapsto \sum_{i = 3}^{2g} \beta_{i} (\langle v, \bar{c}_{i} \rangle \bar{c}_{1} + \langle v, \bar{c}_{1} \rangle \bar{c}_{i}) + \sum_{i = 3}^{2g} \gamma_{i} (\langle v, \bar{c}_{i} \rangle \bar{c}_{2} + \langle v, \bar{c}_{2} \rangle \bar{c}_{i})$$ $$\label{eq commutator2} = \langle v, \sum_{i = 3}^{2g} \beta_{i} \bar{c}_{i} \rangle \bar{c}_{1} + \langle v, \bar{c}_{1} \rangle \sum_{i = 3}^{2g} \beta_{i} \bar{c}_{i} + \langle v, \sum_{i = 3}^{2g} \gamma_{i} \bar{c}_{i} \rangle \bar{c}_{2} + \langle v, \bar{c}_{2} \rangle \sum_{i = 3}^{2g} \gamma_{i} \bar{c}_{i}.$$ Suppose that $u = 0$. Assume that $\sum_{i = 3}^{2g} \beta_{i} \bar{c}_{i} \neq 0$. Then by the nondegeneracy of the symplectic pairing, we can choose $v \in M \otimes {\mathbb{F}}_{2}$ such that $\langle v, \sum_{i = 3}^{2g} \beta_{i} \bar{c}_{i} \rangle = 1$. Then $u(v)$ written as a linear combination using the basis $\{\bar{c}_{1}, ... , \bar{c}_{2g}\}$ has $\bar{c}_{1}$-coefficient equal to $1$, a contradiction because $u(v) = 0$. Therefore, $\sum_{i = 3}^{2g} \beta_{i} \bar{c}_{i} = 0$, which implies that $\beta_{3} = ... = \beta_{2g} = 0$. Now assume that $\sum_{i = 3}^{2g} \gamma_{i} \bar{c}_{i} \neq 0$. Then similarly, we can choose $w \in M \otimes {\mathbb{F}}_{2}$ such that $\langle w, \sum_{i = 3}^{2g} \gamma_{i} \bar{c}_{i} \rangle = 1$ and get that $u(w)$ has $\bar{c}_{2}$-coefficient equal to $1$, a contradiction because $u(w) = 0$. Therefore, $\sum_{i = 3}^{2g} \gamma_{i} \bar{c}_{i} = 0$, which implies that $\gamma_{3} = ... = \gamma_{2g} = 0$, and so $S_{2}$ is linearly independent. We finally claim that if a linear combination of elements in $S_{1}$ is equal to a linear combination of elements in $S_{2}$, then these linear combinations must be trivial. Since the full set of $t_{i}$’s and their commutators coincides with $S_{1} \cup S_{2}$, this implies the statement of the proposition. Let $u \in \mathfrak{sp}_{2g}({\mathbb{F}}_{2})$ be a linear combination of elements in $S_{2}$, written as in (\[eq commutator2\]) with $\beta_{i}, \gamma_{i} \in {\mathbb{F}}_{2}$. Now it is clear from the formula there that for each of the matrices in $S_{2}$, the $(1, j)$th entries are all equal and the $(2, j)$th entries are all equal for $3 \leq j \leq 2g$. Then we see by putting $\bar{c}_{3}, ... , \bar{c}_{2g}$ into this formula that $\beta_{3} = ... = \beta_{2g}$ and $\gamma_{3} = ... = \gamma_{2g}$. But then we have $$\langle \bar{c}_{1}, \sum_{i = 3}^{2g} \beta_{i}\bar{c}_{i} \rangle = \langle \bar{c}_{1}, \sum_{i = 3}^{2g} \gamma_{i}\bar{c}_{i} \rangle = \langle \bar{c}_{2}, \sum_{i = 3}^{2g} \beta_{i}\bar{c}_{i} \rangle = \langle \bar{c}_{2}, \sum_{i = 3}^{2g} \gamma_{i}\bar{c}_{i} \rangle = 0,$$ and we get that the upper left-hand $2 \times 2$ submatrix of $u$ is the $0$ matrix. This is also true of any matrix in $\mathfrak{g}'$, so we know from what was given in (\[eq base case\]) that $t_{1}$, $t_{2}$, and $[t_{1}, t_{2}]$ do not appear when $u$ is written as a linear combination of elements in $S_{1}$, and therefore we have $u \in \mathfrak{g}'$. As was noted above, every matrix in $\mathfrak{g}'$ has all $0$’s in its first and second rows. Thus, for any $v \in M \otimes {\mathbb{F}}_{2}$, when $u(v)$ is written as a linear combination of $\bar{c}_{i}$’s, the $\bar{c}_{1}$-coefficient and the $\bar{c}_{2}$-coefficient are both $0$. Now by the same argument that was used for the previous claim, we have $\beta_{3} = ... = \beta_{2g} = \gamma_{3} = ... = \gamma_{2g} = 0$. Therefore $u = 0$, as desired. Let $N = 2n$ (resp. $N = n + n'$) if $n' \leq n$ (resp. if $n' > n$); that is, $N = n + \max \{n, n'\}$. In order to show that $G$ contains the congruence subgroup $\Gamma(2^{N})$, since $G$ is closed, it suffices to show that the image of $G$ modulo $2^{N + m}$ contains $\Gamma(2^{N}) / \Gamma(2^{N + m})$ for each integer $m \geq 1$. We claim that in fact we only need to show this for $m = 1$. Indeed, for any $m \geq 1$, the restriction of the logarithm map to $\Gamma(2)$ sends each element $T \in \Gamma(2^{N + m - 1})$ to an element of $\mathfrak{sp}_{2g}({\mathbb{Z}}_{2})$ which is equivalent to $T - 1$ modulo $2^{N + m}$ since $(T - 1)^{2} \equiv 0$ modulo $2^{N + m}$. In this way, one verifies that there is an isomorphism from the additive group $\mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$ to $\Gamma(2^{N + m - 1}) / \Gamma(2^{N + m})$, given by sending an element $t \in \mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$ to $1 + 2^{N + m - 1}\tilde{t} \in {\mathrm{Sp}}(M \otimes {\mathbb{Z}}_{2})$, where $\tilde{t}$ is any operator in ${\mathrm{Sp}}(M \otimes {\mathbb{Z}}/ 2^{N + m}{\mathbb{Z}})$ whose image modulo $2$ is $t$. In particular, each $\Gamma(2^{N + m - 1}) / \Gamma(2^{N + m})$ is an elementary abelian group of exponent $2$ and rank $2g^{2} + g$ (this is also known from the proof of [@sato2010abelianization Corollary 2.2]). Moreover, it is easy to check that for each $m \geq 1$, the map sending each matrix in $\Gamma(2^{N})$ to its $2^{m - 1}$th power induces a group isomorphism $\Gamma(2^{N}) / \Gamma(2^{N + 1}) \stackrel{\sim}{\to} \Gamma(2^{N + m - 1}) / \Gamma(2^{N + m})$. It follows that if the image of $G$ modulo $2^{N + 1}$ contains $\Gamma(2^{N}) / \Gamma(2^{N + 1})$, then the image of $G$ modulo $2^{N + m}$ contains $\Gamma(2^{N}) / \Gamma(2^{N + m})$ for all $m \geq 1$. Thus, in order to prove the proposition, it suffices to show that the image of $G$ modulo $2^{N + 1}$ contains $\Gamma(2^{N}) / \Gamma(2^{N + 1})$. As above, let $t_{i}$ denote $T_{c_{i}} - 1$ for $1 \leq i \leq 2g$. Since Lemma \[lemma transvection commutator basis\] says that the image modulo $2$ of $\{t_{i}\}_{1 \leq i \leq 2g} \cup \{[t_{i}, t_{j}]\}_{1 \leq i < j \leq 2g}$ is a basis of $\mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$, it suffices to show that the image of $G$ modulo $2^{N + 1}$ contains the images of the elements in $\{1 + 2^{N}t_{i}\}_{1 \leq i \leq 2g} \cup \{1 + 2^{N}[t_{i}, t_{j}]\}_{1 \leq i < j \leq 2g} \subset \Gamma(2^{2n})$. For ease of notation, we write $n'' = \max \{n, n'\}$. Clearly $G$ is generated by $\{1 + 2^{n}t_{i}\}_{1 \leq i \leq 2g - 1} \cup \{1 + 2^{n'}t_{2g}\}$. We verify using the property $t_{i}^{2} = 0$ that $(1 + 2^{n}t_{i})^{2^{n''}} \equiv 1 + 2^{n + n''}t_{i}$ (mod $2^{n + n'' + 1}$) for $1 \leq i \leq 2g - 1$; $(1 + 2^{n'}t_{2g})^{2^{n + n'' - n'}} \equiv 1 + 2^{n + n''}t_{2g}$ (mod $2^{n + n'' + 1}$); and $$(1 + 2^{n}t_{i})(1 + 2^{n''}t_{j})(1 + 2^{n}t_{i})^{-1}(1 + 2^{n''}t_{j})^{-1} \equiv 1 + 2^{n + n''}[t_{i}, t_{j}] \ (\mathrm{mod} \ 2^{n + n'' + 1})$$ for $1 \leq i < j \leq 2g$. Thus, $G \supset \Gamma(2^{n + n''})$, as desired. We now state and prove another proposition which will be needed only for the last statements of Theorems \[thm main1\] and \[thm main2\], which pertain to the case that the hyperelliptic curve has degree $4$. \[prop open subgroup degree 4\] Assume the notation of Proposition \[prop open subgroup\] and that $g = 1$ and $n' = n$, and let $c_{3} \in M$ be an element which is equivalent modulo $2$ to $a_{1}$. Then the subgroup of $\Gamma(2^{n})$ generated by $G$ and the element $T_{c_{3}}^{2^{n}}$ coincides with $\Gamma(2^{n})$. By the argument used at the beginning of the proof of Proposition \[prop open subgroup\], we only need to show that the images of $T_{c_{1}}^{2^{n}}, T_{c_{2}}^{2^{n}}, T_{c_{3}}^{2^{n}}$ modulo $2^{n + 1}$ generate $\Gamma(2^{n}) / \Gamma(2^{n + 1})$. Using the isomorphism from the additive group $\mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$ to $\Gamma(2^{n}) / \Gamma(2^{n + 1})$ which was established in that proof, we see that it suffices to show that $\{\bar{t}_{1}, \bar{t}_{2}, \bar{t}_{3}\}$ is linearly independent and thus generates the rank-$3$ elementary abelian group $\mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$, where $\bar{t}_{i}$ denotes the image modulo $2$ of $T_{c_{i}} - 1$ for $1 \leq i \leq 3$. But this is clear from noting that $c_{3} \equiv c_{1} + c_{2}$ (mod $2$) and writing out these linear operators as matrices with respect to an ordered basis of $M \otimes {\mathbb{F}}_{2}$ consisting of the images of $c_{1}$ and $c_{2}$. Proof of main theorems and a further result {#S4} =========================================== The main goal of this section is to prove Theorems \[thm main1\] and \[thm main2\]. Our strategy for this is to put together local results with respect to several primes of $K$ using Proposition \[prop Galois action local\] to get several elements in $G_{2}$ and then to use Proposition \[prop open subgroup\] to determine that the subgroup of $G_{2}$ generated by these elements contains a certain congruence subgroup of the symplectic group. This is realized by the following theorem, which can be used in a far more general situation than required for Theorems \[thm main1\] and \[thm main2\] and is therefore a useful result in its own right. \[thm several primes\] Let $J$ be the Jacobian of the hyperelliptic curve $C$ over $K$ of genus $g$ and degree $d'$ with defining equation $y^{2} = \prod_{i = 1}^{d'} (x - \alpha_{i})$ for some elements $\alpha_{i} \in \mathcal{O}_{K}$ for $1 \leq i \leq d'$, and define the $2$-adic Galois image $G_{2}$ as above. Suppose that there are distinct primes $\mathfrak{p}_{1}$, ... , $\mathfrak{p}_{d' - 1}$ of $K$ not lying over $(2)$ and such that for $1 \leq i \leq d' - 1$, the only two $\alpha_{j}$’s which are equivalent modulo $\mathfrak{p}_{i}$ are $\alpha_{i}$ and $\alpha_{d'}$; let $m_{i} \geq 1$ be the maximal integer such that $\mathfrak{p}_{i}^{m_{i}} \mid (\alpha_{d'} - \alpha_{i})$. Let $n = \max \{v_{2}(m_{i})\}_{i = 1}^{d' - 2}$. If $n' := v_{2}(m_{d' - 1}) \leq n$ or if $d' = 2g + 2$, we have $\Gamma(2) \supseteq G_{2} \supsetneq \Gamma(2^{2n + 2})$. Otherwise, we have $\Gamma(2) \supseteq G_{2} \supsetneq \Gamma(2^{n + n' + 2})$. Moreover, if $d' = 4$, then $G_{2} \supseteq \Gamma(2^{\max \{n, n'\} + 1})$. Using the embedding $\bar{K} \hookrightarrow {\mathbb{C}}$ fixed at the beginning of §\[S2\], we identify $T_{\ell}(J)$ with $H_{1}(C({\mathbb{C}}), {\mathbb{Z}}) \otimes {\mathbb{Z}}_{\ell}$ as we did in §\[S2\]. Let $\{a_{1}, ... , a_{g}, b_{1}, ... , b_{g}\}$ be a symplectic basis of $H_{1}(C({\mathbb{C}}), {\mathbb{Z}})$ as in Definition \[dfn symplectic basis\](b) with $(z_{1}, ... , z_{2g + 2}) = (\alpha_{1}, ... , \alpha_{2g + 1}, \infty)$ if $d'$ is odd and $(z_{1}, ... , z_{2g + 2}) = (\alpha_{1}, ... , \alpha_{2g}, \alpha_{2g + 2}, \alpha_{2g + 1})$ if $d'$ is even. Then Proposition \[prop Galois action local\] says that for $1 \leq i \leq 2g$, the image of $G_{K, \mathfrak{p}_{i}} \subset G_{K}$ in ${\mathrm{GSp}}(T_{2}(J))$ contains $T_{c_{i}}^{2m_{i}}$, where $T_{c_{i}} \in {\mathrm{Sp}}(T_{2}(J))$ is the transvection with respect to an element $c_{i} \in T_{2}(J)$ which is equivalent modulo $2$ to the formula given in (\[eq c\_i\]). Meanwhile, if $d'$ is even, the image of $G_{K, \mathfrak{p}_{2g + 1}}$ contains $T_{c_{d'}}^{2m_{2g + 1}}$ for some other element $c_{d'} \in T_{2}(J)$ corresponding to the lift of a loop separating $\{\alpha_{2g + 1}, \alpha_{2g + 2}\}$ from its complement in $\{\alpha_{j}\}_{j = 1}^{2g + 2}$. In any case, we have $T_{c_{1}}^{2m_{1}}, ... , T_{c_{2g}}^{2m_{2g}} \in G_{2}$; note that $m_{i} / 2^{v_{2}(m_{i})} \in {\mathbb{Z}}_{2}^{\times}$ and so by taking suitable powers we get that $T_{c_{1}}^{2^{v_{2}(m_{1}) + 1}}, ... , T_{c_{2g}}^{2^{v_{2}(m_{2g}) + 1}} \in G_{2}$. It then follows from Proposition \[prop open subgroup\] applied to $M := H_{1}(C({\mathbb{C}}), {\mathbb{Z}})$ that $G_{2} \supset \Gamma(2^{2n + 2})$ if $n' \leq n$ or if $d' = 2g + 2$ and $G_{2} \supset \Gamma(2^{n + n' + 2})$ otherwise. If $d' = 4$, then in particular, we have $T_{c_{1}}^{2^{\max\{n, n'\} + 1}}, T_{c_{2}}^{2^{\max\{n, n'\} + 1}}, T_{c_{3}}^{2^{\max\{n, n'\} + 1}} \in G_{2}$ with $c_{1}, c_{2}, c_{3}$ as described above. Clearly we have $c_{1} \equiv a_{1} + b_{1}$, $c_{2} \equiv b_{1}$, and $c_{3} \equiv a_{1}$ (mod $2$), and so Proposition \[prop open subgroup degree 4\] implies that $G_{2} \supseteq \Gamma(2^{\max \{n, n'\} + 1})$. \[rmk index bound\] We note that it is generally not difficult to compute the order of $G / \Gamma(2^{2n + 2})$ or $G / \Gamma(2^{n + n' + 2})$, where $G \subseteq G_{2}$ is the subgroup generated by the powers of transvections given in the proof above. Therefore, one may improve the upper bound for $[\Gamma(2) : G_{2}]$ which directly follows from the statement of Theorem \[thm several primes\]. For example, we have $[\Gamma(2) : G_{2}] \leq 2^{(2n + 1)(2g^{2} + g) - (n + 1)(d' - 1)}$ in the case that $n = n'$. [(Legendre curve)]{} \[ex Legendre\] For any $\lambda \in \mathcal{O}_{K} \smallsetminus \{0, 1\}$, let $E_{\lambda}$ be the elliptic curve over $K$ given by $y^{2} = x(x - 1)(x - \lambda)$. Suppose that there exist (necessarily distinct) primes $\mathfrak{p}_{1}$ and $\mathfrak{p}_{2}$ of $K$ not lying over $(2)$ and integers $m_{1}, m_{2}, \geq 1$ such that $\mathfrak{p}_{1}^{m_{1}}$ exactly divides $(\lambda)$ and $\mathfrak{p}_{2}^{m_{2}}$ exactly divides $(\lambda - 1)$. Then Theorem \[thm several primes\] tells us that the $2$-adic Galois image $G_{2}$ (strictly) contains $\Gamma(2^{v_{2}(m_{1}) + v_{2}(m_{2}) + 2})$. In the case that $m_{1} = m_{2} = 1$ (e.g. $K = {\mathbb{Q}}$, $\lambda = 6$, $\mathfrak{p}_{1} = (3)$, $\mathfrak{p}_{2} = (5)$), we get $\Gamma(2) \supset G_{2} \supsetneq \Gamma(4)$ and can therefore directly compute the precise subgroup $G_{2} \cap {\mathrm{Sp}}(T_{2}(E_{\lambda})) \subset \Gamma(2)$ using the well-known fact that the $4$-division field $K(E_{\lambda}[4])$ is generated over $K$ by $\{\sqrt{-1}, \sqrt{\lambda}, \sqrt{\lambda - 1}\}$. It is also possible to prove the statement of Proposition \[prop Galois action local\], and hence the subgroup $G_{2}$ for this example for the particular cases of $C = E_{\lambda}$ over $\mathcal{K}_{\mathfrak{p}_{1}}$ and over $\mathcal{K}_{\mathfrak{p}_{2}}$ using formulas for generators of $2$-power division fields of $E_{\lambda}$ over $K$ found in [@yelton2015dyadic], as the author has done in [@yelton2015hyperelliptic §3.4]. We now prove Theorems \[thm main1\] and \[thm main2\] together. First assume the notation and hypotheses of Theorem \[thm main1\]. Let $\alpha_{1}, ... , \alpha_{d}$ denote the roots of $f$, and write $L = K(\alpha_{1}, ... , \alpha_{d})$ for the splitting field of $f$ over $K$. Note that ${\mathrm{Gal}}(L / K)$ acts transitively on the $\alpha_{i}$’s since $f$ is irreducible. It then follows from the well-known description of the $2$-division field of a hyperelliptic Jacobian (see for instance [@mumford1984tata Corollary 2.11]) that $G_{K}$ does not fix the $2$-torsion points of $J$ and so $G_{2}$ is not contained in $\Gamma(2)$, while the image of ${\mathrm{Gal}}(\bar{K} / L)$ under $\rho_{2}$ coincides with $G_{2} \cap \Gamma(2)$. The fact that $\mathfrak{p} \nmid (2\Delta)$ implies that the extension $L / K$ is not ramified at $\mathfrak{p}$, and so $\mathfrak{p}$ splits into a product $\mathfrak{p}_{1} ... \mathfrak{p}_{r}$ of distinct primes in $L$, for some integer $r$ dividing $[L : K]$. Then since $\mathfrak{p}^{m}$ exactly divides $(f(\lambda)) = \prod_{i = 1}^{d}(\lambda - \alpha_{i})$, we have $\mathfrak{p}_{i} \mid (\lambda - \alpha_{i})$ for some $i$; we assume without loss of generality that $\mathfrak{p}_{1} \mid (\lambda - \alpha_{1})$. Then $\mathfrak{p}_{1}$ cannot divide $(\lambda - \alpha_{i})$ for any $i \in \{2, ... , d\}$, because otherwise for such an $i$ we would have $\mathfrak{p}_{1} \mid (\alpha_{i} - \alpha_{1}) \mid (\Delta)$, which contradicts the hypothesis that $\mathfrak{p} \nmid (2\Delta)$. It follows that $\mathfrak{p}_{1}^{m}$ exactly divides $(\lambda - \alpha_{1})$. Then by applying elements of ${\mathrm{Gal}}(L / K)$ that take $\alpha_{1}$ to each $\alpha_{i}$, we get other primes lying over $\mathfrak{p}$ whose $m$th powers exactly divide the ideals $(\lambda - \alpha_{i})$; we assume without loss of generality that $\mathfrak{p}_{i}^{m}$ exactly divides $(\lambda - \alpha_{i})$ for $1 \leq i \leq d$. Now since the $\mathfrak{p} \nmid (2\Delta)$ hypothesis implies that none of the $\mathfrak{p}_{i}$’s lie over $(2)$, we can apply Theorem \[thm several primes\] with $K$ replaced by $L$, $d' = d + 1$, $\alpha_{d'} = \lambda$, and $m_{1} = ... = m_{d' - 1} = m$ to get the statement of Theorem \[thm main1\]. Now assume the notation and hypotheses of Theorem \[thm main2\]. Then the argument for proving Theorem \[thm main2\] is the same except that when applying Theorem \[thm several primes\] we choose $\mathfrak{p}_{d' - 1}$ to be a prime of $L$ lying over $\mathfrak{p}'$, and we put $d' = d + 2$, $\alpha_{d'} = \lambda$, $\alpha_{d' - 1} = \lambda'$, $m = m_{1} = ... = m_{d' - 2}$, and $m' = m_{d' - 1}$. Realizing uniform boundedness along one-parameter families {#S5} ========================================================== Fix an irreducible monic polynomial $f \in \mathcal{O}_{K}[x]$ of degree $d \geq 2$. Cadoret and Tamagawa have shown in [@cadoret2012uniform Theorems 1.1 and 5.1] that for all but finitely many $\lambda \in K$, the $\ell$-adic Galois image $G_{\ell, \lambda}$ associated to the Jacobian of the curve given by $y^{2} = f(x)(x - \lambda)$ is open in the $\ell$-adic Galois image $G_{\ell, \eta}$ associated to the generic fiber of the family parametrized by $\lambda$. These theorems also assert that there is some integer $B \geq 1$ depending only on $f$ and $\ell$ such that the index of $G_{\ell, \lambda}$ in $G_{\ell, \eta}$ is bounded by $B$ for all but finitely many $\lambda \in K$. The following theorem recovers the openness result for $\ell = 2$ when $d \geq 4$ and explicitly provides the aforementioned uniform bound when $d$ is even, under the assumption that $K$ has class number $1$. (Note that for the elliptic curve case, where $d \in \{ 2, 3\}$, such openness results are already known from the celebrated Open Image Theorem of Serre given by [@serre1989abelian IV-11], while uniform bounds are given by [@arai2008uniform Theorem 1.3].) It is interesting to note that Faltings’ Theorem is used both in the proof of [@cadoret2012uniform Theorem 1.1] and in our proof of the theorem below. \[thm uniform bounds\] Assume that $\mathcal{O}_{K}$ is a PID. Let $f \in \mathcal{O}_{K}[x]$ be an irreducible monic polynomial of degree $d \geq 3$ with discriminant $\Delta$. For each $\lambda \in K$, let $J_{\lambda}$ denote the Jacobian of the hyperelliptic curve $C_{\lambda}$ with defining equation $y^{2} = f(x)(x - \lambda)$, and write $G_{2, \lambda} \subseteq {\mathrm{GSp}}(T_{2}(J_{\lambda}))$ for the image of the associated $2$-adic Galois representation. a\) If $d \geq 4$, then the Lie subgroup $G_{2, \lambda} \subseteq {\mathrm{GSp}}(T_{2}(J_{\lambda}))$ is open for all but finitely many $\lambda \in K$. b\) If $d = 3$ (resp. if $d \geq 5$), then $G_{2, \lambda} \cap {\mathrm{Sp}}(T_{2}(J_{\lambda})) \supsetneq \Gamma(4)$ for all but finitely many $\lambda \in \mathcal{O}_{K}[(2\Delta)^{-1}] \cdot (K^{\times})^{4}$ (resp. all but finitely many $\lambda \in \mathcal{O}_{K}[(2\Delta)^{-1}] \cdot (K^{\times})^{2}$). c\) If $d = 4$, then we have $G_{2, \lambda} \cap {\mathrm{Sp}}(T_{2}(J_{\lambda})) \supsetneq \Gamma(16)$ for all but finitely many $\lambda \in K$. If $d \geq 6$ is even, then we have $G_{2, \lambda} \cap {\mathrm{Sp}}(T_{2}(J_{\lambda})) \supsetneq \Gamma(4)$ for all but finitely many $\lambda \in K$. Let $\Sigma \subset K^{\times}$ denote the multiplicative subgroup generated by the elements $\xi \in \mathcal{O}_{K}$ such that $\xi$ is divisible only by primes which divide $2\Delta$. We claim that $\Sigma$ is finitely generated. Indeed, there is an obvious map from $\Sigma$ to the free ${\mathbb{Z}}$-module formally generated by the (finite) set of prime ideals of $\mathcal{O}_{K}$ which divide $2\Delta$, and its kernel is the unit group $\mathcal{O}_{K}^{\times}$, which is also well known to be finitely generated. Choose any $\lambda \in K$, which we may write as $\mu / \nu$ for some coprime $\mu, \nu \in \mathcal{O}_{K}$, because $\mathcal{O}_{K}$ is a PID. Let $h(x) = \nu^{d}f(\nu^{-1} x)$, which is a monic polynomial in $\mathcal{O}_{K}[x]$; note that the discriminant of $h$ is equal to $\nu^{2d^{2} - d}\Delta$. Then there is a $K(\sqrt{\nu})$-isomorphism from $C_{\lambda}$ to the hyperelliptic curve $C_{\lambda}'$ whose defining equation is $y^{2} = \nu^{d}f(\nu^{-1}x)(x - \mu) \in \mathcal{O}_{K}[x]$, given by $(x, y) \mapsto (\nu x, \nu^{(d + 1) / 2}y)$. Thus, letting $J_{\lambda}'$ denote the Jacobian of $C_{\lambda}'$, the $2$-adic Tate modules $T_{2}(J_{\lambda})$ and $T_{2}(J_{\lambda}')$ are isomorphic as ${\mathrm{Gal}}(\bar{K} / K(\sqrt{\nu}))$-modules. In light of this, we replace $K$ with $K(\sqrt{\nu})$ and consider the $2$-adic Galois image $G_{2, \lambda}$ associated to $J_{\lambda}'$. Now Theorem \[thm main1\] says that if there is a prime element $\mathfrak{p}$ dividing $\nu^{d}f(\lambda)$ but not $2\nu^{d^{2} - d}\Delta$, then $G_{2, \lambda} \subset {\mathrm{GSp}}(T_{2}(J_{\lambda})) \cong {\mathrm{GSp}}(T_{2}(J_{\lambda}'))$ is open. It follows from the fact that $\mu$ and $\nu$ are coprime that $\nu^{d}f(\lambda)$ is not divisible by any prime element dividing $\nu$, so a prime $\mathfrak{p}$ satisfying the above condition does not divide $2\Delta$. The existence of such a prime is equivalent to the condition that $\nu^{d} f(\lambda) \notin \Sigma$, so to prove part (a) it suffices to show that $f(\lambda) \in \Sigma \cdot (K^{\times})^{d}$ for only finitely many $\lambda \in K$. Note that any such $\lambda$ yields a solution $(x = \lambda, y) \in K \times K$ to an equation of the form $\xi y^{d} = f(x)$ with $\xi \in \Sigma'$, where $\Sigma' \subset \Sigma$ is a set of representatives of elements in $\Sigma / \Sigma^{d}$. If $d \geq 4$, then an application of the Riemann-Hurwitz formula shows that such an equation defines a smooth curve of genus $\geq 2$, and then Faltings’ Theorem implies that there are only finitely many solutions defined over $K$ to each such equation. Therefore, to prove (a) it suffices to show that there are only finitely many choices of $\xi$. But this follows from the fact that $\Sigma / \Sigma^{d}$ is finite because $\Sigma$ is finitely generated. Now assume that $d = 3$ or $d \geq 5$ and that $\lambda \in \mathcal{O}_{K}[(2\Delta)^{-1}] \cdot (K^{\times})^{s}$, with $s = 4$ if $d = 3$ and $s = 2$ otherwise. Then if we write $\lambda = \mu / \nu$ as above, we have $\nu \in \Sigma \cdot (K^{\times})^{s}$. Suppose that $\nu^{d} f(\lambda) \notin \Sigma \cdot (K^{\times})^{s}$, which is equivalent to saying that $f(\lambda) \notin \Sigma \cdot (K^{\times})^{s}$. Then there is a prime element $\mathfrak{p}$ dividing $\nu^{d} f(\lambda)$ but not $2\nu^{d^{2} - d}\Delta$ (so $\mathfrak{p} \nmid 2\Delta$ as before) and such that the maximum integer $m \geq 1$ with $\mathfrak{p}^{m} \mid \nu^{d}f(\lambda)$ satisfies $v_{2}(m) \leq v_{2}(s) - 1$. Then Theorem \[thm main1\] implies that $G_{2, \lambda} \cap {\mathrm{Sp}}(T_{2}(J_{\lambda})) \supsetneq \Gamma(4)$ both when $d = 3$ and when $d \geq 5$. Therefore, to prove (b) it suffices to show that $f(\lambda) \in \Sigma \cdot (K^{\times})^{s}$ for only finitely many $\lambda \in K$. This follows from the same argument as above, once we observe by Riemann-Hurwitz that the curves given by $\xi y^{s} = f(x)$ have genus $\geq 2$. Finally, assume that $d \geq 4$ is even and choose any $\lambda = \mu / \nu \in K$ as before. Let $s = 4$ if $d = 4$ and let $s = 2$ otherwise. Suppose that $\nu^{d} f(\lambda) \notin \Sigma \cdot (K^{\times})^{s}$, which in both cases is equivalent to saying that $f(\lambda) \notin \Sigma \cdot (K^{\times})^{s}$. Then there is a prime element $\mathfrak{p}$ dividing $\nu^{d} f(\lambda)$ but not $2\nu^{d^{2} - d}\Delta$ (so $\mathfrak{p} \nmid 2\Delta$ as before) and such that the maximum integer $m \geq 1$ with $\mathfrak{p}^{m} \mid \nu^{d}f(\lambda)$ satisfies $v_{2}(m) \leq v_{2}(s) - 1$. Then Theorem \[thm main1\] implies that $G_{2, \lambda} \cap {\mathrm{Sp}}(T_{2}(J_{\lambda}))$ strictly contains $\Gamma(16)$ (resp. $\Gamma(4)$). Therefore, to prove (c) it suffices to show that $f(\lambda) \in \Sigma \cdot (K^{\times})^{s}$ for only finitely many $\lambda \in K$, which likewise follows from checking that the curves given by $\xi y^{s} = f(x)$ have genus $\geq 2$. \[rmk not PID\] a\) If we drop the assumption that $\mathcal{O}_{K}$ is a PID in the statement of Theorem \[thm uniform bounds\], then we observe from the proof above that we still get the statement of (a) when “all but finitely many $\lambda \in K$" is replaced by “all but finitely many $\lambda \in \mathcal{O}_{K}$" (and this statement holds for $d \in \{2, 3\}$ as well). In particular, this shows that the elements $\lambda \in \mathcal{O}_{K}$ which satisfy the hypothesis in Theorem \[thm main1\] account for all but finitely many of the elements $\lambda \in \mathcal{O}_{K}$ such that $G_{2, \lambda}$ is open. b\) In any case, the hypothesis “$\mathcal{O}_{K}$ is a PID" may be weakened to “the Hilbert class field tower of $K$ terminates". This follows from the fact that under this hypothesis, there is a finite extension of $K$ with class number $1$, and it clearly suffices to prove the assertions of Theorem \[thm uniform bounds\] when $K$ is replaced with a finite extension of $K$. \[rmk 4-torsion\] Parts (b) and (c) of Theorem \[thm uniform bounds\] say that for a given polynomial $f$ of degree $d \neq 4$, there are many elements $\lambda \in K$ such that $G_{2, \lambda} \supsetneq \Gamma(4)$. In these cases, it is always possible to compute the full structure of $G_{2}$ and determine its index in ${\mathrm{GSp}}(T_{2}(J_{\lambda}))$ by considering the Galois action on the $4$-torsion subgroup of $J$ and using formulas for the generators of the $4$-division field $K(J[4])$ over $K$. Such formulas are provided by [@yelton2015images Proposition 3.1] in the case that $d$ is odd and are found in [@yelton2015hyperelliptic §2.4] in the case that $d$ is even.

--- abstract: 'We classify all possible limits of families of translates of a fixed, arbitrary complex plane curve. We do this by giving a set-theoretic description of the projective normal cone (PNC) of the base scheme of a natural rational map, determined by the curve, from the ${{\mathbb{P}}}^8$ of $3\times 3$ matrices to the ${{\mathbb{P}}}^N$ of plane curves of degree $d$. In a sequel to this paper we determine the multiplicities of the components of the PNC. The knowledge of the PNC as a cycle is essential in our computation of the degree of the ${\text{\rm PGL}}(3)$-orbit closure of an arbitrary plane curve, performed in [@MR2001h:14068].' address: - 'Dept. of Mathematics, Florida State University, Tallahassee FL 32306, U.S.A.' - 'Inst. för Matematik, Kungliga Tekniska Högskolan, S-100 44 Stockholm, Sweden' author: - 'Paolo Aluffi, Carel Faber' bibliography: - 'ghizzIbib.bib' title: 'Limits of PGL(3)-translates of plane curves, I' --- Introduction {#intro} ============ In this paper we determine the possible [*limits*]{} of a fixed, arbitrary complex plane curve ${{\mathscr C}}$, obtained by applying to it a family of translations $\alpha(t)$ centered at a singular transformation of the plane. In other words, we describe the curves in the boundary of the ${\text{\rm PGL}}(3)$-orbit closure of a given curve ${{\mathscr C}}$. Our main motivation for this work comes from enumerative geometry. In [@MR2001h:14068] we have determined the [*degree*]{} of the ${\text{\rm PGL}}(3)$-orbit closure of an arbitrary (possibly singular, reducible, non-reduced) plane curve; this includes as special cases the determination of several characteristic numbers of families of plane curves, the degrees of certain maps to moduli spaces of plane curves, and isotrivial versions of the Gromov-Witten invariants of the plane. A description of the limits of a curve, and in fact a more refined type of information is an essential ingredient of our approach. This information is obtained in this paper and in its sequel [@ghizzII]; the results were announced and used in [@MR2001h:14068]. The set-up is as follows. Consider the natural action of ${\text{\rm PGL}}(3)$ on the projective space of plane curves of a fixed degree. The orbit closure of a curve ${{\mathscr C}}$ is dominated by the closure ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ of the graph of the rational map $c$ from the ${{\mathbb{P}}}^8$ of $3\times3$ matrices to the ${{\mathbb{P}}}^N$ of plane curves of degree $d$, associating to $\varphi\in {\text{\rm PGL}}(3)$ the translate of ${{\mathscr C}}$ by $\varphi$. The boundary of the orbit consists of limits of ${{\mathscr C}}$ and plays an important role in the study of the orbit closure. Our computation of the degree of the orbit closure of ${{\mathscr C}}$ hinges on the study of ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$, and especially of the scheme-theoretic inverse image in ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ of the base scheme ${{\mathscr S}}$ of $c$. Viewing ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ as the blow-up of ${{\mathbb{P}}}^8$ along ${{\mathscr S}}$, this inverse image is the exceptional divisor, and may be identified with the projective normal cone (PNC) of ${{\mathscr S}}$ in ${{\mathbb{P}}}^8$. A description of the PNC leads to a description of the limits of ${{\mathscr C}}$: the image of the PNC in ${{\mathbb{P}}}^N$ is contained in the set of limits, and the complement, if nonempty, consists of easily identified ‘stars’ (that is, unions of concurrent lines). This paper is devoted to a set-theoretic description of the PNC for an arbitrary curve. This suffices for the determination of the limits, but does not suffice for the enumerative applications in [@MR2001h:14068]; these applications require the full knowledge of the PNC [*as a cycle,*]{} that is, the determination of the multiplicities of its different components. We obtain this additional information in [@ghizzII]. The final result of our analysis (including multiplicities) was announced in §2 of [@MR2001h:14068]. The proofs of the facts stated there are given in the present article and its sequel. The main theorem of this paper (Theorem \[mainmain\], in §\[proof\]) gives a precise set-theoretic description of the PNC, relying upon five types of families and limits identified in §\[germlist\]. In this introduction we confine ourselves to formulating a weaker version, focusing on the determination of limits. In [@ghizzII] (Theorem 2.1), we compute the multiplicities of the corresponding five types of components of the PNC. The limits of a curve ${{\mathscr C}}$ are necessarily curves with [*small linear orbit,*]{} that is, curves with infinite stabilizer. Such curves are classified in §1 of [@MR2002d:14084]; we reproduce the list of curves obtained in [@MR2002d:14084] in an appendix at the end of this paper (§\[appendix\]). For another classification, from a somewhat different viewpoint, we refer to [@MR1698902]. For these curves, the limits can be determined using the results in [@MR2002d:14083] (see also §\[boundary\]). The following statement reduces the computation of the limits of an arbitrary curve ${{\mathscr C}}$ to the case of curves with small orbit. \[main\] Let ${{\mathscr X}}$ be a limit of a plane curve ${{\mathscr C}}$ of degree $d$, obtained by applying to it a ${{\mathbb{C}}}((t))$-valued point of ${\text{\rm PGL}}(3)$ with singular center. Then ${{\mathscr X}}$ is in the orbit closure of a star (reproducing projectively the $d$-tuple cut out on ${{\mathscr C}}$ by a line meeting it properly), or of curves with small orbit determined by the following features of ${{\mathscr C}}$: - The linear components of the support ${{{{\mathscr C}}'}}$ of ${{\mathscr C}}$; - The nonlinear components of ${{{{\mathscr C}}'}}$; - The points at which the tangent cone of ${{\mathscr C}}$ is supported on at least $3$ lines; - The Newton polygons of ${{\mathscr C}}$ at the singularities and inflection points of ${{{{\mathscr C}}'}}$; - The Puiseux expansions of formal branches of ${{\mathscr C}}$ at the singularities of ${{{{\mathscr C}}'}}$. The limits corresponding to these features may be described as follows. In cases I and III they are unions of a star and a general line, that we call ‘fans’; in case II, they are supported on the union of a nonsingular conic and a tangent line; in case IV, they are supported on the union of the coordinate triangle and several curves from a pencil $y^c=\rho\, x^{c-b} z^b$, with $b<c$ coprime positive integers; and in case V they are supported on unions of quadritangent conics and the distinguished tangent line. The following picture illustrates the limits in cases IV and V:  A more precise description of the limits is given in §\[germlist\], referring to the classification of these curves obtained in §1 of [@MR2002d:14084] and reproduced in §\[appendix\] of this paper. The proof of Theorem \[main\] (or rather of its more precise form given in Theorem \[mainmain\]) is by an explicit reduction process, and goes along the following lines. The stars mentioned in the statement are obtained by families of translations $\alpha(t)$ (‘germs’) centered at an element $\alpha(0)\not\in{{\mathscr S}}$. To analyze germs centered at points of ${{\mathscr S}}$, we introduce a notion of equivalence of germs (Definition \[equivgermsnew\]), such that equivalent germs lead to the same limit. We then prove that every germ centered at a point of ${{\mathscr S}}$ is essentially equivalent to one with matrix representation $$\begin{pmatrix} 1 & 0 & 0\\ q(t) & t^b & 0\\ r(t) & s(t)t^b & t^c \end{pmatrix}$$ with $0\le b\le c$ and $q$, $r$, and $s$ polynomials. Here, coordinates are chosen so that the point $p=(1:0:0)$ belongs to ${{\mathscr C}}$. Studying the limits obtained by applying such germs to ${{\mathscr C}}$, we identify five specific types of families (the [*marker germs*]{} listed in §\[germlist\]), reflecting the features of ${{\mathscr C}}$ at $p$ listed in Theorem \[main\], and with the stated kind of limit. We prove that unless the germ is of one of these types, the corresponding limit is already accounted for (for example, it is in the orbit closure of a star of the type mentioned in the statement). In terms of the graph of the rational map $c$ mentioned above, we prove that every component of the PNC is hit at a general point by the lift in ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ of one of the five distinguished types of germs. This yields our set-theoretic description of the PNC. In fact, the lifts intersect the corresponding components transversally, and this will be important in our determination of the multiplicities of the components in [@ghizzII]. The procedure underlying the proof of Theorem \[mainmain\] may be applied to any given plane curve, producing a list of its limits. In practice, one needs to find the marker germs for the curve; these determine the components of the PNC. The two examples in §\[twoexamples\] illustrate this process, and show that components of all types may already occur on curves of degree $4$. Here is a simpler example, for a curve of degree $3$. \[exone\] Consider the irreducible cubic ${{\mathscr C}}$ given by the equation $$xyz+y^3+z^3=0\,.$$ It has a node at $(1:0:0)$ and three inflection points. According to Theorem \[mainmain\] and the list in §\[germlist\], the PNC for ${{\mathscr C}}$ has one component of type II and several of type IV. The latter correspond to the three inflection points and the node. A list of representative marker germs for the component of type II and for the component of type IV due to the node may be obtained by following the procedure explained in §\[setth\]: $${\rm II}: \begin{pmatrix} -2 & -t & 0\\ 1 & t & 0\\ 1 & 0 & t^2 \end{pmatrix};\quad {\rm IV}: \begin{pmatrix} 1 & 0 & 0\\ 0 & t & 0\\ 0 & 0 & t^2 \end{pmatrix} \,,\, \begin{pmatrix} 1 & 0 & 0\\ 0 & t^2 & 0\\ 0 & 0 & t \end{pmatrix}\,.$$ The latter two marker germs, corresponding to the two lines in the tangent cone at the node, have the same center and lead to projectively equivalent limits, hence they contribute the same component of the PNC. Equations for the limits of ${{\mathscr C}}$ determined by the germs listed above are $$x(xz+2y^2)=0,\quad y(y^2+xz)=0,\quad \text{and} \quad z(z^2+xy)=0\,,$$ respectively: a conic with a tangent line, and a conic with a transversal line (two limits). The inflection points also contribute components of type IV; the limits in that case are cuspidal cubics. According to Theorem \[main\], all limits of ${{\mathscr C}}$ (other than stars of lines) are projectively equivalent to one of these curves, or to limits of them (cf. §\[boundary\]). Necessary preliminary considerations, and the full statement of the main theorem, are found in §\[prelim\]. The determination of the limits by successive reductions of a given family of curves, proving the result, is worked out in §\[setth\] and §\[typeVcomps\]. In §\[boundary\] we summarize the more straightforward situation for curves with small orbits. Harris and Morrison ([@MR99g:14031], p. 138) pose the [*flat completion problem*]{} for families of embedded curves, asking for the determination of all curves in ${{\mathbb{P}}}^n$ that can arise as flat limits of a family of embedded stable curves over the punctured disc. The present article solves the isotrivial form of this problem, for plane curves. In principle, a solution of the isotrivial flat completion problem for plane curves can already be found in the marvelous article [@ghizz] by Aldo Ghizzetti, dating back to the 1930s. However, Ghizzetti’s results do not lead to a description of the PNC, which is necessary for our application in [@MR2001h:14068], and which is the main result of this paper and of its sequel. Caporaso and Sernesi use our determination of the limits in [@MR2003k:14035] (Theorem 5.2.1). Hacking [@MR2078368] and Hassett [@MR2000j:14045] study the limits of families of nonsingular plane curves of a given degree, by methods different from ours: they allow the plane to degenerate together with the curve. It would be interesting to compare their results to ours. However, there are fundamental differences between the phenomena we study and those addressed in [@MR2078368] and [@MR2000j:14045]; for example, our families are constant in moduli, and our results apply to [*arbitrary*]{} plane curves. By the same token, neither Hacking-stability nor GIT-stability play an important role in our study. Consider the case of a plane curve with an analytically irreducible singularity. The determination of the contribution of the singularity to the PNC of the curve requires both its [*linear*]{} type and [*all*]{} its Puiseux pairs, see §5 of [@MR2001h:14068]. In general, the stability conditions mentioned above require strictly less (cf. Kim-Lee [@MR2090618]). For example, a singularity analytically isomorphic to $y^2=x^5$ on a [*quartic*]{} leads necessarily to a component of type V (cf. Example \[extwo\]), whereas on a quintic, it leads to either a component of type IV or a component of type V, according to the order of contact with the tangent line. For GIT-stability, see also Remark \[GITrem\]. The enumerative problem considered in [@MR2001h:14068], as well as the question of limits of PGL-translates, makes sense for hypersurfaces of projective space of any dimension. The case of configurations of points in ${{\mathbb{P}}}^1$ is treated in [@MR1244973]. The degree of the orbit closure of a configuration of planes in ${{\mathbb{P}}}^3$ is computed in [@MR2455792]. In general, these problems appear to be very difficult. The techniques used in this paper could in principle be used in arbitrary dimension, but the case-by-case analysis (which is already challenging for curves in ${{\mathbb{P}}}^2$) would likely be unmanageable in higher dimension. By contrast, the techniques developed in [@ghizzII] should be directly applicable: once ‘marker germs’ have been determined, computing the multiplicities of the corresponding components of the PNC should be straightforward, using the techniques of [@ghizzII]. [^1] We also thank the referee of this paper and [@ghizzII], for the careful reading of both papers and for comments that led to their improvement. Set-theoretic description of the PNC {#prelim} ==================================== Limits of translates {#rough} -------------------- We work over ${{\mathbb{C}}}$. We choose homogeneous coordinates $(x:y:z)$ in ${{\mathbb{P}}}^2$, and identify ${\text{\rm PGL}}(3)$ with the open set of nonsingular matrices in the space ${{\mathbb{P}}}^8$ parametrizing $3\times 3$ matrices. We consider the right action of ${\text{\rm PGL}}(3)$ on the space ${{\mathbb{P}}}^N={{\mathbb{P}}}H^0({{\mathbb{P}}}^2, \mathcal O(d))$ of degree-$d$ plane curves; if $F(x,y,z)=0$ is an equation for a plane curve ${{\mathscr C}}$, and $\alpha\in {\text{\rm PGL}}(3)$, we denote by ${{\mathscr C}}\circ\alpha$ the curve with equation $F(\alpha(x,y,z))=0$. We will consider families of plane curves over the punctured disk, of the form ${{\mathscr C}}\circ\alpha(t)$, where $\alpha(t)$ is a $3\times 3$ matrix with entries in ${{\mathbb{C}}}[t]$, such that $\alpha(0)\ne 0$, $\det\alpha(t) \not\equiv 0$, and $\det\alpha(0)=0$. Simple reductions show that studying these families is equivalent to studying all families ${{\mathscr C}}\circ\alpha(t)$, where $\alpha(t)$ is a ${{\mathbb{C}}}((t))$-valued point of ${{\mathbb{P}}}^8$ such that $\det\alpha(0)=0$. We also note that if ${{\mathscr C}}$ is a smooth curve of degree $d\ge 4$, then any family of curves of degree $d$ parametrized by the punctured disk and whose members are abstractly isomorphic to ${{\mathscr C}}$, i.e., an isotrivial family, is essentially of this type (cf. [@MR770932], p. 56). The arcs of matrices $\alpha(t)$ will be called [*germs,*]{} and viewed as germs of curves in ${{\mathbb{P}}}^8$. The flat limit $\lim_{t\to 0}\,{{\mathscr C}}\circ \alpha(t)$ of a family ${{\mathscr C}}\circ\alpha(t)$ as $t \to 0$ may be computed concretely by clearing common powers of $t$ in the expanded expression $F(\alpha(t))$, and then setting $t=0$. Our goal is the determination of all possible limits of families as above, for a given arbitrary plane curve ${{\mathscr C}}$. The Projective Normal Cone {#ident} -------------------------- The set of all translates ${{\mathscr C}}\circ\alpha$ is the [*linear orbit*]{} of ${{\mathscr C}}$, which we denote by ${{\mathscr O_{{\mathscr C}}}}$; the complement of ${{\mathscr O_{{\mathscr C}}}}$ in its closure ${\overline{{{\mathscr O_{{\mathscr C}}}}}}$ is the [*boundary*]{} of the orbit of ${{\mathscr C}}$. By the [*limits of ${{\mathscr C}}$*]{} we will mean the limits of families ${{\mathscr C}}\circ\alpha(t)$ with $\alpha(0)\not\in {\text{\rm PGL}}(3)$. For every curve ${{\mathscr C}}$, the boundary is a subset of the set of limits; if $\dim{{\mathscr O_{{\mathscr C}}}}=8$ (the stabilizer of ${{\mathscr C}}$ is finite), then these two sets coincide. If $\dim{{\mathscr O_{{\mathscr C}}}}<8$ (the stabilizer is infinite, and the orbit is [*small,*]{} in the terminology of [@MR2002d:14083] and [@MR2002d:14084]) then there are families with limit equal to ${{\mathscr C}}$; in this case, the whole orbit closure ${\overline{{{\mathscr O_{{\mathscr C}}}}}}$ consists of limits of ${{\mathscr C}}$. The set of limit curves is itself a union of orbits of plane curves; our goal is a description of representative elements of these orbits; in particular, this will yield a description of the boundary of ${{\mathscr O_{{\mathscr C}}}}$. In this section we relate the set of limits of ${{\mathscr C}}$ to the [*projective normal cone*]{} mentioned in the introduction. Points of ${{\mathbb{P}}}^8$, that is, $3\times 3$ matrices, may be viewed as rational maps ${{\mathbb{P}}}^2 \dashrightarrow {{\mathbb{P}}}^2$. The kernel of a singular matrix $\alpha\in {{\mathbb{P}}}^8$ determines a line of ${{\mathbb{P}}}^2$ (if $\operatorname{rk}\alpha=1$) or a point (if $\operatorname{rk}\alpha=2$); $\ker\alpha$ will denote this locus. Likewise, the image of $\alpha$ is a point of ${{\mathbb{P}}}^2$ if $\operatorname{rk}\alpha=1$, or a line if $\operatorname{rk}\alpha=2$. The action map $\alpha \mapsto {{\mathscr C}}\circ\alpha$ for $\alpha\in {\text{\rm PGL}}(3)$ defines a rational map $$c: {{\mathbb{P}}}^8 \dashrightarrow {{\mathbb{P}}}^N\quad.$$ We denote by ${{\mathscr S}}$ the base scheme of this rational map. The closure of the graph of $c$ may be identified with the blow-up ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ of ${{\mathbb{P}}}^8$ along ${{\mathscr S}}$. The support of ${{\mathscr S}}$ consists of the matrices $\alpha$ such that (with notation as above) $F(\alpha(x,y,z))\equiv 0$; that is, matrices whose image is contained in ${{\mathscr C}}$. The [*projective normal cone*]{} (PNC) of ${{\mathscr S}}$ in ${{\mathbb{P}}}^8$ is the exceptional divisor $E$ of this blow-up. We have the following commutative diagram: $$\xymatrix@M=10pt{ E \ar@{->>}[d] \ar@<-.5ex>@{^(->}[r] & {{{{\widetilde{{{\mathbb{P}}}}}}}}^8 \ar@{->>}[d]^\pi \ar@<-.5ex>@{^(->}[r] & {{\mathbb{P}}}^8\times {{\mathbb{P}}}^N \ar@{->>}[d] \\ {{\mathscr S}}\ar@<-.5ex>@{^(->}[r] & {{\mathbb{P}}}^8 \ar@{-->}[r]^c & {{\mathbb{P}}}^N }$$ Therefore, as a subset of ${{\mathbb{P}}}^8\times{{\mathbb{P}}}^N$, the support of the PNC is $$\begin{gathered} |E|=\{(\alpha,{{\mathscr X}})\in {{\mathbb{P}}}^8\times{{\mathbb{P}}}^N : \text{${{\mathscr X}}$ is a limit of ${{\mathscr C}}\circ \alpha(t)$}\\ \text{for some germ $\alpha(t)$ centered at $\alpha\in {{\mathscr S}}$ and not contained in ${{\mathscr S}}$}\}\quad.\end{gathered}$$ \[PNCtolimits\] The set of limits of ${{\mathscr C}}$ consists of the image of the PNC in ${{\mathbb{P}}}^N$, and of limits of families ${{\mathscr C}}\circ \alpha(t)$ with $\alpha=\alpha(0)$ a singular matrix whose image is not contained in ${{\mathscr C}}$. In the latter case: if $\alpha$ has rank 1, the limit consists of a multiple line supported on $\ker\alpha$; if $\alpha$ has rank 2, the limit consists of a star of lines through $\ker\alpha$, reproducing projectively the tuple of points cut out by ${{\mathscr C}}$ on the image of $\alpha$. The PNC dominates the set of limits of families ${{\mathscr C}}\circ \alpha(t)$ for which $\alpha(t)$ is centered at a point of indeterminacy of $c$. This gives the first statement. To verify the second assertion, assume that $\alpha(t)$ is centered at a singular matrix $\alpha$ at which $c$ [*is*]{} defined; $\alpha$ is then a rank-1 or rank-2 matrix such that $F(\alpha(x,y,z))\not\equiv 0$. After a coordinate change we may assume without loss of generality that $$\alpha=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad\text{or}\quad \alpha=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$ and $F(x,0,0)$, resp. $F(x,y,0)$ are not identically zero. These are then the forms defining the limits of the corresponding families, and the descriptions given in the statement are immediately verified in these cases. The second part of Lemma \[PNCtolimits\] may be viewed as the analogue in our context of an observation of Pinkham (‘sweeping out the cone with hyperplane sections’, [@MR0376672], p. 46). \[eluding\] Denote by $R$ the proper transform in ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ of the set of singular matrices in ${{\mathbb{P}}}^8$. Lemma \[PNCtolimits\] asserts that the set of limits of ${{\mathscr C}}$ is the image of the union of the PNC and $R$. A more explicit description of the image of $R$ has eluded us; for a smooth curve ${{\mathscr C}}$ of degree $\ge 5$ these ‘star limits’ have two moduli. It would be interesting to obtain a classification of curves ${{\mathscr C}}$ with smaller ‘star-moduli’. The image of the [*intersection*]{} of $R$ and the PNC will play an important role in this paper. Curves in the image of this locus will be called ‘rank-$2$ limits’; we note that the set of rank-$2$ limits has dimension $\le 6$. Lemma \[PNCtolimits\] translates the problem of finding the limits for families of plane curves ${{\mathscr C}}\circ\alpha(t)$ into the problem of describing the PNC for the curve ${{\mathscr C}}$. Each component of the PNC is a $7$-dimensional irreducible subvariety of ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8\subset {{\mathbb{P}}}^8\times {{\mathbb{P}}}^N$. We will describe it by listing representative points of the component. More precisely, note that ${\text{\rm PGL}}(3)$ acts on ${{\mathbb{P}}}^8$ by right multiplication, and that this action lifts to a right action of ${\text{\rm PGL}}(3)$ on ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$. Each component of the PNC is a union of orbits of this action. For each component, we will list germs $\alpha(t)$ lifting on ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ to germs $\tilde\alpha(t)$ so that the union of the orbits of the centers $\tilde\alpha(0)$ is dense in that component. Marker germs {#germlist} ------------ In a coarse sense, the classification of limits into ‘types’ as in Theorem \[main\] depends on the image of the center $\alpha(0)$ of the family: this will be a subset of ${{\mathscr C}}$ (cf. Lemma \[PNCtolimits\]), hence it will either be a (linear) component of ${{\mathscr C}}$ (type I), or a point of ${{\mathscr C}}$ (general for type II, singular or inflectional for types III, IV, and V). We will now list germs determining the components of the PNC in the sense explained above. We will call such a germ a [*marker germ,*]{} as the center of its lift to ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ (the corresponding [*marker center*]{}) ‘marks’ a component of the PNC. The first two types depend on global features of ${{\mathscr C}}$: its linear and nonlinear components. The latter three depend on local features of ${{\mathscr C}}$: inflection points and singularities of (the support of) ${{\mathscr C}}$. That there are only two global types is due to the fact that the order of contact of a nonlinear component and the tangent line at a general point equals two (in characteristic zero). The three local types are due to linear features at singularities of ${{\mathscr C}}$ (type III), single nonlinear branches at special points of ${{\mathscr C}}$ (type IV), and collections of several matching nonlinear branches at singularities of ${{\mathscr C}}$ (type V). Only type V leads to limits with additive stabilizers, and the absence of further types is due to the fact, shown in [@MR2002d:14084], that in characteristic zero only one kind of curves with small orbit has additive stabilizers (also cf. §\[appendix\]). \[GITrem\] A plane curve with small orbit is not GIT-stable. Whether it is strictly semistable or unstable is not directly related to the questions we are considering here. For example, the curves $xyz$ and $x^2yz$ have similar behavior from the point of view of this paper; yet the former is strictly semistable, the latter is unstable. Similarly, consider the union of a general quartic and a multiple line in general position. This has 8-dimensional orbit; it is stable in degree 5, strictly semistable in degree 6, and unstable in higher degrees. But the multiplicity of the line does not affect the behavior from our point of view in any substantial way. The lesson we draw from these examples is that there is no direct relation between the considerations in this paper and GIT. We should point out that the referee of this paper suggests otherwise, noting that closures of orbits are of interest in both contexts, curves with small orbits play a key role, and the mechanics of finding the limits is somewhat similar in the two situations. The referee asks: [*which marker germs would be relevant in a GIT analysis?*]{} We pass this question on to the interested reader. The terminology employed in the following matches the one in §2 of [@MR2001h:14068]; for example, a [*fan*]{} is the union of a star and a general line. In four of the five types, $\alpha=\alpha(0)$ is a rank-1 matrix and the line $\ker\alpha$ plays an important role; we will call this ‘the kernel line’. [**Type I.**]{} Assume ${{\mathscr C}}$ contains a line, defined by a linear polynomial $L$. Write a generator of the ideal of ${{\mathscr C}}$ as $$F(x,y,z)=L(x,y,z)^m G(x,y,z)$$ with $L$ not a factor of $G$. Type I limits are obtained by germs $$\alpha(t)=\alpha(0)+t\beta(t)\quad,$$ where $\alpha(0)$ has rank 2 and image the line defined by $L$. As we are assuming (cf. §\[rough\]) that $\det\alpha(t)\not\equiv 0$, the image of $\beta(t)$ is not contained in $\operatorname{im}\alpha(0)$, so that the limit $\lim_{t\to 0}L\circ\beta(t)$ is a well-defined line $\ell$. The limit $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ consists of the $m$-fold line $\ell$, and a star of lines through the point $\ker\alpha(0)$. This star reproduces projectively the tuple cut out on $L$ by the curve defined by $G$.  The limit is in general a fan, and degenerates to a star if the $m$-fold line $\ell$ contains the point $\ker\alpha(0)$. Fans and stars are studied in [@MR2002d:14084], and are the only kinds of curves with small orbit that consist of lines; they are items (1) through (5) in our classification of curves with small orbit, see §\[appendix\]. For types II—V we choose coordinates so that $p=(1:0:0)$ is a point of ${{\mathscr C}}$; for types II, IV, and V we further require that $z=0$ is a chosen component $\ell$ of the tangent cone to ${{\mathscr C}}$ at $p$. [**Type II.**]{} Assume that $p$ is a nonsingular, non-inflectional point of the support ${{{{\mathscr C}}'}}$ of ${{\mathscr C}}$, contained in a nonlinear component, with tangent line $z=0$. Let $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0 \\ 0 & t & 0 \\ 0 & 0 & t^2 \end{pmatrix}\quad.$$ Then the ideal of $\lim_{t\to 0}{{\mathscr C}}\circ \alpha(t)$ is generated by $$x^{d-2S}(y^2+\rho x z)^S\quad,$$ where $S$ is the multiplicity of the component in ${{\mathscr C}}$, and $\rho\ne 0$; that is, the limit consists of a (possibly multiple) nonsingular conic tangent to the kernel line, union (possibly) a multiple of the kernel line.  Such curves are items (6) and (7) in the classification reproduced in §\[appendix\]. The extra kernel line is present precisely when ${{\mathscr C}}$ is not itself a multiple nonsingular conic. [**Type III.**]{} Assume that $p$ is a singular point of ${{{{\mathscr C}}'}}$ of multiplicity $m$ in ${{\mathscr C}}$, with tangent cone supported on at least three lines. Let $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0 \\ 0 & t & 0 \\ 0 & 0 & t \end{pmatrix}\quad.$$ Then $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is a fan consisting of a star centered at $(1:0:0)$ and projectively equivalent to the tangent cone to ${{\mathscr C}}$ at $p$, and of a residual $(d-m)$-fold line supported on the kernel line $x=0$.  [**Type IV.**]{} Assume that $p$ is a singular or inflection point of the support of ${{\mathscr C}}$. Germs of type IV are determined by the choice of the line $\ell$ in the tangent cone to ${{\mathscr C}}$ at $p$, and by the choice of a side of a corresponding Newton polygon, with slope strictly between $-1$ and $0$. This procedure is explained in more detail in §\[details\]. Let $b<c$ be relatively prime positive integers such that $-b/c$ is the slope of the chosen side. Let $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix}\quad.$$ Then the ideal of $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is generated by a polynomial of the form $$x^{\overline e}y^fz^e \prod_{j=1}^S(y^c+\rho_j x^{c-b}z^b)\quad,$$ with $\rho_j \ne 0$. The number $S$ of ‘cuspidal’ factors in the limit curve is the number of segments cut out by the integer lattice on the selected side of the Newton polygon.  The germ listed above contributes a component of the PNC unless $b/c=1/2$ and the limit curve is supported on a conic union (possibly) the kernel line. The limit curves arising in this way are items (7) through (11) listed in §\[appendix\]. (In particular, the picture drawn above does not capture the possible complexity of the situation: several cuspidal curves may appear in the limit, as well as all lines of the basic triangle.) These limit curves are studied enumeratively in [@MR2002d:14083]. The limit curves contributing components to the PNC in this fashion are precisely the curves that contain nonlinear components and for which the maximal connected subgroup of the stabilizer of the union of the curve and the kernel line is the multiplicative group ${{\mathbb{G}}}_m$. [**Type V.**]{} Assume $p$ is a singular point of the support of ${{\mathscr C}}$. Germs of type V are determined by the choice of the line $\ell$ in the tangent cone to ${{\mathscr C}}$ at $p$, the choice of a formal branch $z=f(y)=\gamma_{\lambda_0}y^{\lambda_0}+\dots$ for ${{\mathscr C}}$ at $p$ tangent to $\ell$, and the choice of a certain ‘characteristic’ rational number $C>\lambda_0$ (assuming these choices can be made). This procedure is also explained in more detail in §\[details\]. For $a<b<c$ positive integers such that $\frac ca=C$ and $\frac ba= \frac{C-\lambda_0}2+1$, let $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0 \\ t^a & t^b & 0 \\ \underline{f(t^a)} & \underline{f'(t^a)t^b} & t^c \end{pmatrix}$$ where $\underline{\cdots}$ denotes the truncation modulo $t^c$. The integer $a$ is chosen to be the minimal one for which all entries in this germ are polynomials. Then $\lim_{t\to 0}{{\mathscr C}}\circ \alpha(t)$ is given by $$x^{d-2S}\prod_{i=1}^S\left(zx-\frac {\lambda_0(\lambda_0-1)}2 \gamma_{\lambda_0}y^2 -\frac{\lambda_0+C}2 \gamma_{\frac{\lambda_0+C}2}yx-\gamma_C^{(i)}x^2\right)\quad,$$ where $S$ and $\gamma_C^{(i)}$ are defined in §\[details\].  These curves consist of [*at least two*]{} ‘quadritangent’ conics—that is, nonsingular conics meeting at exactly one point—and (possibly) a multiple kernel line. (Again, the picture drawn here does not capture the subtlety of the situation: these limits may occur already for irreducible singularities.) These curves are item (12) in the list in §\[appendix\], and are studied enumeratively in [@MR2002d:14083], §4.1. They are precisely the curves for which the maximal connected subgroup of the stabilizer is the additive group ${{\mathbb{G}}}_a$. Details for types IV and V {#details} -------------------------- [*Type IV:*]{} Let $p=\operatorname{im}\alpha(0)$ be a singular or inflection point of the support of ${{\mathscr C}}$; choose a line in the tangent cone to ${{\mathscr C}}$ at $p$, and choose coordinates $(x:y:z)$ as before, so that $p=(1:0:0)$ and the selected line in the tangent cone has equation $z=0$. The [*Newton polygon*]{} for ${{\mathscr C}}$ in the chosen coordinates is the boundary of the convex hull of the union of the positive quadrants with origin at the points $(j,k)$ for which the coefficient of $x^iy^jz^k$ in the generator $F$ for the ideal of ${{\mathscr C}}$ is nonzero (see [@MR88a:14001], p. 380). The part of the Newton polygon consisting of line segments with slope strictly between $-1$ and $0$ does not depend on the choice of coordinates fixing the flag $z=0$, $p=(1:0:0)$. The limit curves are then obtained by choosing a side of the polygon with slope strictly between $-1$ and $0$, and setting to $0$ the coefficients of the monomials in $F$ [*not*]{} on that side. These curves are studied in [@MR2002d:14083]; typically, they consist of a union of cuspidal curves. The kernel line is part of the distinguished triangle of such a curve, and in fact it must be one of the distinguished tangents. Here is the Newton polygon for the curve of Example \[exone\], with respect to the point $(1:0:0)$ and the line $z=0$:  Setting to zero the coefficient of $z^3$ produces the limit $y(y^2+xz)$. [*Type V:*]{} Let $p=\operatorname{im}\alpha(0)$ be a singular point of the support of ${{\mathscr C}}$, and let $m$ be the multiplicity of ${{\mathscr C}}$ at $p$. Again choose a line in the tangent cone to ${{\mathscr C}}$ at $p$, and choose coordinates $(x:y:z)$ so that $p=(1:0:0)$ and $z=0$ is the selected line. We may describe ${{\mathscr C}}$ near $p$ as the union of $m$ ‘formal branches’, cf. §\[formalbranches\]; those that are tangent to the line $z=0$ (but not equal to it) may be written $$z=f(y)=\sum_{i\ge 0} \gamma_{\lambda_i} y^{\lambda_i}$$ with $\lambda_i\in {{\mathbb{Q}}}$, $1<\lambda_0<\lambda_1<\dots$, and $\gamma_{\lambda_0}\ne 0$. The choices made above determine a finite set of rational numbers, which we call the ‘characteristics’ for ${{\mathscr C}}$ (w.r.t. $p$ and the line $z=0$): these are the numbers $C$ for which there exist two branches ${{\mathscr B}}$, ${{\mathscr B}}'$ tangent to $z=0$ that agree modulo $y^C$, differ at $y^C$, and have $\lambda_0<C$. (Formal branches are called ‘pro-branches’ in [@MR2107253], Chapter 4; the numbers $C$ are ‘exponents of contact’.) Let $S$ be the number of branches that agree with ${{\mathscr B}}$ (and ${{\mathscr B}}'$) modulo $y^C$. The initial exponents $\lambda_0$ and the coefficients $\gamma_{\lambda_0}$, $\gamma_{\frac {\lambda_0+C}2}$ for these $S$ branches agree. Let $\gamma_C^{(1)},\dots,\gamma_C^{(S)}$ be the coefficients of $y^C$ in these branches (so that at least two of these numbers are distinct, by the choice of $C$). Then the limit is defined by $$x^{d-2S}\prod_{i=1}^S\left(zx-\frac {\lambda_0(\lambda_0-1)}2 \gamma_{\lambda_0}y^2 -\frac{\lambda_0+C}2 \gamma_{\frac{\lambda_0+C}2}yx-\gamma_C^{(i)}x^2\right)\quad.$$ This is a union of quadritangent conics with (possibly) a multiple of the distinguished tangent, which must be supported on the kernel line. The main theorem, and the structure of its proof {#proof} ------------------------------------------------ Simple dimension counts show that, for each type as listed in §\[germlist\], the union of the orbits of the marker centers is a set of dimension $7$ in ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8\subset {{\mathbb{P}}}^8\times{{\mathbb{P}}}^N$; hence it is a dense set in a component of the PNC. In fact, marker centers of type I, III, IV, and V have 7-dimensional orbit, so the corresponding components of the PNC are the orbit closures of these points. Type II marker centers are points $(\alpha, {{\mathscr X}})\in {{\mathbb{P}}}^8\times{{\mathbb{P}}}^N$, where $\alpha$ is a rank-1 matrix whose image is a general point of a nonlinear component of ${{\mathscr C}}$. The support of ${{\mathscr X}}$ contains a conic tangent to the kernel line; this gives a 1-parameter family of 6-dimensional orbits in ${{\mathbb{P}}}^8\times{{\mathbb{P}}}^N$, accounting for a component of the PNC. We can now formulate a more precise version of Theorem \[main\]: \[mainmain\] Let ${{\mathscr C}}\subset {{\mathbb{P}}}^2_{{\mathbb{C}}}$ be an arbitrary plane curve. The marker germs listed in §\[germlist\] determine components of the PNC for ${{\mathscr C}}$, as explained above. Conversely, all components of the PNC are determined by the marker germs of type I–V listed in §\[germlist\]. By the considerations in §\[ident\], this statement implies Theorem \[main\]. The first part of Theorem \[mainmain\] has been established above. In order to prove the second part, we will define a simple notion of ‘equivalence’ of germs (Definition \[equivgermsnew\]), such that, in particular, equivalent germs $\alpha(t)$ lead to the same component of the PNC. We will show that any given germ $\alpha(t)$ centered at a point of ${{\mathscr S}}$ either is equivalent (after a parameter change, if necessary) to one of the marker germs, or its lift in ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ meets the PNC at a point of $R$ (cf. Remark \[eluding\]) or of the boundary of the orbit of a marker center. In the latter cases, the center of the lift varies in a locus of dimension $<7$, hence such germs do not contribute components to the PNC. The following lemma allows us to identify easily limits in the intersection of $R$ and the PNC. \[rank2lemma\] Assume that $\alpha(0)$ has rank $1$. If $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ is a star with center on $\ker\alpha(0)$, then it is a rank-2 limit. Assume ${{\mathscr X}}=\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ is a star with center on $\ker\alpha(0)$. We may choose coordinates so that $x=0$ is the kernel line, and the generator for the ideal of ${{\mathscr X}}$ is a polynomial in $x,y$ only. If $$\alpha(t)=\begin{pmatrix} a_{11}(t) & a_{12}(t) & a_{13}(t) \\ a_{21}(t) & a_{22}(t) & a_{23}(t) \\ a_{31}(t) & a_{32}(t) & a_{33}(t) \end{pmatrix}\quad,$$ then ${{\mathscr X}}=\lim_{t\to 0}{{\mathscr C}}\circ\beta(t)$ for $$\beta(t)=\begin{pmatrix} a_{11}(t) & a_{12}(t) & 0 \\ a_{21}(t) & a_{22}(t) & 0 \\ a_{31}(t) & a_{32}(t) & 0 \end{pmatrix}\quad.$$ Since $\alpha(0)$ has rank 1 and kernel line $x=0$, $$\alpha(0)=\begin{pmatrix} a_{11}(0) & 0 & 0 \\ a_{21}(0) & 0 & 0 \\ a_{31}(0) & 0 & 0 \end{pmatrix}=\beta(0)\quad.$$ Now $\beta(t)$ is contained in the rank-2 locus, verifying the assertion. A limit $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ as in this lemma will be called a ‘kernel star’. Sections \[setth\] and \[typeVcomps\] contain the successive reductions bringing a given germ $\alpha(t)$ centered at a point of ${{\mathscr S}}$ into one of the forms given in §\[germlist\], or establishing that it does not contribute a component of the PNC. This analysis will conclude the proof of Theorem \[mainmain\]. Two examples {#twoexamples} ------------ The two examples that follow illustrate the main result, and show that components of all types may already occur on curves of degree 4. Simple translations are used to bring the marker germs provided by §\[germlist\] into the form given here. Consider the reducible quartic ${{\mathscr C}}_1$ given by the equation $$(y+z)(xy^2+xyz+xz^2+y^2z+yz^2)=0\,.$$ It consists of an irreducible cubic with a node at $(1:0:0)$ and a line through the node and the inflection point $(0:1:-1)$. The other inflection points are $(0:1:0)$ and $(0:0:1)$. According to Theorem \[mainmain\] and the list in §\[germlist\], the PNC for ${{\mathscr C}}_1$ has one component of type I, one component of type II, one component of type III, corresponding to the triple point $(1:0:0)$, and four components of type IV: one for each of the inflection points $(0:1:0)$ and $(0:0:1)$, one for the node $(0:1:-1)$ and the tangent line $x=y+z$ to the cubic at that point, and one for the triple point $(1:0:0)$ and the two lines in the tangent cone $y^2+yz+z^2=0$ to the cubic at that point. Here is a schematic drawing of the curve, with features marked by the corresponding types (four points are marked as $\text{IV}_i$, since four different points are responsible for the presence of type IV components):  A list of representative marker germs is as follows: $${\rm I}: \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & -1 & t \end{pmatrix};\quad {\rm II}: \begin{pmatrix} 2 & 0 & 0\\ -3 & t & 0\\ 6 & 0 & t^2 \end{pmatrix};\quad {\rm III}: \begin{pmatrix} 1 & 0 & 0\\ 0 & t & 0\\ 0 & 0 & t \end{pmatrix};$$ and, for type IV: $$\begin{pmatrix} t & 0 & 0\\ 0 & 1 & 0\\ -t & 0 & t^3 \end{pmatrix},\quad \begin{pmatrix} t & 0 & 0\\ -t & t^3 & 0\\ 0 & 0 & 1 \end{pmatrix},\quad \begin{pmatrix} t & 0 & 0\\ 0 & 1 & 0\\ t & -1 & t^3 \end{pmatrix},\quad \begin{pmatrix} 1 & 0 & 0\\ 0 & \rho t & 0\\ 0 & t & t^2 \end{pmatrix},\quad \begin{pmatrix} 1 & 0 & 0\\ 0 & \rho^2 t & 0\\ 0 & t & t^2 \end{pmatrix}$$ (where $\rho$ is a primitive third root of unity). The latter two marker germs have the same center and lead to projectively equivalent limits, hence they contribute the same component of the PNC. The corresponding limits of ${{\mathscr C}}_1$ are given by $$xy^2z,\quad x^2(8y^2-9xz),\quad x(y+z)(y^2+yz+z^2),\quad y(y^2z+x^3),\quad z(yz^2+x^3),$$$$x(y^2z-x^3),\quad y^2(y^2-(\rho+2)xz),\quad \text{and} \quad y^2(y^2-(\rho^2+2)xz),$$ respectively: a triangle with one line doubled, a conic with a double tangent line, a fan with star centered at $(1:0:0)$, a cuspidal cubic with its cuspidal tangent (two limits), a cuspidal cubic with the line through the cusp and the inflection point, and finally a conic with a double transversal line (two limits). Schematically, the limits may be represented as follows:  According to Theorem \[main\], all limits of ${{\mathscr C}}_1$ (other than stars of lines) are projectively equivalent to one of these curves, or to limits of them (cf. §\[boundary\]). \[extwo\] Consider the irreducible quartic ${{\mathscr C}}_2$ given by the equation $$(y^2-xz)^2=y^3z.$$ It has a ramphoid cusp at $(1:0:0)$, an ordinary cusp at $(0:0:1)$, and an ordinary inflection point at $(3^3 5{:}{-}2^6 3^2{:}{-}2^{12})$; there are no other singular or inflection points. The PNC for ${{\mathscr C}}_2$ has one component of type II, two components of type IV, corresponding to the inflection point and the ordinary cusp, and one component of type V, corresponding to the ramphoid cusp. (Note that there is no component of type IV corresponding to the ramphoid cusp.) Representative marker germs for the latter two components are $${\rm IV}: \begin{pmatrix} 0 & t^3 & 0\\ t^2 & 0 & 0\\ 0 & 0 & 1 \end{pmatrix}\quad{\rm and}\quad {\rm V}: \begin{pmatrix} 1 & 0 & 0\\ t^4 & t^5 & 0\\ t^8 & 2t^9 & t^{10} \end{pmatrix}$$ and the corresponding limits of ${{\mathscr C}}_2$ are given by $$z(y^2z-x^3)\quad{\rm and}\quad(y^2-xz+x^2)(y^2-xz-x^2),$$ respectively: a cuspidal cubic with its inflectional tangent and a pair of quadritangent conics. The connected component of the stabilizer of the latter limit is the additive group. The germ with entries $1$, $t$, and $t^2$ on the diagonal and zeroes elsewhere leads to the limit $(y^2-xz)^2$, a double conic; its orbit is too small to produce an additional component of type IV. Proof of the main theorem: key reductions and components of type I–IV {#setth} ===================================================================== Outline {#preamble} ------- In this section we show that, for a given curve ${{\mathscr C}}$, any germ $\alpha(t)$ contributing to the PNC is ‘equivalent’ (up to a coordinate and parameter change, if necessary) to a marker germ as listed in §\[germlist\]. As follows from §\[rough\] and Lemma \[PNCtolimits\], we may assume that $\det\alpha(t)\not\equiv 0$ and that the image of $\alpha(0)$ is contained in ${{\mathscr C}}$. Observe that if the center $\alpha(0)$ has rank 2 and is a point of ${{\mathscr S}}$, then $\alpha(t)$ is already of the form given in §\[germlist\], Type I; it is easy to verify that the limit is then as stated there. This determines completely the components of type I. Thus, we will assume in most of what follows that $\alpha(0)$ has rank 1, and its image is a point of ${{\mathscr C}}$. ### Equivalence of germs \[equivgermsnew\] Two germs $\alpha(t)$, $\beta(t)$ are [*equivalent*]{} if $\beta(t\nu(t))\equiv \alpha(t)\circ m(t)$, with $\nu(t)$ a unit in ${{\mathbb{C}}}[[t]]$, and $m(t)$ a germ such that $m(0)=I$ (the identity). For example: if $n(t)$ is a ${{\mathbb{C}}}[[t]]$-valued point of ${\text{\rm PGL}}(3)$, then $\alpha(t)\circ n(t)$ is equivalent to $\alpha(t)\circ n(0)$. We will frequently encounter this situation. \[stequivnew\] Let ${{\mathscr C}}$ be any plane curve, with defining homogeneous ideal $(F(x,y,z))$. If $\alpha(t)$, $\beta(t)$ are equivalent germs, then the initial terms in $F\circ\alpha(t)$, $F\circ\beta(t)$ coincide up to a nonzero multiplicative constant; in particular, the limits $\lim_{t\to 0}{{\mathscr C}}\circ \alpha(t)$, $\lim_{t\to 0}{{\mathscr C}}\circ \beta(t)$ are equal. If $\alpha$ and $\beta$ are equivalent germs, note that $\alpha(0)= \beta(0)$; by Lemma \[stequivnew\] it follows that, for every curve ${{\mathscr C}}$, $\alpha$ and $\beta$ lift to germs in ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ centered at the same point. ### Summary of the argument The general plan for the rest of this section is as follows: we will show that every contributing $\alpha(t)$ centered at a rank-1 matrix is equivalent (in suitable coordinates, and possibly up to a parameter change) to one of the form $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0\\ 0 & t^b & 0\\ 0 & 0 & t^c \end{pmatrix}\quad\text{or}\quad \begin{pmatrix} 1 & 0 & 0 \\ t^a & t^b & 0 \\ \underline{f(t^a)} & \underline{f'(t^a) t^b} & t^c\end{pmatrix} \quad,$$ where $b\le c$ resp. $a<b\le c$ are positive integers, $z=f(y)$ is a formal branch for ${{\mathscr C}}$ at $(1:0:0)$, and $\underline{\cdots}$ denotes the truncation modulo $t^c$ (cf. §\[germlist\] and §\[details\]). The main theorem will follow from further analyses of these forms, identifying which do [*not*]{} contribute components to the PNC, and leading to the restrictions explained in §\[germlist\] and §\[details\]. Specifically, the germs on the left lead to components of type II, III, and IV (§\[1PS\]); those on the right lead to components of type V. The latter germs require a subtle study, performed in §\[typeVcomps\], leading to the definition of ‘characteristics’ and to the description given in §\[details\] (cf. Proposition \[typeV\]). Linear algebra -------------- ### This subsection is devoted to the proof of the following result. \[keyreduction\] Every germ as specified in §\[preamble\] is equivalent to one which, up to a parameter change, has matrix representation $$\begin{pmatrix} 1 & 0 & 0\\ q(t) & t^b & 0\\ r(t) & s(t)t^b & t^c \end{pmatrix}$$ in suitable coordinates, with $1\le b\le c$ and $q,r,s$ polynomials such that $\deg(q)<b$, $\deg(r)<c$, $\deg(s)<c-b$, and $q(0)=r(0)=s(0)=0$. A refined version of this statement is given in Lemma \[faber\]. We will deal with $3\times 3$ matrices with entries in ${{\mathbb{C}}}[[t]]$, that is, ${{\mathbb{C}}}[[t]]$-valued points of $\operatorname{Hom}(V,W)$, for $V$, $W$ 3-dimensional complex vector spaces with chosen bases. Every such matrix $\alpha(t)$ determines a germ in ${{\mathbb{P}}}^8$. A generator $F$ of the ideal of ${{\mathscr C}}$ will be viewed as an element of ${\text{\rm Sym}}^d W^*$, for $d=\deg{{\mathscr C}}$; the composition $F\circ\alpha(t)$, a ${{\mathbb{C}}}[[t]]$-valued point of ${\text{\rm Sym}}^d V^*$, generates the ideal of ${{\mathscr C}}\circ\alpha(t)$. We will call matrices of the form $$\lambda(t)=\begin{pmatrix} t^a & 0 & 0\\ 0 & t^b & 0\\ 0 & 0 & t^c \end{pmatrix}$$ ‘1-PS’, as they correspond to 1-parameter subgroups of ${\text{\rm PGL}}(3)$. We will say that two matrices $\alpha(t)$, $\beta(t)$ are equivalent if the corresponding germs are equivalent in the sense of Definition \[equivgermsnew\]. The following lemma will allow us to simplify matrix expressions of germs up to equivalence. Define the degree of the zero polynomial to be $-\infty$. \[MPI\] Let $$h_1(t) =\begin{pmatrix} u_1 & b_1 & c_1 \\ a_2 & u_2 & c_2 \\ a_3 & b_3 & u_3 \end{pmatrix}$$ be a matrix with entries in ${{\mathbb{C}}}[[t]]$, such that $h_1(0)=I$, and let $a\le b\le c$ be integers. Then $h_1(t)$ can be written as a product $h_1(t)=h(t)\cdot j(t)$, with $$h(t)=\begin{pmatrix} 1 & 0 & 0 \\ q & 1 & 0 \\ r & s & 1 \end{pmatrix} \quad,\quad j(t)=\begin{pmatrix} v_1 & e_1 & f_1 \\ d_2 & v_2 & f_2 \\ d_3 & e_3 & v_3 \end{pmatrix}$$ where $q$, $r$, $s$ are [*polynomials,*]{} satisfying 1. $h(0)=j(0)=I$; 2. $\deg(q)<b-a$, $\deg(r)<c-a$, $\deg(s)<c-b$; 3. \[refMPI\] $d_2\equiv0\pmod{t^{b-a}}$, $d_3\equiv0\pmod{t^{c-a}}$, $e_3\equiv0\pmod{t^{c-b}}$. Necessarily $v_1=u_1, e_1=b_1$ and $f_1=c_1$. Use division with remainder to write $ v_1^{-1}a_2=D_2t^{b-a}+q $ with $\deg(q)<b-a$, and let $d_2=v_1D_2t^{b-a}$ (so that $qv_1+d_2=a_2$). This defines $q$ and $d_2$, and uniquely determines $v_2$ and $f_2$. (Note that $q(0)=d_2(0)=f_2(0)=0$ and that $v_2(0)=1$.) Similarly, we let $r$ be the remainder of $(v_1v_2-e_1d_2)^{-1}(v_2a_3-d_2b_3)$ after division by $t^{c-a}$; and $s$ be the remainder of $(v_1v_2-e_1d_2)^{-1}(v_1b_3-e_1a_3)$ after division by $t^{c-b}$. Then $\deg(r)<c-a$, $\deg(s)<c-b$ and $r(0)=s(0)=0$; moreover, we have $$v_1r+d_2s\equiv a_3\pmod{t^{c-a}},\qquad e_1r+v_2s\equiv b_3 \pmod{t^{c-b}},$$ so we take $d_3=a_3-v_1r-d_2s$ and $e_3=b_3-e_1r-v_2s$. This defines $r$, $s$, $d_3$ and $e_3$, and uniquely determines $v_3$. \[MPIcorol\] Let $h_1(t)$ be a matrix with entries in ${{\mathbb{C}}}[[t]]$, such that $h_1(0)=I$, and let $a\le b\le c$ be integers. Then there exists a constant invertible matrix $L$ such that the product $$h_1(t)\cdot \begin{pmatrix} t^a & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix}$$ is equivalent to $$\begin{pmatrix} 1 & 0 & 0 \\ q & 1 & 0 \\ r & s & 1 \end{pmatrix}\cdot \begin{pmatrix} t^a & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix}\cdot L$$ where $q$, $r$, $s$ are polynomials such that $\deg(q)<b-a$, $\deg(r)<c-a$, $\deg(s)<c-b$, and $q(0)=r(0)=s(0)=0$. With notation as in Lemma \[MPI\] we have $$j(t)\cdot \begin{pmatrix} t^a & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix} = \begin{pmatrix} v_1 t^a & e_1 t^b & f_1 t^c \\ d_2 t^a & v_2 t^b & f_2 t^c \\ d_3 t^a & e_3 t^b & v_3 t^c \end{pmatrix} = \begin{pmatrix} t^a & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix} \cdot \ell(t)\quad,$$ with $$\ell(t)=\begin{pmatrix} v_1 & e_1 t^{b-a} & f_1 t^{c-a} \\ d_2 t^{a-b} & v_2 & f_2 t^{c-b} \\ d_3 t^{a-c} & e_3 t^{b-c} & v_3 \end{pmatrix}\quad.$$ By (\[refMPI\]) in Lemma \[MPI\], $\ell(t)$ has entries in ${{\mathbb{C}}}[[t]]$ and is invertible; in fact, $L=\ell(0)$ is lower triangular, with 1’s on the diagonal. Therefore Lemma \[MPI\] gives $$h_1(t)\cdot \begin{pmatrix} t^a & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix}= \begin{pmatrix} 1 & 0 & 0 \\ q & 1 & 0 \\ r & s & 1 \end{pmatrix}\cdot \begin{pmatrix} t^a & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix}\cdot \ell(t)\quad,$$ from which the statement follows. The gist of this result is that, up to equivalence, matrices ‘to the left of a 1-PS’ and centered at the identity may be assumed to be lower triangular, and to have polynomial entries, with controlled degrees. ### We denote by $v$ the order of vanishing at $0$ of a polynomial or power series; we define $v(0)$ to be $+\infty$. The following statement is a refined version of Proposition \[keyreduction\]. \[faber\] Let $\alpha(t)$ be a $\/3\times 3$ matrix with entries in ${{\mathbb{C}}}[[t]]$, such that $\alpha(0)\ne 0$ and $\det \alpha(t)\not\equiv 0$. Then there exist constant invertible matrices $H$, $M$ such that $\alpha(t)$ is equivalent to $$\beta(t)=H\cdot \begin{pmatrix} 1 & 0 & 0 \\ q & 1 & 0 \\ r & s & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix} \cdot M\quad,$$ with - $b\le c$ nonnegative integers, $q,r,s$ polynomials; - $\deg(q)<b$, $\deg(r)<c$, $\deg(s)<c-b$; - $q(0)=r(0)=s(0)=0$. If, further, $b=c$ and $q$, $r$ are not both zero, then we may assume that $v(q)<v(r)$. Finally, if $q(t)\not\equiv 0$ then we may choose $q(t)=t^a$, with $a=v(q)<b$ (and thus $a<v(r)$ if $b=c$). By standard diagonalization of matrices over Euclidean domains, every $\alpha(t)$ as in the statement can be written as a product $$h_0(t)\cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix} \cdot k(t)\quad,$$ where $b\le c$ are nonnegative integers, and $h_0(t)$, $k(t)$ are invertible (over ${{\mathbb{C}}}[[t]]$). Letting $H=h_0(0)$, $h_1(t)=H^{-1}\cdot h_0(t)$, and $K=k(0)$, this shows that $\alpha(t)$ is equivalent to $$H\cdot h_1(t) \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix} \cdot K$$ with $h_1(0)=I$, and $K$ constant and invertible. By Corollary \[MPIcorol\], this matrix is equivalent to $$\beta(t)=H\cdot \begin{pmatrix} 1 & 0 & 0 \\ q & 1 & 0 \\ r & s & 1 \end{pmatrix}\cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix} \cdot L\cdot K$$ with $L$ invertible, and $q$, $r$, $s$ polynomials satisfying the needed conditions. Letting $M=L\cdot K$ gives the statement in the case $b<c$. If $b=c$, then the condition that $\deg s<c-b=0$ forces $s\equiv 0$. When $q$ and $r$ are not both $0$, the inequality $v(q)<v(r)$ may be obtained by conjugating with a constant matrix. If $q(t)\not\equiv 0$ and $v(q)=a$, then we can extract its $a$-th root as a power series. It follows that there exists a unit $\nu(t)\in{{\mathbb{C}}}[[t]]$ such that $q(t\nu(t))=t^a$. Therefore, $$\beta(t\nu(t))=H\cdot \begin{pmatrix} 1 & 0 & 0 \\ t^a & 1 & 0 \\ r(t\nu(t)) & s(t\nu(t)) & 1 \end{pmatrix}\cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & \nu(t)^b & 0 \\ 0 & 0 & \nu(t)^c \end{pmatrix} \cdot M\quad.$$ Another application of Corollary \[MPIcorol\] allows us to truncate the power series $r(t\nu(t))$ and $s(t\nu(t))$ to obtain polynomials $\underline r$, $\underline s$ satisfying the same conditions as $r$, $s$, at the price of multiplying to the right of the 1-PS by a constant invertible matrix $\underline K$: that is, $\beta(t\nu(t))$ (and hence $\alpha(t)$) is equivalent to $$H\cdot \begin{pmatrix} 1 & 0 & 0 \\ t^a & 1 & 0 \\ \underline r & \underline s & 1 \end{pmatrix}\cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix} \cdot \left[ \underline K \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & \nu(0)^b & 0 \\ 0 & 0 & \nu(0)^c \end{pmatrix} \cdot M \right]\quad.$$ Renaming $r=\underline r$, $s=\underline s$, and absorbing the factors on the right into $M$ completes the proof of Lemma \[faber\]. The matrices $H$, $M$ appearing in Lemma \[faber\] may be omitted by changing the bases of $W$ and $V$ accordingly. Further, we may assume that $b>0$, since we are already reduced to the case in which $\alpha(0)$ is a rank-1 matrix. This concludes the proof of Proposition \[keyreduction\]. In what follows, we will assume that $\alpha$ is a germ in the standard form given above. Components of type II, III, and IV {#1PS} ---------------------------------- It will now be convenient to switch to affine coordinates centered at the point $(1:0:0)$. We write $$F(1:y:z)=F_m(y,z)+F_{m+1}(y,z)+\cdots +F_d(y,z)\quad,$$ with $d=\deg {{\mathscr C}}$, $F_i$ homogeneous of degree $i$, and $F_m\ne 0$. Thus, $F_m(y,z)$ generates the ideal of the [*tangent cone*]{} of ${{\mathscr C}}$ at $p$. We first consider the case in which $q=r=s=0$, that is, in which $\alpha(t)$ is itself a 1-PS: $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0\\ 0 & t^b & 0\\ 0 & 0 & t^c \end{pmatrix}$$ with $1\le b \le c$. Also, we may assume that $b$ and $c$ are coprime: this only amounts to a reparametrization of the germ by $t \mapsto t^{1/gcd(b,c)}$; the new germ is not equivalent to the old one in terms of Definition \[equivgermsnew\], but clearly achieves the same limit. Germs with $b=c$ $(=1)$ lead to components of type III, cf. §\[germlist\] (also cf. [@MR2001h:14068], §2, Fact 4(i)): \[tgcone\] If $q=r=s=0$ and $b=c$, then $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is a fan consisting of a star projectively equivalent to the tangent cone to ${{\mathscr C}}$ at $p$, and of a residual $(d-m)$-fold line supported on $\ker\alpha$. The composition $F\circ\alpha(t)$ is $$F(x:t^by:t^bz)=t^{bm}x^{d-m}F_m(y,z)+t^{b(m+1)}x^{d-(m+1)} F_{m+1}(y,z)+\cdots+ t^{dm}F_d(y,z)\quad.$$ By definition of limit, $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ has ideal $(x^{d-m}F_m(y,z))$, proving the assertion. The case $b<c$ corresponds to the germs of type II and type IV in §\[germlist\]. We have to prove that contributing germs of this type are precisely those satisfying the further restrictions specified there: specifically, $-b/c$ must be a slope of one of the Newton polygons for ${{\mathscr C}}$ at the point. We first show that $z=0$ must be a component of the tangent cone: If $q=r=s=0$ and $b<c$, and $z=0$ is not contained in the tangent cone to ${{\mathscr C}}$ at $p$, then $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is a rank-2 limit. The condition regarding $z=0$ translates into $F_m(1,0)\ne 0$. Applying $\alpha(t)$ to $F$, we find: $$F(x:t^by:t^cz)=t^{bm}x^{d-m} F_m(y,t^{c-b}z)+t^{b(m+1)} x^{d-(m+1)} F_{m+1}(y,t^{c-b}z)+\cdots$$ Since $F_m(1,0)\ne 0$, the dominant term on the right-hand-side is $x^{d-m}y^m$. This proves the assertion, by Lemma \[rank2lemma\]. Components of the PNC that arise due to 1-PS with $b<c$ may be described in terms of the [*Newton polygon*]{} for ${{\mathscr C}}$ at $(0,0)$ relative to the line $z=0$, which we may now assume to be part of the tangent cone to ${{\mathscr C}}$ at $p$. The Newton polygon for ${{\mathscr C}}$ in the chosen coordinates is the boundary of the convex hull of the union of the positive quadrants with origin at the points $(j,k)$ for which the coefficient of $x^iy^jz^k$ in the equation for ${{\mathscr C}}$ is nonzero (see [@MR88a:14001], p. 380). The part of the Newton polygon consisting of line segments with slope strictly between $-1$ and $0$ does not depend on the choice of coordinates fixing the flag $z=0$, $p=(0,0)$. \[Newtonsides\] Assume $q=r=s=0$ and $b<c$. - If $-b/c$ is not a slope of the Newton polygon for ${{\mathscr C}}$, then the limit $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is supported on (at most) three lines; these curves do not contribute components to the PNC. - If $-b/c$ is a slope of a side of the Newton polygon for ${{\mathscr C}}$, then the ideal of the limit $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ is generated by the polynomial obtained by setting to $0$ the coefficients of the monomials in $F$ [*not*]{} on that side. Such polynomials are of the form $$G=x^{\overline e}y^fz^e \prod_{j=1}^S(y^c+\rho_j x^{c-b}z^b) \quad.$$ For the first assertion, simply note that under the stated hypotheses only one monomial in $F$ is dominant in $F\circ\alpha(t)$; hence, the limit is supported on the union of the coordinate axes. A simple dimension count shows that such limits may span at most a 6-dimensional locus in ${{\mathbb{P}}}^8\times{{\mathbb{P}}}^N$, and it follows that such germs do not contribute a component to the PNC. For the second assertion, note that the dominant terms in $F\circ\alpha(t)$ are precisely those on the side of the Newton polygon with slope equal to $-b/c$. It is immediate that the resulting polynomial can be factored as stated. If the point $p=(1:0:0)$ is a singular or an inflection point of the support of ${{\mathscr C}}$, and $b/c\ne 1/2$, we find the type IV germs of §\[germlist\]; also cf. [@MR2001h:14068], §2, Fact 4(ii). The number $S$ of ‘cuspidal’ factors in $G$ is the number of segments cut out by the integer lattice on the selected side of the Newton polygon. If $b/c=1/2$, then a dimension count shows that the corresponding limit will contribute a component to the PNC (of type IV) unless it is supported on a conic union (possibly) the kernel line. If $p$ is a [*nonsingular, non-inflectional*]{} point of the support of ${{\mathscr C}}$, then the Newton polygon consists of a single side with slope $-1/2$; these are the type II germs of §\[germlist\]. Also cf. [@MR2001h:14068], Fact 2(ii). Components of type V {#typeVcomps} ==================== Having dealt with the 1-PS case in the previous section, we may now assume that $$\tag{$\dagger$} \alpha(t)=\begin{pmatrix} 1 & 0 & 0\\ q(t) & t^b & 0\\ r(t) & s(t)t^b & t^c \end{pmatrix}$$ with the conditions listed in Lemma \[faber\], and further [*such that $q,r$, and $s$ do not all vanish identically.*]{} Our task is to show that contributing germs of this kind must in fact be of the form specified in §\[germlist\] and §\[details\]. We will show that a germ $\alpha(t)$ as above leads to a rank-2 limit (and hence does not contribute a component to the PNC) unless $\alpha(t)$ and certain formal branches (cf. [@MR88a:14001] and [@MR1836037], Chapter 6 and 7) of the curve are closely related. More precisely, we will prove the following result. \[standardform\] Let $\alpha(t)$ be as specified above, and assume that $\lim_{t\to 0}\mathcal C\circ\alpha(t)$ is not a rank-2 limit. Then ${{\mathscr C}}$ has a formal branch $z=f(y)$, tangent to $z=0$, such that $\alpha$ is equivalent to a germ $$\begin{pmatrix} 1 & 0 & 0 \\ t^a & t^b & 0 \\ \underline{f(t^a)} & \underline{f'(t^a) t^b} & t^c\end{pmatrix} \quad,$$ with $a<b<c$ positive integers. Further, it is necessary that $\frac ca\le \lambda_0+2(\frac ba-1)$, where $\lambda_0>1$ is the (fractional) order of the branch. For a power series $g(t)$ with fractional exponents, we write here $\underline{g(t)}$ for its truncation modulo $t^c$. (The truncations appearing in the statement are in fact polynomials.) The proof of the proposition requires the analysis of several cases. We will first show that under the hypothesis that $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is not a rank-2 limit we may assume that $q(t)\not\equiv 0$, and this will allow us to replace it with a power of $t$; next, we will deal with the $b=c$ case; and finally we will see that if $b<c$ and $\alpha(t)$ is not in the stated form, then the limit of every irreducible branch of ${{\mathscr C}}$ is a star with center $(0:0:1)$. This will imply that the limit of ${{\mathscr C}}$ is a kernel star in this case, proving the assertion by Lemma \[rank2lemma\]. This analysis is carried out in §\[formalbranches\]–\[Eop\]. In §\[charaV\] we determine germs of the form given in Proposition \[standardform\] that can lead to components of type V, obtaining the description given in §\[germlist\]. In §\[quadritangent\] we complete the proof of Theorem \[mainmain\], recovering the description given in §\[details\] of the limits obtained along these germs. Limits of formal branches {#formalbranches} ------------------------- In this subsection we recall the notion of formal branches and define the ‘limit’ of a formal branch. The limit of a curve ${{\mathscr C}}$ will be expressed in terms of the limits of its formal branches. Choose affine coordinates $(y,z)=(1:y:z)$ so that $p=(0,0)$, and let $\Phi(y,z)=F(1:y:z)$ be the generator for the ideal of ${{\mathscr C}}$ in these coordinates. Decompose $\Phi(y,z)$ in ${{\mathbb{C}}}[[y,z]]$: $$\Phi(y,z)=\Phi_1(y,z)\cdot\cdots\cdot \Phi_r(y,z)$$ with $\Phi_i(y,z)$ irreducible power series. These define the [*irreducible branches*]{} of ${{\mathscr C}}$ at $p$. Each $\Phi_i$ has a unique tangent line at $p$; if this tangent line is [*not*]{} $y=0$, by the Weierstrass preparation theorem we may write (up to a unit in ${{\mathbb{C}}}[[y,z]]$) $\Phi_i$ as a monic polynomial in $z$ with coefficients in ${{\mathbb{C}}}[[y]]$, of degree equal to the multiplicity $m_i$ of the branch at $p$ (cf. for example [@MR1836037], §6.7). If $\Phi_i$ [*is*]{} tangent to $y=0$, we may likewise write it as a polynomial in $y$ with coefficients in ${{\mathbb{C}}}[[z]]$; [*mutatis mutandis,*]{} the discussion which follows applies to this case as well. Concentrating on the first case, let $$\Phi_i(y,z)\in {{\mathbb{C}}}[[y]][z]$$ be a monic polynomial of degree $m_i$, defining an irreducible branch of ${{\mathscr C}}$ at $p$, not tangent to $y=0$. Then $\Phi_i$ splits (uniquely) as a product of linear factors over the ring ${{\mathbb{C}}}[[y^*]]$ of power series with [*rational nonnegative*]{} exponents: $$\Phi_i(y,z)=\prod_{j=1}^{m_i} \left(z-f_{ij}(y)\right)\quad,$$ with each $f_{ij}(y)$ of the form $$f(y)=\sum_{k\ge 0} \gamma_{\lambda_k} y^{\lambda_k}$$ with $\lambda_k\in {{\mathbb{Q}}}$, $1\le \lambda_0<\lambda_1<\dots$, and $\gamma_{\lambda_k}\ne 0$. We call each such $z=f(y)$ a [*formal branch*]{} of ${{\mathscr C}}$ at $p$. The branch is [*tangent to $z=0$*]{} if the dominating exponent $\lambda_0$ is $>1$. The terms $z-f_{ij}(y)$ in this decomposition are the Puiseux series for ${{\mathscr C}}$ at $p$. We will need to determine $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ as a union of ‘limits’ of the individual formal branches at $p$. The difficulty here resides in the fact that we cannot perform an arbitrary change of variable in a power series with fractional exponents. In the case in which we will need to do this, however, $\alpha(t)$ will have the following special form: $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0 \\ t^a & t^b & 0 \\ r(t) & s(t)t^b & t^c \end{pmatrix}$$ with $a<b\le c$ positive integers and $r(t)$, $s(t)$ polynomials (satisfying certain restrictions, which are immaterial here). The difficulty we mentioned may be circumvented by the following [*ad hoc*]{} definition. \[branchlimit\] The [*limit*]{} of a formal branch $z=f(y)$, along a germ $\alpha(t)$ as above, is defined by the dominant term in $$(r(t)+s(t)t^by+t^cz)-f(t^a)-f'(t^a)t^by-f''(t^a)t^{2b}\frac{y^2}2 -\cdots$$ where $f'(y)=\sum \gamma_{\lambda_k}\lambda_k y^{\lambda_k-1}$ etc. By ‘dominant term’ we mean the coefficient of the lowest power of $t$ after cancellations. This coefficient is a polynomial in $y$ and $z$, giving the limit of the branch according to our definition. This definition behaves as expected: that is, the limit of ${{\mathscr C}}$ is the union of the limits of its individual formal branches. This fact will be used several times in the rest of the paper, and may be formalized as follows. \[limitlemma\] Let $\Phi(y,z)\in {{\mathbb{C}}}[[y]][z]$ be a monic polynomial, $$\Phi(y,z)=\prod_i \left(z-f_i(y)\right)$$ a decomposition over ${{\mathbb{C}}}[[y^*]]$, and let $\alpha(t)$ have the special form above. Then the dominant term in $\Phi\circ\alpha(t)$ is the product of the limits of the formal branches $z=f_i(y)$ along $\alpha$, as in Definition \[branchlimit\]. Let $m$ be the multiplicity of ${{\mathscr C}}$ at $p=(0,0)$. For simplicity, we assume that no branches of ${{\mathscr C}}$ are tangent to the line $y=0$, leaving to the reader the necessary adjustments in the presence of such branches. We write the generator $F$ for the ideal of ${{\mathscr C}}$ as a product of formal branches $F =\prod_{i=1}^m (z-f_i(y))$. We will focus on the formal branches that are tangent to the line $z=0$, which may be written explicitly as $$z=f(y)=\sum_{k\ge 0} \gamma_{\lambda_k} y^{\lambda_k}$$ with $\lambda_k\in {{\mathbb{Q}}}$, $1<\lambda_0<\lambda_1<\dots$, and $\gamma_{\lambda_k}\ne 0$. Now we begin the proof of Proposition \[standardform\]. Reduction to $q\ne 0$ --------------------- The first remark is that, under the assumptions that $q$, $r$, and $s$ do not all vanish, we may in fact assume that $q(t)$ is not identically zero. \[qnot0\] If $\alpha(t)$ is as in $(\dagger)$, and $q= 0$, then $\lim_{t\to 0}\mathcal C\circ\alpha(t)$ is a rank-2 limit. (Sketch.) Assume $q=0$, and study the action of $\alpha(t)$ on individual monomials $x^Ay^Bz^C$ in an equation for ${{\mathscr C}}$: $$m_{ABC}:=x^A y^B (r(t) x+s(t) t^b y+t^c z)^C t^{bB}\quad.$$ There are various possibilities for the vanishing of $r$ and $s$, but the dominant terms in $m_{ABC}$ are always kernel stars, which are rank-2 limits by Lemma \[rank2lemma\]. Reduction to $b<c$ ------------------ By Lemma \[qnot0\] and the last part of Lemma \[faber\] we may replace $\alpha(t)$ with an equivalent germ $$\begin{pmatrix} 1 & 0 & 0 \\ t^a & t^b & 0 \\ r(t) & s(t)t^b & t^c\end{pmatrix}$$ with $a<b\le c$, and $r(t)$, $s(t)$ polynomials of degree $<c$, $<(c-b)$ respectively and vanishing at $t=0$. Next, we have to show that if $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ is not a rank-2 limit then $b<c$ and $r(t)$, $s(t)$ are as stated in Proposition \[standardform\]. \[b=c\] Let $\alpha(t)$ be as above. If $b=c$, then $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is a rank-2 limit. Decompose $F(1:y:z)$ in ${{\mathbb{C}}}[[y,z]]$: write $F(1:y:z)=G(y,z)\cdot H(y,z)$, where $G(y,z)$ collects the branches that are [*not*]{} tangent to $z=0$. If $b=c$, then necessarily $s=0$: $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0 \\ t^a & t^b & 0 \\ r(t) & 0 & t^b \end{pmatrix}\quad,$$ and further $a<v(r)$ (cf. Lemma \[faber\]). The reader can verify that the limits of the branches collected in $G$ are supported on the kernel line $x=0$. The limit of each (formal) branch collected in $H(y,z)$ may be computed as in Definition \[branchlimit\], and is found to be given by a homogeneous equation in $x$ and $z$ only: that is, a $(0:1:0)$-star. It follows that the limit of ${{\mathscr C}}$ is again a kernel star, hence a rank-2 limit by Lemma \[rank2lemma\]. End of the proof of Proposition \[standardform\] {#Eop} ------------------------------------------------ By Lemma \[b=c\], we may now assume that $\alpha$ is given by $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0\\ t^a & t^b & 0\\ r(t) & s(t)t^b & t^c \end{pmatrix}$$ with the usual conditions on $r(t)$ and $s(t)$, and further $a<b<c$. The limit of ${{\mathscr C}}$ under $\alpha$ is analyzed by studying limits of formal branches. \[otherbranches\] The limits of formal branches that are not tangent to the line $z=0$ are necessarily $(0:0:1)$-stars. Further, if $a<v(r)$, then the limit of such a branch is the kernel line $x=0$. \[tangentbranches\] The limit of a formal branch $z=f(y)$ tangent to the line $z=0$ is a $(0:0:1)$-star unless - $r(t)\equiv f(t^a)\pmod{t^c}$; - $s(t)\equiv f'(t^a)\pmod{t^{c-b}}$. The limit of the branch is given by the dominant terms in $$r(t)+s(t)t^by+t^cz=f(t^a)+f'(t^a)t^by+\dots$$ If $r(t)\not\equiv f(t^a)\pmod{t^c}$, then the weight of the branch is necessarily $<c$, so the ideal of the limit is generated by a polynomial in $x$ and $y$, as needed. The same reasoning applies if $s(t)\not\equiv f'(t^a)\pmod{t^{c-b}}$. To verify the condition on $\frac ca$ stated in Proposition \[standardform\], note that the limit of the formal branch $z=f(y)$ is now given by the dominant term in $$r(t)+s(t) t^b y+t^c z=f(t^a)+f'(t^a)t^b y+\frac{f''(t^a)t^{2b}y^2}2 +\dots:$$ the dominant weight will be less than $c$ (causing the limit to be a $(0:0:1)$-star) if $c>2b+v(f''(t^a))=2b+a(\lambda_0-2)$. The stated condition follows at once, completing the proof of Proposition \[standardform\]. Characterization of type V germs {#charaV} -------------------------------- In the following, we will replace $t$ by a root of $t$ in the germ obtained in Proposition \[standardform\], if necessary, in order to ensure that the exponents appearing in its expression are relatively prime integers; the resulting germ determines the same component of the PNC. In order to complete the characterization of type V germs given in §\[germlist\], we need to determine the possible triples $a<b<c$ yielding germs contributing components of the PNC. This determination is best performed in terms of $B=\frac ba$ and $C=\frac ca$. Let $$z=f(y)=\sum_{k\ge 0} \gamma_{\lambda_k} y^{\lambda_k}$$ with $\lambda_k\in {{\mathbb{Q}}}$, $1<\lambda_0<\lambda_1<\dots$, and $\gamma_{\lambda_k}\ne 0$, be a formal branch tangent to $z=0$. Every choice of such a branch and of a rational number $C=\frac ca>1$ determines a truncation $$f_{(C)}(y):=\sum_{\lambda_k<C} \gamma_{\lambda_k} y^{\lambda_k}\quad.$$ The choice of a rational number $B=\frac ba$ satisfying $1<B<C$ and $B\ge \frac{C-\lambda_0}2+1$ determines now a germ as prescribed by Proposition \[standardform\]: $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0 \\ t^a & t^b & 0 \\ \underline{f(t^a)} & \underline{f'(t^a) t^b} & t^c\end{pmatrix}$$ (choosing the smallest positive integer $a$ for which the entries of this matrix have integer exponents). Observe that the truncation $\underline{f(t^a)}=f_{(C)}(t^a)$ is identically 0 if and only if $C\le\lambda_0$. Also observe that $\underline{f'(t^a)t^b}$ is determined by $f_{(C)}(t^a)$, as it equals the truncation to $t^c$ of ${(f_{(C)})}'(t^a)t^b$. \[abc\] If $C\le\lambda_0$ or $B\ne \frac{C-\lambda_0}2+1$, then $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ is a rank-2 limit. We deal with the different cases separately. \[Clelambda0\] If $C\le\lambda_0$, then $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ is a $(0:1:0)$-star. If $C=\frac ca\le\lambda_0$, then $f_{(C)}(y)=0$, so $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0 \\ t^a & t^b & 0 \\ 0 & 0 & t^c\end{pmatrix}\quad.$$ The statement follows by computing the limit of individual formal branches, using Definition \[branchlimit\]. By Lemma \[rank2lemma\], the limits obtained in Lemma \[Clelambda0\] are rank-2 limits, so the first part of Proposition \[abc\] is proved. As for the second part, the limit of a branch tangent to $z=0$ depends on whether the branch truncates to $f_{(C)}(y)$ or not. These cases are studied in the next two lemmas. Recall that, by our choice, $B\ge \frac{C-\lambda_0}2+1$. \[nottrunc\] Assume $C>\lambda_0$, and let $z=g(y)$ be a formal branch tangent to $z=0$, such that $g_{(C)}(y)\ne f_{(C)}(y)$. Then the limit of the branch is supported on a kernel line. The limit of the branch is determined by the dominant terms in $$\underline{f(t^a)}+\underline{f'(t^a)t^b}y+t^cz=g(t^a)+g'(t^a) t^by+\dots .$$ As the truncations $g_{(C)}$ and $f_{(C)}$ do not agree, the dominant term is independent of $z$. Under our hypotheses on $B$ and $C$, it is found to be independent of $y$ as well, as needed. \[dominant\] Assume $C>\lambda_0$, and let $z=g(y)$ be a formal branch tangent to $z=0$, such that $g_{(C)}(y)= f_{(C)}(y)$. Denote by $\gamma_C^{(g)}$ the coefficient of $y^C$ in $g(y)$. - If $B> \frac{C-\lambda_0}2+1$, then the limit of the branch $z=g(y)$ by $\alpha(t)$ is the line $$z=(C-B+1)\gamma_{C-B+1} y+\gamma_C^{(g)}\quad.$$ - If $B= \frac{C-\lambda_0}2+1$, then the limit of the branch $z=g(y)$ by $\alpha(t)$ is the conic $$z=\frac {\lambda_0(\lambda_0-1)}2\gamma_{\lambda_0}y^2+\frac {\lambda_0+C}2\gamma_{\frac{\lambda_0+C}2}y+\gamma_C^{(g)}\quad.$$ Rewrite the expansion whose dominant terms give the limit of the branch as: $$t^c z=(g(t^a)-\underline{f(t^a)})+(g'(t^a)t^b-\underline{f'(t^a)t^b})y +\frac{g''(t^a)}2 t^{2b} y^2+\dots$$ The dominant term has weight $c=Ca$ by our choices; if $B> \frac{C-\lambda_0}2+1$ then the weight of the coefficient of $y^2$ exceeds $c$, so it does not survive the limiting process, and the limit is a line. If $B= \frac{C-\lambda_0}2+1$, the term in $y^2$ is dominant, and the limit is a conic. The explicit expressions given in the statement are obtained by reading the coefficients of the dominant terms. We can now complete the proof of Proposition \[abc\]: \[completion\] Assume $C>\lambda_0$. If $B>\frac{C-\lambda_0}2+1$, then the limit $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is a rank-2 limit. We will show that the limit is necessarily a kernel star, which gives the statement by Lemma \[rank2lemma\]. As $B>1$, the coefficient $\gamma_{C-B+1}$ is determined by the truncation $f_{(C)}$, and in particular it is the same for all formal branches with that truncation. Since $B>\frac{C-\lambda_0}2+1$, by Lemma \[dominant\] the branches considered there contribute lines through the fixed point $(0:1:(C-B+1)\gamma_{C-B+1})$. We are done if we check that all other branches contribute a kernel line $x=0$: and this is implied by Lemma \[otherbranches\] for branches that are not tangent to $z=0$ (note $a<v(r)$ for the germs we are considering), and by Lemma \[nottrunc\] for formal branches $z=g(y)$ tangent to $z=0$ but whose truncation $g_{(C)}$ does not agree with $f_{(C)}$. Quadritangent conics {#quadritangent} -------------------- We are ready to complete the proof of Theorem \[mainmain\], by determining the limits of the last contributing germs. These have been reduced to the form listed as type V in §\[germlist\] (up to a coordinate change, and replacing $t$ by a root of $t$): $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0 \\ t^a & t^b & 0 \\ \underline{f(t^a)} & \underline{f'(t^a)t^b} & t^c \end{pmatrix}$$ for some branch $z=f(y)=\gamma_{\lambda_0}y^{\lambda_0}+\dots$ of ${{\mathscr C}}$ tangent to $z=0$ at $p=(0,0)$, and further satisfying $C>\lambda_0$ and $B=\frac{C-\lambda_0}2+1$ for $B=\frac ba$, $C=\frac ca$. Type V components of the PNC will arise depending on the limit $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$, which we now determine. \[quadconics\] If $C>\lambda_0$ and $B=\frac{C-\lambda_0}2+1$, then the limit $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ consists of a union of quadritangent conics, with distinguished tangent equal to the kernel line $x=0$, and of a multiple of the distinguished tangent line. Both $\gamma_{\lambda_0}$ and $\gamma_{\frac{\lambda_0+C}2}$ are determined by the truncation $f_{(C)}$ (since $C>\lambda_0$); hence the equations of the conics $$z=\frac {\lambda_0(\lambda_0-1)}2\gamma_{\lambda_0}y^2+\frac {\lambda_0+C}2\gamma_{\frac{\lambda_0+C}2}y+\gamma_C$$ contributed (according to Lemma \[dominant\]) by different branches with truncation $f_{(C)}$ may only differ in the coefficient $\gamma_C$. It is immediately verified that all such conics are tangent to the kernel line $x=0$, at the point $(0:0:1)$, and that any two distinct such conics meet only at the point $(0:0:1)$; thus they are necessarily quadritangent. Finally, the branches that do not truncate to $f_{(C)}(y)$ must contribute kernel lines, by Lemmas \[otherbranches\] and \[nottrunc\]. The degenerate case in which only one conic arises corresponds to germs not contributing components of the projective normal cone, by dimension considerations. A component is present as soon as there are two or more conics, that is, as soon as two branches contribute distinct conics to the limit. This leads to the description given in §\[germlist\]. We say that a rational number $C$ is ‘characteristic’ for ${{\mathscr C}}$ (with respect to $z=0$) if at least two formal branches of ${{\mathscr C}}$ (tangent to $z=0$) have the same nonzero truncation, but different coefficients for $y^C$. \[typeV\] The set of characteristic rationals is finite. The limit $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ obtained in Lemma \[quadconics\] determines a component of the projective normal cone precisely when $C$ is characteristic. If $C\gg 0$, then branches with the same truncation must in fact be identical, hence they cannot differ at $y^C$, hence $C$ is not characteristic. Since the set of exponents of any branch is discrete, the first assertion follows. The second assertion follows from Lemma \[quadconics\]: if $C>\lambda_0$ and $B=\frac{C-\lambda_0}2+1$, then the limit is a union of a multiple kernel line and conics with equation $$z=\frac {\lambda_0(\lambda_0-1)}2\gamma_{\lambda_0}y^2+\frac {\lambda_0+C}2\gamma_{\frac{\lambda_0+C}2}y+\gamma_C\quad:$$ these conics are different precisely when the coefficients $\gamma_C$ are different, and the statement follows. Proposition \[typeV\] leads to the procedure giving components of type V explained in §\[details\] (also cf. [@MR2001h:14068], §2, Fact 5), concluding the proof of Theorem \[mainmain\]. Boundaries of orbits {#boundary} ==================== We have now completed the set-theoretic description of the PNC determined by an arbitrary plane curve ${{\mathscr C}}$. As we have argued in §\[prelim\], this yields in particular a description of the boundary of ${{\mathscr O_{{\mathscr C}}}}$. In this section we include a few remarks aimed at making this description more explicit. If $\dim{{\mathscr O_{{\mathscr C}}}}=8$, then the boundary of ${{\mathscr O_{{\mathscr C}}}}$ consists of the image of the union of the PNC and of the proper transform $R$ of the complement of ${\text{\rm PGL}}(3)$ in ${{\mathbb{P}}}^8$ (cf. Remark \[eluding\]). Curves in the image of $R$ are stars (Lemma \[PNCtolimits\]). Curves in the image of the components of the PNC belong to the orbit closures of the limits of the marker germs listed in §\[germlist\]. We have proved that this list is exhaustive; therefore, the boundary of a given curve ${{\mathscr C}}$ may be determined (up to stars) by identifying the marker germs for ${{\mathscr C}}$, and taking the union of the orbit closures of the (finitely many) corresponding limits. This reduces the determination of the curves in the boundary of the orbit of a given curve to the determination for curves with [*small*]{} orbit (i.e., of dimension $\le 7$). We note that, for a curve ${{\mathscr C}}$ with small orbit, some components of the PNC will in fact dominate ${\overline{{{\mathscr O_{{\mathscr C}}}}}}$: indeed, in this case ${{\mathscr C}}$ has positive dimensional stabilizer in ${\text{\rm PGL}}(3)$; the limit of a germ centered at a singular matrix and otherwise contained in the stabilizer is ${{\mathscr C}}$ itself. This germ can be chosen to be equivalent to a marker germ, identifying a component of the PNC which dominates the orbit closure. As mentioned in the introduction, the boundary for a curve with small orbit may be determined by a direct method. Indeed, for such a curve we have constructed in [@MR2002d:14083] explicit sequences of blow-ups at nonsingular centers which resolve the indeterminacies of the basic rational map, and hence dominate the corresponding graph ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$. The boundary of the curve may be determined by studying the image in ${{\mathbb{P}}}^N$ of the various exceptional divisors of these blow-ups. The result may be summarized by indicating which types of curves with small orbits are in the boundary of a given curve with small orbit. Figure \[figure1\] expresses part of this relation in terms of the representative pictures for curves with small orbit shown in §\[appendix\]. The five columns represent curves with orbits of dimension 7, 6, 5, 4, 2 respectively. Arrows indicate specialization: for example, the figure indicates that the boundary of the orbit of the union of a conic and a tangent line contains stars, but not single conics. Stars with more than three lines are not displayed, to avoid cluttering the picture; the three kinds of curves displayed in the leftmost column all degenerate to such stars, the only exceptions being the special cases of the second picture given by the union of a conic and a transversal line, and by a single cuspidal cubic. The situation illustrated here is precisely what one would expect from naive considerations; it is confirmed by the study of the blow-ups mentioned above. Slightly more refined phenomena (for example, involving multiplicities of the components) are not represented in this figure; in general, they can be easily established by applying the results of this paper or by analyzing the blow-ups of [@MR2002d:14083]. ![[]{data-label="figure1"}](pictures/smallpichor3) We close by pointing out one such phenomenon. In general, the union of a set of quadritangent conics and a tangent line can specialize to the union of a conic and a tangent line in two ways: (i) by type II germs aimed at a general point of one of the conic components, and (ii) by a suitable type IV germ aimed at the tangency point. The multiplicity of the conic in the limit is then the multiplicity of the selected component in case (i), and the sum of the multiplicities of all conic components in case (ii). If the curve consists solely of quadritangent conics, it degenerates to a multiple conic in case (ii). This possibility occurs in the boundary of the orbit of the quartic curve from Example \[extwo\], represented in Figure \[figure2\]. We have omitted the set of stars of four distinct lines also in this figure; in this case, it is a $6$-dimensional union of $5$-dimensional orbits. ![[]{data-label="figure2"}](pictures/exapic) Appendix: curves with small linear orbits {#appendix} ========================================= For the convenience of the reader, we reproduce here the description of plane curves with small linear orbits given in [@MR2002d:14084]. That reference contains a proof that this list is exhaustive, and details on the stabilizer of each type of curve (as well as enumerative results for orbits of curves consisting of unions of lines, items (1)–(5) in the following list). Let ${{\mathscr C}}$ be a curve with small linear orbit. We list all possibilities for ${{\mathscr C}}$, together with the dimension of the orbit ${{\mathscr O_{{\mathscr C}}}}$ of ${{\mathscr C}}$. The irreducible components of the curves described here may appear with arbitrary multiplicities. 1. ${{\mathscr C}}$ consists of a single line; $\dim{{\mathscr O_{{\mathscr C}}}}=2$. 2. ${{\mathscr C}}$ consists of 2 (distinct) lines; $\dim{{\mathscr O_{{\mathscr C}}}}=4$. 3. ${{\mathscr C}}$ consists of 3 or more concurrent lines; $\dim{{\mathscr O_{{\mathscr C}}}}=5$. (We call this configuration a [*star*]{}.) 4. ${{\mathscr C}}$ is a triangle (consisting of 3 lines in general position); $\dim{{\mathscr O_{{\mathscr C}}}}=6$. 5. ${{\mathscr C}}$ consists of 3 or more concurrent lines, together with 1 other (non-concurrent) line; $\dim{{\mathscr O_{{\mathscr C}}}}=7$. (We call this configuration a [*fan*]{}.) 6. ${{\mathscr C}}$ consists of a single conic; $\dim{{\mathscr O_{{\mathscr C}}}}=5$. 7. ${{\mathscr C}}$ consists of a conic and a tangent line; $\dim{{\mathscr O_{{\mathscr C}}}}=6$. 8. ${{\mathscr C}}$ consists of a conic and 2 (distinct) tangent lines; $\dim{{\mathscr O_{{\mathscr C}}}}=7$. 9. ${{\mathscr C}}$ consists of a conic and a transversal line and may contain either one of the tangent lines at the 2 points of intersection or both of them; $\dim{{\mathscr O_{{\mathscr C}}}}=7$. 10. ${{\mathscr C}}$ consists of 2 or more bitangent conics (conics in the pencil $y^2+\lambda x z$) and may contain the line $y$ through the two points of intersection as well as the lines $x$ and/or $z$, tangent lines to the conics at the points of intersection; again, $\dim{{\mathscr O_{{\mathscr C}}}}=7$. 11. ${{\mathscr C}}$ consists of 1 or more (irreducible) curves from the pencil $y^b+\lambda z^a x^{b-a}$, with $b\ge 3$, and may contain the lines $x$ and/or $y$ and/or $z$; $\dim{{\mathscr O_{{\mathscr C}}}}=7$. 12. ${{\mathscr C}}$ contains 2 or more conics from a pencil through a conic and a double tangent line; it may also contain that tangent line. In this case, $\dim{{\mathscr O_{{\mathscr C}}}}=7$. The last case is the only one in which the maximal connected subgroup of the stabilizer is the additive group ${{\mathbb{G}}}_a$; this fact was mentioned in §\[germlist\]. The following picture represents schematically the curves described above.   [^1]: [**Acknowledgments.**]{} Work on this paper was made possible by support from Mathematisches Forschungsinstitut Oberwolfach, the Volkswagen Stiftung, the Max-Planck-Institut für Mathematik (Bonn), Princeton University, the Göran Gustafsson foundation, the Swedish Research Council, the Mittag-Leffler Institute, MSRI, NSA, NSF, and our home institutions. We thank an anonymous referee of our first article on the topic of linear orbits of plane curves, [@MR94e:14032], for bringing the paper of Aldo Ghizzetti to our attention.

--- abstract: | The [*Reeb space*]{} of a function or a map on a manifold is defined as the space of all the connected components of inverse images. A Reeb space represents the manifold compactly. In fact, such stuffs are fundamental and useful tools in geometric theory of Morse functions and more general maps not so ill: in other words, a branch of the global singularity theory. The author has been interested in the following problem first established and explicitly solved by Sharko: can we construct an explicit good function inducing a given graph as the Reeb space ([*Reeb graph*]{})? Such problems have been explicitly solved by several researchers and after that the author has set problems of new types and solved them. A [*pseudo quotient map*]{} on a differentiable manifold is a surjective continuous map onto a lower dimensional polyhedron. More precisely, a [*pseudo quotient map*]{} is defined as a map locally regarded as the natural quotient map onto the Reeb space defined from a differentiable map of a given class. They were first defined by Kobayashi and Saeki in 1996 as useful objects in the theory of global singularity related to generic maps into the plane and later the author has used these objects in new explicit situations starting from redefining. In this paper, we consider classes of continuous or differentiable maps on manifolds, introduce the [*Reeb graphs*]{} of the maps and redefine pseudo quotient maps onto graphs in a bit new way. Last, we attack and solve a new problem on construction before. address: 'Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku Fukuoka 819-0395, Japan' author: - Naoki Kitazawa title: Maps on manifolds onto graphs locally regarded as a quotient map onto a Reeb space and construction problem --- Introduction {#sec:1} ============ Reeb spaces and graphs and differentiable functions realizing given graphs as Reeb graphs ----------------------------------------------------------------------------------------- The [*Reeb space*]{} of a continuous map of a suitable class on a topological space is the space of all the connected components of inverse images. For a differentiable function, consider the set of all the points in the Reeb space coinciding with the set of all the connected components of inverse images including [*singular points*]{}: a [*singular point*]{} of a smooth map is a point at which the rank of the differential drops. For Morse functions, functions with finitely many singular points on closed manifolds and functions of several suitable classes, the spaces are graphs such that the vertex sets are the sets defined before. They are called the [*Reeb graphs*]{} of the maps. They seem to have been first defined in [@reeb]. Reeeb graphs and spaces are fundamental and important in the algebraic and differential topological theory of Morse functions and their generalizations, or in other words, the theory of global singularity. We introduce several terminologies and a problem on construction of good functions inducing Reeb graphs isomorphic to given graphs. The [*singular set*]{} of a differentiable map is defined as the set of all the singular points. A [*singular value*]{} is a point in the target manifold such that the inverse image contains a singular point and a [*regular value*]{} is a point in the target manifold which is not a singular value. The [*singular value set*]{} is the image of the singular set. \[prob:1\] Can we construct a differentiable function with good geometric properties inducing a given graph as the Reeb graph? We do not fix a manifold on which we construct a desired function. A problem of this type was first considered and explicitly solved by Sharko ([@sharko]). [@batistacostamezasarmiento], [@martinezalfaromezasarmientooliveira], [@masumotosaeki] and [@michalak] are important studies related to this. Later the author has set and solved explicit cases in [@kitazawa4] and [@kitazawa5]. We comment on studies of the author. Different from the other studies, conditions on inverse images of regular values are posed and manifolds appearing there may not be spheres, for example. Pseudo quotient maps -------------------- A [*pseudo quotient map*]{} on a differentiable manifold is a surjective continuous map onto a lower dimensional polyhedron. and defined as a map locally regarded as the natural quotient map onto the Reeb space of a differentiable map of a suitable class. They were first defined by Kobayashi and Saeki in 1996 ([@kobayashisaeki]) as useful objects in the theory of global singularity related to generic maps of dimensions larger than $2$ into the plane. Later the author has used these objects in new explicit situations starting from redefining in [@kitazawa2] and [@kitazawa3] for example. The content of the present paper -------------------------------- In this paper, we perform the following. - We consider classes of continuous or differentiable maps on differentiable manifolds for maps of which we introduce the [*Reeb graphs*]{} and we introduce the Reeb graphs of these maps. This is a refinement of the definition of a Reeb graph which has not appeared. - We redefine pseudo quotient maps on differentiable manifolds of a class before. - We set a construction problem and give an answer (Theorem \[thm:1\]) with several terminologies needed. These are a problem of a new type and a result of a new type. The author is a member of the project Grant-in-Aid for Scientific Research (S) (17H06128 Principal Investigator: Osamu Saeki) “Innovative research of geometric topology and singularities of differentiable mappings” (https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17H06128/ ) and supported by this project. Classes of continuous or differentiable maps on differentiable manifolds and the Reeb graphs of the maps of these classes ========================================================================================================================= Let $(r,s)$ be a pair of non-negative integers satisfying $r>s$ or a pair such that $r=\infty$ and that $s$ is a non-negative integer. Let $X$ be a $C^r$ manifold of dimension $m>1$ and $Y$ be a $C^r$ manifold of dimension $1$. Let $A \subset X$ be a measure zero set. A map $f:(X,A) \rightarrow N$ between the $C^r$ manifolds is said to be a [*$(C^r,C^s)$*]{} map if $f$ is of class $C^r$ at any point in $X-A$ and of class $C^s$ at any point in $A$. $A$ is called the [*measure zero set*]{} of $f$ Consider the Reeb space $W_f$ of the map $f$. Let $V$ be the set of all the points including singular points or points in $A$. If we can regard $W_f$ as a graph whose vertex set is $V$, then we call the graph the [*Reeb graph*]{} of $f$. A pseudo quotient map of a class of maps ======================================== Let $(r,s)$ be a pair of non-negative integers satisfying $r>s$ or a pair such that $r=\infty$ and that $s$ is a non-negative integer. Two $C^r$ maps $c_1:X_1 \rightarrow Y_1$ and $c_2:X_2 \rightarrow Y_2$ are said to be [*$C^s$ equivalent*]{} if a pair $({\phi}_X,{\phi}_Y)$ of $C^s$ diffeomorphisms satisfying ${\phi}_Y \circ c_1=c_2 \circ {\phi}_X$ exists. Let $X_1$ and $X_2$ be differentiable manifolds of dimension $m>1$ and $Y$ be a graph. Let $r$ be a positive integer or $\infty$. Two continuous maps $c_1:X_1 \rightarrow Y$ and $c_2:X_2 \rightarrow Y$ are said to be [*$C^r$-PL equivalent*]{} if there exists a $C^r$ diffeomorphism $\phi$ satisfying $c_1=c_2 \circ {\phi}_X$ exists. Let $m>2$ be a positive integer. Let $\mathcal{C}$ be a class of $(C^r,C^s)$ maps from $m$-dimensional differentiable manifolds into $1$-dimensional ones whose Reeb spaces are regarded as Reeb graphs. A continuous map $q$ on an $m$-dimensional differentiable manifold onto a graph is said to be a [*pseudo quotient map*]{} of the class $\mathcal{C}$ if the following hold. 1. At each point $p$ in the interior of an edge, consider a small closed interval $C_p$ including the point, $q {\mid}_{q^{-1}(C_p)}:q^{-1}(C_p) \rightarrow C_p$ is $C^r$-PL equivalent to a $C^r$ trivial bundle, appearing locally around a closed interval in the interior of an edge in the Reeb graph of a map of the class $\mathcal{C}$. 2. At each vertex $p$, consider a small regular neighborhood $C_p$ including the point, $q {\mid}_{q^{-1}(C_p)}:q^{-1}(C_p) \rightarrow C_p$ is $C^r$-PL equivalent to a PL map $q_{C_p}$ over a small regular neighborhood of a vertex in the Reeb graph of a map $f_p$ of $\mathcal{C}$ such that the domain includes no point in the measure zero set of the map $f_p$, or $C^s$-PL equivalent to a PL map over a small regular neighborhood of a vertex in the Reeb graph of a map of $\mathcal{C}$. A pseudo quotient map of a class $\mathcal{C}$ is said to be [*realized*]{} as a quotient map of the class $\mathcal{C}$ if by identifying the target graphs suitably, we can regard the original map as $C^r$-PL equivalent to the quotient map onto the Reeb graph of a map of the class. Examples will be presented in the next section. The main theorem and its proof ============================== We introduce the main theorem, related to Problem \[prob:1\], A [*fold*]{} map is a $C^{\infty}$ map at each singular point which is represented as a Morse function and an identity map on a $C^{\infty}$ manifold. A [*special generic*]{} map is a fold map at each singular point which is represented as a natural height function of an unit open ball and an identity map on a $C^{\infty}$ manifold. Precise explanations on these maps are in [@golubitskyguillemin], [@saeki] and [@saeki2], for example. For an integer $n>1$, let $L_n$ be a $1$-dimensional polyhedron represented as $${\bigcup}_{k=0}^{n-1} \{(r \cos \frac{2k\pi}{n}, r \sin \frac{2k\pi}{n}) \mid 0 <r \leq 1 \} \bigcup \{(0,0)\} \subset {\mathbb{R}}^2.$$ ![$L_n$.[]{data-label="fig:1"}](ln.eps){width="30mm"} Let $q$ be a continuous map from a $C^s$ manifold of dimension $m>1$ onto $L_n \subset {\mathbb{R}^2}$. Suppose that for a transformation ${\phi}_{n}$ on $L_n$ defined as $${\phi}_{n}(r \cos \frac{2k\pi}{n}, r \sin \frac{2k\pi}{n})=(r \cos \frac{2(k+1)\pi}{n}, r \sin \frac{2(k+1)\pi}{n})$$ there exists a $C^s$ diffeomorphism ${\Phi}_{n}$ satisfying ${\phi}_n \circ q=q \circ {\Phi}_n$ and that for one ${\phi}_{r}$ on $L_n$ defined as $${\phi}_{r}(r \cos \frac{2k\pi}{n}, r \sin \frac{2k\pi}{n})=(r \cos \frac{-2k\pi}{n}, r \sin \frac{-2k\pi}{n})$$ there exists a $C^s$ diffeomorphism ${\Phi}_{r}$ satisfying ${\phi}_r \circ q=q \circ {\Phi}_r$. Then $q$ is said to be [*$D_n$-symmetric*]{}. \[thm:1\] There exist a class $\mathcal{C}$ of $(C^r,C^0)$ maps whose Reeb spaces are regarded as Reeb graphs and a class ${\mathcal{Q}}_{\mathcal{C}}$ of pseudo quotient maps of the class satisfying the following. 1. For maps of the class $\mathcal{C}$, inverse images of regular values are disjoint unions of standard spheres. 2. For maps of the class ${\mathcal{Q}}_{\mathcal{C}}$, around each vertex of degree $n>1$, the map is regarded as a $D_n$-symmetric map onto $L_n$ by applying a suitable identification. 3. For any finite connected graph which is not a single point, we can construct a map of the class ${\mathcal{Q}}_{\mathcal{C}}$ onto the graph. 4. For a map of the class ${\mathcal{Q}}_{\mathcal{C}}$, if for the target graph, for each vertex, the degree is at most $3$, then the map is realized as a map into $S^1$ or $\mathbb{R}$ of the class $\mathcal{C}$. In the proof, we first perform local construction around each vertex (Step 1–3) and in Step 4, we complete the construction by constructing remaining parts. Last, we give the strict definitions of $\mathcal{C}$ and $\mathcal{Q}_{\mathcal{C}}$. We will see that this completes the proof except for the fourth condition. Last we discuss the fourth condition. Step 1 Around a vertex of degree $2$.\ We consider a trivial $C^r$ bundle over $[-1,1]$ whose fiber is a standard sphere. We compose a surjective function over $[-1,1]$ defined by $t \mapsto t^3$. $0 \in [-1,1]$ is the vertex of degree $2$. The other points are not in the vertex set. The map is regarded as a $D_2$-symmetric map onto $L_2$.\  \ Step 2 Around a vertex of degree $n \geq 3$.\ We consider a $C^{\infty}$ map on a $C^{\infty}$ manifold of dimension $m>1$ into the plane whose image is the bounded domain surrounded by the five arcs as FIGURE \[fig:2\]. We construct the map as a special generic map such that the restriction to the singular set is embedding and the singular value set is the disjoint union of two thick arcs. Note that inverse images of regular values are standard spheres ($m>2$) or two point sets ($m=2$). For more precise facts on special generic maps into the plane, see [@saeki2] for example. ![A special generic map into the plane.[]{data-label="fig:2"}](zahyoone.eps){width="30mm"} Let $f$ be a $C^{\infty}$ function whose value is $1$ on the interval $x \leq 0$ and which is strictly increasing on the interval $x \geq 0$. We can do in the step before so that by composing a $C^{\infty}$ map $T$ defined as $$T(x,y):= \begin{cases} (x,y) & (y \leq 0) \\ (f(y)x,y) & (y \geq 0) \end{cases}$$ the resulting image is as FIGURE \[fig:3\]: the arc connecting two thick arcs is defined as a part of a quadratic curve of a form $x^2-y^2=a$. We can determine $c(t)$ and we can naturally determine a $C^{\infty}$ function $c$ on $[0,t]$ based on quadratic curves of this form. We can extend this to the bounded domain except two arcs in the bottom as a $C^{\infty}$ map. However, we can extend to the arcs as a continuous map: we define the map so that the values on these two arcs are $(0,-1)$. Thus we can define a map $C$ on the bounded domain, Through the steps, we can define a $(C^r,C^0)$ function on the given $m$-dimensional manifold to a closed interval $[-1,c(t)]$ where we identify $x$ with $(0,x)$ for $-1 \leq x \leq c(t)$. ![A deformation of the special generic map into the plane of FIGURE \[fig:2\].[]{data-label="fig:3"}](zahyotwo.eps){width="30mm"} We consider the map on an $m$-dimensional manifold into the plane obtained in the explanation of FIGURE \[fig:2\] and $n$-copies of such a map. We deform these maps by scaling suitably and attach the maps as FIGURE \[fig:4\] on the arcs corresponding to ones including $(0,-1)$ in the original image and the inverse images. $D_k$ are the images of the maps: maps are suitably scaled so that $(0,-1)$ goes to $(0,0)$ and that the angle formed by the pairs of the arcs including $(0,0)$ are equal, for example. We can obtain a local map around the vertex satisfying the first two conditions. Note that the measure zero set of the local function is obtained by identifying the equations of $n$-copies of $S^{m-1}$. ![Attacing copies of a map in FIGURE \[fig:2\].[]{data-label="fig:4"}](zahyothree.eps){width="30mm"}  \ Step 3 Around a vertex of degree $1$.\ We consider a natural height function on a unit disc of dimension $m>1$ whose image is $[0,1]$ and $0$ or $1$ is the only one vertex. The function is also a Morse function with just $1$ singular point in the interior.\  \ Step 4 Completing the construction.\ For the interior of each edge, we construct a trivial $C^r$ bundle whose fiber is a standard sphere. Last we glue all the constructed local maps together to obtain a global map.\  \ Step 5 Define $\mathcal{C}$ and $\mathcal{Q}_{\mathcal{C}}$.\ Through Steps 1–4, we construct a map for arbitrary finite graphs which are not single points. Last, we explain about the classes $\mathcal{C}$ and $\mathcal{Q}_{\mathcal{C}}$. $\mathcal{Q}_{\mathcal{C}}$ are naturally defined as a class locally of a form of the obtained forms. We define $\mathcal{C}$ as a map such that around each singular value corresponding to a vertex of degree not $1$, the map is represented as the composition of a presented local map onto a regular neighborhood of a graph, regarded as a map into the plane, a homeomorphism on the plane satisfying the following (we define such a map as an [*almost smooth generalized rotation with reflection*]{}) and a projection onto a straight line $\mathbb{R}$ including $(0,0)$. 1. The homeomorphism is $C^{\infty}$ on ${\mathbb{R}}^2-\{(0,0)\}$. 2. For each point not $(0,0)$, the homeomorphism preserves the distance between the point and $(0,0)$. 3. The homeomorphism maps each straight line originating from $(0,0)$ to another straight line originating from $(0,0)$. For around each singular value corresponding to a vertex of degree $0$, we define the class so that the local form is as in Step 3: a natural height function on a unit disc. This completes the proof except the fourth condition. For each finite graph which is not a single point or which has no vertex of degree larger than $3$, then we can orient the graph so that we can construct a continuous map into $S^1$ satisfying the following. 1. On each edge the map is injective. 2. The orientation of each edge canonically induced from a canonical orientation of $S^1$ is compatible with the defined orientation. If the graph has no loop, then we can replace $S^1$ by $\mathbb{R}$. For a map of the class $\mathcal{Q}_{\mathcal{C}}$, if for the target graph, for each vertex, the degree is at most $3$, then we can orient the graph as this and we can construct a local function compatible with the definition of the class $\mathcal{C}$ and the orientation. This is oweing to the definitions of a $D_2$-symmetric and a $D_3$-symmetric map and an almost smooth generalized rotation with reflection. We can consider a transformation by an almost smooth generalized rotation with reflection to construct a local function compatible with the desired orientation. See FIGURE \[fig:4.5\] for the case of a vertex of degree $2$. ![Almost smooth generalized rotations with reflections and projections (around a vertex of degree $2$: arrows indicate local orientations of the graphs induced naturally from local functions).[]{data-label="fig:4.5"}](rot.eps){width="30mm"} This completes the proof. Last we present another example of classes of maps and pseudo quotient maps of the class. A [*standard-spherical*]{} Morse function is a Morse function such that the following hold ([@kitazawa]). 1. At distinct singular points the values are distinct. 2. Inverse images of regular values are disjoint unions of standard spheres. 3. A vertex of the Reeb graph such that the inverse image includes a singular point not giving a local extremum is a vertex of degree $3$. We consider pseudo quotient maps of the class of such functions. We regard these functions as $C^{\infty}$ functions whose measure zero sets are empty. We present local forms of these pseudo quotient maps with several inverse images in FIGURE \[fig:5\]. ![Local forms of the pseudo quotient maps with several inverse images.[]{data-label="fig:5"}](sphelocal.eps){width="30mm"} We investigate the local form around a vertex of degree $3$ of a pseudo quotient map of the class of the functions with the inverse image. Consider a small regular neighborhood of the just one non-manifold point in the inverse image of the vertex. See FIGURE \[fig:6\]. ![The local form around a vertex of degree $3$ of a pseudo quotient map of the class of the functions with the inverse image (a regular neighborhood of the just one non-manifold point in the inverse image of the vertex).[]{data-label="fig:6"}](localspheind.eps){width="15mm"} If we remove the inverse image of the vertex, then the resulting space has $4$ connected components: two of them are in the upper part and the others are in the lower part. Moreover, the former two components are mapped onto an interval and the latter two ones are mapped onto the disjoint union of two intervals in the graph. This leads us to the fact that a pseudo quotient map whose target is as FIGURE \[fig:7\] cannot be realized as a quotient map of the class. Arrows indicate natural local orientations induced from canonically obtained local functions. Note that around a vertex of degree $3$, the local map cannot be regared as a $D_3$-symmetric map. ![The graph represents the target of a pseudo quotient map of the class of standard-spherical functions: arrows indicate local orientations induced canonically from local functions.[]{data-label="fig:7"}](sphenewex.eps){width="15mm"} For local forms of such functions, see also [@saeki3] for example. Last, compare this case with Theorem \[thm:1\]. [25]{} E. B. Batista, J. C. F. Costa and I. S. Meza-Sarmiento, *Topological classification of circle-valued simple Morse Bott functions*, Journal of Singularities, Volume 17 (2018), 388–402. R. Bott, *Nondegenerate critical manifolds*, Ann. of Math. 60 (1954), 248–261. M. Golubitsky and V. Guillemin, *Stable Mappings and Their Singularities*, Graduate Texts in Mathematics (14), Springer-Verlag(1974). N. Kitazawa, *Lifts of spherical Morse functions*, submitted to a refereed journal, arxiv:1805.05852. N. Kitazawa, *Generalizations of Reeb spaces of special generic maps and applications to a problem of lifts of smooth maps*, submitted to a refereed journal, arxiv:1805.07783. N. Kitazawa, *A new explicit way of obtaining special generic maps into the $3$-dimensional Euclidean space*, arxiv:1806:04581. N. Kitazawa, *On Reeb graphs induced from smooth functions on $3$-dimensional closed orientable manifolds with finite singular values*, submitted to a refereed journal, arxiv:1902.08841. N. Kitazawa, *On Reeb graphs induced from smooth functions on closed or open surfaces*, submitted to a refereed journal, arxiv:1908.04340. M. Kobayashi and O. Saeki, *Simplifying stable mappings into the plane from a global viewpoint*, Trans. Amer. Math. Soc. 348 (1996), 2607–2636. J. Martinez-Alfaro, I. S. Meza-Sarmiento and R. Oliveira, *Topological classification of simple Morse Bott functions on surfaces*, Contemp. Math. 675 (2016), 165–179. Y. Masumoto and O. Saeki, *A smooth function on a manifold with given Reeb graph*, Kyushu J. Math. 65 (2011), 75–84. L. P. Michalak, *Realization of a graph as the Reeb graph of a Morse function on a manifold*. to appear in Topol. Methods Nonlinear Anal., Advance publication (2018), 14pp, arxiv:1805.06727. G. Reeb, *Sur les points singuliers d'une forme de Pfaff complétement intègrable ou d'une fonction numérique*, Comptes Rendus Hebdomadaires des Séances de I'Académie des Sciences 222 (1946), 847–849. O. Saeki, *Notes on the topology of folds*, J. Math. Soc. Japan Volume 44, Number 3 (1992), 551–566. O. Saeki, *Topology of special generic maps of manifolds into Euclidean spaces*, Topology Appl. 49 (1993), 265–293. O. Saeki, *Topology of singular fibers of differntiable maps*, Lecture Notes in Math., Vol. 1854, Springer-Verlag, 2004. V. Sharko, *About Kronrod-Reeb graph of a function on a manifold*, Methods of Functional Analysis and Topology 12 (2006), 389–396.

--- abstract: 'The possibility that like-charges can attract each other under the mediation of mobile counterions is by now well documented experimentally, numerically, and analytically. Yet, obtaining exact results is in general impossible, or restricted to some limiting cases. We work out here in detail a one dimensional model that retains the essence of the phenomena present in higher dimensional systems. The partition function is obtained explicitly, from which a wealth of relevant quantities follow, such as the effective force between the charges or the counterion profile in their vicinity. Isobaric and canonical ensembles are distinguished. The case of two equal charges screened by an arbitrary number $N$ of counterions is first studied, before the more general asymmetric situation is addressed. It is shown that the parity of $N$ plays a key role in the long range physics.' author: - Gabriel Téllez - Emmanuel Trizac title: 'Screening like-charges in one-dimensional Coulomb systems: Exact results' --- Introduction ============ Coulombic effects are often paramount in soft matter systems, where the large dielectric constant of the solvent (say water) invites ionizable groups at the surface of macromolecules to dissociate [@KeHP01; @Levin02; @Messina09]. While a realistic treatment requires considering three dimensional systems, interesting progress has been achieved for lower dimensional problems where the key mechanisms can be studied in greater analytical detail [@Janco81; @Forrester98; @Samaj03]. In particular, a one dimensional model was introduced in the 1960s by Lenard and Prager independently, for which a complete thermodynamic solution was provided [@Len61; @Pra61; @EL62]. This model has been further studied in Ref. [@DHNP09], but it turns out that some interesting features have been overlooked in relation with the like-charge attraction phenomenon [@Levin02; @Varenna]. This striking non mean-field effect, relevant for strongly coupled charged matter [@Netz01; @Varenna] is the thread in our study. The paper is organized as follows. The model is first defined in section \[sec:equal-charges\]. It mimics the screening of charged colloids. The Coulomb potential in one dimension between two charges $q$ and $q'$ located along a line with coordinates $\widetilde{x}$ and $\widetilde{x}'$ is $$v(\widetilde{x},\widetilde{x}')=-qq'|\widetilde{x}-\widetilde{x}'| \,.$$ Therefore, the electric field created by one particle is of constant magnitude. This fact simplifies the study of the equilibrium statistical mechanics of such systems, and allows to obtain some of its properties by simple arguments. Furthermore, it also allows for an explicit computation of the partition function [@Len61; @Pra61]. The system under scrutiny can be envisioned as a collection of parallel charged plates, able to move along a perpendicular axis. The salient properties of this system can be obtained by simple arguments which we present in section \[sec:equal-charges\], followed afterwards by a more technical analysis where the explicit calculation of the partition function is performed, first in the isobaric and then in the canonical ensemble. After having presented the symmetric case, section \[sec:different-charges\] will generalize the investigation to the situations where the two screened charges are different. Noteworthy is that parity of the particle number<span style="font-variant:small-caps;"></span> considerations will play an important role in the remainder. Screening of two equal charges by counterions only {#sec:equal-charges} ================================================== Consider two charges $q$ along a line located at $\widetilde{x}=0$ and $\widetilde{x}=\widetilde{L}$. Between the charges there are $N$ counterions of charge $e=-2q/N$ between them. Consider the equilibrium thermal properties of this system at a temperature $T$, and as usual define $\beta=1/(k_B T)$ with $k_B$ the Boltzmann constant. This simple model mimics the screening and effective interaction between two charged colloids in a counterion solution, without added salt. In one dimension, $\beta e^2$ has dimensions of inverse length, therefore it is convenient to use rescaled units in which all distances are measured in units of $1/(\beta e^2)$: $x=\beta e^2 \widetilde{x}$. It is also convenient to work with a dimensionless pressure $P=\widetilde{P}/e^2$ where $\widetilde{P}$ is the pressure (equal to the force, in one dimensional systems). The potential energy (dimensionless, measured in units of $k_B T$) of the system is $$\label{eq:pot} U=-\sum_{1\leq i < j \leq N} |x_i-x_j| + \left(\frac{N}{2}\right)^2 L.$$ Before presenting the technical analysis, we start by simple and more quantitative considerations. Possibility of attraction between like-charges {#sec:like-charge-attraction} ---------------------------------------------- ### A heuristic argument The possibility of attraction between the two $+q$ charges at $0$ and $L$ is related to the parity of $N$. If $N$ is odd, $N=2p+1$, then $p$ counterions will form a double layer around each charge $q$. This will form two compound objects with charge $q(1-2p/N)=q/N$ each one, located around $0$ and $L$. There will be in addition one counterion between these two object, which is essentially free, as the electric field created by the charges located on each side around $0$ and $L$ cancel each other. When $L$ is large enough, consider figure \[fig:1Dmodel-odd-attract\]. The right side of the system composed of one charge $q$ and $p$ counterions has charge $q/N$. The left side which, for the sake of the argument, has the free counterion plus the compound charge, exhibits a total charge $-q/N$. Thus the force exerted by the left side on the right side is $\widetilde{P}\to -q^2/N^2=-e^2/4$, an attractive force. Thus one expects that $P \to -1/4$, for $L\to\infty$. ![An odd number of mobile counterions screening two like-charges. The $N$ mobile ions (counter-ions) have charge $-2q/N$ and the confining objects have charge $q$, so that the whole system is electro-neutral. Here, $N=2p+1$ is odd, so that a single ion (referred to as the misfit since the net electric force acting on it vanishes) “floats” in between the two screened boundaries which attract, each, $p$ ions in their vicinity (see also Fig. \[fig:1Dmodel-odd\]). This single free counterion provides the binding mechanism responsible for long range attraction. In the canonical treatment, $L$ is held fixed, while in the isobaric situation, it is a fluctuating quantity. \[fig:1Dmodel-odd-attract\]](1Dmodel-odd-attract){width="70.00000%"} On the other hand, if $N$ is even, there will not be a free counterion between the layers, which will be completely neutral, thus one expects that $P\to 0^{+}$ when $L\to\infty$, as shown in figure \[fig:1Dmodel-even\]. ![An even number of counterions screening two like-charges ($N=2p$). At large distance, the two double-layers (made up of an ion $q$ and $p$ counter-ions) decouple since they are neutral. No misfit ion is present to mediate attraction, and the pressure is repulsive at all distances. \[fig:1Dmodel-even\]](1Dmodel-even){width="70.00000%"} ### Beyond heuristics The previous intuition, providing a large distance attraction for odd $N$, can be substantiated by a simple calculation. Use will be made here of the contact theorem [@HBL79; @contact2; @contact3; @DHNP09; @MaTT15], an exact relation between the force exerted on the charge $q$, and the ionic density at contact (stemming from the mobile charges $-2q/N$). Such a relation is particularly useful for discussing the like-charge attraction phenomenon [@Netz01; @SaTr11; @rque9]. The argument allowing to get the contact density is two-fold, and goes as follows. ![Upon regrouping the $p+1$ leftmost counterions in Fig. \[fig:1Dmodel-odd-attract\], one obtains an ion with charge $-q-q/N$. This newly defined system has the same large distance pressure as that of Fig. \[fig:1Dmodel-odd-attract\]. \[fig:argument\_trick\]](1Dmodel-odd-effective-ion){width="70.00000%"} First, we argue that at large $L$, the $p$ counterions that are closest to each boundary remain in their vicinity, while the middle free counterion (the misfit in Figs. \[fig:1Dmodel-odd-attract\] and \[fig:1Dmodel-odd\]), which does not feel any electric field by symmetry, tends to be unbounded and no longer contributes to the pressure (discarding $1/L$ terms). In a second step, we thus compute the contact density in a system of an isolated charge $+q$, with a double-layer of $p$ ions in the vicinity (the total charge of this composite object, shown on the right hand-side of Fig. \[fig:1Dmodel-odd-attract\]) is $q/N$. The solution to this problem is not immediate, but can be found by a convenient mapping onto a more convenient problem, shown in Fig. \[fig:argument\_trick\]. As illustrated in the figure, we regroup the $p+1$ leftmost counterions in a single ion, having charge $-q(1+1/N)$. At large distances, this regroupment does not influence the distribution of counterions around the rightmost ion $+q$, and thus leaves the large $L$ pressure unaffected. The next important argument is that the pressure can be equivalently computed from the contact density at the rightmost, or leftmost charge $+q$. It is thus simpler to perform the calculation in the newly defined regrouped system (left hand side of Fig. \[fig:argument\_trick\]). The regrouped ion with charge $-q(1+1/N)$ is in the electric field of the charge $q$ on its left, and of the composite system on its right having charge $q/N$. This amounts to a field $q(1-1/N)$. Hence, the electric potential energy reads $q^2(1-1/N) \tilde{x}(1+1/N)$. The corresponding Boltzmann weight gives the density of the regrouped ion $$\rho(\tilde{x}) \,=\, \beta q^2\left(1-\frac{1}{N^2}\right) \, \exp\left[-\beta q^2 \tilde{x} \left(1-\frac{1}{N^2}\right) \right]$$ where due account was taken of normalization ($\int \rho\,d\tilde{x}=1$). The contact density $\rho(0)=\beta q^2(1-1/N^2)$ finally yields the pressure through the contact theorem $\beta \widetilde P = \rho(0) - \beta q^2$. We get here $\widetilde P=-q^2/N^2$ (or equivalently $P=-1/4$), a result which by construction holds in the large $L$ limit. The reason for a non vanishing pressure at large distance is that the $p$ counter-ions cannot exactly screen the charge of an ion $q$. It is no longer the case when $N$ is even, in which case $P\to 0$ for $L\to\infty$. The present results will be fully corroborated by direct partition function calculations. ### Correction to large distance asymptotics and crossover pressure Returning to the case when $N=2p+1$ is odd, we can also estimate the first correction to the pressure for large $L$. Consider that $L$ is fixed (canonical ensemble) and large. Since the system is somehow equivalent to two double layers with a free counterion in between, this counterion will contribute to the pressure (denoted as $P_c$ in the canonical, fixed-$L$ ensemble) with a correction $1/L$. This estimate can be made more quantitative. The available space for the free counterion is not $L$, but it is rather $L$ minus the space occupied by the diffuse counterion layers, given by $\langle x_p \rangle_{\infty}$ the thermal average position of the $p$-th counterion if they have been ordered $x_1<x_2< \cdots < x_p < x_{p+1} < \cdots < x_{2p+1}$, in the limit $L\to\infty$. Thus $$P_c = -\frac{1}{4} + \frac{1}{L-2\langle x_p \rangle_{\infty}} + o\left(\frac{1}{L}\right) \,.$$ This is illustrated in figure \[fig:1Dmodel-odd\]. In the following section, we evaluate explicitly $\langle x_p \rangle_{\infty}$ and find $$\label{eq:xp} \langle x_p \rangle_{\infty} = \frac{p}{p+1} = \frac{N-1}{N+1} \,.$$ Then, for large $L$, we expect $$\label{eq:pLinfty} P_c = -\frac{1}{4} + \frac{1}{L-2\, \frac{N-1}{N+1}}+ o\left(\frac{1}{L}\right) \,.$$ ![An odd number of counterions screening two like-charges. The free “misfit” ion is singled out. \[fig:1Dmodel-odd\]](1Dmodel-odd){width="70.00000%"} In the other limiting case $L\to 0$, the result is [@DHNP09] $P_c=N/L$, that can be understood as all the $N$ counterions are squeezed in a small distance $L$. Thus we see that the pressure is positive (repulsive force) for small separations $L\to 0$ then changes to negative pressure (attractive force) for large $L$. We will show in the following section that the $o(1/L)$ corrections in (\[eq:pLinfty\]) are actually exponentially small, in the canonical ensemble, therefore equation (\[eq:pLinfty\]) gives a fairly good approximation for the pressure for a large set of values of the separation $L$. From this, one can estimate the distance $L^*$, at which the effective force between the two charges becomes attractive $$\label{eq:Lstar_cano} L^* \simeq 4 + 2 \langle x_p \rangle_{\infty} = 4+ 2\,\frac{N-1}{N+1} \,.$$ Figure \[fig:PvsL-canon\] shows the pressure $P_c$ as a function of $L$, for $N=25$ and for $N=26$ particles. For $N=25$ (odd) the pressure changes its sign at $L^*=4+2*24/26\simeq 5.85$, while for $N=26$ the pressure is always positive. Summarizing, in the case of odd $N$, the possibility of having an effective attraction for large separations $L$ is due to the sharing of the “free” ion which leads to the creation of opposite charges objects (ions $q$ plus their counterion clouds). Although the analytical results presented here are valid only for this one-dimensional model, the same physical mechanism has also been observed in three dimensional systems [@MHK00; @Kim14]. It can also be surmised that in situation of odd $N$ where the free counterion has a varying charge, attraction will be all the stronger as the charge will increase in absolute value. In addition, the very mechanism brought to the fore here indicates that at mean-field level, where the discrete nature of ions is discarded, attraction should be suppressed, which indeed is the case [@lca1; @lca2; @lca3]. ![ \[fig:PvsL-canon\] The (canonical) pressure $P_c$ as a function of the separation $L$, for $N=25$ (continuous bottom line) and $N=26$ (dashed upper line). For $N$ odd the pressure becomes negative at large distances. ](PvsL-canon){width="50.00000%"} Explicit exact calculation of the partition function ---------------------------------------------------- ### Preliminary observations The equilibrium thermodynamics of the one-dimensional two-component Coulomb gas was solved simultaneously but independently by Lenard [@Len61] and Prager [@Pra61]. In the present model, only one type of identical particles (the counterions) are present. It is convenient to order the particles as $0\leq x_1 \leq \cdots \leq x_N \leq L$. Then, rearranging the terms in (\[eq:pot\]), the potential energy of the system can be written as $$\label{eq:Uodd} U=\frac{N^2 L}{4}-2\sum_{j=0}^{p-1}(p-j)(x_{2p+1-j}-x_{1+j}) \quad \text{for\ }N=2p+1\text{\ odd},$$ and $$U=\frac{N^2 L}{4}-\sum_{j=0}^{p-1}(2p-2j-1)(x_{2p-j}-x_{1+j}) \quad \text{for\ }N=2p\text{\ even}.$$ Notice that in the case $N=2p+1$, the particle with position $x_{p+1}$ does not appear in the potential energy. It is the free counterion (misfit) discussed in the previous section, whose role is crucial for the possibility of like-charge attraction. The canonical configuration integral is $$\label{eq:Z} Z_c(N,L)=\int_{0}^{L} dx_N \int_0^{x_N} dx_{N-1}\cdots \int_{0}^{x_3} dx_2 \int_{0}^{x_2} dx_1 \, e^{-U} \,.$$ As mentioned by Lenard in his seminal paper [@Len61] “the (configuration) integral is elementary (because) the class of functions consisting of exponential of linear functions is closed under the operation of indefinite integral (...) however the task of evaluating (it) is not trivial”. For small $N$ one can compute by hand $Z_c$, and for larger given values of $N$ it can be obtained numerically with the aid of a computer algebra system software program. By inspection of the integral (\[eq:Z\]), one can deduce that $Z_c$ is a linear combination of products of exponentials of $L$ and linear functions of $L$. One can also deduce the argument of each exponential function of $L$ by keeping track of the factor that multiplies each $x_k$ in the integral (\[eq:Z\]). These come from the explicit term in $U$ (for instance, for $x_{j+1}$ it is $2(p-j)$ in the case $N$ odd), but after each successive integration, the factor of $x_{k}$ will by added to the one of $x_{k+1}$ due to the upper limit of integration. Taking that into account, one realizes that the exponentials of $L$ in $Z_c$ are of the form $\exp(-(j+\frac{1}{2})^2 L)$ in the case $N$ odd, and $\exp(-j^2 L)$ in the case $N$ even. Thus, the configuration canonical integral is expected to be of the form $$Z_c(N,L)=\sum_{j=0}^{p} e^{-(j+\frac{1}{2})^2 L } ( A_j L + B_j) \quad \text{for\ }N=2p+1\text{\ odd},$$ and $$Z_c(N,L)=\sum_{j=0}^{p} e^{-j^2 L } ( C_j L + D_j) \quad \text{for\ }N=2p\text{\ even}.$$ The non trivial task is to evaluate explicitly the coefficients $A_j$, $B_j$, $C_j$ and $D_j$. This is done in section \[sec:canonical-exact\]. ### Previous results In [@DHNP09], the present system was studied, but an exact analytical explicit evaluation of the partition function for an arbitrary number of particles was not achieved. Rather, an interesting reformulation of this model was proposed, by mapping it into a quantum mechanical problem, following a technique put forward by Edwards and Lenard [@EL62]. It was shown in [@DHNP09] that the configuration integral is given by $$Z_c(N,L)=b(N/2,N/2,L)$$ where $b(n,N/2,x)$ is the solution of a set of $N$ coupled elementary linear differential equations $$\frac{db(n,N/2,x)}{dx}=-(n^2/2)\, b(n,N/2,x)+b(n-1,N/2,x)$$ with the initial condition $b(n,N/2,0)=\delta_{n,-N/2}$. Integrating this equation one has $$\label{eq:Dean-system} b(n,N/2,x_n)=\int_0^{x_{n}} e^{-(n^2/2) (x_{n}-x_{n-1})} b(n-1,N/2,x_{n-1}) \, dx_{n-1} \,.$$ Then, starting from the known $b(-N/2,N/2,x_1)$ one has to perform successively $N$ integrals (\[eq:Dean-system\]) to obtain $b(N/2,N/2,L)$ and the configuration integral. This task is equivalent to performing directly the $N$ integrals of the configuration integral (\[eq:Z\]). Thus, unfortunately, the method proposed in [@DHNP09] does not provide any computational advantage over a direct numerical evaluation of the partition function. Here, our goal is to obtain an explicit analytical expression for the configuration integral for an arbitrary number of particles $N$. Using Lenard [@Len61] and Prager [@Pra61] method, we will first compute the partition function of the constant pressure ensemble $$Z_P(N,P)=\int_{0}^{\infty} e^{-P L} Z_c(N,L)\, dL$$ which is the Laplace transform of the canonical configuration integral $Z_c$. This is a straightforward application of the technique of Lenard and Prager, and it is actually much simpler than the complete work presented in [@Len61; @Pra61], since all particles are identical and we will not have to deal with the combinatorial problem of studying the different configurations of charges. Then, we shall invert the Laplace transform to obtain the canonical, constant “volume” $L$, configuration integral $Z_c(N,L)$. Since we are interested in finite systems, the results from the canonical ensemble and the constant pressure ensemble will differ, and it is of interest to compare them. ### Evaluation of the diffuse layer size $\langle x_p \rangle_{\infty}$ To introduce the technique used to compute the partition function, we undertake in this section a preliminary, simpler task, based on the same technique: the exact evaluation of the diffuse layer size $\langle x_p \rangle_{\infty}$. This quantity appeared in the discussion of section \[sec:like-charge-attraction\]. Consider here that $L\to\infty$ and $N=2p+1$. The double layer composed by the charge $q$ at $L$ and its corresponding $p$ counterions are thereby ‘sent to infinity’. The remaining $p+1$ counterions, however, still feel the electric field created by this far charged double layer. The potential energy part which depends on the position of the remaining counterions is $$U_{\infty}=2\sum_{j=0}^{p-1}(p-j) x_{1+j} \,.$$ We wish to evaluate $$\langle x_p \rangle_{\infty}=\frac{\int_{0<x_1<x_2<\cdots<x_p} x_p\, e^{-U_{\infty}} \,\prod_{k=1}^{p}dx_k}{\int_{0<x_1<x_2<\cdots<x_p} e^{-U_{\infty}} \,\prod_{k=1}^{p}dx_k} \,.$$ Let $$F(s)=\int_{0<{x}_1<{x}_2<\cdots<{x}_p} \, e^{-U_{\infty}- s x_p/2}\,d{x}_1\ldots d{x}_p \,.$$ Then $\langle {x}_p \rangle_{\infty}=-2\,d\ln F(s)/ds|_{s=0}$. Following Lenard [@Len61] and Prager [@Pra61] it is convenient to re-write the potential energy as $$U_{\infty} = \frac{1}{2} \left[\sum_{j=1}^{p} \left( (p-j+1)^2+(p-j+2)^2 \right)(x_j-x_{j-1}) -x_p\right]$$ with the convention that $x_0=0$. Let us define $$\label{eq:fj} f_j({x})=e^{- \left[ (p-j+1)^2+(p-j+2)^2 \right] {x}/2 }\, H({x})$$ where $H({x})$ is the Heaviside step function. Then $$F(s)=\int_{0}^{\infty} d{x}_1 \cdots \int_{0}^{\infty} d{x}_p \prod_{j=1}^{p} f_{j}({x}_{j}-{x}_{j-1}) \, e^{-(s-1){x}_p/2}$$ We notice that $F(s)$ is the Laplace transform (evaluated at $(s-1)/2$) of the $p$-fold convolution product $f_1*f_2*\cdots * f_p$. The Laplace transform ${\cal L} f_{j}$ of $f_j$ is elementary $${\cal L} f_{j}\left(\frac{s-1}{2}\right) =\frac{2}{(p-j+1)^2+(p-j+2)^2+s-1}= \frac{2}{2(p-j+1)(p-j+2)+s}$$ Then $$F(s)=\prod_{j=1}^p\frac{2}{2(p-j+1)(p-j+2)+s} =\prod_{k=1}^p\frac{2}{2 k (k+1)+s}$$ Computing the derivative of $\ln F(s)$ we obtain $$\begin{aligned} \langle {x}_p \rangle_{\infty}&=& -2\left.\frac{d\ln F(s)}{ds}\right|_{s=0}=\sum_{j=1}^p \frac{1}{(p-j+1)(p-j+2)} \nonumber \\ &=& \sum_{k=1}^p\frac{1}{k(k+1)}= \sum_{k=1}^p\left(\frac{1}{k}-\frac{1}{k+1}\right) \nonumber\\ &=&\left(1-\frac{1}{p+1}\right)= \frac{p}{p+1} \,.\end{aligned}$$ Thus proving (\[eq:xp\]). ### Isobaric ensemble {#sec:isobaric-exact} Consider now the finite system with $L<\infty$. We will detail the calculations in the case $N=2p+1$ odd, the case $N$ even can be obtained by a simple adaptation of the same technique. As it was done in the previous section, it is convenient to re-write the potential energy (\[eq:Uodd\]) as $$U=-\frac{L}{4} +\frac{1}{2} \sum_{j=1}^{p+1} \left( (p-j+1)^2+(p-j+2)^2 \right) \left(x_{2p-j+3}-x_{2p-j+2} + x_j - x_{j-1} \right)$$ where, by convention, we defined $x_0=0$ and $x_{2p+2}=L$. With $f_j$ defined in (\[eq:fj\]), we notice again that the canonical partition function is a convolution product of $2p+2$ functions $f_j$ $$\label{eq:Zc_convol} Z_c(2p+1,L)= e^{L/4} \left( \mathop{{\scalebox{1.3}{\raisebox{-0.2ex}{$\ast$}}}}_{j=1}^{p+1} f_j * f_j \right) (L) \,.$$ The isobaric partition function $Z_P$ is the Laplace transform of $Z_c$, and we have $$\begin{aligned} Z_P(2p+1,P)&=& \prod_{j=1}^{p+1} \left( {\cal L}f_j \left( P-\frac{1}{4} \right)\right)^2 \nonumber\\ &=& \prod_{k=0}^{p} \frac{4}{[2 k (k+1)+s]^2} = \prod_{k=0}^{p} \frac{1}{\left[\left(k+\frac{1}{2}\right)^2+P\right]^2} \label{eq:ZP_odd} \end{aligned}$$ where $s=(4P+1)/2$. Factoring $\left(k+\frac{1}{2}\right)^2+P=(k+\frac{1}{2}-i\sqrt{P})(k+\frac{1}{2}+i\sqrt{P})=|k+\frac{1}{2}+i\sqrt{P}|^2$, the above product can be expressed in terms of Gamma functions $$Z_P(2p+1,P)= \left( \frac{1}{P+\frac{1}{4}} \right)^2 \left| \frac{\Gamma(\frac{3}{2}+i\sqrt{P})}{\Gamma(p+\frac{3}{2}+i\sqrt{P})} \right|^4 \,.$$ The average length of the system is given by the usual thermodynamic relation $$\begin{aligned} \label{eq:Lave} \langle L \rangle & = & -\frac{\partial \ln Z_P}{\partial P} = \frac{2}{P+ \frac{1}{4}} +\sum_{k=1}^{p} \frac{2}{\left(k+\frac{1}{2}\right)^{2}+ P} \\ &=& \frac{2}{P+ \frac{1}{4}} + \frac{2}{\sqrt{P}} \Im m\hspace{-1mm} \left[ \psi\left(p+\frac{3}{2}+i\sqrt{P}\right) -\psi\left(\frac{3}{2}+i\sqrt{P}\right) \hspace{-0.5mm}\right]\end{aligned}$$ where $\psi(z)=d\ln\Gamma(z)/dz$. We can notice that this expression has a pole for $P=-1/4$, from which we obtain the behavior when $\langle L\rangle\to\infty$, $P\to-1/4$, in agreement with the general discussion of section \[sec:like-charge-attraction\]. When $N$ is even this pole is absent (see below). If $N=2p$ is even, similar calculations lead to $$\label{eq:Zc_even} Z_c(2p,L)=e^{L/4} f_{p+\frac{3}{2}} * \left( \mathop{{\scalebox{1.3}{\raisebox{-0.2ex}{$\ast$}}}}_{j=1}^{p} f_{j+\frac{1}{2}} * f_{j+\frac{1}{2}} \right) (L)$$ and $$\label{eq:ZP_even} Z_P(2p,P)= \frac{1}{P}\prod_{k=1}^{p} \frac{1}{(k^2+P)^2} = \frac{1}{P} \left|\frac{\Gamma(1+i\sqrt{P})}{\Gamma(p+1+i\sqrt{P})}\right|^4 \,.$$ Notice an important difference in the analytic structure of the partition function in the case $N$ odd (\[eq:Zc\_convol\])–(\[eq:ZP\_odd\]) and $N$ even (\[eq:Zc\_even\])–(\[eq:ZP\_even\]): for $N$ even, there is a single function $f_{p+3/2}$ in the convolution product, leading to a pole of order one for $P=0$, in contrast to the case $N$ odd, where the functions $f_{p+1}$ appear twice in the convolution product and the pole for the smallest value of $|P|$ is of order two and it is for $P=-1/4$, rather than $P=0$. In the case $N$ even, the term $f_{p+1}*f_{p+1}$ corresponds to the coupling of the left diffuse layer with the free counterion and the coupling of this same free counterion with the right diffuse layer. On the other hand in the case $N$ odd, the term $f_{p+3/2}$ corresponds to the direct coupling of the left and right diffuse layers. The average length, for $N=2p$ even, is $${\langle L \rangle}= \frac{1}{P} + \sum_{k=1}^p \frac{2}{k^2 + P} \,.$$ We note that ${\langle L \rangle}\to \infty$ when $P\to 0^{+}$, in contrast to what happens when $N$ is odd, where ${\langle L \rangle}\to \infty$ when $P\to -1/4$. ### Canonical ensemble {#sec:canonical-exact} We return to the case $N=2p+1$ odd. To compute the canonical partition function, we need to invert the Laplace transform computed in the previous section $$Z_c(2p+1,L)= {\cal L}^{-1}\left ( \prod_{k=0}^{p} \frac{1}{\left[\left(k+\frac{1}{2}\right)^2+P\right]^2} \right) (L).$$ This rather technical part of the analysis is presented in Appendix \[app:A\], where it is shown that $$\label{eq:ZexactNimpar} Z_c(2p+1,L)= \sum_{j=0}^{p} \left[\frac{2j+1}{(p-j)!(p+j+1)!}\right]^2 e^{-(j+\frac{1}{2})^2 L} \left[ L+\frac{2}{2j+1}\left( \sum_{k=p-j+1}^{p+j+1} \frac{1}{k} -\frac{1}{2j+1} \right) \right] \,.$$ From this expression, we obtain the canonical pressure $P_c=\frac{d\ln Z_c}{dL}$, $$\label{eq:PexactNimpar} P_c=-\frac{ {\displaystyle \sum_{j=0}^{p}} \frac{4\left(j+\frac{1}{2}\right)^4}{\left[(p-j)!(p+j+1)!\right]^2} \left[ L+\frac{2}{2j+1}\left( {\displaystyle \sum_{k=p-j+1}^{p+j+1} } \frac{1}{k} -\frac{3}{2j+1} \right) \right] e^{-(j+\frac{1}{2})^2 L} }{ {\displaystyle \sum_{j=0}^{p} } \left[\frac{2j+1}{(p-j)!(p+j+1)!}\right]^2 \left[ L+\frac{2}{2j+1}\left( {\displaystyle \sum_{k=p-j+1}^{p+j+1}} \frac{1}{k} -\frac{1}{2j+1} \right) \right] e^{-(j+\frac{1}{2})^2 L} } \,.$$ For $N=2p$ even, the results are $$Z_c(2p,L)= \frac{1}{(p!)^4}- \sum_{j=1}^p \frac{(2 j)^2 e^{-j^2 L}}{[(p+j)!(p-j)!]^2} \left[ L + \frac{1}{j}\left( \sum_{k=p-j+1}^{p+j} \frac{1}{k} - \frac{1}{2j} \right) \right] \,,$$ and $$P_c=\frac{ {\displaystyle\sum_{j=1}^p} \frac{4j^4 e^{-j^2 L}}{[(p+j)!(p-j)!]^2} \left[ L + \frac{1}{j}\left( {\displaystyle\sum_{k=p-j+1}^{p+j}} \frac{1}{k} -\frac{3}{2j} \right) \right]} {\frac{1}{(p!)^4}- {\displaystyle \sum_{j=1}^p } \frac{(2 j)^2 e^{-j^2 L}}{[(p+j)!(p-j)!]^2} \left[ L + \frac{1}{j}\left( {\displaystyle\sum_{k=p-j+1}^{p+j}} \frac{1}{k} -\frac{1}{2j} \right) \right]} \,.$$ ### Limiting cases and comparison between the ensembles With the exact expressions obtained above, we can prove rigorously the limiting behavior of the pressure when $L\to\infty$ and $L\to 0$ discussed in section \[sec:like-charge-attraction\]. Let us consider first the case $N=2p+1$ odd. In the canonical ensemble, the behavior of the pressure $P_c$ when $L\to \infty$, is obtained from the term $j=0$ of (\[eq:ZexactNimpar\]), confirming the prediction (\[eq:pLinfty\]) of section \[sec:like-charge-attraction\]. Furthermore, we realize that the next to next to leading order correction is exponentially small $$P_c = -\frac{1}{4} + \frac{1}{L-2\, \frac{p}{p+1}} -2 \left(\frac{3p}{p+2}\right)^2 e^{-2L} \left(1+O(L^{-1})\right) + O\left(e^{-6 L}\right) \,.$$ In contrast, when $N=2p$, the pressure tends to 0 exponentially fast when $L\to\infty$ $$P_c = \frac{ 4p^2 e^{-L}}{(p+1)^2} \left(L+ \frac{2p+1}{p(p+1)}-\frac{3}{2}\right) +O\left( e^{-2L} \right) \,.$$ The behavior of the pressure is different in the isobaric ensemble. Consider again first the case $N=2p+1$. From (\[eq:Lave\]), we already know that when $P=-1/4$, $\langle L \rangle \to\infty$. Denoting $s=(4P+1)/2$, one can expand (\[eq:Lave\]) for small $s$ and invert the relation to obtain $P$ as a function of $\langle L\rangle$ when $\langle L\rangle\to\infty$. For instance, to order $O(s)$, Eq. (\[eq:Lave\]) is $$\langle L \rangle= \frac{4}{s}+\frac{2p}{p+1}- s S(p) + o(s) \,,$$ where $$S(p)=\sum_{k=1}^{p} \frac{1}{[k(k+1)]^2} =2{\cal H}^{(2)}_p - \frac{p(3p+4)}{(p+1)^2} \,,$$ with ${\cal H}_{p}^{(r)} = \sum_{k=1}^{p} k^{-r} $ the harmonic numbers. Inverting that relation, up to order $O(\langle L \rangle^{-3})$, gives $$P=-\frac{1}{4} + \frac{2}{{\langle L \rangle}-2\, \frac{p}{p+1}} - \frac{8S(p)}{({\langle L \rangle}-2\, \frac{p}{p+1})^3}+o\left(\frac{1}{({\langle L \rangle}-2\, \frac{p}{p+1})^{3}}\right) \,.$$ Notice a factor 2 of difference in the next to leading order correction (the $O({\langle L \rangle}^{-1})$ term) in the pressure in the isobaric ensemble and the canonical ensemble. Furthermore, in the isobaric ensemble the next to next to leading order corrections are algebraic and not exponential as in the canonical ensemble. For $N=2p$, the behavior of the pressure, in the isobaric ensemble, when ${\langle L \rangle}\to\infty$, is $$P=\frac{1}{{\langle L \rangle}- 2 {\cal H}_p^{(2)}} - \frac{2 {\cal H}_p^{(4)}}{[{\langle L \rangle}-2 {\cal H}_p^{(2)}]^3} +O\left({\langle L \rangle}^{-4}\right) \,.$$ Notice again the different behavior with respect to the canonical ensemble. Here in the isobaric ensemble, the pressure vanishes as $1/{\langle L \rangle}$, whereas in the canonical ensemble it vanishes exponentially fast, as $e^{-L}$. Let us study the other limiting behavior of the pressure, for small separations $L$. Let us focus on the case $N=2p+1$ first. It is not completely straightforward to obtain the behavior of the pressure in the canonical ensemble when $L\to 0$ directly from expression (\[eq:PexactNimpar\]). Rather, it is better to return to (\[eq:Zc\_convol\]), and notice that if $L\to 0$, then the convolution product $f_j*f_j$ behaves as $$f_j*f_j(x) = x H(x) + O(x^2)$$ which is independent of $j$. Then, $$\left( \mathop{{\scalebox{1.3}{\raisebox{-0.2ex}{$\ast$}}}}_{j=1}^{p+1} f_j * f_j \right) (x) = \frac{x^{2p+1}}{(2p+1)!} + O(x^{2p+2})$$ and $$Z_c(2p+1,L)= \frac{L^{N}}{N!} + O(L^{N+1}) \,.$$ We deduce that the pressure behaves as $$\label{eq:PcanoL0} P_c\sim \frac{N}{L} \quad\text{when\ }L\to 0 \,,$$ a result already noticed in [@DHNP09]. Eq. (\[eq:PcanoL0\]) also holds when $N=2p$. In the isobaric ensemble, when $N=2p+1$, if ${\langle L \rangle}\to 0$, then, necessarily, $s=(4P+1)/2\to\infty$ in (\[eq:Lave\]). Expanding that equation to order $O(s^{-2})$, one obtains $$\label{eq:PisoL0} P = \frac{N+1}{{\langle L \rangle}} - \frac{N(N+2)}{12} +O({\langle L \rangle}) \quad\text{when\ }{\langle L \rangle}\to 0 \,.$$ This result also holds true for $N=2p$. Notice again the difference between the canonical (\[eq:PcanoL0\]) and isobaric ensemble (\[eq:PisoL0\]), where the leading term changes from $N/L$ to $(N+1)/L$. When $N=2p+1$ is odd, the pressure changes of sign when $L$ varies. It is positive for $L\to 0$ and negative for $L\to \infty$. We already obtained an approximation of the value $L^*$ of $L$ when this occurs in the canonical ensemble, see (\[eq:Lstar\_cano\]), up to exponentially small corrections. In the isobaric ensemble, one just has to put $P=0$ in (\[eq:Lave\]) to obtain the exact value $$\label{eq:Lstar_isop} \langle L^* \rangle =8 \left( 1+ \sum_{k=1}^{p} \frac{1}{(2k+1)^2} \right) = \pi^2-2\psi'(p+3/2) \,.$$ For this quantity, the predictions from the canonical ensemble (\[eq:Lstar\_cano\]) and the isobaric ensemble (\[eq:Lstar\_isop\]) are again different. Figure \[fig:PvsL-iso-cano\] shows the pressure as a function of the separation, for $N=15$, in the isobaric ensemble and the canonical ensemble. Notice that the pressure from the canonical ensemble is smaller that the one in the isobaric ensemble for the same separation. Figure \[fig:LstarvsN\] shows the value of $L^*$ for which the pressure changes of sign as a function of $N$, when $N$ is odd, in both ensembles. Notice again that in the canonical ensemble, the change of sign of the pressure occurs for smaller values $L^*$ of the separation than in the isobaric ensemble. ![ \[fig:PvsL-iso-cano\] The pressure $P$ as a function of the separation $L$, for $N=15$. The top continuous line represents the result from the isobaric ensemble, and the dotted bottom line those from the canonical ensemble. ](PvsL-iso-cano){width="7cm"} ![ \[fig:LstarvsN\] The value of the separation $L^*$ for which the pressure vanishes and changes sign as a function of $N$ for $N$ odd. The filled squares represent the results from the isobaric ensemble, and the filled disks, their canonical counterpart.](LstarvsN){width="7cm"} Screening of two unequal charges {#sec:different-charges} ================================ In this section we consider a generalization of the previous model, where the two charges located at $x=0$ and at $x=L$ are $q_1$ and $q_2$, respectively, which can be eventually different. The overall system should be neutral, therefore $q_1+q_2=-Ne$, $e$ being charge of one counterion. It is convenient to introduce the notation $Q_1$ and $Q_2$ such that $q_1=-eQ_1$ and $q_2=-eQ_2$. The electroneutrality relation is $Q_1+Q_2=N$. The charge asymmetry can be characterized by the quantity $a =Q_1-Q_2$, which allows to write $Q_1=(N+a)/2$ and $Q_2=(N-a)/2$. The potential energy of the system is now $$\label{eq:pot-a} U(N,L,Q_1,Q_2)=-\sum_{1\leq i < j \leq N} |{x}_i-{x}_j| + a \sum_{i=1}^{N} {x}_i + (Q_2)^2 {L} \,.$$ The overall effect of the charge asymmetry is to introduce a global electric field proportional to $a$ (the term in $\sum_i x_i$). Isobaric ensemble {#sec:isobaric-exact-a} ----------------- Adapting the ideas of section \[sec:isobaric-exact\] to the present case, we can obtain the isobaric partition function. Once again, the results differ depending on the parity of the number of counterions $N$. For $N=2p+1$ odd, $$\label{eq:ZP-Nodd-a} Z_P(2p+1,P,Q_1,Q_2)=\prod_{k=0}^{p}\frac{1}{\left[(k+\frac{1-a}{2})^2+ P\right] \left[(k+\frac{1+a}{2})^2+ P\right]}$$ while for $N=2p$ even, $$\label{eq:ZP-Neven-a} Z_P(2p,P,Q_1,Q_2)=\frac{1}{\left(\frac{a}{2}\right)^2+P} \prod_{k=1}^{p}\frac{1}{\left[(k-\frac{a}{2})^2+ P\right] \left[(k+\frac{a}{2})^2+P\right]} \,.$$ The above formulas highlight the difference between the two cases, depending on the parity of $N$. However both formulas can be summarized in a single one as $$\begin{aligned} \label{eq:ZP-a} Z_P(N,P,Q_1,Q_2)&=\prod_{\ell=0}^{N}\frac{1}{(\ell-\frac{N-|a|}{2})^2+P} =\prod_{l=0}^{N}\frac{1}{(\ell-Q_{<})^2+P} \nonumber\\ &=\prod_{y\in\{-Q_<,-Q_{<}+1,\ldots, Q_{>}-1,Q_{>}\}}\frac{1}{y^2+P} \,.\end{aligned}$$ where we defined $$\label{eq:Qinf} Q_{<}=\frac{N-|a|}{2}=\min\left(Q_1,Q_2\right) \quad\text{and}\quad Q_{>}=\frac{N+|a|}{2}=\max\left(Q_1,Q_2\right)\,.$$ Taking the derivative of (\[eq:ZP-a\]) with respect to $P$, we obtain the relation between the average length ${\langle L \rangle}$ of the system and the pressure $P$ in the isobaric ensemble $$\label{eq:Lave-a} {\langle L \rangle}= \sum_{\ell=0}^{N} \frac{1}{(\ell-Q_{<})^2+P} \,.$$ If $a\not\in\mathbb{Z}$ is not an integer ($q_1$ and $q_2$ are not integer multiples of $-e/2$), or $|a|> N$ ($q_1$ and $q_2$ have opposite signs), then $Z_P$ has simple poles. But when $a\in\mathbb{Z}$ is an integer and $|a| \leq N$, the partition function $Z_P$ turns out to have some double poles. This corresponds to the case when $2Q_1$ and $2Q_2$ are both positive integers. In that case it is best to reorder the products in (\[eq:ZP-a\]) to make those double poles more apparent. The result depends on the parity of $2Q_1$ and $2Q_2$ (both have the same parity). If $2Q_1$ and $2Q_2$ are odd, then $Q_1$ and $Q_2$ are half integers: $Q_1={\left\lfloor Q_1 \right\rfloor}+\frac{1}{2}$ and $Q_2={\left\lfloor Q_2 \right\rfloor}+\frac{1}{2}$. The notation ${\left\lfloor x \right\rfloor}$ denotes the floor function of $x$ (largest integer less or equal than $x$). The isobaric partition function (\[eq:ZP-a\]) becomes $$\label{eq:ZP-Na-odd} Z_P(N,P,Q_1,Q_2)= \prod_{\ell=0}^{{\left\lfloor Q_{<} \right\rfloor}}\frac{1}{\left[(\ell+\frac{1}{2})^2+P\right]^2} \prod_{\ell={\left\lfloor Q_{<} \right\rfloor}+1}^{{\left\lfloor Q_{>} \right\rfloor}}\frac{1}{(\ell+\frac{1}{2})^2+P} \,,$$ and the corresponding equation of state is $$\label{eq:Lave-Na-odd} {\langle L \rangle}= \sum_{\ell=0}^{{\left\lfloor Q_{<} \right\rfloor}}\frac{2}{(\ell+\frac{1}{2})^2+P}+ \sum_{\ell={\left\lfloor Q_{<} \right\rfloor}+1}^{{\left\lfloor Q_{>} \right\rfloor}}\frac{1}{(\ell+\frac{1}{2})^2+P} \,.$$ When $Q_1$ and $Q_2$ are positive integers, these expressions become $$\label{eq:ZP-Na-even} Z_P(N,P,Q_1,Q_2)=\frac{1}{P} \prod_{\ell=1}^{Q_{<}}\frac{1}{\left(\ell^2+P\right)^2} \prod_{\ell=Q_{<}+1}^{Q_{>}}\frac{1}{\ell^2+P} \,,$$ and $$\label{eq:Lave-Na-even} {\langle L \rangle}=\frac{1}{P}+ \sum_{\ell=1}^{Q_{<}}\frac{2}{\ell^2+P} +\sum_{\ell=Q_{<}+1}^{Q_{>}}\frac{1}{\ell^2+P} \,.$$ Canonical ensemble: the partition function ------------------------------------------ To compute the canonical partition function, one has to perform the inverse Laplace transform of the expressions obtained in the last section. From the above discussion, it is clear that the results will have a different analytical structure depending on whether the isobaric partition function has simple or double poles, that is, depending on whether $a$ is an integer or not. If $a$ is not an integer, or $|a|>N$, all poles of $Z_P$ are simple poles, and we obtain from (\[eq:ZP-a\]): $$\begin{aligned} Z_c(N,{L},Q_1,Q_2)&= \sum_{j=0}^{N} (-1)^{j}e^{-(j-\frac{N-|a|}{2})^2 L} \frac{(2j-N+|a|)\,\Gamma(j-N+|a|)}{j!\,(N-j)!\,\Gamma(j+|a|+1)} \nonumber\\ &= \sum_{j=0}^{N} (-1)^{j}e^{-(j-Q_{<})^2 L} \frac{2(j-Q_{<})\,\Gamma(j-2Q_{<})}{j!\,(N-j)!\,\Gamma(j+|Q_1-Q_2|+1)} \,. \label{eq:Zc-a}\end{aligned}$$ This formula is valid whenever $2Q_1$ and $2Q_2$ are not integers, or if $Q_1$ and $Q_2$ have opposite signs ($Q_{<}<0$ and $Q_{>}>0$). If $a$ is an integer, with $|a| \leq N$, then using (\[eq:ZP-Na-odd\]), we obtain, when $Q_1$ and $Q_2$ are half integers, $$\begin{aligned} \label{eq:Zc-Na-odd} Z_c(N,{L},Q_1,Q_2)&= \sum_{j=0}^{{\left\lfloor Q_{<} \right\rfloor}} \frac{(2j+1)^2\,e^{-(j+\frac{1}{2})^2 {L}}} {({\left\lfloor Q_1 \right\rfloor}+1+j)!({\left\lfloor Q_1 \right\rfloor}-j)! ({\left\lfloor Q_2 \right\rfloor}+1+j)!({\left\lfloor Q_2 \right\rfloor}-j)!} \nonumber\\ &\hspace{-2.9cm} \times \left[ L -\frac{1}{2j+1} \left( \psi( {\left\lfloor Q_1 \right\rfloor}-j) -\psi({\left\lfloor Q_1 \right\rfloor}+1+j)+ \psi({\left\lfloor Q_2 \right\rfloor}-j) -\psi({\left\lfloor Q_2 \right\rfloor}+1+j) +\frac{2}{2j+1} \right) \right] \nonumber\\ &+ \sum_{j={\left\lfloor Q_{<} \right\rfloor}+1}^{{\left\lfloor Q_{>} \right\rfloor}} \frac{e^{-(j+\frac{1}{2})^2{L}/4} (j-{\left\lfloor Q_{<} \right\rfloor}-1)!\,(2j+1)(-1)^{j-{\left\lfloor Q_{<} \right\rfloor}-1} }{({\left\lfloor Q_{<} \right\rfloor}+1+j)!({\left\lfloor Q_{>} \right\rfloor}+1+j)!({\left\lfloor Q_{>} \right\rfloor}-j)!} \,, \nonumber\\\end{aligned}$$ and when $Q_1$ and $Q_2$ are integers, $$\begin{aligned} \label{eq:Zc-Na-even} Z_c(N,{L},Q_1,Q_2)&= \sum_{j=1}^{Q_{<}} \frac{-(2j)^2\,e^{-j^2 {L}}} {(Q_1+j)!(Q_1-j)!(Q_2+j)!(Q_2-j)!} \nonumber\\ &\hspace{-2.5cm} \times \left[ L -\frac{1}{2j} \left( \psi(Q_1+1-j) -\psi(Q_1+1+j)+ \psi(Q_2+1-j) -\psi(Q_2+1+j) +\frac{1}{j} \right) \right] \nonumber\\ &+ \sum_{j=Q_{<}+1}^{Q_{>}} \frac{e^{-j^2{L}} (j-Q_{<}-1)!\,(2j)(-1)^{j-Q_{<}} }{(Q_{<}+j)!(Q_{>}+j)!(Q_{>}-j)!} + \frac{1}{\left(Q_1!Q_2!\right)^2} \,. \end{aligned}$$ The two previous results (\[eq:Zc-Na-odd\]) and (\[eq:Zc-Na-even\]) show the different analytical structure of the two cases which depend on the parity of $2Q_1$ and $2Q_2$, in particular the existence of a term independent of $L$ in the case $2Q_1$ and $2Q_2$ even, and the form of the argument of the exponentials $e^{-j^2 L}$ (for $2Q_1$ even), as opposed to $e^{-(j+\frac{1}{2})^2 L}$ (for $2Q_1$ odd). However, both results (\[eq:Zc-Na-odd\]) and (\[eq:Zc-Na-even\]) can be subsumed in a single formula as follows. Let us define $$\begin{aligned} \label{eq:ALj} A_j(N,{L},Q_1,Q_2)&=\frac{(2(Q_{<}-j))^2 (-1)^{2Q_{>}+1}} {(2Q_{<}-j)!j!(N-j)!(|Q_1-Q_2|+j)!} \\ &\quad \times \left[ L - \frac{ \psi({\scriptstyle j+1})-\psi({\scriptstyle 2Q_{<}-j+1}) +\psi({\scriptstyle j+|Q_1-Q_2|+1})-\psi({\scriptstyle N-j+1}) +\scriptstyle \frac{1}{Q_{<}-j} }{2(Q_{<}-j)} \right] \,, \nonumber\end{aligned}$$ for $j\neq Q_{<}$, and, when $Q_{<}$ is an integer, define $$\label{eq:ALj-part} A_{Q_{<}}(N,{L},Q_1,Q_2)= \frac{1}{\left(Q_1!Q_2!\right)^2} \,.$$ Also, let $$\label{eq:Dj} D_j(N,Q_1,Q_2) =\frac{j!\,2(j+Q_{<}+1)(-1)^{j+2Q_{>}+1}}{(2Q_{<}+j+1)!(N+j+1)!(|Q_1-Q_2|-j-1)!} \,.$$ Then, both results (\[eq:Zc-Na-odd\]) and (\[eq:Zc-Na-even\]) are equivalent to $$Z_c(N,{L},Q_1,Q_2)= \sum_{j=0}^{{\left\lfloor Q_{<} \right\rfloor}} A_j(N,L,a)\, e^{-(Q_{<}-j)^2 L} +\sum_{j=0}^{|Q_1-Q_2|-1} D_j(N,a)\, e^{-(j+Q_{<}+1)^2 {L}} \,. \label{eq:Zc-a-general}$$ Canonical ensemble: asymptotic behavior of the pressure ------------------------------------------------------- For small separations $L$, the results (\[eq:PcanoL0\]), $P_c\sim N/L$ (canonical) and (\[eq:PisoL0\]), $P\sim (N+1)/{\langle L \rangle}$ (isobaric), still hold independently of the charge asymmetry $a$. Thus, the effective interaction is always repulsive at short distance, irrespective of the charges $q_1$ and $q_2$, even in the case where these charges are opposite. Indeed, the pressure is dominated here by the entropy cost for confining the ions in a narrow domain. The behavior for large separations $L$ will depend on whether the charges $q_1$ and $q_2$ are multiples of $e$ or not, and their relative signs. There are four cases to consider. **Opposite charges.** First, suppose that $q_1 q_2 < 0$, the charges at the edges have opposite signs. This corresponds to the case $|a|> N$, and the canonical partition function is obtained with Eq. (\[eq:Zc-a\]). From that expression, we deduce that for $L$ large, the leading order is given by the term $j=0$ of that sum. Therefore, the effective force is attractive and given by $$\label{eq:PcQ1Q2neg} P_c\sim - (Q_{<})^2 \,,\qquad L\to\infty\,, \,$$ where here $Q_{<}=(N-|a|)/2<0$ corresponds to the charge of the edge particle which has the same sign as the small ions. This result can actually be obtained by simple arguments. The small ions will be repelled by the particle with charge corresponding to $Q_{<}$ and attracted to the other edge where there is a particle with charge $-eQ_{>}$ with $Q_{>}=(N+|a|)/2>0$. By electroneutrality, the charge of the compound object formed by the small ions and $-eQ_{>}$ will be $eQ_{<}$. The effective force between this object and the other opposite charge $-eQ_{<}$ is repulsive, equal to $-(eQ_{<})^2$, thus recovering (\[eq:PcQ1Q2neg\]). Application of the contact theorem of course yields the same result, since the density of counterions vanishes at contact with $Q_{<}$ (a similar effect was reported in [@PaTr11; @SaTr14]). **Like-charges that are not integer multiples of $-e$.** To discuss this situation, we keep in mind that $Q_1>0$ and $Q_2>0$ are not integers. The small ions of charge $e$ will be divided into two parts that will try to screen the charges $q_1$ and $q_2$. A number ${\left\lfloor Q_1 \right\rfloor}$ of counterions will partially screen $q_1$ and ${\left\lfloor Q_2 \right\rfloor}$ ions will partially screen the other charge $q_2$. Each edge, with its screening cloud of counterions, will have a charge $-e (Q_1-{\left\lfloor Q_1 \right\rfloor})=-e \{ Q_1 \}$ and $-e (Q_2-{\left\lfloor Q_2 \right\rfloor})=-e \{ Q_2 \}$ respectively, where $\{ x \}:=x-{\left\lfloor x \right\rfloor}$ denotes the fractional part of $x$. However, since $Q_1$ and $Q_2$ are not integers, we have ${\left\lfloor Q_1 \right\rfloor}+{\left\lfloor Q_2 \right\rfloor}=N-1$: there is still one counterion to take into consideration. This counterion feels the electric field created by the charge difference $-e (\{Q_1\}-\{Q_2\})$, therefore it will be attracted to the edge which has the largest remaining charge (in the sense of the largest between $\{Q_1\}$ and $\{ Q_2 \}$). To fix the ideas suppose $\{Q_1\}>\{Q_2\}$. The remaining ion will become part of the screening cloud of $q_1$, and the charge of that compound object will be $-e(\{Q_1\}-1)$. Then the effective force between the two edges will be $e^2 (\{Q_1\}-1) \{Q_2\} = -e^2 \{Q_2\}^2$, the last equality coming from the fact that $\{Q_1\}+\{Q_2\}=1$. Summarizing, in general we expect an attractive force at large separations given by $$\label{eq:PcQnotint} P_c\sim -\left(\min\left( \{ Q_1 \}, \{ Q_2 \} \right)\right)^2\,, \qquad L\to \infty\,.$$ This can be verified by identifying the largest argument of the exponentials in the canonical partition function (\[eq:Zc-a\]) or, equivalently, the largest pole of the isobaric partition function (\[eq:ZP-a\]). The poles of the isobaric partition function are $-(\ell - Q_{<})^2$, with $\ell$ going from 0 to $N$. Then, one can notice that $\ell - Q_{<}$ varies from $-Q_{<}<0$ up to $Q_{>}>0$ by integer steps of 1. From this one-dimensional array of points, we are interested in the one that is the closest to 0. That is precisely $\min\left( \{ Q_1 \}, \{ Q_2 \} \right)$, in agreement with (\[eq:PcQnotint\]). One can also notice from (\[eq:Zc-a\]) that in the canonical ensemble, the next to leading order correction to (\[eq:PcQnotint\]) is exponentially small of order $O(e^{-|\{ Q_1 \} - \{ Q_2 \} | L})$. **Like-charges that are half-integer multiples of $-e$.** A degenerate case of the previous situation is when $Q_1$ and $Q_2$ are half-integers, that is $\{Q_1\}=\{Q_2\}=\frac{1}{2}$. In this case the canonical partition function is given by (\[eq:Zc-Na-odd\]) instead of (\[eq:Zc-a\]). The leading order is still given by (\[eq:PcQnotint\]), specifically $P_c\sim -1/4$. But the correction to leading order is not exponentially small, it can be read from the term $j=0$ of (\[eq:Zc-Na-odd\]) $$\label{eq:Pc-half-int} P_c=-\frac{1}{4}+\frac{1}{L-L_1-L_2 } + O(e^{-2L}) \,,$$ with $$\label{eq:L1L2} L_m=1-\psi(Q_m+\frac{1}{2}+1)+\psi(Q_m+\frac{1}{2})= \frac{Q_m-\frac{1}{2}}{Q_m+\frac{1}{2}}= \frac{{\left\lfloor Q_m \right\rfloor}}{{\left\lfloor Q_m \right\rfloor}+1} \,, \quad m=1,\, 2\,.$$ We find here the generalization of the charge-symmetric case ($Q_1=Q_2=p+\frac{1}{2}$) discussed in section \[sec:equal-charges\]. Each charge $q_1$ and $q_2$ is screened by ${\left\lfloor Q_1 \right\rfloor}$ and ${\left\lfloor Q_2 \right\rfloor}$ ions. The remaining counterion is free to roam in a region of size $L-L_1-L_2$, and with zero electric field. This ion contributes to the pressure with a term $\frac{1}{L-L_1-L_2}$. Here $L_1=\langle x_{{\left\lfloor Q_1 \right\rfloor}}\rangle_{\infty}$ is the size of the screening layer of ${\left\lfloor Q_1 \right\rfloor}$ counterions formed around $q_1$ and $L_2=\lim_{L\to\infty} (L-\langle x_{N+1-{\left\lfloor Q_2 \right\rfloor}} \rangle)$ the size of the layer of ${\left\lfloor Q_2 \right\rfloor}$ counterions formed around $q_2$ (compare (\[eq:L1L2\]) to (\[eq:xp\]), when ${\left\lfloor Q_1 \right\rfloor}={\left\lfloor Q_2 \right\rfloor}=p$). **Like-charges that are natural integer multiples of $-e$.** In this case, the screening is not frustrated as in all the previous situations. Simply $Q_1$ counterions will screen the charge $q_1$ forming a neutral object, and similarly around $q_2$ there will be a screening cloud of $Q_2$ counterions. Since both objects with their screening clouds are neutral, the effective force between them is expected to be $P_c\to 0^{+}$. This can be verified from the expression for the partition function applicable here, Eq. (\[eq:Zc-Na-even\]). If $L\to\infty$, we have $$\label{eq:Zceven-Linfty} Z_c=\frac{1}{Q_1!^2 Q_2!^2} \left.- \frac{4 e^{-L}}{Q_1!^2Q_2!^2} \frac{Q_1}{Q_1+1} \frac{Q_2}{Q_2+1} \left[ L +\frac{1}{2} \left( \frac{2Q_1+1}{Q_1(Q_1+1)} - \frac{2Q_2+1}{Q_2(Q_2+1)}-1\right)\right]\right] +O(e^{-4L})\,.$$ Therefore, $$\label{eq:Pcasym-int} P_c= 4e^{-L} \frac{Q_1}{Q_1+1} \frac{Q_2}{Q_2+1} \left[ L +\frac{1}{2} \left( \frac{2Q_1+1}{Q_1(Q_1+1)} - \frac{2Q_2+1}{Q_2(Q_2+1)}-3\right)\right]+O(e^{-2L}) \,.$$ Density profile --------------- With the above results, we can obtain an explicit expression for the density profile of counterions $$\label{eq:density-def} n({x})=\frac{\sum_{k=1}^{N} \int_{x_1< \cdots <x_{k-1}<x_k=x<x_{k+1}<\cdots<x_N} e^{-U(N,L,Q_1,Q_2)} \, \prod_{j=1,j\neq k}^{N} dx_j }{Z_c(N,{L},Q_1,Q_2)} \,.$$ Notice that due to the fact that each particle only feels a constant electric field proportional to the difference between the number of charges which are at its left and right sides, the potential energy has the following property $$\label{eq:Upot-divide} U(N,L,Q_1,Q_2)= U(k-1,x_k,Q_1,Q_2-(N-k+1)) +U(N-k,L-x_k,Q_1-k,Q_2) \,.$$ This can be interpreted as follows. If the particle at position ${x}_k$ is fixed, the system decouples into two independent systems, one of size ${x}_k$ with $k-1$ particles, and the other one of size ${L}-{x}_k$ with $N-k$ particles, with the appropriate charges at each boundary (obtained by summing the charges at the left side and right sides of ${x_k}$ of the original system). Then, the computation of the integrals in (\[eq:density-def\]) simply yields the product of the two partition functions of each subsystem, $$\label{eq:density} n({x})=\frac{\sum_{k=1}^{N} Z_c(k-1,{x}, Q_1,Q_2-N+k-1) Z_c(N-k,{L}-{x}, Q_1 -k , Q_2) }{Z_c(N,{L},Q_1,Q_2)} \,,$$ where each $Z_c$ should be replaced by its appropriate corresponding expression from (\[eq:Zc-a\]) or (\[eq:Zc-a-general\]). ### Contact density and pressure From this expression we can verify the known relation between the contact density at $x=0$ (or $x=L$) and the pressure [@HBL79]. Indeed, notice that $$\label{eq:n0} n(0)=\frac{Z_c(N-1,L,Q_1-1,Q_2)}{Z_c(N,L,Q_1,Q_2)} \,.$$ On the other hand, from Eq. (\[eq:Zc-a\]) we can verify that $$\label{eq:dZ} \frac{\partial Z_c(N,L,Q_1,Q_2)}{\partial L}=Z_c(N-1,L,Q_1-1,Q_2) -\left(Q_1\right)^2 \,,$$ where this last relation was obtained by writing $-(j-\frac{N-a}{2})^2=(N-j)(j+a)-((N+a)/2)^2$ in (\[eq:Zc-a\]), and recalling that $Q_1=(N+a)/2$. Therefore, we find $$\label{eq:Pcontact} P_c=n(0)-(Q_1)^2=n(L)-(Q_2)^2 \,.$$ The last equality is obtained using the same argument on $x=L$ in $n(x)$. ### Asymptotic behavior of the density Let us consider the case $a=0$, ie. $Q_1=Q_2=N/2$. Figure \[fig:density\] shows a plot of the density profile for $N=25$ and $N=26$. Notice that in the case $N=26$ even, the density falls off quickly to zero far from the boundaries $x=0$ and $x=L$. On the other hand, when $N=25$ is odd, the density does not fall to zero, but goes to a non-vanishing value shown by the horizontal line. This corresponds to the density of the free counterion, responsible for the effective attraction between the two charges $q_1$ and $q_2$ as discussed earlier. ![ \[fig:density\] The density profile for $N=25$ and $N=26$ counter-ions and $L=10$. Notice that in the case where the number of counter-ions is odd, $N=25$, the density far from the edges converges to a non zero value $1/(L-2\langle x \rangle_{\infty})$, here close to $0.124$.](density){width="11cm"} To quantify this behavior, consider expression (\[eq:density\]) for the density in the case $N=2p+1$, and $Q_1=Q_2=p+\frac{1}{2}$, $$\label{eq:densitya0} n({x})=\frac{\sum_{k=1}^{N} Z_c(k-1,{x}, p+\frac{1}{2}, k-p-\frac{3}{2}) Z_c(2p+1-k,{L}-{x}, p-k+\frac{1}{2}, p+\frac{1}{2} ) }{Z_c(2p+1,{L},p+\frac{1}{2},p+\frac{1}{2})} \,.$$ In this sum, the partition function $Z_c(k-1,{x}, p+\frac{1}{2}, k-p-\frac{3}{2})$ corresponds to a system with charges $-e(p+\frac{1}{2})$ and $-e(k-p-\frac{3}{2})$ at its boundaries. If $k\leq p$, these two charges carry opposite signs, therefore, $Z_c(k-1,{x}, p+\frac{1}{2}, k-p-\frac{3}{2})$ is given by Eq. (\[eq:Zc-a\]). Then, if $1\ll x \ll L$, $Z_c(k-1,{x}, p+\frac{1}{2}, k-p-\frac{3}{2})=O(e^{-(p-k+\frac{3}{2})^2 x})$. On the other hand, the second partition function, $Z_c(2p+1-k,{L}-{x}, p-k+\frac{1}{2}, p+\frac{1}{2} )$, corresponds to a system with charges $-e(p-k+\frac{1}{2})$ and $-e(p+\frac{1}{2})$ at its edges. If $k\leq p$, these two charges carry the same sign and are half integers multiples of $e$, therefore $Z_c(2p+1-k,{L}-{x}, p-k+\frac{1}{2}, p+\frac{1}{2} )$ should be obtained by using Eq. (\[eq:Zc-Na-even\]). In particular one can notice that if $1\ll x \ll L$, then $Z_c(2p+1-k,{L}-{x}, p-k+\frac{1}{2}, p+\frac{1}{2} )=O(e^{-(L-x)/4})$. Therefore, in the sum (\[eq:densitya0\]) all terms with $k \leq p$ decay exponentially fast when $x$ is far from the boundaries: they are of order $O\left( e^{- ((p-k+\frac{3}{2})^2-\frac{1}{4})x}\right)$. The same argument could be applied to all the terms with $k\geq p+2$, with the roles of $Z_c(k-1,{x}, p+\frac{1}{2}, k-p-\frac{3}{2})$ and $Z_c(2p+1-k,{L}-{x}, p-k+\frac{1}{2}, p+\frac{1}{2} )$ interchanged. Then, only one term in the sum (\[eq:densitya0\]) survives, it corresponds to $k=p+1$, which is precisely the index of the position of the free counterion. In this term, both $Z_c(k-1,{x}, p+\frac{1}{2}, k-p-\frac{3}{2})$ and $Z_c(2p+1-k,{L}-{x}, p-k+\frac{1}{2}, p+\frac{1}{2} )$ with $k=p+1$, correspond to a system with charges $-e(p+\frac{1}{2})$ and $e/2$ at its edges (notice the opposite signs), and those partition functions should both be computed using (\[eq:Zc-a\]). The leading order of these partition functions, when $1\ll x \ll L$, is $$\begin{aligned} Z_c(p,x,p+\frac{1}{2},-\frac{1}{2})\sim \frac{e^{-x/4}}{p!(p+1)!} &\quad\text{and}\quad Z_c(p,L-x,-\frac{1}{2}, p+\frac{1}{2})\sim \frac{e^{-(L-x)/4}}{p!(p+1)!}\end{aligned}$$ while the leading order of the denominator of (\[eq:densitya0\]) is $$Z_c(2p+1,L,p+\frac{1}{2},p+\frac{1}{2}) \sim \frac{e^{-L/4}}{(p!(p+1)!)^2}\left(L-2\frac{p}{p+1}\right) \,.$$ This gives $$n(x)\sim \frac{1}{L-2\frac{p}{p+1}}=\frac{1}{L-2\langle x_p \rangle_{\infty}} \,,\quad \text{for\ } 1\ll x\ll L\,.$$ This is the analytical confirmation of the intuitive analysis of section \[sec:like-charge-attraction\] where it was explained that when $N$ is odd, there is one free ion roaming between the two charges with an available space equal to $L-2\langle x_p \rangle_{\infty}$, as shown in figure \[fig:1Dmodel-odd\]. In the case where $N$ is even, a similar analysis shows that all terms of the sum (\[eq:density\]) fall of exponentially fast when $x$ is far from the boundaries. The large $N$ limit ------------------- It is interesting to consider the limit $N\to\infty$. Due to the electroneutrality condition $q_1+q_2+eN=0$, one needs to consider different situations: whether $q_1$ and $q_2$ are kept finite, then necessarily the charge of the counterions $e$ should vanish as $1/N$. Then we notice that this is also a mean field regime. The other possible limit is to consider that $e$ has a non vanishing finite value, then $q_1$ and/or $q_2$ should go to infinity as $N$. ### Mean field limit, $N\to\infty$ and $e\to0$. Momentarily, it is best to return to dimensional units $\widetilde{L}$ and $\widetilde{P}$: the rescaling by $e^2$ is not appropriate here, because $e\to0$. Consider the equation of state (\[eq:Lave-a\]) derived in the isobaric ensemble, which now reads $$\begin{aligned} \label{eq:Lave-a-dim} \beta \langle \widetilde{L} \rangle &= \sum_{\ell=0}^{N} \frac{1}{(e\ell+q_{<})^2+\widetilde{P}} \sim\frac{1}{e}\int_{q_{<}}^{-q_{>}} \frac{dy}{y^2+P} \,,\end{aligned}$$ where $q_{<}=-eQ_{<}$ and $q_{>}=-eQ_{>}$. Since $e\to0$, one can recognize a Riemann sum and replace it by an integral. This finally leads to $$\label{eq:Lave-PB} \beta e \langle \widetilde{L} \rangle \sqrt{\widetilde{P}} = \arctan\frac{q_1}{\sqrt{\widetilde{P}}}+ \arctan\frac{q_2}{\sqrt{\widetilde{P}}} \,.$$ We recover here the implicit relation between $\langle \widetilde{L} \rangle$ and $\widetilde{P}$ from the mean field theory as described by the Poisson–Boltzmann equation [@LP99; @KTNBFP08]. Indeed, referring for instance to [@KTNBFP08], where the mean field regime of the present problem was considered, Eq. (\[eq:Lave-PB\]) can be directly obtained from a simple linear combination of Eqs. (16) and (17) of [@KTNBFP08]. Notice that the interesting effects, such as like-charge attraction, stemming from the discrete nature of the charges, are lost in this mean field limit. Like-charges will always have a repulsive effective interaction in the mean field regime [@lca1; @lca2; @lca3]. A related comment is that the asymptotic negative pressure reported for odd $N$ in section \[sec:equal-charges\], $\widetilde P = -q^2/N^2$, vanishes in the limiting process addressed here. It should be noted that the present limit is also the thermodynamic limit, since we have to remember that $e$ is of order $1/N$, therefore in the left hand side of (\[eq:Lave-PB\]) $\langle\widetilde{L}\rangle$ should be of order $N$. To make this more apparent, introduce the average distance per ion $\langle \widetilde{\ell} \rangle = \langle\widetilde{L}\rangle/N$ (inverse of the density), then (\[eq:Lave-PB\]) becomes $$\label{eq:Lave-PB-b} \beta (q_1+q_2) \langle \widetilde{\ell} \rangle \sqrt{\widetilde{P}} = \arctan\frac{q_1}{\sqrt{\widetilde{P}}}+ \arctan\frac{q_2}{\sqrt{\widetilde{P}}} \,.$$ ### Limit $N\to\infty$ and $e$ fixed. In this situation, the charges at the edges $q_1$ and $q_2$ should be of order $N$, or at least one of them. Consider the case when both $Q_1>0$ and $Q_2>0$ are of order $N$. Then, when $N\to\infty$, Eq. (\[eq:Lave-a\]) can be put in the following form by shifting the index of the summation by ${\left\lfloor Q_{<} \right\rfloor}$, $${\langle L \rangle}=\sum_{\ell=-\infty}^{\infty} \frac{1}{(\ell-\{Q_{<}\})^2+P} \,.$$ Notice that by shifting the index $\ell$ by one, we can replace $\{Q_{<}\}$ by $\{Q_{>}\}$ if necessary. One can then write $$\label{eq:Lave-inf} {\langle L \rangle}=\sum_{\ell=-\infty}^{\infty} \frac{1}{(\ell-\min(\{Q_{1}\},\{Q_{2}\}))^2+P} \,.$$ Notice that in this analysis, the limit depends on how $Q_1$ and $Q_2$ are taken to infinity, and assumes that the fractional part of them is kept fixed as $N$ is increased. To cover the whole range of values for ${\langle L \rangle}$ from 0 to $+\infty$, it is necessary that $P$ covers the range from $-\min(\{Q_{1}\},\{Q_{2}\})^{2}$ to $+\infty$. We recover the same phenomenology as in the case $N$ finite, when ${\langle L \rangle}\to\infty$, $P\to-\min(\{Q_{1}\},\{Q_{2}\})^{2}$. So, the pressure can become attractive, except in the case where $Q_1$ and $Q_2$ are integers. Eq. (\[eq:Lave-inf\]) can be made more explicit in two particular cases. When $Q_1$ and $Q_2$ are integers, $${\langle L \rangle}=\sum_{\ell=-\infty}^{\infty} \frac{1}{\ell^2+P}= \frac{\pi\coth(\pi\sqrt{P})}{\sqrt{P}} \,,$$ and when $Q_1$ and $Q_2$ are half integers, $${\langle L \rangle}=\sum_{\ell=-\infty}^{\infty} \frac{1}{(\ell+\frac{1}{2})^2+P}= \frac{\pi\tanh(\pi\sqrt{P})}{\sqrt{P}} \,.$$ When $Q_1$ and $Q_2$ are not integers, the value of ${\langle L \rangle}$ for which the pressure changes of sign is given by putting $P=0$ in (\[eq:Lave-inf\]) $$\label{eq:LstarNinfty} \langle L^{*} \rangle=\sum_{\ell=-\infty}^{\infty} \frac{1}{(\ell-\min(\{Q_1\},\{Q_2\}))^2}=\psi'(\{Q_1\})+\psi'(\{Q_2\}) \,.$$ When $Q_1$ and $Q_2$ are half-integers this reduces to $\langle L^{*} \rangle=\pi^{2}$. Conclusion ========== We have studied a simple one-dimensional system as a model to understand the effective interaction between charged particles that are screened by counterions only. This model evidences the possibility of attraction between two like-charges at large separation. The physical phenomenon behind this attraction is a frustration of the screening process due to the discrete nature of the electric charges. More specifically, if the two like-charges are not integers multiples of the charge of the counterions, a perfect screening of the charges is not possible, and there will be a “misfit” counterion, responsible for the over-screening of one of the like-charges, leading to an effective attractive force. A by-product is that in the mean-field limit where discreteness effects are washed out, no like-charge attraction is possible, a well-known phenomenon. The present model is in addition interesting from a purely theoretical perspective, since it is exactly solvable: it is possible to compute explicitly its partition functions (isobaric and canonical), the pressure (effective force) and the density profile of the counterions. Although the specific exact results and expression for the effective force are particular to this one-dimensional model, the physical mechanism responsible for the attraction between like-charges could also be applicable for three dimensional situations [@MHK00]. In particular the case $N=1$ leads to an equation of state that is equivalent to that found under strong coupling for three dimensional planar interfaces, screened by point counter-ions interacting through the standard $1/r$ Coulomb potential [@Netz01; @SaTr11; @Varenna]. **Acknowledgements.** This work was supported by an ECOS Nord/COLCIENCIAS-MEN-ICETEX action of Colombian and French cooperation. G. T. acknowledges support from Fondo de Investigaciones, Facultad de Ciencias, Universidad de los Andes, project “Apantallamiento y atracción de cargas similares en sistemas de Coulomb de una dimensión”, 2015-2. Two equal charges: canonical expressions {#app:A} ======================================== The inverse Laplace transform can be computed with integral inversion formula which can be evaluated using the residue theorem $${\cal L}^{-1}\left ( \prod_{k=0}^{p} \frac{1}{\left[\left(k+\frac{1}{2}\right)^2+P\right]^2} \right) (L) = \sum_{j=0}^{p} \mathop\text{Res}_{P=-(j+\frac{1}{2})^2} \frac{ e^{PL}}{\prod_{k=0}^{p}\left[\left(k+\frac{1}{2}\right)^2+P\right]^2}$$ Each residue is straightforward to compute $$\mathop\text{Res}_{P=-(j+\frac{1}{2})^2} \frac{e^{PL}}{ \prod_{k=0}^{p}\left[\left(k+\frac{1}{2}\right)^2+P\right]^2} = \frac{e^{-\left(j+\frac{1}{2}\right)^2L}}{ \prod_{k=0,k\neq j}^{p}\left[\left(k+\frac{1}{2}\right)^2- \left(j+\frac{1}{2}\right)^2\right]^2} \left( L - \sum_{l=0, l\neq j}^{p} \frac{2}{\left(l+\frac{1}{2}\right)^2-\left(j+\frac{1}{2}\right)^2} \right) \,.$$ Writing $$\frac{1}{(k+\frac{1}{2})^2-(j+\frac{1}{2})^2}= \frac{1}{(k-j)(k+j+1)}= \frac{1}{2j+1}\left(\frac{1}{k-j}-\frac{1}{k+j+1}\right),$$ the above product and sum can be simplified $$\frac{1}{\prod_{k=0,k\neq j}^{p}\left[\left(k+\frac{1}{2}\right)^2- \left(j+\frac{1}{2}\right)^2\right]} =\frac{(-1)^{j} (2j+1)}{(p-j)!(p+j+1)!} \,,$$ and $$\begin{aligned} \sum_{l=0, l\neq j}^{p} \frac{1}{\left(l+\frac{1}{2}\right)^2-\left(j+\frac{1}{2}\right)^2} &=& \frac{2}{2j+1}\left( \frac{1}{2j+1}-\sum_{k=p-j+1}^{p+j+1} \frac{1}{k} \right) \\ &=&\frac{2}{2j+1}\left( \frac{1}{2j+1}+\psi(p-j+1)-\psi(p+j+2) \right) \,. \nonumber\end{aligned}$$ Gathering all results, the exact explicit result for the canonical partition function is found in the form of Eq. (\[eq:ZexactNimpar\]). [00]{} Electrostatic Effects in Soft Matter and Biophysics, edited by P. Kekicheff, C. Holm, and R. Podgornik, (Kluwer Academic, Dordrecht, 2001). Y. Levin, Rep. Prog. Phys. [**65**]{}, 1577 (2002). R. Messina, J. Phys.: Condens. Matter [**21**]{} 113102 (2009). B. Jancovici, Phys. Rev. Lett. [**46**]{}, 386 (1981). P. J. Forrester, Phys. Rep. [**301**]{}, 235 (1998). L. Šamaj, J. Phys. A: Math. Gen. [**36**]{}, 5913 (2003). A. Lenard, J. Math. Phys. **2**, 682 (1961). S. Prager, Adv. Chem. Phys. **4**, 201 (1961). S. Edwards and A. Lenard, J. Math. Phys. **3**, 778 (1962). D. S. Dean, R. R. Horgan, A. Naji, R. Podgornik, J. Chem. Phys. **130**, 094504 (2009). E. Trizac and L. Šamaj, Proceedings of the International School of Physics Enrico Fermi [**184**]{}, 61 (2013), edited by C. Bechinger, F. Sciortino and P. Ziherl. R. R. Netz, Eur. Phys. J. E [**5**]{}, 557 (2001). D. Henderson, L. Blum and J. L. Lebowitz, J. Electroanal. Chem. **102**, 315 (1979). S. L. 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--- abstract: | We demonstrate a five-bit nuclear-magnetic-resonance quantum computer that distinguishes among various functions on four bits, making use of quantum parallelism. Its construction draws on the recognition of the sufficiency of linear coupling along a chain of nuclear spins, the synthesis of a suitably coupled molecule, and the use of a multi-channel spectrometer. address: - | Institut für Organische Chemie, J. W. Goethe-Universität, Marie-Curie-Str. 11,\ D-60439 Frankfurt, Germany - | Biological Chemistry and Molecular Pharmacology, Harvard Medical School,\ 240 Longwood Avenue, Boston, MA 02115, USA - | Gordon McKay Laboratory, Division of Engineering and Applied Sciences,\ Harvard University, Cambridge, MA 02138, USA - 'Bruker Analytik GmbH, Silberstreifen, D-76287 Rheinstetten, Germany' - | Institut für Organische Chemie und Biochemie, Technische Universität München,\ Lichtenbergstr. 4, D-85748 Garching, Germany author: - 'R. Marx' - 'A. F. Fahmy' - 'John M. Myers' - 'W. Bermel' - 'S. J. Glaser' title: | Realization of a 5-Bit NMR Quantum Computer\ Using a New Molecular Architecture --- = 9in Introduction ============ While quantum computers of two bits have been implemented [@bitsa], as have nuclear-magnetic-resonance (NMR) quantum computers of three bits [@bitsc], extending the number of bits has not proved easy. We report the implementation of an NMR quantum computer having five bits, involving the use of a linear coupling pattern [@Brueschweiler], synthesis of a molecule having five usable spin-active nuclei with predominantly linear spin-spin coupling, and the development of radio-frequency (r.f.) pulse sequences to act as quantum logic gates for the molecule synthesized. Techniques to suppress unwanted couplings between nuclear spins are described, as are techniques to avoid perturbing some nuclear spins while manipulating others. Results are presented of a test of the five-bit computer on a problem of Deutsch and Jozsa to distinguish one class of mathematical function from another [@deutsch]. Definition of an [****]{}-bit NMR computer ========================================== An $n$-bit quantum computer is called on to do three things: 1) accept an instruction to prepare a starting state and prepare that state; 2) accept instructions for and implement quantum gates (from which more general unitary transformations of the state can be composed); and 3) measure the state and yield an outcome. The connection to computation with classical computers depends on the recognition, due to Bennett[@bennett2], that all classical computations can be made reversible. Any terminating reversible computation is a permutation of the inputs, which is unitary, and thus belongs to the class of transformation performable on a quantum computer. (For issues of possibly nonterminating programs, see [@myers].)=-1 In theory, a variant of the quantum computer is the expectation-value quantum computer (EVQC), which in place of an outcome of a measurement yields the expectation value [@gradientPPS1; @gradientPPS2]. NMR quantum computing was born of the recognition that an EVQC can be approximated by use of an NMR spectrometer containing a liquid sample, the molecules of which have $n$ atoms with a nuclear spin of 1/2 (and possibly other atoms, either spinless or having spins not used) [@gradientPPS1; @gradientPPS2; @logicalPPS]. Because tumbling of the molecules decouples each molecule from all the others, the sample can be described by a density matrix for the nuclear spins of the atoms of a single molecule [@Ernst], with only the spin-degrees of freedom, corresponding to the desired Hilbert space of dimension $2^n$. NMR spectrometers sense only the traceless part of the density matrix, so in place of matter in a pure state, an NMR computer can use a liquid sample described by a density matrix proportional to a sum of a pure state and any multiple of the unit matrix. Such a density matrix, called a [*pseudopure*]{} state [@gradientPPS1], plays a role in the 5-bit quantum computer. Acting as an $n$-bit EVQC, a suitable NMR spectrometer allows the preparation of a pseudopure starting state, the programming and execution of r.f. pulse sequences that implement quantum gates, and the determination of expectation values visible in NMR spectra. To perform the unitary operations required of a quantum computer, a sufficient set of quantum gates consists of all single-spin operations and all controlled-not gates that act on one nuclear spin under the control of another nuclear spin. Single-spin gates are implemented by selective r.f. pulses. Controlled-not gates between nuclei having spin-spin coupling will be described, along with techniques to avoid unwanted influences on other spins. A key feature of the present design of the NMR quantum computer is the reliance on a chain of linear coupling and the use of swap gates to implement a controlled-not in which a spin $j$ controls spin $k$, where $j$ and $k$ have no direct spin-spin coupling [@Brueschweiler]. This allows use in NMR quantum computers of a molecule having a simpler coupling pattern, and eases the problem of unwanted influences on spins. Design of Test ============== The proof of the pudding is in the eating: the 5-bit NMR computer to be described was tested on the Deutsch-Jozsa problem for functions of 4 bits[@deutsch], in the form described in [@cleve], modified for efficiency with NMR as described by Jones and Mosca [@bitsb]. (A recent simplification [@collins], unused here, would permit working with functions of 5 bits.) The problem is to decide whether a function program selected from a set of possible programs computes one kind of function or another. Specifically, the problem is to distinguish programs for balanced functions from programs for constant functions, where the functions are from $\{0,1\}^4$ to $\{0,1\}$. (A function is constant if its value is independent of its argument, and is called balanced if the value for half the arguments is $1$ while the value is $0$ for the other half.) The test actually made was to distinguish between programs for one constant and one balanced function, defined as follows: $$f_{0}(\vec{x}) \stackrel{\rm def}{=} 0$$ and $$f_{b}(\vec{x}) \stackrel{\rm def}{=} x_{1}\oplus x_{2}\oplus x_{3}\oplus x_{4}\label{eq:fb}$$ for all $\vec{x}$, where $\vec{x} \stackrel{\rm def}{=} (x_{1},x_{2},x_{3},x_{4})$, and “$\oplus$” is addition modulo 2. Also, several controlled-not (CNOT) gates were tested, along with a variety of 1-bit operators. The balanced function chosen, $f_{b}$, has the nice property of being implementable also in classical reversible gates with no work bits. Used to solve this problem, a quantum computer is a resource used both to specify the function under test and to determine what it is. In order to separate these two uses, one can view the quantum computer as used alternately by a [*specifier*]{} of the function and a [*decision maker*]{}, two players of a game in which: (A) the decision maker prepares the starting state; (B) the specifier runs the function program;[^1] and (C) the decision maker makes a measurement independent of the function program, and interprets the result to decide the function class. On an NMR quantum computer, (A) the decision maker starts a play by using r.f. pulses and magnetic-field gradients (independent of the function to be specified) to put the liquid sample in the pseudopure state having a density matrix with a traceless part proportional to $$\rho_{i} \stackrel{\rm def}{=}16 |00001\rangle\langle 00001|-\frac{1}{2} {\bf 1} = 16 I_1^{\alpha}I_2^{\alpha}I_3^{\alpha}I_4^{\alpha}I_5^{\beta}- \frac{1}{2}{\bf 1}\label{eq:rhoi}$$ in terms of the polarization operators $I_k^{\alpha} = (\frac{1}{2}{\bf 1}+I_{kz})$ and $I_k^{\beta} = (\frac{1}{2}{\bf 1}-I_{kz})$ usual to NMR [@Ernst; @Brueschweiler]. Then the decision maker applies a unitary transform $U_{90}$ by use of a hard $90^{\circ}$ $y$-pulse which for this particular state has the same effect as the Hadamard transform on each spin [@bitsb]. $$\begin{aligned} \rho_{i} \stackrel{U_{90}}{\rightarrow} \rho_0 &=& U_{90} \rho_i U_{90}^{\dag}\nonumber\\ & = &16 \left(\frac{1}{2}{\bf 1}+I_{1x}\right)\left(\frac{1}{2}{\bf 1} +I_{2x}\right)\left(\frac{1}{2}{\bf 1}+I_{3x}\right)\nonumber\\ &&\mbox{}\times \left(\frac{1}{2}{\bf 1}+I_{4x}\right)\left(\frac{1}{2}{\bf 1}-I_{5x}\right) -\frac{1}{2}{\bf 1}.\label{eq:rho0}\end{aligned}$$ \(B) The specifier chooses a function $f$ from one of the set of functions undergoing test, here $f_{0}$ or $f_{b}$, and runs the quantum version of a program to compute $f$; this program is a sequence of gates, each a unitary transformation implemented by an r.f. pulse sequence. The total program implements a unitary transformation $U(f)$, defined by its action on basis vectors $|\vec{x},\ x_5\rangle$: $$U(f)|\vec{x},\ x_5\rangle = |\vec{x},\ x_5\oplus f(\vec{x})\rangle. \label{eq:ufunc}$$ The transform $U(f)$ produces the density matrix with traceless part proportional to $\rho_f$: $$\rho_0 \stackrel{U(f)}{\rightarrow} \rho_f.$$ \(C) The decision maker reads out the NMR spectrum which depends on $\rho_f$. The spectrum differs according to whether $f$ is balanced or constant, and thus tells the decision maker the function class, with only one function evaluation, a large saving over classical computation, which could require 9 evaluations for functions of four bits. In theory, for the case $f = f_0$, $U(f)$ is specified to be $U(f_0)$, which by Eqs. (1) and (5) turns out to be the identity matrix, so one should have $\rho_f = \rho_0$. The spectrometer detects only the terms of the righthand side of Eq. (\[eq:rho0\]) that are linear in $I_{x}$, so for a spectrometer adjusted to give an upward peak for $I_{x}$, the resulting spectrum is in theory $\frac{|\,|\,|\,|\, } {\;\;\;\;\;\;\;|}$, which has, from left to right, positive peaks for spins 1 to 4 and a negative peak for spin 5. For the balanced function, $f = f_{b}$ (Eq. (\[eq:fb\])), $U(f_b)$ is defined by $U(f_b)|\vec{x},\ x_5\rangle = |\vec{x},\ x_{1}\oplus x_{2}\oplus x_{3}\oplus x_{4}\oplus x_5\rangle$. A unitary operator that is simpler to implement, that has the same effect on the fifth (value) bit, and that allows the distinction between constant and balanced functions is $\tilde{U}(f_b)$ defined by $$\begin{aligned} \tilde{U}(f_b)|\vec{x},\ x_5\rangle &=& |x_{1},\ x_{1} \oplus x_{2},\ x_{1} \oplus x_{2}\oplus x_{3},\ x_{1}\oplus x_{2}\nonumber\\ &&\mbox{}\oplus x_{3}\oplus x_{4}, x_{1}\oplus x_{2}\oplus x_{3}\oplus x_{4}\oplus x_5\rangle,\label{eq:Vb}\end{aligned}$$ which we implemented by sequential application of the gates (CNOT)$_{12}$, (CNOT)$_{23}$, (CNOT)$_{34}$, and (CNOT)$_{45}$. The spectrum calculated for the density matrix $\tilde{\rho}_b$ obtained by transforming $\rho_0$ with $\tilde{U}(f_b)$ is $\frac{|\,|\,|\,\;\; }{\;\;\;\;\;|\,|}$ ([*vide infra*]{}). Realization and Test of a 5-bit NMR Computer ============================================ A 5-bit NMR quantum computer requires a molecule having 5 spin-active nuclei, with long relaxation times. Large separation of resonance frequencies of the nuclei allows rapid selective control of the spins. For frequency separation, it is desirable to use different atomic species for different spins, which requires a multi-channel NMR spectrometer. Our NMR experiments were performed using a BRUKER AVANCE 400 spectrometer with five independent r.f.channels and a QXI probe (H,C-F,N). The lock coil was also used for deuterium decoupling utilizing a lock switch. A linear path of spin-spin couplings is sufficient for all computations [@Brueschweiler]. Given the availability of a 5-channel spectrometer, we chose as the “hardware” of our NMR quantum computing experiments the molecule BOC-($^{13}$C$_2$-$^{15}$N-$^{2}$D$_2^\alpha$-glycine)-fluoride which contains an isolated coupling network consisting of five nuclei, each having spin 1/2: the amide $^{1}$H, the $^{15}$N, the aliphatic $^{13}$C$^\alpha$, the carbonyl $^{13}$C$^\prime$, and the $^{19}$F nuclear spin (see Fig. 1). For simplicity, we will refer to these spins (and the corresponding bits) as 1, 2, 3, 4 and 5, respectively. All spins are heteronuclear, except for C$^\alpha$ and C$^\prime$ which however have a relatively large chemical shift difference. The five-spin system is well isolated from the protons of the BOC protecting group which are separated by more than four chemical bonds. In addition, the deuterium spins (D) which are attached to C$^\alpha$ can be fully decoupled from the spins of interest using standard heteronuclear decoupling techniques [@dec1; @dec2; @dec3]. The substance was synthesized starting from commercially available $^{13}$C and $^{15}$N labeled glycine (see Appendix A) and was dissolved in deuterated dimethyl-sulfoxide (DMSO-D$_6$). NMR experiments were performed at a magnetic field of about 9.4 Tesla and a sample temperature of 27$^\circ$ C. The experimentally determined $T_2$ relaxation times for spins 1–5 were 250 ms, 490 ms, 450 ms, 590 ms, and 260 ms, respectively. Resonance frequencies $\nu_k$ and scalar coupling constants $J_{kl}$ are summarized in Table I. Except for the $J_{23}$ coupling constant of 13.5 Hz, the spin chain is connected by one-bond coupling constants $J_{k\{k+1\}}$ larger than 60 Hz. In the multiple rotating frame (see Ref.  and Appendix B) the precession frequency of each individual spin is 0, which considerably simplifies implementation, because only coupling terms need to be considered (and manipulated). 1.5 -1.5The experimental implementation of the propagator $U(f_0)$ corresponding to $f_0$ is trivial because by Eq. (\[eq:ufunc\]) the propagator is the unit operator, implemented by doing nothing. In contrast, the construction of the pulse sequence to implement the series of CNOT-gates that define the unitary transformation $\tilde{U}(f_b)$ of Eq. (\[eq:Vb\]) for the balanced function $f_b$ (Eq. (\[eq:fb\])) requires attention. The goal is to create robust pulse sequence elements that minimize the effects of experimental imperfections. The pulse sequence elements shown in Fig. 2 A-D were designed specifically for the coupling topology of our 5-spin system to implement the unitary operators corresponding to  ([CNOT]{})$_{12}$, ([CNOT]{})$_{23}$, ([CNOT]{})$_{34}$, and ([CNOT]{})$_{45}$.[^2] During these CNOT gates that act on two directly coupled spins $k$ and $l$, only the couplings $J_{kl}$ are active, while the effect of all other couplings in the spin system are refocused by cyclic pulse sequences [@Ernst; @Waugh; @Quant]. Figure 3 shows schematically the pulse sequence actually used for the propagator $\tilde{U}(f_b)$ for the balanced function $f_b$; this sequence benefited from applying simple rules for pulse cancellation (see Appendix B.2). The NMR implementation of the Deutsch-Jozsa algorithm starts with the preparation of the pseudopure state $\rho_{i}$ of Eq. (\[eq:rhoi\]). The preparation of such a pseudopure state by a single pulse sequence requires a non-unitary transformation of the thermal equilibrium density operator [@gradientPPS1; @bounds]. This can be achieved using spatial averaging [@gradientPPS1; @Brueschweiler] or temporal averaging [@gradientPPS1; @temporalPPS]. In the basis formed by Cartesian product operators [@Ernst], $\rho_{i}$ can be expressed as a linear combination of 31 terms that only consist of $z$ spin operators: $$\begin{aligned} \rho_{i}&=& \sum_{n=1}^5 s_n \, I_{nz} +\sum_{m<n} s_n \, 2 I_{mz} I_{nz} + \sum_{l<m<n} s_n\, 4 I_{lz} I_{mz} I_{nz} \nonumber \\ &&\mbox{}+ \sum_{k<l<m<n} s_n\, 8 I_{kz} I_{lz} I_{mz} I_{nz} - 16 I_{1z} I_{2z} I_{3z} I_{4z} I_{5z},\nonumber\\ \label{eq:terms}\end{aligned}$$ where $s_n=-1$ if $n=5$ and $s_n=1$ otherwise. It is straightforward to create each of these terms from the thermal equilibrium density operator, using standard building blocks of high-resolution NMR [@INEPT]. In principle, temporal averaging could be realized by repeating steps (A)–(C) of the game for all 31 terms in Eq.(\[eq:terms\]) and by summing up the resulting spectra. However, because currently available NMR spectrometers require a distinct experiment to detect each spin species ($^{1}$H, $^{15}$N, $^{13}$C and $^{19}$F) (see Appendix B.3), a total of 124 NMR experiments would be required for each function $f$ in order to include all terms in the temporal averaging. A detailed analysis shows that of the 31 terms that constitute the pseudopure state $\rho_{i}$, only the five linear terms $I_{kz}$ and the four bilinear terms $2I_{kz} I_{\{k+1\}z}$ are transformed into detectable operators by the propagator $U_{90}$ (to create $\rho_0$) followed by the propagators $U(f_0)$ or $\tilde{U}(f_b)$, as the case may be (see Table II and Appendix C). As pointed out previously [@bitsc], preparing just the linear terms $I_{kz}$ suffices in some cases of the Deutsch-Jozsa problem to distinguish constant from balanced functions, because in these cases a balanced function gives a vanishing signal for at least one of the input spins. However, in the presence of experimental imperfections, it is desirable to identify a balanced function based on the sign reversal of the signal of at least one of the input spins, rather than by the lack of a signal. For the special case of the balanced function $f_b$ that was chosen for this demonstration experiment, this can be achieved by including also the bilinear terms $2I_{kz} I_{\{k+1\}z}$ as starting operators (see Table II). Samples described by these linear and bilinear terms of $\rho_i$ were prepared (see Appendix B.4) to demonstrate experimental control of the five-spin system and to execute cases of the Deutsch-Jozsa algorithm. For each function ($f_0$ and $f_b$) the following three sets of experiments were performed (see experimental spectra in Fig. 4). Set 1 (first row of curves from the bottom in Fig. 4): preparation of the linear terms $I_{kz}$ (with algebraic signs as specified in Eq. (8) and Table II), application of $U_{90}$ and $U(f)$, and detection of spin $k$ for $k=1,\ \dots,\ 5$; set 2 (second row in Fig. 4): preparation of the bilinear terms $2 I_{kz} I_{\{k+1\}z}$ (with algebraic signs as specified in Eq. (8) and Table II), application of $U_{90}$ and $U(f)$, and detection of spin $k$ for $k=1,\ \dots,\ 4$; and set 3 (third row in Fig. 4): preparation of the bilinear terms $2 I_{\{k-1\}z} I_{kz}$ (with algebraic signs as specified in Eq. (8) and Table II), application of $U_{90}$ and $U(f)$, and detection of spin $k$ for $k=2,\ \dots,\ 5$. The observed spectra shown in Fig. 4 correspond closely to the theoretical predictions (see Table II). For the constant function $f_0$, only the experiments of set 1 yield detectable signals. For the balanced function $f_b$, the experiments of set 1 only yield a detectable signal for spin 5, whereas for spins 1–4 detectable signals are only obtained in the experiments of set 2. As expected, only spurious signals are detected for the experiments of set 3. The amplitude of these spurious signals is typically on the order of 4% compared to the full signals. As expected, all the signals of spins 1–4 are positive for the constant function whereas the signal of spin 4 is inverted by the propagator $\tilde{U}(f_b)$ corresponding to the balanced function. For $f_b$ the signal amplitudes reach only between 55% and 70% of the amplitudes found for $f_0$. This signal loss can be attributed mainly to relaxation and experimental imperfections during the sequence that implements $\tilde{U}(f_b)$ (Fig. 3), which has an overall duration of 51.4 ms. Through combined synthetic, analytic, and spectroscopic work, a five-bit NMR quantum computer was built and shown to implement superposition, quantum interference, and designed unitary transformations. Although obstacles had to be overcome, none were fundamental, and quantum computers with more than five bits will be built. Lots of interesting questions have been raised for future work pertaining to the constraints and opportunities for linking molecular architecture, spectrometer design, and algorithms for NMR quantum computing. S.J.G. acknowledges support by the Fonds der Chemischen Industrie and the DFG. R.M. is supported by a stipend of the Fonds der Chemischen Industrie and the Bundesministerium für Bildung und Forschung (BMBF). We thank C. Griesinger, M. Grundl, R. Kerssebaum, B. Luy, R. Mayr-Stein, M. Kettner, M. Reggelin, H. Schwalbe, and A. Tüchelmann for valuable discussions and technical assistance. A.F.F. thanks G. Wagner (Harvard Medical School) for support and encouragement and acknowledges support from National Science Foundation. J.M.M. thanks T. T. Wu (Harvard University) for many critical insights. Synthesis of Molecule ===================== We purchased 250 mg of $^{13}$C$_2$-$^{15}$N-glycine from Martek Biosciences Corporation, 6480 Dobbin Road, Columbia, Maryland 21045. The labeled glycine was fully deuterated by treatment with a solution of NaOD in D$_2$O at 140$^\circ$C. The product was dissolved in water for reprotonation while retaining the deuterium atoms in alpha-position. The resulting $^{13}$C$_2$-$^{15}$N-$^2$D$_2^\alpha$-glycine was protected in a standard reaction with di-[*tert*]{}.-butyl-dicarbonate (BOC-anhydride) as reagent (O. Keller, W. E. Keller, G. van Look and G. Wersin, Org. Synth. [**63**]{}, 160 (1985)). Finally the carboxylic acid was converted by cyanuric fluoride into the desired acyl fluoride: BOC-($^{13}$C$_2$-$^{15}$N-$^{2}$D$_2^\alpha$-glycine)-fluoride (L. A. Carpino, E. M. E. Mansour and D. Sadat-Aalaee, J. Org. Chem., 2611 (1991)). The substance dissolved in DMSO-D$_6$ at room temperature shows NMR spectra that weaken with a half-life of about a week, indicative of reactions not yet determined. The solution was stable during storage at a temperature of $-30^\circ$ C. NMR pulse sequences =================== For the preparation of the elements of a pseudopure state and the implementation of quantum gates, robust r.f. pulse sequences are desirable. Pulse-sequence parameters with negligible experimental errors are the durations of r.f. pulses and of delays. In addition, the phases of r.f. pulses and of the receiver can be controlled with negligible errors. The most important experimental imperfections are r.f. amplitude errors that result from miscalibrations and from the r.f. field inhomogeneity created by the r.f. coils. In addition to the use of compensating schemes, such as super cycles and composite pulses [@Ernst], experimental imperfections can be reduced by designing pulse sequence elements with a minimum number of r.f.pulses. For example, pulses to refocus frequency offset terms in homonuclear spin systems with different chemical shifts can be eliminated by implementing the experiments in the multiple-rotating frame in which the precession frequency of each individual spin is 0 (see section B.1). More generally, pulses can often be eliminated or replaced by phase adjustments with negligible errors (see section B.2). For the available spectrometer, the experimental pulse parameters are summarized in section B.3. The preparation of the elements of the pseudopure state $\rho_i$ is discussed in section B.4. Implementation of experiments in the multiple rotating frame ------------------------------------------------------------ For heteronuclear spins with resonance frequencies $\nu_k$ and $\nu_l$ in the laboratory frame, the spins are irradiated on-resonance and the observed signals are demodulated by the determined resonance frequencies. If only a single r.f. channel is available for several homonuclear spins, on-resonance irradiation of several homonuclear spins can be achieved using phase-modulation of the r.f. pulses. The reference phase of each pulse applied to spin $k$ must be adjusted such that it matches the desired phase in the corresponding rotating frame ([*vide infra*]{}). In addition, the phases of the detected signals need to be corrected for the relative phases that have been acquired by the respective rotating frames during the course of the experiment. In our case with the two homonuclear spins C$^{\alpha}$ (spin \#3) and C$^\prime$ (spin \#4), the transmitter frequency of the carbon r.f. channel was set to the C$^{\alpha}$ resonance frequency. In order to simplify the combination of different quantum gates, the durations of the pulse sequences for each gate were chosen to be integer multiples of $\Delta=1/\vert \nu_3-\nu_4\vert=81.75\ \mu$s. Hence, the rotating frames are aligned at the end of each gate. Simplifying pulse sequences --------------------------- Some quantum gates, such as (CNOT)$_{kl}$, require $z$ rotations of individual spins which can be implemented using composite r.f.pulses [@compz]. However, these pulses can be avoided if $z$ rotations (by angle $\varphi$) are implemented by a corresponding negative rotation of the respective rotating frame of reference. In practice, this results in an additional phase shift (by angle $- \varphi$) of all following r.f. pulses that are applied to this spin and of the receiver phase of this spin. Furthermore, $180_\vartheta^\circ$ pulses (with arbitrary phase $\vartheta$) are required in some cases to refocus the evolution due to $J$ couplings. In order to undo the rotation caused by these pulses, additional $180_\vartheta^\circ$ or $180_{-\vartheta}^\circ$ pulses are often needed at the beginning or at the end of these quantum gates. An appropriate choice of the position and phase $\vartheta$ of these pulses often makes it possible to cancel two pulses from adjacent gates (e.g., $180^\circ_x$ and $180^\circ_{-x}$) or to absorb a 180$^\circ$ pulse into the phase of an adjacent 90$^\circ$ pulse (e.g., a $180^\circ_{x}$ pulse preceded or followed by a $90^\circ_{-x}$ pulse is equivalent to a single $90^\circ_{x}$ pulse). Even if r.f. pulses cannot be completely eliminated, the accumulation of small flip angle errors can be avoided by a proper choice of pulse phases which is common practice in the design of modern NMR multiple pulse sequences [@Ernst]. For example, the so-called MLEV-4 expansion [@dec1; @dec2; @dec3] $180^\circ_{x}$$180^\circ_{-x}$$180^\circ_{-x}$$180^\circ_{x}$ (used here, e.g., for spin 5 decoupling during spin 3- and spin 4-selective $90^\circ$ pulses) is preferable to $180^\circ_{x}$$180^\circ_{-x}$$180^\circ_{x}$$180^\circ_{-x}$ or to $180^\circ_{x}$$180^\circ_{x}$$180^\circ_{x}$$180^\circ_{x}$. Experimental pulse parameters ----------------------------- Due to their large frequency separation, selective pulses for spins 1 ($^{1}$H), 2 ($^{15}$N) and 5 ($^{19}$F) could be implemented by simple square pulses. The durations of 90$^\circ$ pulses were 8.85 $\mu$s, 41 $\mu$s and 11.75 $\mu$s, respectively. For spins 3 ($^{13}$C$^\alpha$) and 4 ($^{13}$C$^\prime$) the following shaped pulses with minimal durations and optimal selectivity were chosen based on numerical simulations and experimental optimizations: 90$^\circ$ pulses were implemented as e-SNOB pulses [@esnob], not for the usual 270$^\circ$, but for a $90^\circ$ rotation with a duration of 224 $\mu$s; selective 180$^\circ$ pulses were implemented as Gaussian pulses [@gauss] with a duration of 250 $\mu$s and a truncation level of 20%. The application of these shaped e-SNOB and Gaussian pulses on C$^\alpha$ has a nonresonant effect [@McCoy] on C$^\prime$ which corresponds to experimentally determined $z$ rotations of $\varphi_e=-4^\circ$ and $\varphi_g=-18^\circ$, respectively. Conversely, a shaped e-SNOB pulse applied to C$^\prime$ leads to a $z$ rotation of $-\varphi_e $ for C$^\alpha$. In all experiments these phase shifts were taken into account by adjusting the phases of the following pulse and the receiver phases (see Fig. 3). (Note that the phases of the two selective Gaussian 180$^\circ$ pulses applied to spin 3 in the period $\tau_{45}$ is not corrected because their absolute phases are arbitrary, c.f. Appendix B.2.) During the spin 3- or spin 4-selective 180$^\circ$ pulses, the evolution due to the strong $J_{35}$ and $J_{45}$ couplings is automatically refocused. As this is not the case for spin 3- or spin 4-selective 90$^\circ$ pulses, spin 5 was actively decoupled during these pulses (see Fig. 3). As commercial high-resolution NMR spectrometers are commonly not equipped with multiple receivers, it was not possible to simultaneously detect the signals of different spin species. Moreover, the application of any given pulse sequence required four different pulse programs because the routing of the r.f. channels (for the creation of $^{1}$H, $^{15}$N, $^{13}$C, $^{19}$F and $^{2}$D pulses) depends on the detected spin species ($^{1}$H, $^{15}$N, $^{13}$C or $^{19}$F). Due to this technical limitation, each spin species had to be detected in a separate experiment for every term of the initial density operator $\rho_i$. However, this made it possible to use standard heteronuclear decoupling techniques to simplify the detected signals and to significantly increase the signal-to-noise ratio of the experiments. During spin 1 detection, spins 2 and 3 were decoupled with an r.f. amplitude $\nu_{rf}=\gamma B_{rf}/(2 \pi)$ of $0.6$ kHz and $0.4$ kHz, respectively. During spin 2 detection, spins 1 and 3 were decoupled with an r.f. amplitude of $2.3$ kHz and $0.4$ kHz, respectively. During spin 3 detection, spins 1, 2 and 5 were decoupled with an r.f. amplitude of $2.3$ kHz, $0.6$ kHz and $2.0$ kHz, respectively, and during spin 4 detection, spin 5 was decoupled with an r.f. amplitude of $2.0$ kHz. In all these cases, the WALTZ-16 decoupling sequence [@dec1; @dec2; @dec3] was used. In principle, also the $J_{34}$ coupling could be effectively eliminated during detection of spin 3 or spin 4 using time-shared decoupling. However, this was not possible with our experimental setup because more than five separate r.f. channels would have been required. From the resulting doublets (with splitting $J_{34}$) an apparent singlet was created by merging the two doublet components [@sattler]. During spin 5 detection, spins 3 and 4 were simultaneously decoupled using a double-selective G3-MLEV sequence [@dec1; @dec2; @dec3] with an r.f. amplitude of 6 kHz. In addition, during all experiments deuterium decoupling was applied using a WALTZ-16 sequence with $\nu_{rf}=0.5$ kHz. In order to approximate a constant sample temperature of about 27$^\circ$ C in spite of the additional sample heating effected by the decoupling sequences, 16 dummy scans were used prior to signal acquisition of spins 1 and 5, whereas 4 dummy scans were used prior to signal acquisition of spins 2, 3, and 4. Nevertheless, the linewidths of the experimental signals shown in Fig. 4 were slightly increased by residual sample heating effects and imperfections of the decoupling sequences. Pulse sequences for the preparation of the terms of $\rho_{i}$ and $\rho_{0}$ ----------------------------------------------------------------------------- In order to improve the sensitivity of the experiments and to filter out signals from impurities in the sample, individual Cartesian product operator terms of $\rho_{i}$ were created using sequential INEPT transfer steps [@INEPT] starting from $^1$H magnetization, corresponding to the operator $I_{1z}$. The term $I_{1z}$ was prepared from the thermal equilibrium density operator by applying spin 2, 3, 4, and 5 selective $90^\circ$ pulses followed by a pulsed field gradient of the static magnetic field. An X-filter element [@x-filt] was used to select $^{1}$H spins that are coupled to $^{15}$N. For the preparation of other terms of $\rho_{i}$ (starting from $I_{1z}$), the phase $\phi_a$ of the first 90$^\circ$ pulse applied to spin 1 was subject to a two-step phase cycle. In addition, the phase $\phi_b$ of the 90$^\circ$ pulses for the implementation of $U_0$ (see Eq. (4)) was also subject to an independent phase cycle. Overall, this resulted in a four-step phase cycle with the pulse phases $\phi_a=\{0^\circ,180^\circ, 0^\circ, 180^\circ\}$, $\phi_b=\{90^\circ, 90^\circ, 270^\circ, 270^\circ\}$ and the relative receiver phases $\phi_{\rm rec}=\{0^\circ,180^\circ, 0^\circ, 180^\circ\}$. Density operator terms ====================== For all 31 Cartesian product operator terms in $\rho_i$ (Eq. (\[eq:terms\])), the corresponding terms in $\rho_0$ and $\tilde{\rho}_b$ are summarized in Table III. The transformation $ \rho_0 \stackrel{\tilde{U}(f_b)}{\longrightarrow} \tilde{\rho}_b $ of the unitary operator corresponding to the balanced function $f_b$ is composed of four consecutive unitary transformations corresponding to $({\rm CNOT})_{kl}$ quantum gates: $$\rho_0 \ \stackrel{({\rm CNOT})_{12}}{\longrightarrow} \ \rho^\prime \ \stackrel{({\rm CNOT})_{23}}{\longrightarrow} \ \rho^{\prime \prime} \ \stackrel{({\rm CNOT})_{34}}{\longrightarrow} \ \rho^{\prime \prime \prime} \ \stackrel{({\rm CNOT})_{45}}{\longrightarrow} \ \tilde{\rho}_b.$$ The transformations of the individual $({\rm CNOT})_{kl}$ gates can be derived using the following rules [@bitsc]: $$\begin{aligned} I_{kx} & \stackrel{({\rm CNOT})_{kl}}{\longrightarrow}& 2 I_{kx}I_{lx},\\ 2 I_{kx}I_{lx}& \stackrel{({\rm CNOT})_{kl}}{\longrightarrow}& I_{kx},\\ I_{lx}& \stackrel{({\rm CNOT})_{kl}}{\longrightarrow}& I_{lx}.\end{aligned}$$ The terms of the intermediate operators $\rho^\prime$, $\rho^{\prime\prime}$ and $\rho^{\prime\prime\prime}$ are also given in Table III for completeness. 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A [**119**]{}, 171 (1996). E. Wörgötter, G. Wagner, and K. Wüthrich, J. Am. Chem. Soc. [**108**]{}, 6162 (1986). [^1]: These two moves must be iterated for a classical computer, but not in the quantum solution of the Deutsch-Jozsa problem, giving the quantum computer a large advantage over the classical computer. [^2]: During each pulse sequence shown in Fig. 2 only the coupling $J_{kl}$ is active which is required in order to implement (${\rm CNOT})_{kl}$. The effects of other non-zero couplings (see Table I) are effectively eliminated, except for $J_{13}$ in the sequence implementing ([CNOT]{})$_{45}$ (Fig. 2 D). Although it would be straightforward to remove also this coupling, this would require additional pulses which can be avoided because in our spin system the coupling $J_{13}=2.7$ has a negligible effect during the relatively short duration $\Delta_{45}=1/(2 J_{45})=1.39$ ms of this gate.

--- abstract: 'We study the two-dimensional kinetic Ising model below its equilibrium critical temperature, subject to a square-wave oscillating external field. We focus on the multi-droplet regime where the metastable phase decays through nucleation and growth of [*many*]{} droplets of the stable phase. At a critical frequency, the system undergoes a genuine non-equilibrium phase transition, in which the symmetry-broken phase corresponds to an asymmetric stationary limit cycle for the time-dependent magnetization. We investigate the universal aspects of this dynamic phase transition at various temperatures and field amplitudes via large-scale Monte Carlo simulations, employing finite-size scaling techniques adopted from equilibrium critical phenomena. The critical exponents, the fixed-point value of the fourth-order cumulant, and the critical order-parameter distribution all are consistent with the universality class of the two-dimensional [*equilibrium*]{} Ising model. We also study the cross-over from the multi-droplet to the strong-field regime, where the transition disappears.' address: | $^1$School of Computational Science and Information Technology, Florida State University, Tallahassee, Florida 32306-4120\ $^2$Center for Materials Research and Technology and Department of Physics, Florida State University, Tallahassee, Florida 32306-4350 author: - 'G. Korniss,$^1$[^1] C. J. White,$^{1,2}$ P. A. Rikvold,$^{1,2}$ and M. A. Novotny$^1$' title: 'Dynamic Phase Transition, Universality, and Finite-size Scaling in the Two-dimensional Kinetic Ising Model in an Oscillating Field' --- =10000 Introduction ============ Metastability and hysteresis are widespread phenomena in nature. Ferromagnets are common systems that exhibit these behaviors [@EWIN1881; @WARB1881; @EWIN1882; @STEI1892; @AHARONI], but there are also numerous other examples ranging from ferroelectrics [@BEALE; @RAO91] to electrochemical adsorbate layers [@SMEL; @MITCHELL00] to liquid crystals [@CHENG96]. A simple model for many of these real systems is the kinetic Ising model (in either the spin or the lattice-gas representation). For example, it has been shown to be appropriate for describing magnetization dynamics in highly anisotropic single-domain nanoparticles and uniaxial thin films [@HE; @JIANG; @SUEN; @group2]. The system response to a single reversal of the “external field” is fairly well understood [@switch]. In sufficiently large systems below the equilibrium critical temperature, $T_c$, the order parameter changes its value through the nucleation and growth of [*many*]{} droplets, inside which it has the equilibrium value consistent with the value of the applied field, as shown in Fig. \[decay\_conf\]. This is the multi-droplet regime of phase transformation [@switch; @phase_transformation]. The well-known Avrami’s law [@Avrami] describes this process of homogeneous nucleation followed by growth quite accurately up to the time when the growing droplets coalesce and the stable phase becomes the majority phase [@RAMOS99]. The intrinsic time scale of the system is given by the metastable lifetime, $\langle\tau\rangle$, which is defined as the average first-passage time to zero magnetization. It is a measure of the time it takes for the system to escape from the metastable region of the free-energy landscape. In this paper we will use the magnetic language in which the order parameter is the magnetization, $m$, and its conjugate field is the external magnetic field, $H$. Analogous interpretations, e.g., using the terms polarization and electric field for ferroelectric systems [@BEALE; @RAO91], and coverage and chemical potential for adsorption problems [@SMEL; @MITCHELL00], are straightforward. It is natural next to ask, “what is the response to an oscillating external field?” The hysteretic behavior in ferromagnets has attracted significant experimental interest, mainly focused on the characteristic behavior of the hysteresis loop and its area. Its dependence on the field amplitude and frequency has been intensively studied and its scaling behavior (power law versus logarithmic) is still under investigation, both experimentally [@HE; @JIANG; @SUEN] and theoretically [@JUNG90; @RAO90; @TOME90; @Mendes91; @Zimmer93; @LO90; @SIDES98a; @SIDES98b; @SIDES99]. For the kinetic Ising ferromagnet in two dimensions it has been recently shown [@SIDES98a; @SIDES98b; @SIDES99] that the true behavior is in fact a crossover, approaching a logarithmic frequency dependence only for extremely low frequencies. An important aspect of hysteresis in bistable systems, which is the focus of the present paper, is the dynamic competition between the two time scales in the system: the half-period of the external field $t_{1/2}$ (proportional to the inverse of the driving frequency) and the metastable lifetime $\langle\tau\rangle$. For low frequencies, a complete decay of the metastable phase almost always occurs in each half-period, just like it does after a single field reversal. Consequently, the time-dependent magnetization reaches a limit cycle which is symmetric about zero \[Fig. \[m\_series\](a)\]. For high frequencies, however, the system does not have enough time to switch during one half-period, and the symmetry of the hysteresis loop is broken. The magnetization then reaches an asymmetric limit cycle \[Fig. \[m\_series\](b)\]. Avrami’s law [@Avrami; @RAMOS99] is a good approximation when the majority of the droplets do not overlap. Thus, it can be employed to estimate the time-dependent magnetization and the dynamic order parameter (period-averaged magnetization) in the low-frequency (see the Appendix) and in the high-frequency [@SIDES99] limits. However, it cannot describe the “critical regime” where $t_{1/2}$ becomes comparable to $\langle\tau\rangle$, and which is dominated by coalescing droplets. This symmetry breaking between the symmetric and asymmetric limit cycles has been the subject of intensive research over the last decade. It was first observed during numerical integration of a mean-field equation of motion for the magnetization of a ferromagnet in an oscillating field [@TOME90; @Mendes91]. Since then, it has been observed and studied in numerous Monte Carlo (MC) simulations of kinetic Ising systems [@LO90; @SIDES99; @SIDES98; @ACHA95; @ACHA97C; @ACHA97D; @ACHA98; @BUEN00], as well as in further mean-field studies [@Zimmer93; @ACHA95; @ACHA97D; @ACHA98; @BUEN98]. It may also have been experimentally observed in ultrathin films of Co on Cu(001) [@JIANG]. The results of these studies suggest that this symmetry breaking corresponds to a genuine continuous non-equilibrium phase transition. For recent reviews see Refs. [@ACHA94; @CHAK99]. Associated with the transition is a divergent time scale (critical slowing down) [@ACHA97D] and, for spatially extended systems, a divergent correlation length [@SIDES99; @SIDES98]. Estimates for the critical exponents and the universality class of the transition have recently become available [@SIDES99; @SIDES98; @UGA99] after the successful application of finite-size scaling techniques borrowed from equilibrium critical phenomena [@FISH72; @BIND92; @BIND90; @LANDAU76]. The purpose of the present paper is to extend preliminary results [@UGA99] and to provide more accurate estimates of the exponents for two-dimensional kinetic Ising systems in a square-wave oscillating field. The use of the square-wave field tests the universality of the dynamic phase transition (DPT) [@SIDES99; @SIDES98], and it also significantly increases computational speed, compared to the more commonly used sinusoidal field. We further explore the universal aspects of the transition by varying the temperature and field amplitude within the multi-droplet regime, and we study the cross-over to the strong-field regime where the transition disappears. In obtaining our results, we rely on dynamic MC simulations. Computational methods are always helpful, especially when theoretical ideas are largely missing. There are cases, however, when even the use of standard equilibrium techniques, such as finite-size scaling requires some insight and building analogies between equilibrium and non-equilibrium systems [@SIDES99; @SIDES98]. This is the case for our present study. No effective “Hamiltonian” was known before the completion of this work for the dynamic order-parameter (in the coarse-grained sense), from which the long-distance behavior of the model could be derived. This is a typical difficulty when dealing with systems far from equilibrium [@DDS; @MARRO_DICKMAN]. Recently, however, a coarse-grained Hamiltonian has been derived [@Fuji] for the dynamic order-parameter, supporting our results for the DPT. Similar to the previous work for sinusoidly oscillating fields [@SIDES99; @SIDES98], we perform large-scale simulations and finite-size scaling to investigate the universal properties of the DPT. The remainder of the paper is organized as follows. In Sec. II we define the model and the observables of interest. Section III contains the Monte Carlo results and analyses. Conclusions and outlook are given in Sec. IV. Model and Relevant Observables ============================== To model spatially extended bistable systems in two dimensions, we study a nearest-neighbor kinetic Ising ferromagnet on a $L$$\times$$L$ square lattice with periodic boundary conditions. The model is defined by the Hamiltonian $$\label{eq:Hamil} {\cal H } = -J \sum_{ {\langle ij \rangle}} {s_{i}s_{j}} - H(t) \sum_{i} {s_{i}} \;,$$ where $s_{i}$$=$$\pm 1$ is the state of the $i$th spin, $J > 0$ is the ferromagnetic interaction, $\sum_{ {\langle ij \rangle} }$ runs over all nearest-neighbor pairs, $\sum_{i}$ runs over all $L^{2}$ lattice sites, and $H(t)$ is an oscillating, spatially uniform applied field. The magnetization per site, $$\label{eq:m(t)} {m(t)} = \frac{1}{L^2} \sum_{i=1}^{L^2} {s_{i}(t)} \;,$$ is the density conjugate to $H(t)$. The temperature $T$ is fixed below its zero-field critical value $T_{\rm c}$ ($J/k_{\rm B} T_{\rm c}$$=$$ \ln(1+\sqrt 2)/2$ [@Onsager], where $k_{\rm B}$ is Boltzmann’s constant), so that the magnetization for $H$$=$$0$ has two degenerate spontaneous equilibrium values, $\pm m_{\rm sp}(T)$. For nonzero fields the equilibrium magnetization has the same sign as $H$, while for $H$ not too strong, the opposite magnetization direction is [*metastable*]{} and decays slowly towards equilibrium with time, as described in Sec. I. The dynamic used in this study, as well as in Refs. [@SIDES99; @SIDES98], is the Glauber single-spin-flip MC algorithm with updates at randomly chosen sites. Note that the random sequential update scheme corresponds to independent Poisson arrivals for the update attempts (discrete events) at each site. Thus, the arrival pattern is strongly asynchronous. The time unit is one MC step per spin (MCSS). Each attempted spin flip from ${s_{i}}$ to ${-s_{i}}$ is accepted with probability $$\label{eq:Glauber} W(s_{i} \rightarrow -s_{i}) = \frac{ \exp(- \beta \Delta E_{i})}{1 + \exp(- \beta \Delta E_{i})} \; .$$ Here $\Delta E_{i}$ is the energy change resulting from acceptance, and $\beta \! = \! 1/k_{\rm B}T$. For the largest system studied ($L$$=$$512$) we used a scalable massively parallel implementation of the algorithm for this [*asynchronous*]{} dynamics [@Luba; @PAR_JCP; @PAR_PRL; @UGA00]. The parallel discrete-event scheme ensures that the underlying dynamic is not changed (that is, the update attempts are identical, independent Poisson arrivals at each site), while a substantial amount of parallelism is exploited. The parallel implementation [@PAR_JCP] was carried out on a Cray T3E, employing up to 256 processing elements. The dynamic order parameter is the period-averaged magnetization [@TOME90], $$\label{eq:Qeq} Q = \frac{1}{2 t_{1/2}} \oint m(t) dt \;,$$ where $t_{1/2}$ is the half-period of the oscillating field, and the beginning of the period is chosen at a time when $H(t)$ changes sign. Although the phase of the field does not influence the results reported in this paper, the choice made here is convenient in studies of the hysteresis loop-area distributions and consistent with Refs. [@SIDES98a; @SIDES98b; @SIDES99; @SIDES98]. Analogously we also define the local order parameter $$\label{eq:Qlocaleq} Q_{i} = \frac{1}{2 t_{1/2}} \oint s_{i}(t) dt \;,$$ which is the period-averaged spin at site $i$. For slowly varying fields the probability distribution of $Q$ is sharply peaked at zero [@SIDES99; @SIDES98]. We shall refer to this as the [*dynamically disordered phase*]{}. It is illustrated by the evolution of the magnetization in Fig. \[m\_series\](a) and by the $Q$$\approx$$0$ time series in Fig. \[Q\_series\]. For rapidly oscillating fields the distribution of $Q$ becomes bimodal with two sharp peaks near $\pm m_{\rm sp}(T)$, corresponding to the broken symmetry of the hysteresis loops [@SIDES99; @SIDES98]. We shall refer to this as the [*dynamically ordered phase*]{}. It is illustrated in Fig. \[m\_series\](b) and by the $Q$$\approx$$m_{\rm sp}$$\sim$${\cal O}(1)$ time series in Fig. \[Q\_series\]. Near the DPT we use finite-size scaling analysis of MC data to estimate the critical exponents that characterize the transition. We also keep track of the normalized period-averaged internal energy (in units of $J$) [@ACHA97D], $$E=-\frac{1}{2 t_{1/2}} \oint \frac{1}{L^2}\sum_{ {\langle ij \rangle}} {s_{i}(t)s_{j}(t)} dt \;, \label{energy_def}$$ since it also exhibits important characteristics of the DPT. Previous studies of the DPT have used an applied field which varies sinusoidally in time. While sinusoidal or linear saw-tooth fields are the most common in experiments and are necessary to obtain a vanishing loop area in the low-frequency limit [@SIDES98a; @SIDES98b; @SIDES99], the wave form of the field should not affect universal aspects of the DPT. This should be so because the transition essentially depends on the competition between two time scales: the half-period $t_{1/2}$ of the applied field, and the average time it takes the system to leave the metastable region near one of its two degenerate zero-field equilibria when a field of magnitude $H_0$ and sign opposite to the magnetization is applied. This [*metastable lifetime*]{}, $\langle \tau(T,H_0) \rangle$, is estimated as the average first-passage time to zero magnetization. In the present paper we use a [*square-wave*]{} field of amplitude $H_0$. This has significant computational advantages over the sinusoidal field variation since we can use two look-up tables to determine the acceptance probabilities: one for $H \!=\! + H_0$ and one for $H \!=\! - H_0$. In terms of the dimensionless half-period, $$\label{eq:Theta} \Theta = t_{1/2} \left/ \langle \tau(T,H_0) \rangle \right. \;,$$ the DPT should occur at a critical value $\Theta_{\rm c}$ of order unity. Although $\Theta$ can be changed by varying either $t_{1/2}$, $H_0$, or $T$, in a first approximation we expect $\Theta_{\rm c}$ to depend only weakly on $H_0$ and $T$. This expectation will be confirmed in Sec. IV by simulations carried out at several values of $H_0$ and $T$ for different system sizes. In many studies of the DPT the transition has been approached by changing $H_0$ or $T$ [@LO90; @ACHA95; @ACHA97C; @ACHA98]. While this is correct in principle, $\langle \tau(T,H_0) \rangle$ depends strongly and nonlinearly on its arguments [@switch]. We therefore prefer changing $t_{1/2}$ at constant $H_0$ and $T$ [@SIDES99; @SIDES98], as this gives more precise control over the distance from the transition. We focus on systems which are not only larger than the critical droplet, but also significantly larger than the typical droplet separation [@switch]. In this regime [*many*]{} supercritical droplets form and contribute to the decay of the metastable phase (the KJMA or Avrami theory for homogeneous nucleation [@Avrami; @RAMOS99]), as seen in Fig. \[decay\_conf\]. This is the only regime where the DPT is expected to exist. For small systems one observes subtle finite-size effects, not related to the DPT but rather to the [*stochastic*]{} single-droplet decay mode [@switch; @tric]. In the single-droplet regime, subject to a periodic applied field, the system exhibits stochastic resonance [@SIDES98b]. Simulation results ================== Signs of the dynamic phase transition ------------------------------------- We performed extensive simulations and finite-size scaling analysis of the data on square lattices with $L$ between $64$ and $512$ at $T$$=$$0.8T_c$ and $H_0$$=$$0.3J$. We also investigated the universality of the DPT within the multi-droplet regime for various fields and temperatures below the equilibrium critical temperature, using smaller systems with $L$ from $64$ to $128$. Typical runs near the DPT consist of $2$$\times$$10^5$ full periods. For example, at $T$$=$$0.8T_c$ and $H_0$$=$$0.3J$, where the critical half-period of the field is about $70$ MCSS, this corresponds to $2.8$$\times$$10^7$ MCSS. Away from the transition point, an order of magnitude shorter runs were sufficient to obtain high-quality statistics. The system was initialized with all spins up and the square-wave external field started with the half-period in which $H$$=$$-H_0$. After some relaxation the system magnetization would reach a limit cycle \[Fig. \[m\_series\]\] (except for thermal fluctuations). In other words, $Q$ \[Eq. (\[eq:Qeq\])\] (together with other period-averaged quantities) becomes a [*stationary*]{} stochastic process \[Fig. \[Q\_series\]\]. We discarded the first $1000$ periods of the time series to exclude transients from the stationary-state averages. For large half-periods ($\Theta\gg\Theta_{\rm c}$) the magnetization switches every half-period \[Fig. \[m\_series\](a)\] and $Q$$\approx$$0$, while for small half-periods ($\Theta\ll\Theta_{\rm c}$) the magnetization does not have time to switch during a single half-period \[Fig. \[m\_series\](b)\], resulting in $|Q|$$\approx$$m_{\rm sp}$, as can be seen from the time series in Fig. \[Q\_series\]. The transition between the high- and low-frequency regimes is characterized by large fluctuations in $Q$ near $\Theta_{\rm c}$ \[Fig. \[Q\_series\]\]. To illustrate the spatial aspects of the transition we also show configurations of the local order parameter $\{Q_i\}$ in Fig. \[local\_conf\]. Below $\Theta_{\rm c}$ \[Fig. \[local\_conf\](a)\] the majority of spins spend most of their time in the $+1$ state, i.e., in the metastable phase during the first half-period, and in the stable equilibrium phase during the second half-period (except for equilibrium fluctuations). Thus, most of the $Q_i$$\approx$$+1$. Droplets of $s_i$$=$$-1$ that nucleate during the negative half-period and then decay back to $+1$ during the positive half-period show up as roughly circular gray spots in the figure. Since the spins near the center of such a droplet become negative first and revert to positive last, these spots appear darkest in the middle. Also, for not too large lattices one occasionally observes the full reversal of an ordered configuration $\{Q_i\}$$\rightarrow$$\{-Q_i\}$, typical of finite, spatially extended systems undergoing symmetry breaking. Above $\Theta_{\rm c}$ \[Fig. \[local\_conf\](c)\] the system follows the field in every half-period (with some phase lag) and $Q_i$$\approx$$0$ at all sites $i$. Near $\Theta_{\rm c}$ \[Fig. \[local\_conf\](b)\] there are large clusters of both $Q_i$$\approx$$+1$ and $Q_i$$\approx$$-1$ separated by “interfaces” where $Q_i$$\approx$$0$. These large-scale structures remain reasonably stationary over several periods. For finite systems in the dynamically ordered phase the probability density of $Q$ becomes bimodal. Thus, to capture symmetry breaking, one has to measure the average norm of Q as the order parameter, i.e., $\langle |Q| \rangle$ [@BIND92]. Figure \[Q\_raw\](a) shows that this order parameter is of order unity for $\Theta<\Theta_{\rm c}$ and vanishes for $\Theta>\Theta_{\rm c}$, except for finite-size effects. To characterize and quantify this transition in terms of critical exponents we employ the well-known technique of finite-size scaling [@FISH72; @BIND92; @BIND90; @LANDAU76]. The quantity analogous to the susceptibility is the scaled variance of the dynamic order parameter [@SIDES99; @SIDES98], $$X^{Q}_{L}=L^2 \left(\langle Q^2 \rangle_L -\langle |Q|\rangle_L ^2 \right) \;. \label{eq:XQ}$$ Note that for our system the field conjugate to $Q$ and a corresponding fluctuation-dissipation theorem are not known, hence we cannot measure the susceptibility directly. For finite systems $X_L$ has a characteristic peak near $\Theta_{\rm c}$ \[see Fig. \[Q\_raw\](b)\] which increases in height with increasing $L$, while no finite-size effects can be observed for $\Theta\ll\Theta_{\rm c}$ and $\Theta\gg\Theta_{\rm c}$. This implies the existence of a divergent length scale, possibly the correlation length which governs the long-distance behavior of the local order-parameter correlations $\langle Q_i Q_j \rangle$. The location of the maximum in $X^Q_L$ also shifts with $L$, which gives further important information about the critical exponents. =5.5cm =5.5cm =5.5cm =5.5cm =5.5cm =5.5cm The normalized stationary time-displaced autocorrelation function of the order parameter, $$C^{Q}_{L}(n) = \frac{\langle Q(j) Q(j+n)\rangle - \langle Q(j) \rangle^{2}} {\langle Q^{2}(j)\rangle - \langle Q(j) \rangle^{2}} \;, \label{auto_corr}$$ provides further insights into the DPT as the system exhibits critical slowing down \[Fig. \[crit\_slow\]\]. This can be seen as increasing correlation times with increasing system sizes. In Sec. III.D we provide a quantitative analysis of the correlation times. We also measured the period-averaged internal energy \[Eq. (\[energy\_def\])\] and its fluctuations [@ACHA97D] $$X^{E}_{L}=L^2 \left(\langle E^2 \rangle_L - \langle E \rangle_L ^2 \right) \;, \label{eq:XE}$$ as can be seen in Fig. \[E\_raw\]. The peaks of these fluctuations exhibit a slow increase with the system size (compared to the order-parameter fluctuations), as one may anticipate by analogy with the equilibrium heat capacity. =5.5cm =5.5cm Finite-size scaling ------------------- Scaling laws and finite-size scaling for equilibrium systems with an a-priori known Hamiltonian can be systematically derived using the concepts of the free energy and the renormalization group [@GOLD]. The kinetic Ising model with the explicitly time-dependent Hamiltonian, Eq. (\[eq:Hamil\]), is driven far from equilibrium. Although the order-parameter distribution $P(Q)$ is stationary, the effective Hamiltonian controlling its fixed-point behavior has not been known until recently [@Fuji] (after the completion of this study). Motivated by the similarity of the finite-size effects shown in Figs. \[Q\_raw\]-\[E\_raw\] to those characteristic of a typical continuous phase transition, we borrow the corresponding scaling assumptions from equilibrium finite-size scaling. For our model the quantity analogous to the reduced temperature in equilibrium systems (i.e., the distance from the [*infinite*]{}-system critical point) is $$\theta= \frac{|\Theta-\Theta_{\rm c}|}{\Theta_{\rm c}}\;. \label{eq:reduced_Theta}$$ Finite-size scaling theory provides simple scaling relations for the observables for finite systems in the critical regime [@BIND92; @BIND90]: $$\begin{aligned} \langle |Q|\rangle _L & = & L^{-\beta /\nu} {\cal F}_{\pm}(\theta L^{1/\nu}) \label{full_scaling_Q}\\ X^Q_L & = & L^{\gamma /\nu} {\cal G}_{\pm}(\theta L^{1/\nu}) \label{full_scaling_XQ} \\ X^E_L & = & c_1\ln\left( L{\cal J}_{\pm}(\theta L^{1/\nu})\right) \;, \label{full_scaling_XE}\end{aligned}$$ where ${\cal F}_{\pm}$, ${\cal G}_{\pm}$, and ${\cal J}_{\pm}$ are scaling functions and the $+$ ($-$) index refers to . The logarithmic scaling in $X^E_L$ is motivated by the very slow divergence of the scaled period-averaged energy variance \[Fig. \[E\_raw\](b)\]. The above formulation of scaling is explicitly based on the infinite-system critical point $\Theta_{\rm c}$, which can be estimated with far greater accuracy than the location of the maximum of the order-parameter fluctuations for the individual finite system sizes. We use the fourth-order cumulant intersection method [@BIND92; @BIND90] to estimate the value of $\Theta_c$ at which the transition occurs in an [*infinite*]{} system. In order to do this, we plot $$U_L=1 - \frac{\langle Q^4\rangle_L}{3\langle Q^2\rangle_L^2} \; \label{eq:cumulant}$$ as a function of $\Theta$ for several system sizes as shown in Fig. \[fig\_cumul\]. For the largest system ($L$=$512$) the statistical uncertainty in $U_L$ was too large to use it to obtain estimates for the crossing. Our estimate for the dimensionless critical half-period, based on the remaining five system sizes, is $\Theta_{\rm c}$$=$$0.918$$\pm$$0.005$ with a fixed-point value $U^*$$=$$0.611$$\pm$$0.003$ for the cumulant \[Fig \[fig\_cumul\](b)\]. Then at $\Theta_c$ the scaling forms Eqs. (\[full\_scaling\_Q\]-\[full\_scaling\_XE\]) yield $$\begin{aligned} \langle |Q|\rangle _L & \propto & L^{-\beta /\nu} \label{scaling_Q}\\ X^Q_L & \propto & L^{\gamma /\nu} \label{scaling_XQ} \\ X^E_L & \propto & c_2 + c_1\ln(L) \;, \label{scaling_XE}\end{aligned}$$ which enable us to estimate the exponent ratios $\beta/\nu$ and $\gamma/\nu$, and to directly check the postulated logarithmic divergence in the period-averaged energy fluctuations. Plotting $\langle |Q|\rangle _L$ and $X^Q_L$ at $\Theta_c$ and utilizing a weighted linear least-squares fit to the logarithmic data yields $\beta/\nu$$=$$0.126\pm 0.005$ \[Fig. \[fit\](a)\] and $\gamma/\nu$$=$$1.74\pm 0.05$ \[Fig. \[fit\](b)\]. Note that these values are extremely close (within statistical errors) to the corresponding ratios for the [*equilibrium*]{} two-dimensional Ising universality class, $\beta/\nu$$=$$1/8$$=$$0.125$ and $\gamma/\nu$$=$$7/4$$=$$1.75$. Further, the straight line in Fig. \[fit\](c) indicates the slow logarithmic divergence of $X^E_L$ at the critical point. In addition to the scaling at $\Theta_c$, we also checked the divergences of the peaks of the fluctuations, $(X^Q_L)_{\rm peak}$ and $(X^E_L)_{\rm peak}$, since they asymptotically should follow the same scaling laws, Eqs. (\[scaling\_XQ\]) and (\[scaling\_XE\]), respectively. The measured exponent $\gamma/\nu$$=$$1.78\pm 0.05$ for $(X^Q_L)_{\rm peak}$ and the logarithmic divergence for $(X^E_L)_{\rm peak}$ agree to within the statistical errors with the results obtained at $\Theta_c$, as can be seen in Fig. \[fit\](b) and (c), respectively. From the finite-system shifting of the transition one can estimate the correlation-length exponent $\nu$ by tracking the shift in the location of the maximum in $X^Q_L$: $$|\Theta_{c}(L)-\Theta_{c}| \propto L^{-1/\nu}\;, \label{scaling_Tc}$$ where $\Theta_{c}(L)$ is the location of the peak for finite systems. However, the precision of this method for our data is very poor, due to limited resolution in finding the locations of the maxima and consequently the large relative errors in $|\Theta_{c}(L)-\Theta_{c}|$. Excluding the smallest (due to strong corrections to scaling) and the largest systems (due to very poor resolution and extremely large statistical error), we obtain $\nu$$=$$0.87\pm0.4$, but the large error estimate obviously implies rather poor accuracy \[Fig. \[Tc\_scale\]\]. =5.5cm =5.5cm =5.5cm =5.5cm =5.5cm =5.5cm To obtain a more complete picture of how well the scaling relations in Eqs. (\[full\_scaling\_Q\]) and (\[full\_scaling\_XQ\]) hold, we plot $\langle |Q|\rangle _L L^{\beta/\nu}$ \[Fig. \[full\_collapse\](a)\] and $X^Q_LL^{-\gamma/\nu}$ \[Fig. \[full\_collapse\](b)\] vs $\theta L^{1/\nu}$ [@LANDAU76]. For the exponent ratios we used $\beta/\nu$$=$$1/8$$=$$0.125$, and $\gamma/\nu$$=$$7/4$$=$$1.75$, since our estimate for those (within small statistical errors) implied that they take on the equilibrium two-dimensional Ising universal values. Most importantly, we used various values of $\nu$ between $0.5$ and $1.2$ to find the best data collapse as observed visually, since our estimate for this exponent was far from reliable. The “optimal” value obtained this way (by showing scaling plots to group members who did not know the particular values of $\nu$ used), and used in Fig. \[full\_collapse\](a) and (b), is $\nu$$=$$0.95\pm0.15$. Full scaling plots using the exact Ising exponents are also shown in Fig. \[full\_collapse\](c) and (d), and they result in similarly good data collapse. Order-parameter histograms at criticality ----------------------------------------- We devote this subsection to analyzing the universal characteristics of the full order-parameter distribution, $P(Q)$, at the critical point. This distribution is bimodal for finite systems if observed for sufficiently long times (Fig. \[series\_hist\]) [@endnote1]. It is more convenient to focus on the distribution of $|Q|$, avoiding the effect of the insufficient number of switching events between the two symmetry-broken phases for large systems, which causes the skewness in Fig. \[series\_hist\](b). Figure \[scaled\_hist\](a) shows the order-parameter distributions $P_{L}(|Q|)$ at the critical point for various system sizes. Finite-size scaling arguments [@BIND92; @BIND90] suggest that at $\Theta_{\rm c}$ $$P_{L}(|Q|)=L^{\beta/\nu}{\cal P}(L^{\beta/\nu} |Q|) \;. \label{scaling_PQ}$$ Thus, the scaled distributions, $L^{-\beta/\nu}P_{L}(|Q|)$ vs $x$$=$$|Q|L^{\beta/\nu}$, should fall on the same curve ${\cal P}(x)$ for different system sizes. Again, we used $\beta/\nu=1/8$. The quality of the data collapse is quite impressive \[Fig. \[scaled\_hist\](b)\], with deviations mainly observed for the smallest $L$ and the largest values of $|Q|$ \[Fig. \[scaled\_hist\](c)\], possibly as a result of corrections to scaling. What we find somewhat surprising, is that the distribution appears to be identical (except for stronger corrections to scaling at the DPT) to that of the equilibrium two-dimensional Ising model on a square lattice with periodic boundary conditions at criticality, [*without*]{} a need for any additional scaling parameters. We checked this by performing standard equilibrium two-dimensional Ising simulations with Glauber dynamics and system sizes ranging from $L$$=$$64$ to $L$$=$$128$, and also by comparing our scaled DPT order-parameter histograms to the high-precision two-dimensional equilibrium Ising MC data of Ref. [@Janke] (Fig. \[scaled\_hist\]). We had expected the [*shapes*]{} of the distributions to be identical for the DPT and equilibrium Ising model, as a consequence of the identical values for the cumulant fixed-point value $U^*$. However, it is not obvious to us why the microscopic length scales in the DPT and the equilibrium Ising model also appear to be identical, as evidenced by the absence of the need for an additional scaling, $L$$\rightarrow$$L/a$ ($a$ being the microscopic length scale in the DPT). =5.5cm =5.5cm =5.5cm =5.5cm =5.5cm =5.5cm Critical slowing down --------------------- Computing the stationary autocorrelation function given by Eq. (\[auto\_corr\]), we already pointed out that at $\Theta_{\rm c}$ the correlation time increases fast with system size \[Fig. \[crit\_slow\]\]. Correlation times are typically extracted from an exponential decay as $$C^{Q}_{L}(n) \propto e^{-n/\tau^{Q}_{L}} \;, \label{exp_decay}$$ and they are expected to be finite for finite systems. The correlation time $\tau^{Q}_{L}$ is also well defined in the $L$$\rightarrow$$\infty$ limit [*away*]{} from the transition. However, it diverges with $L$ at the transition point as $$\tau^{Q}_{L} \propto L^{z} \;, \label{power_law}$$ where $z$ is the dynamical critical exponent. For not too late times we had reasonable statistics including the larger systems (up to $L$$=$$256$) to fit the usual exponential decay \[Fig. \[scale\_crit\_slow\](a)\]. Then plotting the correlation times $\tau^{Q}_{L}$ vs $L$ yields the dynamic exponent $z=1.91\pm0.15$, as shown in Fig. \[scale\_crit\_slow\](b). This value is within two standard deviations of most estimates for the dynamic exponent of the two-dimensional equilibrium Ising model with local dynamics [@dynamic_z]. =5.5cm =5.5cm =5.5cm =5.5cm Universality for various temperatures and fields and cross-over to the strong-field regime ------------------------------------------------------------------------------------------ The underlying ingredient for the spatially extended bistable systems exhibiting a DPT is the local metastability (and the corresponding characteristic time spent in the metastable “free-energy well”) in the presence of an external field. This, in turn, provides a competition between time scales if the system is driven by a periodic field. Based on this, we expect that sufficiently large systems (in which many droplets contribute to the decay of the metastable phase) exhibit the DPT at a half-period $t_{1/2}$ comparable to the metastable lifetime $\langle\tau(T,H_0)\rangle$. In other words, we expect the critical dimensionless half-period to be of order one, $\Theta_{\rm c}\sim{\cal O}(1)$. To test this expectation, we performed simulations at $T$$=$$0.8T_c$ for field amplitudes ranging from $0.3J$ to infinity with system sizes $L$$=$$64$, $90$, and $128$. \[$H_0$$=$$\infty$ corresponds to the Glauber spin-flip probabilities Eq. (\[eq:Glauber\]) being equal to $0$ (1) depending on whether the spin is parallel (anti parallel) to the external field, with no influence from the configuration of the neighboring spins.\] We further performed runs at $T$$=$$0.9T_c$, $T$$=$$0.6T_c$, and $T$$=$$0.5T_c$ for various field amplitudes and system sizes $L=64$ and $90$. The typical run length was $2$$\times$$10^4$ periods. The purpose of these runs was to explore the universal nature of the DPT in the multi-droplet regime, and the crossover to the strong-field regime where the DPT should disappear. In the strong-field regime the droplet picture breaks down since the individual spins are decoupled. Thus, the metastable phase no longer exists, and the decay of the phase having opposite sign to the external field approaches a simple exponential form (which becomes exact in the $H_0$$\rightarrow$$\infty$ limit). In the Appendix we show that under these conditions the system magnetization always relaxes to a symmetric limit cycle with $Q$$=$$0$ for [*all*]{} frequencies, thus, no DPT can exist. Figures \[strongfield\_cross\](a), (b), and (c) show the order parameter vs the dimensionless half-period for $L=64$ and a range of field amplitudes at $T$$=$$0.8T_c$, $T$$=$$0.6T_c$, and $T$$=$$0.5T_c$, respectively. The typical order-parameter profile where the system exhibits the DPT prevails up to some temperature dependent cross-over field amplitude $H_{\times}(T)$ \[filled symbols in Fig. \[strongfield\_cross\]\]. For $H_{0}>H_{\times}(T)$ the underlying decay mechanism belongs to the strong-field regime, and correspondingly the DPT disappears as expected. =5.5cm =5.5cm =5.5cm =5.5cm =5.5cm =5.5cm One can trace this crossover from the multi-droplet to the strong-field regime by plotting $\Theta_{\rm c}$ vs $H_{0}/J$ \[Fig. \[theta\_cross\](a)\] and vs $\langle\tau\rangle$ \[Fig. \[theta\_cross\](b)\] for fixed temperatures. Here again we employed the fourth-order cumulant intersection method using $L$$=$$64$ and $L$$=$$90$ to identify the infinite-system transition point, $\Theta_{\rm c}$. The crossover to the strong-field regime is indicated by the drop in $\Theta_{\rm c}$ for large fields (small lifetimes). =5.5cm =5.5cm =5.5cm =5.5cm =5.5cm =5.5cm The DPT in the multidroplet regime ($H_{0}$$<$$H_{\times}(T)$) appears to be universal, as can be seen from the scaled order-parameter plot in Fig. \[univ\_scale\]. Note that both graphs contain $28$ DPT data sets: three different temperatures, with four different field amplitudes for each, and at least two different system sizes ($L=64,90$, and $128$ at $T$$=$$0.8T_c$, and $L=64$ and $90$ at $T$$=$$0.6T_c$ and $T$$=$$0.5T_c$) for all these parameters! While the slopes in the asymptotic scaling regime appear to be the same, the small parallel shift may be the result of the non-universal critical amplitudes at different temperatures and fields, or simply our inaccuracy in determining $\Theta_{\rm c}(T,H_{0})$, due to the relatively short runs. On the other hand, as expected, the system shows no singularity and $\langle |Q|\rangle$$\stackrel{L\rightarrow\infty}{\longrightarrow}$$0$ for [*any*]{} non-zero frequency in the strong-field regime, $H_{0}$$>$$H_{\times}(T)$ (see the Appendix). Here, the finite-size effects simply reflect the central-limit theorem, i.e., $\langle Q^2\rangle$$\sim$${\cal O}(1/L^2)$, or $\langle |Q|\rangle$$\sim$${\cal O}(1/L)$. Figures \[no\_DPT\](a) and (b) illustrate this at $T$$=$$0.8T_c$, $H_{0}$$=$$1.5J$ and at $T$$=$$0.8T_c$, $H_{0}$$=$$\infty$, respectively. One also expects that a similar behavior prevails at $T$$>$$T_{\rm c}$ for [*any*]{} field amplitude, i.e., $H_{\times}(T)$ vanishes for $T$$>$$T_{\rm c}$. We checked this for $T$$=$$1.1T_c$ and $H_{0}$$=$$0.05J$, and the results confirm the $\langle |Q|\rangle$$\sim$${\cal O}(1/L)$ scaling of the order parameter for all frequencies, showing none of the characteristic finite-size effects of a DPT \[Fig. \[no\_DPT\](c)\]. This result is in distinct disagreement with older work such as Refs. [@ACHA95; @ACHA94], which report observations of the DPT at temperatures considerably above $T_{\rm c}$. For smaller system sizes, however, one may observe nontrivial resonance [@NEDA]. =5.5cm =5.5cm =5.5cm =5.5cm =5.5cm =5.5cm Due to the expected complications of a divergent [*equilibrium*]{} correlation length, we did not perform simulations at $T_{\rm c}$. However, we conjecture that $H_{\times}(T)$ vanishes at $T_{\rm c}$. We expect this conjecture to be extremely difficult to prove or disprove numerically, due to very large and possibly complicated finite-size effects. Conclusions and Outlook ======================= In this paper we have studied the hysteretic response of a spatially extended bistable system exhibiting a DPT. Our model system is the two-dimensional kinetic Ising ferromagnet below its equilibrium critical temperature subject to a periodic square-wave applied field. The results indicate that for field amplitudes and temperatures such that the metastable phase decays via the multi-droplet mechanism, the system undergoes a continuous dynamic phase transition when the half-period of the field, $t_{1/2}$, is comparable to the metastable lifetime, $\langle\tau(T,H_0)\rangle$. Thus, the critical value $\Theta_{\rm c}$ of the dimensionless half-period defined in Eq. (\[eq:Theta\]) is of order unity. As $\Theta$ is increased beyond $\Theta_{\rm c}$, the order parameter $\langle|Q|\rangle$ (the expectation value of the norm of the period-averaged magnetization) vanishes \[Fig. \[Q\_raw\](a)\], displaying singular behavior at the critical point, as shown in Fig. \[Q\_raw\](b) and (c). The characteristic finite-size effects in the order parameter and its fluctuations indicate that there is a divergent correlation length associated with the transition. We used standard finite-size scaling techniques adopted from the theory of equilibrium phase transitions. We estimated $\Theta_{\rm c}$ and the critical exponents $\beta$, $\gamma$, and $\nu$ from relatively high precision data for system sizes between $L=64$ and $512$ at $T$$=$$0.8T_{\rm c}$ and $H_{0}$$=$$0.3J$. Our best estimates are $\beta/\nu$$=$$0.126\pm 0.005$, $\gamma/\nu$$=$$1.74\pm 0.05$, and $\nu$$=$$0.95\pm 0.15$. These values agree within statistical errors with those previously obtained with a sinusoidally oscillating field [@SIDES99; @SIDES98], providing strong evidence that the shape of the field oscillation does not affect the universal aspects of the DPT. Observing the stationary autocorrelation function we also saw that at the transition point the system exhibits critical slowing down governed by the dynamic exponent $z$$=$$1.91\pm0.15$. This is also very close to the corresponding exponent $z$$=$$2.12(5)$, measured in standard two-dimensional Ising simulations with local dynamics [@dynamic_z]. Our best values for the exponent ratios $\beta/\nu$ and $\gamma/\nu$ are given with relatively high confidence, while for $\nu$ it is rather poor. In this sense tracking down the exponent $\nu$ and obtaining an accurate estimate for it remains elusive. Note, however, that we could only rely on the standard (single spin-flip) MC algorithm, since we had to preserve the underlying dynamics. Using more sophisticated algorithms to avoid critical slowing down as seen in Figs. \[crit\_slow\] and \[scale\_crit\_slow\], would require an underlying “Hamiltonian” for the corresponding [*local*]{} order parameter $\{Q_i\}$, which is not yet known. While in a coarse-grained/universal sense a $\phi^4$ Hamiltonian is supported by our data, it does not point to any one particular microscopic Hamiltonian for the microscopic order parameter $\{Q_i\}$. However, a $\phi^4$ coarse-grained Hamiltonian for $\{Q_i\}$ has been recently derived starting from the time-dependent Ginzburg-Landau equation for the magnetization[@Fuji]. Of the known universality classes, our exponent estimates for the DPT are closest (and within the statistical errors) to those of the the two-dimensional equilibrium Ising model: $\beta/\nu$$=$$1/8$$=$$0.125$, $\gamma/\nu$$=$$7/4$$=$$1.75$, and $\nu$$=$$1$. Consequently, our measured exponent ratios satisfy the hyperscaling relation $$2(\beta/\nu)+\gamma/\nu = 1.99\pm 0.05 \approx d \;,$$ where $d$$=$$2$ is the spatial dimension [@endnote2]. Further, the fixed-point value of the fourth-order cumulant, $U^*$$=$$0.611$$\pm$$0.003$, is also extremely close to that of the Ising model, $U^*$$=$$0.6106901(5)$ [@BLOTE]. These findings provide conclusive evidence that the DPT indeed corresponds to a non-trivial fixed point. We tested the full data collapse for the scaled order parameter and its variance, as shown in Fig. \[full\_collapse\], and it confirmed the existence of the universal scaling functions given by Eqs. (\[full\_scaling\_Q\]) and (\[full\_scaling\_XQ\]). Also, at the critical frequency, the order-parameter distributions follow finite-size scaling predictions, Eq. (\[scaling\_PQ\]), as shown in Fig. \[scaled\_hist\]. More surprisingly, the critical DPT order-parameter distributions fall on that of the two-dimensional equilibrium Ising model at the critical temperature (except for stronger corrections to scaling), [*without*]{} any additional fitting of the underlying microscopic length scale. While our finite-size scaling data clearly indicate the existence of a divergent length scale, we did not measure the correlation length for the [*local*]{} order parameter directly. Future studies may include extracting the correlation length, $\xi_Q$, from the $\langle Q_i Q_j\rangle$ correlations (or from the corresponding structure factor). This approach would also provide another way to measure the exponent $\nu$ by plotting $\xi_Q$ vs $\theta$ in the critical regime for large systems, and assuming $\xi_Q$$\sim$$\theta^{-\nu}$. We also studied the universal aspects of the DPT at other temperatures and fields. Shorter runs at $T$$=$$0.8T_{\rm c}$, $T$$=$$0.6T_{\rm c}$, and $T$$=$$0.5T_{\rm c}$ for field amplitudes $H_{0}$$<$$H_{\times}(T)$ also confirmed scaling and the universality of the DPT \[Fig. \[univ\_scale\]\]. The condition for the field amplitude implies that the system only exhibits a DPT in the multi-droplet regime. For $H_{0}$$>$$H_{\times}(T)$ strong-field behavior governs the decay of the magnetization, and the DPT disappears, as indicated by Fig. \[theta\_cross\] and Fig. \[no\_DPT\](a),(b). We also found that the high-temperature phase is qualitatively similar to the strong-field regime in that there is no sign of a DPT for $T$$>$$T_{\rm c}$ \[Fig. \[no\_DPT\](c)\]. One may ask how general the phenomenon of a DPT is in spatially extended bistable systems, subject to a periodic applied “field” which drives the system between its metastable and stable “wells.” It is possible that having up-down symmetry of the period-averaged “magnetization” is sufficient for possessing a Hamiltonian at the coarse-grained level, even if the system is driven far away from equilibrium and is microscopically irreversible [@GRINSTEIN]. Future research can address this question by studying other systems (not necessarily ferromagnets) that exhibit hysteresis. We thank S. W. Sides, S. J. Mitchell, and G. Brown for stimulating discussions. We would like to thank W. Janke for providing us with data from Ref. [@Janke] for comparison of the critical order-parameter distributions. We acknowledge support by the US Department of Energy through the former Supercomputer Computations Research Institute, by the Center for Materials Research and Technology at Florida State University, and by the US National Science Foundation through Grants No. DMR-9634873, DMR-9871455, and DMR-9981815. This research also used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098. Low-frequency and Strong-field Mean-field Approximation {#low-frequency-and-strong-field-mean-field-approximation .unnumbered} ======================================================= For simplicity, in the following we assume that the magnetization decays from $m$$=$$+1$ to $m$$=$$-1$ after a [*single*]{} field reversal ($H_0$$\rightarrow$$-H_0$). This is a good approximation below the equilibrium critical temperature ($m_{\rm sp}$$\approx$$1$), and for any temperature when $H_0$$\rightarrow$$\infty$. Further, we assume that the volume fraction of meta- or unstable spins follows a simple monotonic decay, $\hat{\varphi}(t)$. In terms of the volume fraction of [*positive*]{} spins, $\phi(t)$, the magnetization can be written as $$m(t) = 2\phi(t) - 1 \;.$$ Subject to a square-wave field, $$H(t) = \left\{ \begin{array}{ll} -H_0 & 0 \leq t <t_{1/2} \\ +H_0 & t_{1/2} \leq t <2t_{1/2} \end{array} \right. \;,$$ in the first (second) half-period the volume fraction of the positive (negative) spins decays according to $\hat{\varphi}(t)$. Thus, in each period (measuring time $t$ from the beginning of the period) $$\phi(t) \approx \left\{ \begin{array}{ll} \phi(0)\hat{\varphi}(t) & 0\leq t \leq t_{1/2} \\ 1-\left[1-\phi(t_{1/2})\right ]\hat{\varphi}(t-t_{1/2}) & t_{1/2} \leq t \leq 2t_{1/2} \end{array} \right. \;.$$ Using this approximation, one directly obtains a linear mapping $$\phi_{n+1} = 1-[1-\phi_n\hat{\varphi}(t_{1/2})] \hat{\varphi}(t_{1/2}) \;,$$ where $\phi_n$$\equiv$$\phi(2nt_{1/2})$ is the volume fraction of the positive spins at the beginning of the $n$th period, $n=0,1,2,\ldots$. The stationary value of this quantity is $$\phi^* = \lim_{n\rightarrow\infty}\phi_n = \frac{1}{1+\hat{\varphi}(t_{1/2})} \;.$$ Consequently, the magnetization reaches a stationary limit cycle $$m(t) \approx \left\{ \begin{array}{ll} \frac{2}{1+\hat{\varphi}(t_{1/2})}\hat{\varphi}(t) - 1 & 0 \leq t \leq t_{1/2} \\ 1 - \frac{2}{1+\hat{\varphi}(t_{1/2})}\hat{\varphi}(t-t_{1/2}) & t_{1/2} \leq t \leq 2t_{1/2} \end{array} \right. \;.$$ In this limit cycle the magnetization oscillates about zero, $$m(0) = - m(t_{1/2}) = \frac{1-\hat{\varphi}(t_{1/2})}{1+\hat{\varphi}(t_{1/2})} \; %\stackrel{\Theta\gg 1}{\approx} 1 \;,$$ and the symmetry of the magnetization, $m(t\pm t_{1/2})$$=$$-m(t)$ implies $$Q = \frac{1}{2 t_{1/2}} \oint m(t) dt = 0\;.$$ This corresponds to the symmetric (dynamically disordered) phase. Note that this symmetric phase is always reached when $\hat{\varphi}$ decreases monotonically from unity at $t$$=$$0$ to zero as $t$$\rightarrow$$\infty$ In the multi-droplet regime, the volume fraction of the metastable phase decays according to Avrami’s law [@Avrami; @RAMOS99]. In the low-frequency limit, $t_{1/2}$$\gg$$\langle\tau\rangle$ ($\Theta$$\gg$$1$), each half-period almost always contains a [*complete*]{} metastable decay \[Fig. \[m\_series\](a)\]. Avrami’s law for the metastable volume fraction in each half-period can then be directly applied using $\hat{\varphi}(t)$$=$$\varphi_{\rm ms}(t)$$\approx$$e^{-(\ln2)t^3/\langle\tau\rangle^3}$ [@Avrami; @RAMOS99]. This functional form for $\varphi_{\rm ms}(t)$ breaks down when $t_{1/2}$ becomes comparable to $\langle\tau\rangle$ ($\Theta$$\approx$$1$), thus this simple mean-field approximation cannot predict any instability related to the DPT. In the $H_0$$\rightarrow$$\infty$ limit the individual spins become decoupled. 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--- abstract: | During the final growth phase of giant planets, accretion is thought to be controlled by a surrounding circumplanetary disk. Current astrophysical accretion disk models rely on hydromagnetic turbulence or gravitoturbulence as the source of effective viscosity within the disk. However, the magnetically-coupled accreting region in these models is so limited that the disk may not support inflow at all radii, or at the required rate. Here, we examine the conditions needed for self-consistent accretion, in which the disk is susceptible to accretion driven by magnetic fields or gravitational instability. We model the disk as a Shakura-Sunyaev $\alpha$ disk and calculate the level of ionisation, the strength of coupling between the field and disk using Ohmic, Hall and Ambipolar diffusevities for both an MRI and vertical field, and the strength of gravitational instability. We find that the standard constant-$\alpha$ disk is only coupled to the field by thermal ionisation within $30\,R_J$ with strong magnetic diffusivity prohibiting accretion through the bulk of the midplane. In light of the failure of the constant-$\alpha$ disk to produce accretion consistent with its viscosity we drop the assumption of constant-$\alpha$ and present an alternate model in which $\alpha$ varies radially according to the level magnetic turbulence or gravitoturbulence. We find that a vertical field may drive accretion across the entire disk, whereas MRI can drive accretion out to $\sim200\,R_J$, beyond which Toomre’s $Q=1$ and gravitoturbulence dominates. The disks are relatively hot ($T\gtrsim800\,$K), and consequently massive ($M_{\text{disk}}\sim0.5\,M_J$). author: - | Sarah L. Keith $^{1,2}$[^1] and Mark Wardle$^{1}$\ $^{1}$Department of Physics & Astronomy and MQ Research Centre in Astronomy, Astrophysics & Astrophotonics, Macquarie University,\ NSW 2109, Australia\ $^{2}$Jodrell Bank Centre for Astrophysics, The University of Manchester, Alan Turing Building, Manchester, M13 9PL, United Kingdom bibliography: - 'references.bib' date: 'Accepted Year Month date Day. Received Year Month Day; in original form Year Month Day' title: Accretion in giant planet circumplanetary disks --- \[firstpage\] accretion discs – magnetic fields – MHD – planets and satellites: formation Introduction ============ Gas giant planets form within a protoplanetary disk surrounding a young star [@1985prpl.conf..981L]. Those orbiting within $\sim100\,$au of the star form through the aggregation of a $\sim15M_{\text{Earth}}$ solid core and subsequent gas capture from the surrounding disk [@1996Icar..124...62P; @2009ApJ...695L..53B]. During the initial slow accretion phase the protoplanet envelope is thermally supported and distended. However, once the envelope mass reaches the core mass gas accretion accelerates rapidly and, unable to maintain thermal equilibrium, the envelope collapses [@1996Icar..124...62P; @2009Icar..199..338L]. This ‘run-away’ gas accretion ends once the planet is massive enough that it accretes faster than gas can be replenished into its vicinity. Infalling gas has too much angular momentum to fall directly onto the contracted planet, and so an accretion disk, the circumplanetary disk, forms around the planet [@1982Icar...52...14L; @2009MNRAS.397..657A]. In contrast to the icy conditions implied by satellite systems around Solar System giant planets, circumplanetary disks are likely initially hot and convective [@1989oeps.book..723C]. Most of the protoplanet’s mass is delivered during run-away accretion and so the circumplanetary disk must support a high inflow rate during this phase. The formation of Jupiter consistent with the giant planet formation time-scale inferred from the life-time of protoplanetary disks (life-time$\sim3\times 10^6 $years; ) suggests an inflow rate of $\dot{M}\sim10^{-6}M_J/$year. Models of the accretion phase of a circumplanetary disk include self-luminous disks , Shakura-Sunyaev $\alpha$ disks (@2002AJ....124.3404C [-@2002AJ....124.3404C], [-@2006Natur.441..834C]; [@2005AA...439.1205A; @2013arXiv1306.2276T]), time-dependent disks with MRI-Gravitational instability limit cycles [@2011ApJ...740L...6M; @2012ApJ...749L..37L], and hydrodynamical simulations ([@1999ApJ...526.1001L], [@2002AA...385..647D], [@2003ApJ...599..548D], ). The evolution of the disk associated with the contraction of the proto-planetary envelope and changes in the mode of accretion from the protoplanetary disk have also been addressed [@2010AJ....140.1168W]. The angular momentum transport mechanism is key in determining the disk structure and evolution, however little work has been done to model the disk self-consistently with the accretion mechanism. The $\alpha$-model invokes a source of viscosity (typically hydromagnetic turbulence is suggested) however there is no guarantee that the resulting disk complies with the conditions required for viscosity, hydromagnetic or otherwise. An exception is the time-dependent gravo-magneto outbursting cycles modelled by @2012ApJ...749L..37L, however numerical simulations suggest disks rapidly evolve away from a gravitationally unstable state. There are a variety of candidates for the accretion mechanism, including magnetic forces, gravitational instability, thermally-driven hydrodynamical instabilities, torque from spiral waves generated by satellitesimals \[see and @TurnerPPVI (in preparation) for a review\], and stellar forcing . Magnetic fields and gravitational instability are generally considered the most promising mechanisms within the protoplanetary disk. Magnetically-driven accretion may result from hydromagnetic turbulence produced by the magnetorotational instability (MRI; @1991ApJ...376..214B [@1995ApJ...440..742H]), centrifugally driven disk winds associated with large-scale vertical fields [@1982MNRAS.199..883B; @1993ApJ...410..218W], magnetic braking [@2004ApJ...616..266M], or large-scale toroidal fields [@2000prpl.conf..589S]. MRI turbulence has been modelled extensively (e.g., [@1996ApJ...457..355G; @2004ApJ...605..321S; @2007ApJ...659..729T; @2012MNRAS.420.2419F; @2012MNRAS.422.2737W; @2013ApJ...763...99P]) and simulations of MRI transport in protoplanetary disks indicate $\alpha\sim10^{-3}$, where $\alpha$ is the Shakura-Sunyaev viscosity parameter . Gravitational instability occurs in massive disks and may cause fragmentation or gravitoturbulence [@1964ApJ...139.1217T; @2001ApJ...553..174G]. Certain conditions are required for these mechanisms to be effective. For example, magnetic processes can only act in sufficiently ionised ‘active’ regions, where the evolution of the magnetic field is coupled to the motion of the disk. If the ionisation fraction is too low, magnetic diffusivity decouples their motion (e.g. ). In protoplanetary disks, magnetic coupling is strong enough to permit MRI accretion in two regions: (i) layers above the midplane where cosmic rays, and stellar X-rays and UV photons penetrate, and (ii) close to the star where the disk is hot and thermally ionised [@1996ApJ...457..355G]. Gravitational instability requires strong self-gravity such that Toomre’s stability parameter $Q\lesssim 1$, and quasi-steady gravitoturbulent accretion further requires a cooling time-scale in excess of $\sim30$ orbital time-scales ([@2012MNRAS.427.2022M], [@2012MNRAS.421.3286P]). Existing steady-state model circumplanetary disks are not massive enough for gravitational instability, and so testing for self-consistent accretion has focussed on identifying regions which are susceptible to the MRI. @2011ApJ...743...53F determined the thickness of the magnetically-uncoupled Ohmic midplane ‘dead zone’ of an $\alpha$ disk for the ionisation by cosmic rays. They find that the dead zone extends up at least $2.5$ scale heights (for plasma $\beta=10^4$) with the presence of grains extending this region to even greater heights. These results agree with the recent paper by @2013arXiv1306.2276T which includes ionisation from X-rays, radioactive decay, turbulent mixing, thermal ionisation as well as cosmic rays and accounting for Ambipolar and Ohmic diffusion. They find that $\alpha$ disks are magnetically coupled in surface layers above $\sim3$ scale heights unless the disk is dusty and is shielded from X-rays. They also consider magnetic coupling in the Jovian analogue to the Minimum Mass Solar Nebula-the Minimum Mass Jovian Nebula (MMJN; [@2003Icar..163..198M]), finding that dust must be removed for magnetically coupled surface layers. They find that thermal ionisation in actively supplied disks may permit coupling within the inner $4\,R_J$ of the midplane, although suggest a larger thermally ionised region ($r\lesssim65\,R_J$). Either way, we conclude that current $\alpha$ models of circumplanetary disks are not necessarily susceptible to the magnetically driven accretion assumed at all radii, and that magnetically active surface layers may be too high above the midplane to carry the required accreting column. In this paper, we probe the viability of self-consistent steady-state accretion through the circumplanetary disk midplane, with accretion driven by magnetic fields and gravitoturbulence. We model the disk as Shakura-Sunyaev $\alpha$ disk and solve for the disk structure self-consistently with the opacity using the @2009ApJ...694.1045Z opacity-law (§\[sec:disk\_structure\]). In §\[sec:thermal\_ionisation\] we calculate the ionisation level produced by thermal ionisation, cosmic rays, and radioactive decay, and also consider the effectiveness of turbulent mixing , and Joule heating in resistive MRI regions [@2005ApJ...628L.155I; @2012ApJ...760...56M]. We determine the magnetic field strength needed for accretion by an MRI or large-scale vertical field (§\[sec:B\_field\]), and calculate Ohmic, Hall and Ambipolar diffusivities to determine the strength of magnetic coupling (§\[sec:magnetic\_diffusivity\]). Motivated by the failure of the standard constant-$\alpha$ disk (§\[sec:const\_alpha\_model\]) to produce magnetic coupling consistent with the assumed viscosity profile we present an alternate $\alpha$ disk (§\[sec:thermally\_ionised\_model\]) in which the level of magnetic transport (i.e., $\alpha$) varies radially consistent with the level of viscosity proceed by either magnetic forces or gravitational instability, as per the @2002ApJ...577..534S prescription for $\alpha$ for non-ideal magnetic transport. We present the results in §\[sec:results\], with a summary and discussion of findings in §\[sec:discussion\]. Disk structure {#sec:disk_structure} ============== We model a circumplanetary disk as an axisymmetric, cylindrical, radiative, thin disk surrounding a protoplanet of mass $M$, in orbit around a star of mass $M_*$, at an orbital distance $d$. The disk extends out to a radius $r=R_H/3$ around the planet, where $$\begin{aligned} R_H&=&d\left(\frac{M}{3 M_*}\right)^{\frac{1}{3}}\nonumber\\ &\approx&743\,R_J \left(\frac{d}{5.2\,\text{au}}\right) \left(\frac{M}{M_J}\right)^{\frac{1}{3}}\left(\frac{M_*}{M_\odot}\right)^{-\frac{1}{3}}\end{aligned}$$ is the Hill radius, $R_J$ is the radius of Jupiter, $M_J$ is the mass of Jupiter, and $M_\odot$ is the mass of the Sun [@1998ApJ...508..707Q; @2011MNRAS.413.1447M]. The scale height, $H$, is determined by a balance between thermal pressure, the planet’s gravity, and self-gravity of the disk. Toomre’s $Q$ quantifies the strength of self-gravity, [@1964ApJ...139.1217T] $$\begin{aligned} Q&=&\frac{c_s \Omega}{\pi G \Sigma}\nonumber\\ &\approx&5.3\times10^{3}\,\left(\frac{T}{10^3\,\text{K}}\right)^{\frac{1}{2}}\left(\frac{\Sigma}{10^2\text{g\,cm}^{-2}}\right)^{-1}\left(\frac{M}{M_J}\right)^{\frac{1}{2}}\nonumber\\ &&\times\left(\frac{r}{10^2\,R_J}\right)^{-\frac{3}{2}}, \label{eq:Q}\end{aligned}$$ with $Q\gg1$ for negligible self-gravity and $Q\ll1$ for strong self-gravity. Here, $\Sigma$ is the column density, $\Omega$ is the Keplerian angular velocity, $$\label{eq:keplerian} \Omega=\sqrt{\frac{G M}{r^3}}\approx5.9\times10^{-7}\,\text{s}^{-1}\,\left(\frac{r}{10^2\,R_J}\right)^{-\frac{3}{2}}\left(\frac{M}{M_J}\right)^{\frac{1}{2}},$$ $c_s=\sqrt{kT/m_n}\approx1.9$kms$^{-1}\sqrt{T/1000\,\text{K}}$ is the isothermal sound speed with $m_n=2.34 m_p$ the mean neutral particle mass for a H/He gas at temperature $T$, $m_p$ the proton mass, and $k$ is Boltzmann’s constant. Solving for the scale height for arbitrary $Q$ is complex \[e.g, see [@1978AcA....28...91P]\], and so we adopt the simplified equation of vertical equilibrium (c.f., [@2002ApJ...580..987K]) $$\Omega^2 H^2 + \pi G H \Sigma - c_s^2 = 0, \label{eq:Hrelation_withselfgravity}$$ with solution $$H=\frac{2Q}{1+\sqrt{1+4Q^2}}\frac{c_s}{\Omega}. \label{eq:h_self_gravity}$$ This reduces to the standard approximations $$\begin{aligned} \frac{H}{r} &=& \frac{c_s}{r\Omega}\nonumber\\ &\approx&0.45\,\left(\frac{T}{10^3\,\text{K}}\right)^{\frac{1}{2}}\left(\frac{r}{10^2\,R_J}\right)^{\frac{1}{2}}\left(\frac{M}{M_J}\right)^{-\frac{1}{2}} \label{eq:H_noselfgravity}\end{aligned}$$ for low mass disks (i.e., $M_{\text{disk}}\ll M_J$) where self-gravity is negligible, and $$\begin{aligned} \frac{H}{r}&=&\frac{c_s^2}{\pi G\Sigma r}=Q\frac{c_s}{r\Omega}\nonumber\\ &\approx&2.4\times10^{-2}\,\left(\frac{T}{10^2\,\text{K}}\right)\left(\frac{\Sigma}{10^6\,\text{g}\,\text{cm}^{-2}}\right)^{-1}\nonumber\\ &&\times\left(\frac{r}{10^2\,R_J}\right)^{-1}\end{aligned}$$ for massive, cool, self-gravitating disks. From this we estimate the vertically-averaged neutral mass density $$\begin{aligned} \rho&=&\frac{\Sigma}{2H},\nonumber\\ &\approx&6.2\times10^{-9}\,\text{g cm}^{-3}\, \left(\frac{\Sigma}{10^2\,\text{g cm}^{-2}}\right)\left(\frac{T}{10^3\,\text{K}}\right)^{-\frac{1}{2}}\nonumber\\ &&\times\left(\frac{r}{10^2\,R_J}\right)\left(\frac{M}{M_J}\right)^{-\frac{1}{2}}, \label{eq:density}\end{aligned}$$ and the associated number density, $ n=\rho/m_n\approx 2.6\times10^{15}\,\text{cm}^{-3}\,\left(\rho/10^{-8}\,\text{g\,cm}^{-3}\right)$. The thermal structure of the disk is governed by dissipation driven by the inflow. We use the standard plane-parallel stellar atmosphere model [@1990ApJ...351..632H], $$\sigma T^4=\frac{3}{8} \tau \sigma T_s^{4}, \label{eq:radiative_transfer}$$ to calculate the midplane temperature $T$ from the surface temperature $T_s$ and optical depth, $\tau$, from the midplane to the surface. Gravitational binding energy released during infall results in a surface temperature $$\begin{aligned} T_s&=& \left( \frac{3 \dot{M}\Omega^2}{8\pi \sigma} \right)^{\frac{1}{4}}\nonumber\\ &\approx&82\,\text{K}\left(\frac{\dot{M}}{10^{-6}\,M_J/\text{year}}\right)^{\frac{1}{4}}\left(\frac{M}{M_J}\right)^{\frac{1}{4}}\left(\frac{r}{10^2\,R_J}\right)^{-\frac{3}{4}}, \label{eq:surface_temp}\end{aligned}$$ where $\dot{M}$ is the inflow rate, and $\sigma$ the Stefan-Boltzmann constant. We consider a uniform, steady, inward mass flux throughout the disk. Shock heating of infalling material colliding with the disk contributes additional heating, however it is negligible compared to that of the viscous dissipation \[i.e., flux ratio: $F_\text{infall}/F_\text{viscous}<10^{-4}$; @1981Icar...48..353C\]. Similarly, irradiation from the hot young planet \[$T_J=500$K determined from pure contraction of the young planet; e.g. \] and the accretion hot spot \[$T_{\text{hotspot}}=3300$K calculated using equation (3.3) in @1977MNRAS.178..195P\] is also negligible with $F_\text{planet}/F_\text{viscous}<10^{-4}$ and $F_\text{hotspot}/F_\text{viscous}<10^{-2}$ determined using equation (21) from @2013arXiv1306.2276T. Equations (\[eq:radiative\_transfer\]) and (\[eq:surface\_temp\]) are applicable in optically-thick regions of the disk (i.e., where optical depth $\tau\gg1$). This is appropriate for the midplane, as the high column density favours a large optical depth: $$\label{eq:optical_depth} \tau=\kappa\Sigma/2\gg1.$$ @1994APJ...427..987B @2009ApJ...694.1045Z ----------------------------- -------------------- -------- -------- -- ---------------------------- --------------------- -------- -------- Opacity Regime $\kappa_i$ $a$ $b$ Opacity Regime $\kappa_i$ $a$ $b$ Ice grains $2\times 10^{-4}$ 0 2 Grains $5.3\times10^{-2}$ $0$ $0.74$ Evaporation of ice grains $2\times10^{16}$ 0 $-7$ Grain evaporation $1.0\times10^{145}$ $1.3$ $-42$ Metal grains 0.1 0 $1/2$ Water vapour $1.0\times10^{-15}$ $0$ $4.1$ Evaporation of metal grains $2 \times 10^{81}$ 1 $-24$ $1.1\times10^{64}$ $0.68$ $-18$ Molecules $10^{-8}$ $2/3 $ 3 Molecules $5.1\times10^{-11}$ $0.50$ $3.4$ H scattering $10^{-36}$ $1/3$ 10 H scattering $8.9\times10^{-39}$ $0.38$ $11$ Bound–free and free–free $1.5\times10^{20}$ 1 $-5/2$ Bound–free and free–free $1.1\times10^{19}$ $0.93$ $-2.4$ Electron scattering 0.348 0 0 Electron scattering $0.33$ $0$ $0$ Molecules and H scattering 1.4 0 3.6 This regime is given in the footnote of Table 1 in @2009ApJ...694.1045Z. The dominant sources of opacity in this regime are molecular lines and H scattering (Z. Zhu 2013, private communication). To calculate the opacity, $\kappa$, we use the analytic Rosseland mean opacity law presented in @2009ApJ...694.1045Z. This is a piecewise power-law fit to the @2007ApJ...669..483Z [@2008ApJ...684.1281Z] opacity law. We give this in Table \[table:opacity\_law\], re-expressed as a function of temperature and density, using the ideal gas law[^2]. This model features nine opacity regimes, incorporating the effects of dust grains, molecules, atoms, ions and electrons. The transition temperature $T_{j\rightarrow k}$ between regimes $j$ and $k$, as a function of density, is obtained by equating the opacity in neighbouring regimes (i.e., $\kappa_j=\kappa_{k}$), and is $$T_{j\rightarrow k}=\left(\frac{\kappa_{i,j}}{\kappa_{i,k}}\right)^{\frac{1}{b_{k}-{b_j}}}\rho^{\frac{a_j-a_{k}}{b_{k}-b_j}} \label{eq:transition_temperatures}$$ with two additional constraints: 1. use Grains opacity for $T<794\,K$, and 2. use Molecules and H scattering opacity for $2.34\times10^4\kappa^{0.279}\,K<T<10^4\,K$. As the opacity law is complex we show the temperature and density boundaries for each opacity regime in Fig. \[fig:opacity\_boundaries\]. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![\[fig:opacity\_boundaries\] Temperature and density boundaries of the @2009ApJ...694.1045Z opacity regimes, given in Table \[table:opacity\_law\], calculated with equation (\[eq:transition\_temperatures\]).](fig1.pdf "fig:"){width="48.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- For comparison, we also give the @1994APJ...427..987B opacity law in Table \[table:opacity\_law\]. This opacity law underestimates the opacity for temperatures $T\sim$1500–3000K because it neglects contributions from TiO and water lines longward of 5$\mu$m . The discrepancy is greatest at $\sim1700$K where the @1994APJ...427..987B opacity is a factor $\sim500$ too low, as compared with the @2009ApJ...694.1045Z model. We solve for the local structure (i.e., $\Sigma$ and $T$) simultaneously with the opacity, at each radius. Following @1997ApJ...486..372B, we solve for the radial temperature profile by combining equations (\[eq:Q\]), (\[eq:keplerian\]), (\[eq:h\_self\_gravity\]), (\[eq:density\])–(\[eq:optical\_depth\]) and the opacity law in Table \[table:opacity\_law\], to give $$T^{4-b}=\frac{9 \dot{M}\kappa_i}{2^{a+7}\pi \sigma} \Omega^{2} H^{-a}\Sigma^{a+1}, \label{eq:temperature_density_relation}$$ with $a$, $b$, and $\kappa_i$ specified for each opacity regime. This relationship allows us to describe the disk temperature and column density self consistently, when one or the other is specified. At a given radius, we solve this equation within each opacity regime, and determine whether the resulting temperature and density fall within the limits of that regime. Solutions which do not fall within these limits are discarded. The solution is not necessarily unique, as the disk may satisfy the conditions of multiple opacity regimes (e.g., [@1994APJ...427..987B; @2007ApJ...669..483Z]). Conservation of angular momentum provides the closing relation by specifying the accreting column needed to drive the inflow caused by turbulence, $\dot{M}=2\pi \nu\Sigma$ [^3] . A common approach to modelling the turbulent viscosity $\nu$ is to adopt the $\alpha$-viscosity prescription, in which uncertainties in the form of the viscosity are gathered into a single parameter $\alpha\lesssim1$ , $$\nu=\alpha c_s H. \label{eq:alpha_prescription}$$ Observational estimates of $\alpha$, derived from the inferred mass accretion rates of T-Tauri stars, and the time dependent behaviour of FU Orionis outbursts, dwarf nova, and X-ray transients, indicate $\alpha\sim0.001-0.1$, while numerical magnetohydrodynamical shearing box simulations yield $\alpha\sim 0.01$–$10^{-3}$ \[see @2007MNRAS.376.1740K and references therein\]. This results in an accreting column $$\begin{aligned} \Sigma&=&\frac{\dot{M}}{2\pi \alpha c_s H}\label{eq:mdot_alpha_relation}\\ &\approx&1.6\times10^2 \text{g cm}^{-2} \left(\frac{\dot{M}}{10^{-6} M_J/\text{year}}\right)\left(\frac{\alpha}{10^{-3}}\right)^{-1}\nonumber\\ &&\times\left(\frac{M}{M_J}\right)^{\frac{1}{2}}\left(\frac{T}{1000\,\text{K}}\right)^{-1}\left(\frac{r}{10^2\,R_J}\right)^{-\frac{3}{2}}\end{aligned}$$ for negligible self-gravity. Degree of ionisation {#sec:thermal_ionisation} ==================== In this section we calculate the level of ionisation at the midplane of the circumplanetary disk. The disk is too dense for the penetration of cosmic rays and X-rays down to the midplane, and so the primary sources of ionisation are thermal ionisation and decaying radionuclides. We also consider two further ionising mechanisms produced by the action of MRI turbulence - the transport of ionisation from MRI active surface layers to the midplane by eddies, and ionisation from electric fields generated by MRI turbulence. Thermal ionisation {#sec:thermal_ionisation} ------------------ Ionisation leads to the production of electrons, ions (with atomic number $j$), and charged dust grains with associated number density $n_e$, $n_{i,j}$, $n_g$, mass $m_e$, $m_{i,j}$, $m_g$, and charge $-q$, $+q$, $Z_g q$ respectively. Here, the grain mass and charge represent the mean value. From this we define the total ion number density $n_i\equiv\sum_j n_{i,j}$, and average ion mass $m_i\equiv \left(n_i^{-1}\sum_j n_{i,j} m_{i,j}^{-1/2}\right)^{-2}$, where the summation runs over each ion species. To calculate the level of thermal ionisation we use the Saha equation $$\label{eq:saha} \frac{n_e n_{i,j}}{n_{j}}= g_e \left(\frac{2\pi m_e k T}{h^3}\right)^{\frac{3}{2}} \exp\left(-\frac{\chi_j}{k T}\right),$$ where $n_{j}$ is the number density of neutrals with atomic number $j$, $\chi_j$ is the ionisation potential of the $j^{\text{th}}$ ion species, $g_e=2$ is the statistical weight of an electron, and $h$ is Planck’s constant. Table \[table:ion\_species\] gives the atomic weight and first ionisation energy of five key contributing elements: hydrogen, helium, sodium, magnesium, and potassium [@CRC]. The exponential factor in the Saha equation gives rise to switch on/off behaviour in thermal ionisation, such that the bulk of atoms are ionised in a narrow temperature band around their ionisation temperature. Potassium has the lowest ionisation energy and is first to be ionised with an ionisation temperature of $T\sim10^3$K. --------------- --------- --------------- ----------------------- --------------------- ---------------------- ----------- Atomic number Element Atomic weight Logarithmic abundance Abundance Ionisation potential Depletion (amu) (eV) (dex) 1 H 1.01 12.00 9.21$\times10^{-1}$ 13.60 0 2 He 4.00 10.93 $7.84\times10^{-2}$ 24.59 0 11 Na 22.98 6.24 1.60$\times10^{-6}$ 5.14 $\delta$ 12 Mg 24.31 7.60 $3.67\times10^{-5}$ 7.65 $\delta$ 19 K 39.10 5.03 9.87$\times10^{-8}$ 4.34 $\delta$ --------------- --------- --------------- ----------------------- --------------------- ---------------------- ----------- We use solar photospheric abundances to model the elemental composition of the disk, as given in Table \[table:ion\_species\] . However heavy elements are encorporated into grains, reducing their gas phase abundance. We allow for depletion onto grains through a depletion factor $\delta$ (c.f., [@2000ApJ...543..486S]). The degree of depletion varies greatly between elements, however we make the simplifications that the abundance of elements other than hydrogen and helium are reduced by a constant factor, $10^{\delta}$. Grain depletion in the Orion nebula has been determined by comparing the abundances in the HII region (gas only) with that of Orion O stars (gas+dust; @1998MNRAS.295..401E). Magnesium, a key grain constituent, is depleted at the level $\delta_{\text{Mg}}=-0.92$, which we adopt for all depleted elements. The abundance of the $j^{\text{th}}$ element is related to its logarithmic form, accounting for depletion onto grains: $X_j=\log_{10}(n_{j}/n_H)+12-\delta$, where the logarithmic abundance of hydrogen is defined to be $X_H=12$. The abundance is then $x_j=10^{X_j}/(\sum_i 10^{X_i})$, for which we take the logarithmic abundances of the remaining elements from . Dust grains also act to reduce the ionisation fraction by soaking up electrons, acquiring charge through the competitive sticking of electrons and ions to their surface. The net charge is found through the balance of preferential sticking of electrons due to their higher thermal velocity, with the subsequent Coulomb repulsion that develops. The average charge acquired by a dust grain is [@1987ApJ...320..803D] $$Z_g=\psi\tau-\frac{1}{1+\sqrt{\tau_0/\tau}} \label{eq:grain_charge}$$ where $$\begin{aligned} \tau=\frac{a_g k T}{q^2},\\ \tau_0 \equiv \frac{8m_e}{\pi\mu m_p},\\ \mu\equiv \left( \frac{n_e s_e}{n_i}\right)^2\left(\frac{m_i}{m_p}\right),\end{aligned}$$ where $s_e$ is the electron sticking coefficient, $a_g$ the grain radius, and $\psi$ is the solution to the transcendental equation [@1941ApJ....93..369S]: $$1-\psi=\left(\mu \frac{m_p}{m_e}\right)^{\frac{1}{2}} e^{\psi}. \label{eq:trancendental}$$ We solve this using the second order approximation [@ArmstrongKulesza] $$\psi=1-\ln(1+y)+\frac{\ln(1+y)}{1+\ln(1+y)}\ln[(1+y^{-1}) \ln(1+y)]$$ with $y\equiv e \sqrt{\mu m_p/m_e}$. Charge fluctuations are small, with most grains having charge within one unit about this mean [@1979ApJ...232..729E]. Measurements and analytical estimates of the electron sticking coefficient suggest $s_e$ is in the range $10^{-3}$–$1$ [@1980PASJ...32..405U; @2010PhRvB..82l5408H]. As an approximation, we maximise the impact of grain charge removal by adopting $s_e\sim1$. We adopt a constant gas to dust mass ratio ratio $\rho_d/\rho\equiv f_{dg}=10^{-2}$, grain size $a_g=0.1\mu$m, and grain bulk density $\rho_b=3$g cm$^{-3}$ [@1994ApJ...421..615P]. This leads to a grain number density $$\begin{aligned} n_g&=&\frac{ m_n f_{dg} n}{\frac{4}{3}\pi a_g^3 \rho_b}\nonumber\\ &\approx&3.1\times10^{3}\,\text{cm}^{-3}\left(\frac{n}{10^{15}\,\text{cm}^{-3}}\right)\left(\frac{f_{dg}}{10^{-2}}\right)\nonumber\\ &&\times\left(\frac{a_g}{0.1\,\mu\text{m}}\right)^{-3}\left(\frac{\rho_b}{3\,\text{g cm}^{-3}}\right)^{-1}.\end{aligned}$$ Grain evaporation, which removes grain species, will cause spatial variation of these properties. For instance, very few grains would be present where the temperature exceeds the vaporisation temperature of iron ($T\sim1500$K at $\rho\sim10^{-7}\,$gcm$^{-3}$; [@1994ApJ...421..615P]). However, we find that removing grains in this region (i.e., $f_{dg}=0$ for $r<7\,R_J$), or indeed uniformly across the disk (i.e., $f_{dg}=0$ for all $r$), has no effect on the boundary of the magnetically-coupled region owing to the overwhelming effectiveness of thermal ionisation here. The final condition needed to determine the ionisation level is charge neutrality, $$n_i-n_e +Z_g n_g=0 \label{eq:charge}.$$ To solve equations (\[eq:saha\])–(\[eq:charge\]), we use Powell’s Hybrid Method for root finding [@Powell1970], with the routine `fsolve` from the Python library `scipy.optimize` [@scipython]. This method is a modified form of Newton’s Method, which checks that the residual is improved before accepting a Newton step. This optimisation allows for convergence despite the steep gradients caused by the exponential factor in the Saha equation. Ionisation by decaying radionuclides, cosmic rays and X-rays ------------------------------------------------------------ Cosmic rays and the decay of radionuclides are the primary sources of ionisation in the outer disk where it is too cool for thermal ionisation. The short-lived radioisotope $^{26}$Al is the main contributor to ionisation by decaying radionuclides, yielding an ionisation rate $\zeta_R=7.6\times10^{-19}$s$^{-1}$ [@2009ApJ...690...69U]. Cosmic ray ionisation occurs at a rate $\zeta_{\text{CR}}=10^{-17}\,\text{s}^{-1}\exp(-\Sigma/\Sigma_{\text{CR}})$, where $\Sigma_{\text{CR}}=96\,\text{g}\,\text{cm}^{-2}$ is the attenuation depth of cosmic rays. X-rays from the young star will also ionise the surface layers \[with $\zeta_{\text{XR}}=9.6\times10^{-17}\,\text{s}^{-1}\exp(-\Sigma/\Sigma_{\text{XR}})$ at the orbital radius of Jupiter for a star with Solar luminosity [@1999ApJ...518..848I; @2008ApJ...679L.131T]\], however the X-ray attenuation depth is so small ($\Sigma_{\text{XR}}=8\,\text{g}\,\text{cm}^{-2}$) that X-rays do not reach the midplane and do not contribute to midplane ionisation or accretion \[in contrast with *surface* ionisation calculations by @2013arXiv1306.2276T\]. Calculating the ionisation resulting from radioactive decay involves solving the coupled set of reaction rate equations for electrons, metal ions (number density $n_i$ with metal abundance $x_m$), and grains subject to charge neutrality. Molecular ions are the first ions produced as part of the reaction scheme, however, charge transfer to metals is so rapid that metal ions are more abundant [@2011ApJ...743...53F]. We model the metals as a single species, adopting the mass, $m_i$, and abundance, $x_i$, of the most abundant metal - magnesium . Free electrons and ions are formed through ionisation, and are removed through recombination (rate coefficient $k_{ei}$) and capture by grains (rate coefficients $k_{eg}$, $k_{ig}$ for electrons and ions, respectively). These processes are described by the following rate equations: $$\begin{aligned} \frac{dn_i}{dt} &=& \zeta n - k_{ei}n_{i}n_{e} -k_{ig}n_{g} n_{i}, \label{eq:ion_radioactive_decay}\\ \frac{dn_{e}}{dt} &=& \zeta n - k_{ei}n_{i}n_{e} - k_{e g }n_{g} n_{e}, \label{eq:electron_radioactive_decay}\\ \frac{d Z_g}{dt}&=& k_{ig} n_{i} -k_{e g} n_{e}, \label{eq:grain_radioactive_decay}\\ 0&=&n_i-n_e+Z_g n_g, \label{eq:neutrality_radioactive_decay}\end{aligned}$$ for which we have neglected grain charge fluctuations (see for example, @1980PASJ...32..405U [@2011ApJ...743...53F]). Anticipating that the resulting ionisation fraction will be low, we make the following simplifications: (i) the average grain charge will be low and so we approximate $Z_g\approx0$ in calculating the rate coefficients $k_{ig}, k_{eg}$ and (ii) recombination is inefficient such that charge capture by grains dominates and we set $k_{ei}=0$. The charge capture rate co-efficients for neutral grains are $$\begin{aligned} k_{ig}&=&\pi a_g^2 \sqrt{\frac{8k_b T}{\pi m_i}}\nonumber\\ &\approx & 3.0\times10^{-5}\,\text{cm}^{3}\,\text{s}^{-1}\,\left(\frac{T}{10^3\text{\,K}}\right)^{\frac{1}{2}}\left(\frac{a_g}{0.1\mu\,\text{m}}\right)^2\nonumber\\ &&\times\left(\frac{m_i}{24.3\,m_p}\right)^{-\frac{1}{2}},\\ \label{eq:kmg} k_{eg}&=&\pi a_g^2 \sqrt{\frac{8k_b T}{\pi m_e}}\nonumber\\ &\approx& 6.2\times10^{-3}\,\text{cm}^{3}\,\text{s}^{-1}\,\left(\frac{T}{10^3\text{\,K}}\right)^{\frac{1}{2}}\left(\frac{a_g}{0.1\mu\,\text{m}}\right)^2. \label{eq:keg}\end{aligned}$$ Under these conditions the equilibrium electron and ion number density fractions are $$\begin{aligned} \frac{n_e}{n}&=&\frac{\zeta }{k_{eg} n_{g}},\nonumber\\ &\approx& 5.2\times10^{-20}\,\left(\frac{T}{10^3\text{\,K}}\right)^{-\frac{1}{2}}\left(\frac{n}{10^{15}\,\text{cm}^{-3}}\right)^{-1}\nonumber\\ &&\times\left(\frac{\zeta}{10^{-18}\,\text{s}^{-1}}\right)\left(\frac{\rho_b}{3\,\text{g\,cm}^{-3}}\right)\left(\frac{f_{dg}}{10^{-2}}\right)^{-1}\nonumber\\ &&\times\left(\frac{a_g}{0.1\mu\,\text{m}}\right),\\\label{eq:ne_fraction} \frac{n_i}{n}&=&\frac{k_{eg}}{k_{ig}}\frac{n_e}{n_n},\nonumber\\ &\approx&1.1\times10^{-17}\,\left(\frac{T}{10^3\text{\,K}}\right)^{-\frac{1}{2}}\left(\frac{n}{10^{15}\,\text{cm}^{-3}}\right)^{-1}\nonumber\\ &&\times\left(\frac{\zeta}{10^{-18}\,\text{s}^{-1}}\right)\left(\frac{\rho_b}{3\,\text{g\,cm}^{-3}}\right)\left(\frac{f_{dg}}{10^{-2}}\right)^{-1}\nonumber\\ &&\times\left(\frac{a_g}{0.1\mu\,\text{m}}\right)\left(\frac{m_i}{24.3\,m_p}\right)^{\frac{1}{2}}.\label{eq:ni_fraction}\end{aligned}$$ We insert these values into equation (\[eq:neutrality\_radioactive\_decay\]) to calculate an improved estimate of the grain charge: $$\begin{aligned} Z_g&=&-\frac{n_i}{n_g}\nonumber\\ &\approx&-3.5\times 10^{-6} \,\left(\frac{T}{10^3\text{\,K}}\right)^{-\frac{1}{2}}\left(\frac{n}{10^{15}\,\text{cm}^{-3}}\right)^{-1}\nonumber\\ &&\times\left(\frac{\zeta}{10^{-18}\,\text{s}^{-1}}\right)\left(\frac{\rho_b}{3\,\text{g\,cm}^{-3}}\right)^2\left(\frac{f_{dg}}{10^{-2}}\right)^{-2}\nonumber\\ &&\times\left(\frac{a_g}{0.1\mu\,\text{m}}\right)^4\left(\frac{m_i}{24.3\,m_p}\right)^{\frac{1}{2}}.\label{eq:grain_charge_rd}\end{aligned}$$ Charge capture by grains has removed a large fraction of the free electrons and so the average grain charge is small (validating our initial estimate, $Z_g\approx0$), and simply traces the ion density: To calculate the charge resulting from the combined efforts of thermal ionisation, decay of radionuclides, and external ionisation sources we add the contributions linearly. A complete treatment would address the non-linear effects associated with using the combined charge particle population, rather than treating the populations as independent. However, as the drop-off of the radial thermal ionisation profile is so steep, the contribution of decaying radionuclides and cosmic rays within $r\lesssim55\,R_J$ is insignificant when compared to thermal ionisation. Similarly, thermal ionisation is highly inefficient beyond this distance, and so charge production is by radioactive decay and cosmic rays. Ionisation from MRI turbulence ------------------------------ The action of MRI turbulence in the disk offer two further ionising mechanisms, which we describe below. We do not calculate the level of ionisation produced by these mechanisms, but rather determine their effectiveness within the circumplanetary disk. Eddies within MRI active surface layers caused by cosmic ray ionisation may penetrate into the underlying dead zone, transporting ionised material with them . Turbulent mixing may deliver enough ionisation into the dead zone for magnetic coupling and reactivation of the dead zone [@2007ApJ...659..729T]. The vertical mixing time-scale for diffusion through a scale height is $$\tau_D=\frac{H^2}{\nu}=(\alpha\Omega)^{-1},$$ which is $1000$ dynamical times for Shakura-Sunyaev viscosity parameter $\alpha=10^{-3}$. However, free charges are removed through recombination and grain charge capture which lowers the ionisation fraction. From equation (\[eq:electron\_radioactive\_decay\]), we find that charges are removed on a time-scale $$\tau_R=\left( k_{ei}\overline{n_i}+k_{eg}\overline{n_g}\right )^{-1},$$ where the ion and grain number densities are vertically averaged along the path. We calculate the grain charge capture rate $k_{eg}$ for neutral grains, and the ion number density using the height averaged cosmic ray and constant radioactive decay ionisation rates assuming that ion capture by grains is small. We use a vertically uniform temperature, however we find no qualitative difference in the results using midplane or surface temperatures. For turbulent mixing to be effective in delivering ionisation to the midplane, it must be at least as rapid as charge removal (i.e., $\tau_D\gtrsim\tau_R$). Thus, we determine the effectiveness of midplane ionisation from active surface layers by comparing the charge removal and vertical mixing time-scales in §\[sec:results\]. Ionisation is also produced through currents generated by the action of the MRI turbulent field [@2005ApJ...628L.155I]. The electric field, $E$, associated with the MRI may be able to accelerate electrons to high enough energies that they are able to ionise hydrogen in some regions. Such MRI ‘sustained’ regions occur within the minimum mass solar nebula, reducing the vertical extent of the dead zone away from the midplane [@2012ApJ...760...56M]. Here we determine if self-sustained MRI occurs in circumplanetary disks. Joule heating is the primary mechanism for converting work done by shear \[work per unit volume $W_{\text{S}}=(3/2)\alpha\Omega p$\] into the electron kinetic energy. The work dissipated per unit volume by Joule heating of an equipartition current \[i.e., the current $J_{\text{eq}}=cB_{\text{eq}}/(4 \pi H)$ associated with an equipartition field over a length scale $H$\], is $W_J=f_{\text{fill}} f_{\text{sat}} J_{\text{eq}} E$. Here $c$ is the speed of light, $f_{\text{fill}}$ is the filling factor representing the fraction of the total volume contributing to Joule heating, and $f_{\text{sat}}$ is the ratio of the saturation current in MRI unstable regions to the equipartition current. @2012ApJ...760...56M performed three dimensional shearing box simulations to determine the time, space, and ensemble averaged filling factor and MRI saturation current, finding $f_{\text{fill}}=0.264$ and $f_{\text{sat}}=13.1$. The total energy available for ionisation through Joule heating is limited to the work done by shear (i.e., $W_J \le W_S$), and so the electric field strength cannot exceed [@2012ApJ...760...56M][^4] $$E=\frac{3\alpha c_s B_{\text{eq}}}{4 c f_{\text{fill}} f_{\text{sat}}} \left(\frac{2Q}{1+\sqrt{1+4Q^2}}\right). \label{eq:selfsustain}$$ Given this restriction, we calculate the maximum electron kinetic energy, $\epsilon$, available from Joule heating [@2005ApJ...628L.155I], $$\epsilon=0.43 q E l \sqrt{m_n/m_e}$$ where $l=1/(n \langle\sigma_{en}\rangle)\approx1\,\text{cm}\,(10^{15}\,\text{cm}^{-3}/n)$ is the electron mean free path, and $\langle\sigma_{en}\rangle=10^{15}\,\text{cm}^2$ is the momentum transfer rate co-efficient between elections and neutrals. For ionisation to be effective, the electron energy, $\epsilon$ must exceed the ionisation threshold of neutral particles within the disk. Magnetic field strength {#sec:B_field} ======================= Further to a possible proto-planetary dynamo field (e.g., Jupiter’s present day surface field is $4.2$G; ), the disk may accrete its own field from the protoplanetary disk [@1998ApJ...508..707Q; @2013arXiv1306.2276T]. As both MRI and vertical fields have been modelled extensively in protoplanetary disks, we consider both field geometries in driving accretion in circumplanetary disks. We calculate the magnetic field strength, $B$, required to drive accretion at the inferred accretion rate, $\dot{M}=10^{-6} M_J/\text{year}$. Three dimensional stratified and unstratified shearing box, and global MRI simulations with a net vertical flux indicate that during accretion the MRI magnetic field saturates with [@1995ApJ...440..742H; @2004ApJ...605..321S; @2011ApJ...730...94S; @2013ApJ...763...99P] $$\alpha\approx0.5\beta^{-1}=0.5\frac{B^2}{8\pi p}, \label{eq:alpha_B_relation}$$ where $\beta\equiv 8\pi p/B^2$ is the plasma beta, and $p=c_s^2\rho$ is the pressure. This leads to a magnetic field strength $$B_{\text{MRI}}= \sqrt{16\pi \alpha c_s^2\rho}, \label{eq:alpha_magnetic_field}$$ which can be directly determined by the inflow rate as $$\begin{aligned} B_{\text{MRI}}&=&\left(\frac{\dot{M}\Omega^2}{c_s}\right)^{\frac{1}{2}}\left(\frac{1+\sqrt{1+4Q^2}}{Q}\right)\nonumber\\ &\approx&0.66\text{\,G}\left(\frac{\dot{M}}{10^{-6} M_J/\text{year}}\right)^{\frac{1}{2}}\left(\frac{M}{M_J}\right)^{1/2}\left(\frac{T}{10^3\,\text{K}}\right)^{-\frac{1}{4}}\nonumber\\ &&\times\left(\frac{r}{10^2\,R_J}\right)^{-\frac{3}{2}}\left(\frac{Q^{-1}+\sqrt{Q^{-2}+4}}{2}\right). \label{eq:BMRI}\end{aligned}$$ The equipartition field, $B_{\text{eq}}=\sqrt{8\pi p}$, defines the maximum field that the disk can support before magnetic pressure dominates over thermal pressure. From equation (\[eq:alpha\_B\_relation\]) we see that the MRI field is sub-equipartition, satisfying $$\frac{B_{\text{MRI}}}{B_{\text{eq}}}= \frac{v_a}{\sqrt{2}c_s}=\sqrt{2\alpha} \label{eq:equipartition_ratio}$$ which is constant for a given $\alpha$, and where the Alfvén speed is $$\begin{aligned} v_a&=&\frac{B}{\sqrt{4\pi \rho}},\nonumber\\ &\approx&8.9\times10^{-2}\,\text{km\,s}^{-1}\,\left(\frac{B}{1\text{\,G}}\right)\left(\frac{\rho}{10^{-9}\text{g\,cm}^{-3}}\right)^{-\frac{1}{2}}. \end{aligned}$$ Large-scale fields acting through disk winds and jets may also drive angular momentum transport and have been studied in the context of protoplanetary disks (e.g., @1993ApJ...410..218W [@1994ApJ...429..781S; @2013arXiv1301.0318B]). Magnetically-driven outflows have also been proposed for circumplanetary disks . If a vertical field drives the inflow the field strength must be at least $$\begin{aligned} B_{\text{V}}&=&\sqrt{\frac{\dot{M} \Omega}{2r}},\nonumber\\ &\approx&0.16\text{\,G}\left(\frac{\dot{M}}{10^{-6} M_J/\text{year}}\right)^{\frac{1}{2}}\left(\frac{M}{M_J}\right)^{\frac{1}{4}}\left(\frac{r}{10^2\,R_J}\right)^{-5/4}.\end{aligned}$$ Magnetic coupling {#sec:magnetic_diffusivity} ================= We are now in a position to calculate the level of magnetic diffusivity within the disk to identify which regions of the disk are subject to magnetically-driven transport. A minimum level of interaction between the disk and the magnetic field is needed for magnetically-controlled accretion. Collisions disrupt the gyromotion of charged species around the magnetic field. Collisions between the electrons, ions, and neutrals occur at a rate $\nu_{ij}$ (for colliding species $i$ with $j$), with [@2008MNRAS.385.2269P] $$\begin{aligned} \nu_{\text{ei}}&=&1.6\times10^{-2}\,\text{s}^{-1}\,\left(\frac{T}{10^3\,\text{K}}\right)^{-\frac{3}{2}}\left(\frac{n_e}{10\,\text{cm}^{-3}}\right)\nonumber \\ &&\times\left(\frac{n_n}{10^{15}\,\text{cm}^{-3}}\right),\\ \nu_{\text{en}}&=&6.7\times10^{6}\,\text{s}^{-1}\,\left(\frac{T}{10^3\,\text{K}}\right)^{-\frac{1}{2}}\left(\frac{\rho_n}{10^{-9}\,\text{g\,cm}^{-3}}\right),\nonumber\\\\ \nu_{\text{in}}&=&3.4\times10^{5}\,\text{s}^{-1}\,\left(\frac{\rho_n}{10^{-9}\,\text{g\,cm}^{-3}}\right),\end{aligned}$$ where $\rho_n=\rho-(\rho_i+\rho_e)$, and $n_n=\rho_n/m_n$ are the mass and number density of neutral particles, respectively. Electron–ions collisions are the dominant source of drag in the highly ionised inner region, however neutral drag dominates across the remainder of the disk. The Hall parameter for a species $j$, $\beta_j$, quantifies the relative strength of magnetic forces and neutral drag. It is the ratio of the gyrofrequency to the neutral collision frequency , $$\beta_j=\frac{|Z_j|eB}{m_j c} \frac{1}{\nu_{jn}}.$$ The Hall parameter is large, $\beta_j\gg1$, when magnetic forces dominate the equation of motion, and small, $\beta_j\ll1$, when neutral drag decouples the motion from the field. The Hall parameters for ions, electrons, and grains are $$\begin{aligned} \beta_i&\approx&4.6\times10^{-3}\left(\frac{B}{1\,G}\right)\left(\frac{n}{10^{15}\,\text{cm}^{-3}}\right)^{-1},\\ \beta_e&\approx&1.1 \left(\frac{B}{1\,G}\right)\left(\frac{n}{10^{15}\,\text{cm}^{-3}}\right)^{-1}\left(\frac{T}{10^3\,K}\right)^{-\frac{1}{2}}, \\ \beta_g&\approx&3.1\times10^{-3}\,Z_g\left(\frac{B}{1\,\text{G}}\right)\left(\frac{n}{10^{15}\,\text{cm}^{-3}}\right)^{-1}\left(\frac{T}{10^3\,K}\right)^{-\frac{1}{2}}\nonumber\\ &&\times\left(\frac{a_g}{0.1\,\mu\text{m}}\right)^{\frac{1}{2}}\left(\frac{\rho_b}{3\,\text{g\,cm}^{-3}}\right)^{\frac{1}{2}}.\end{aligned}$$ Ions and grains, being the more massive particles, have a lower gyrofrequency, and hence a lower Hall parameter. Thus, neutral collision are more effective at decoupling ions and grains than electrons. This leads to three regimes, according to the neutral density: (a) Ohmic regime, high density: electron–ion or neutral collisions are so frequent as to decouple both electrons and ions (i.e., $\beta_i\ll\beta_e\ll1$). (b) Hall regime, intermediate density: neutral collisions decouple ions, but the electrons remain tied to the field (i.e., $\beta_i \ll 1 \ll \beta_e$). (c) Ambipolar regime, low density: both the ions and electrons are coupled to the magnetic field, and drift through the neutrals. (i.e., $1\ll\beta_i\ll\beta_e$). In each regime collisions produce magnetic diffusivity which affects the evolution of the magnetic field through the induction equation: $$\begin{aligned} \frac{\partial \bold{B}}{\partial t}&=&\nabla(\bold{v}\times \bold{B})-\nabla\times[\eta_O(\nabla\times \bold{B})+\eta_H(\nabla\times \bold{B})\times \hat{\bold{B}}]\nonumber\\ &&-\nabla\times[\eta_A(\nabla\times \bold{B})_{\perp}], \label{eq:induction}\end{aligned}$$ where $\bold{v}$ is the fluid velocity. The Ohmic ($\eta_O$), Hall ($\eta_H$), and Ambipolar diffusivities ($\eta_A$) are \[[@2008MNRAS.385.2269P], Wardle & Pandey (in preparation)\] $$\begin{aligned} \label{eq:ohmic} \eta_O&=&\frac{m_e c^2}{4\pi e^2n_e} (\nu_{\text{en}}+\nu_{\text{ei}})\nonumber\\ &\approx& 1.9\times10^{17}\text{cm}^{2}\, \text{s}^{-1}\,\left[\left(\frac{T}{10^3\,\text{K}}\right)^{\frac{1}{2}}\left(\frac{n_e}{10\,\text{cm}^{-3}}\right)^{-1} \right .\nonumber\\ &&\left .\times \left(\frac{10^{-9}\,\text{g\,cm}^{-3}}{\rho}\right)+ 2.4\times10^{-9}\,\left(\frac{T}{10^3\,\text{K}}\right)^{-\frac{3}{2}} \right ]\text{,}\end{aligned}$$ $$\begin{aligned} \eta_H&=&\frac{cB}{4\pi e n_e}\left(\frac{1+\beta_g^2-\beta_i^2P}{1+\left(\beta_g+\beta_i P\right)^2}\right)\nonumber\\ &\approx&5.0\times10^{17}\,\text{cm}^{2}\,\text{s}^{-1}\,\left(\frac{B}{1\,\text{G}}\right)\left(\frac{n_e}{10\,\text{cm}^{-3}}\right)^{-1}\nonumber\\ &&\times\left(\frac{1+\beta_g^2-\beta_i^2P}{1+\left(\beta_g+\beta_i P\right)^2}\right), \label{eq:hall}\end{aligned}$$ $$\begin{aligned} \eta_A&=&\left(\frac{B^2}{4\pi\rho_i\nu_\text{ni}}\right)\left(\frac{\rho_n}{\rho}\right)^2\left(\frac{1+\beta_g^2+\left(1+\beta_i\beta_g\right)P}{1+\left(\beta_g+\beta_iP\right)^2}\right)\nonumber\\ &\approx&6.0\times10^{16}\,\text{cm}^{2}\,\text{s}^{-1}\left(\frac{B}{1\text{\,G}}\right)^{2}\left(\frac{\rho_n}{\rho}\right)^2\left(\frac{n_i}{10\,\text{cm}^{-3}}\right)^{-1}\nonumber\\ &&\times\left(\frac{\rho}{10^{-9}\,\text{g\,cm}^{-3}}\right)^{-1}\left(\frac{1+\beta_g^2+\left(1+\beta_i\beta_g\right)P}{1+\left(\beta_g+\beta_iP\right)^2}\right), \label{eq:ambipolar}\end{aligned}$$ where $P=n_g\,\vline Z_g\vline/n_e$ is the Havnes parameter. The magnetic field couples to the motion of the disk in regions of low magnetic diffusivity \[i.e., where $|\nabla\times(\bold{v}\times \bold{B})| \gg |\nabla\times[\eta (\nabla\times\bold{B})]|$, for each diffusivity, $\eta$\]. For MRI fields we require that the turbulent magnetic field grows faster than dissipation can destroy it such that [@2002ApJ...577..534S; @2013ApJ...764...65M] $$\begin{aligned} \eta &<& v_{a,z}^2/\Omega\nonumber\\ &\approx&1.3\times10^{14}\,\text{cm}^{2}\,\text{s}^{-1}\,\left(\frac{B}{1\,\text{G}}\right)^2\left(\frac{\rho}{10^{-9}\text{\,g\,cm}^{-3}}\right)^{-1}\nonumber\\ &&\times\left(\frac{r}{10^2\,R_J}\right)^{\frac{3}{2}}\left(\frac{M}{M_J}\right)^{-\frac{1}{2}} \label{eq:MRI_criterion}\end{aligned}$$ for each diffusivity $\eta=\eta_O$, $\eta_H$, and $\eta_A$. This condition is equivalent to the condition $\Lambda>1$, where $\Lambda=v_{a,z}^2/(\eta \Omega)$ is the Elsasser number. The coupling condition uses the Alfvén speed for the vertical component of the magnetic field. We calculate the vertical field component as $B_z\sim B_{\text{MRI}}/\sqrt{28}$, using results from @2004ApJ...605..321S. If, instead, a vertical (rather than turbulent) field is responsible for angular momentum transport (e.g., through the action of a disk wind or jet), the condition is more relaxed as we only require that the magnetic field couples to the shear, with $$\begin{aligned} \eta &<& c_s^2/\Omega\nonumber\\ &\approx &6.1\times10^{16}\,\text{cm}^{2}\,\text{s}^{-1}\,\left(\frac{T}{10^3\,\text{K}}\right)\left(\frac{r}{10^2\,R_J}\right)^{\frac{3}{2}}\left(\frac{M}{M_J}\right)^{-\frac{1}{2}} \label{eq:LS_criterion}\end{aligned}$$ for each diffusivity. Magnetic interaction still occurs for diffusivity at, or above the coupling threshold, however coupling is weak in these conditions and the connection between the dynamics of the disk and field is diminished. Disk models {#sec:disk_models} =========== We consider four circumplanetary disk models in this paper. We present two Shakura-Sunyaev $\alpha$ disks developed for this work: (i) a constant-$\alpha$ model in which the viscosity parameter is radially uniform (§\[sec:const\_alpha\_model\]), and (ii) a self-consistent accretion model in which the level of angular momentum transport is consistent with the strength of magnetic coupling or gravitational instability at all radii (§\[sec:thermally\_ionised\_model\]). For comparison we also describe two key circumplanetary disk models in the literature: (iii) the Minimum Mass Jovian Nebula (§\[sec:MMJN\]), and (iv) the Canup & Ward $\alpha$ disk (§\[sec:canup\_ward\]). Constant-$\alpha$ model {#sec:const_alpha_model} ----------------------- Here we take the traditional approach, adopting the $\alpha$-viscosity prescription with a radially-uniform $\alpha$. This allows for direct comparison with existing steady state circumplanetary disk models which adopt a constant $\alpha$. We take $\alpha=10^{-3}$ in keeping with the results of simulations (with net zero magnetic flux). However, the disk may accrete a net field which enhances transport, and so we also consider $\alpha=10^{-2}$. To obtain the radial temperature profile for this model we insert equations (\[eq:H\_noselfgravity\]) and (\[eq:mdot\_alpha\_relation\]) into equation (\[eq:temperature\_density\_relation\]), yielding [@1997ApJ...486..372B] $$T^{\frac{3}{2}a-b+5}=\frac{9\kappa_i}{2^{2a+8}\sigma } \left(\frac{\mu m_p}{k}\right)^{\frac{3}{2}a+1} \alpha^{-(a+1)}\left(\frac{\dot{M}}{\pi}\right)^{a+2} \left(\frac{GM}{r^3}\right)^{a+\frac{3}{2}}. \label{eq:const_alpha_temp_density_relationship}$$ We calculate all other properties, such as column density, by inserting this temperature profile into the relations given in §\[sec:disk\_structure\]. Self-consistent accretion model {#sec:thermally_ionised_model} ------------------------------- The constant-$\alpha$ model implicitly assumes that the angular momentum transport mechanism operates at all radii, and to the right degree. Ionisation by cosmic rays and decaying radionuclides is insufficient to couple the disk and magnetic field [@2011ApJ...743...53F], and thermal ionisation is only active in the inner disk where $T\gtrsim10^3\,$K. Without gravitoturbulence from gravitational instability, or magnetically driven transport, which relies on magnetic coupling, little if any viscosity is produced throughout the bulk of the disk (i.e., $\alpha\approx0$). Thus, equation (\[eq:mdot\_alpha\_relation\]) is invalid across the majority of the disk. Motivated by the inconsistency of the constant-$\alpha$ disk, we present an enhanced steady-state $\alpha$ disk in which the level of angular momentum transport (i.e., $\alpha$) driven by magnetic fields or gravitoturbulence is consistent with the level of magnetic coupling and strength of gravitational instability at all radii. To achieve this we divide the disk into three regions according to the mode of transport: 1. Saturated magnetic transport - the inner disk is hot enough for significant thermal ionisation allowing for strong magnetic coupling (i.e., $\eta_O, \eta_H, \eta_A$ are well below than the coupling threshold) and Toomre’s $Q\gg1$. Magnetically-driven angular momentum transport is maximally efficient and $\alpha$ saturates at its maximum value, which we take as $\alpha_{\text{sat}}=10^{-3}$. In this region the disk is identical to the constant-$\alpha$ disk. 2. Marginally coupled magnetic transport - in the majority of the disk, magnetic diffusivity exceeds the coupling threshold while self-gravity is still too weak for gravitoturbulence (i.e., Toomre’s $Q>1$). In this intermediate region accretion is magnetically driven, although at a reduced efficiency. @2002ApJ...577..534S determined the saturation level for MRI turbulence, and hence $\alpha$, for Ohmic and Ohmic+Hall MHD simulations in the non-linear regime (i.e., $\eta\Omega/v_{a,z}^2<1$; see their Fig. 20). They find that $\alpha$ is proportional to the ratio of the coupling threshold, $v_{a,z}^2/\Omega$, to Ohmic diffusivity. By extension we also assume that the effective $\alpha$ for non-turbulent accretion (i.e., for a vertical field) also adjusts according to the level of resistivity, using the analogous coupling threshold, $c_s^2/\Omega$. Thus, in this regime for the two modes of magnetic transport, we take $\alpha$ to be $$\label{eq:alphaSS02} \alpha = \left\{ \begin{array}{lr} \alpha_{\text{sat}} v_{a,z}^2/\left(\eta_O\Omega\right) & \text{for an MRI field,}\\ \alpha_{\text{sat}} c_s^2/\left(\eta_O\Omega\right) & \text{for a vertical field,} \end{array} \right.$$ which is at most $\alpha_{\text{sat}}$ [@2002ApJ...577..534S]. 3. Gravoturbulent transport - in the outer disk magnetic coupling at the level required by equation (\[eq:alphaSS02\]) would result in a gravitationally unstable disk with Toomre’s $Q<1$, and so self-gravitational forces dominate. The cooling timescale determines whether the disk fragments or enters a gravoturbulent state. We find that the cooling time-scale is much longer than the dynamical time-scale, $\Omega^{-1}$, [@2007ApJ...662..642R; @2009ApJ...695L..53B] with $$\begin{aligned} \Omega t_{\text{cool}}&=& \frac{\Sigma c_s^2 \Omega}{\sigma T_s^4}\nonumber\\ &=&\frac{8 c_s^3}{3G \dot {M}Q}\nonumber\\ &\sim& 1.9\times10^5\, \left(\frac{T}{120\,\text{K}}\right)^{\frac{3}{2}}\left(\frac{\dot{M}}{10^{-6}\,M_J\text{/year}}\right)^{-1}Q^{-1},\label{eq:t_cool}\end{aligned}$$ \[using equations (\[eq:Q\]) and (\[eq:surface\_temp\]), for a minimum midplane temperature $T=120\,$K set by the temperature of the Solar Nebula at the present day orbital radius of Jupiter according to the Minimum Mass Solar Nebula [@1981PThPS..70...35H]\] and so gravitoturbulence rather than fragmentation occurs [@2012MNRAS.427.2022M]. Either by the slow build up of surface density from inflow onto the disk coupled with heating by dissipation of turbulence [@2001ApJ...553..174G] or by time dependent evolution of gravitationally-unstable disks [@2011MNRAS.410..994F; @2013ApJ...767...63S], the disk likely evolves towards a state with $Q\sim1$. Thus, in this region we take $Q=1$. We solve for the disk profile by inserting equation (\[eq:h\_self\_gravity\]), the scale height with self-gravity, into equation (\[eq:temperature\_density\_relation\]) requiring one final relation to close the set of equations. Each region has its own closing equation to account for the differences in the mode of transport : 1. In the saturated magnetic transport region, we use equation (\[eq:mdot\_alpha\_relation\]) with constant $\alpha=\alpha_{\text{sat}}$, inverted to give the surface density as a function of temperature. 2. In the marginally coupled magnetic transport region we solve for the midplane temperature numerically using `fsolve` from the Python library `scipy.optimize` [@scipython]. The solution is determined so that $\alpha$ calculated by inverting equation (\[eq:mdot\_alpha\_relation\]) is consistent with that from equation (\[eq:alphaSS02\]). To achieve this, at each iteration of the temperature solver we calculate the surface density, scale height and Q through equations (\[eq:Q\]), (\[eq:h\_self\_gravity\]) and (\[eq:temperature\_density\_relation\]) numerically using `fsolve`. These allow us to determine $\alpha$ from equation (\[eq:mdot\_alpha\_relation\]), and to also calculate the resulting ionisation fraction, magnetic field, and diffusivity (according to §\[sec:thermal\_ionisation\], §\[sec:B\_field\], and §\[sec:magnetic\_diffusivity\] respectively) for determining $\alpha$ from equation (\[eq:alphaSS02\]). Necessarily, $\alpha$ varies radially \[i.e., $\alpha\rightarrow\alpha(r)$\]. 3. In the Gravoturbulent region, we set $Q=1$ and invert equation (\[eq:Q\]) to give the surface density as a function of temperature. We post-calculate $\alpha(r)$ using equation (\[eq:mdot\_alpha\_relation\]). We solve the complete set of equations using the routine `fsolve` from the Python library `scipy.optimize` [@scipython]. Minimum Mass Jovian Nebula {#sec:MMJN} -------------------------- The Minimum Mass Jovian Nebula (MMJN) is an adaptation of the Minimum Mass Solar Nebula used for modelling the Solar nebula [@1977ApSS..51..153W; @1981PThPS..70...35H]. The MMJN is produced by smearing out the solid mass of the satellites to form a disk, and augmenting it with enough gas to bring the composition up to solar (i.e., $f_{dg}\sim10^{-2}$). We use the surface density for the MMJN given in @2003Icar..163..198M which follows a $\Sigma\propto r^{-1}$ profile, except in a transition region ($20R_J<r<26R_J)$ where the profile steepens, $$\Sigma=\left\{ \begin{array}{lr} 5.1\times10^5 \text{\,g\,cm}^{-2}\left(\frac{r}{14\,R_J}\right)^{-1} &r<20\,R_J,\\ 3.6\times10^{5}\text{\,g\,cm}^{-2}\left(\frac{r}{20\,R_J}\right)^{-13.5} & 20\,R_J<r<26\,R_J,\\ 3.1\times10^3 \text{\,g\,cm}^{-2}\left(\frac{r}{87\,R_J}\right)^{-1} &26\,R_J<r<150\,R_J. \end{array} \right.$$ We use the opacity ($\kappa=10^{-4}\,\text{cm}^{2}\,\text{g}^{-1}$; appropriate for absorption by hydrogen molecules) and temperature profile given by @1982Icar...52...14L, $$T = \left(240\,\text{K}\left(\frac{r}{15\,R_J}\right)^{-1}+(130\,\text{K})^{4}\right)^{1/4}.$$ The temperature profile follows $T\propto r^{-1}$ in the optically-thick inner regions, and is matched to the temperature of the ambient nebula ($T_{\text{neb}}=130\,$K) at the outer edge of the disk. Canup & Ward $\alpha$ disk {#sec:canup_ward} -------------------------- Canup & Ward (2002, 2006) model the circumplanetary disk as a steady-state, thin, axisymmetric, constant–$\alpha$ disk. They adopt the @1974MNRAS.168..603L surface density model, and use the plane-parallel stellar atmosphere model to calculate the midplane temperature. Heating sources are viscous dissipation, the ambient stellar nebula ($T_{\text{neb}}=150\,$K), and the hot young planet. The midplane temperature and density are solved self-consistently for a uniform opacity, however a range of opacities ($\kappa=10^{-4}$–1 cm$^{2}$ g$^{-1}$) are considered to account for uncertainty in the population of sub-micron grains. A range of inflow rates ($\dot{M}=10^{-8}$–$10^{-4} M_J/$year), and viscosity parameters ($\alpha=10^{-4}$–$10^{-2}$), are considered to model the disk at both early and late times. However, a low inflow rate ($\dot{M}= 2\times10^{-7}M_J/$year) is needed to match the ice line with the present day location of Ganymede and to ensure solid accretion is slow enough to account for Callisto’s partially differentiation. This indicates that the disk must be ‘gas-starved’ as compared with the MMJN. We calculate this disk model using the method given in [@2002AJ....124.3404C][^5], with parameters taken from @2006Natur.441..834C (i.e., $\alpha=6.5\times10^{-3}$, $\dot{M}=10^{-6} M_J/$year, and $\kappa=0.1$ cm$^{2}$g$^{-1}$). Results {#sec:results} ======= We are now in a position to apply the tools developed in §\[sec:disk\_structure\]–§\[sec:magnetic\_diffusivity\] to the models described in §\[sec:disk\_models\]. All figures are shown for a protoplanet in orbit around a solar-mass star at the current orbital distance of Jupiter (i.e., $M_*=1M_{\odot}$, and $d=5.2$au), calculated with the standard parameter set $\alpha=10^{-3}$, $\dot{M}=10^{-6}M_J/$year, and $M=M_J$, unless otherwise stated. Disk structure {#disk-structure} -------------- ------------------------------------ ------------------------------------ {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} ------------------------------------ ------------------------------------ Fig. \[fig:models\] shows the radial disk structure for each model. The constant-$\alpha$ disk, MMJN, and Canup & Ward disks are shown as the solid, long-dashed, and dot-dashed curves, respectively. The self-consistent accretion disk is shown for both an MRI (dotted curve) and vertical field (short-dashed curve). The curves are labelled $\alpha$, MMJN, CW, MRI, and V respectively. The temperature profiles are shown in the top-left panel. The temperature profile for the constant-$\alpha$ and self-consistent accretion disks follow a power law with index changes at the transitions between opacity regimes. The self-consistent accretion disk profiles follow the constant-$\alpha$ profile out to $\sim30\,R_J$ where the temperature, and thermal ionisation level is high enough for good magnetic coupling. The stronger coupling requirement for an MRI field makes for a slightly hotter and more dense disk than for accretion driven by a vertical field, and so the disk is gravoturbulent beyond $200\,R_J$, where the temperature profile steepens. There is no corresponding gravoturbulent region for the self-consistent accretion disk with vertical field. Nevertheless, the self-consistent accretion disk is remarkably similar when either the MRI or verticals used for drive accretion. The profile for constant-$\alpha$ disk follows $T\propto r^{-1.1}$ in the outer regions where the opacity is primary from grains \[i.e., $a=0, b=0.74$; see equation (\[eq:const\_alpha\_temp\_density\_relationship\])\]. Of the parameter set $\alpha$, $\dot{M}$ and $M$, the temperature profile is most sensitive to changes in the inflow rate. An order of magnitude change in $\dot{M}$ only corresponds to a factor $\sim3$ change in the temperature across most of the disk, with little effect beyond $\sim40\,R_J$.The profiles are multivalued in the region $r\sim2$–$5\,R_J$, with a characteristic ‘S-shape’. Here the disk satisfies conditions for multiple opacity regimes, with the radially increasing, unstable branch corresponding to the H-scattering opacity regime. The viscous-thermal instability associated with this feature has been used to model outbursts in circumstellar disks surrounding T-Tauri stars - most notably FU Orionis outbursts by @1997ApJ...486..372B. The constant-$\alpha$ and self-consistent accretion disks are hotter than the Canup & Ward and MMJN disks, which aim to model a later phase of the disk when the opacity is from ice grains (and necessarily lower; see the opacity profile in bottom-right panel of Fig. \[fig:models\]), and the disk is cool enough to form icy satellites. As inflow from the protoplanetary disk tapers, the disk cools, consistent with the evolution to an icy state recorded by the Solar System giant-planet satellite systems. For example, reducing the inflow rate by a factor of ten lowers the temperature to only $370\,$K at the disk outer edge. The column density profile is shown in the top-right panel. The profile for the constant-$\alpha$ disk is generally shallow, decreasing by only a factor of $\sim12$ between the inner and outer edge. Like the @2002AJ....124.3404C disk, the column density is low compared with the MMJN, and so the disk is ‘gas starved’. Consequently, the disk mass is also low, with $M_{\text{disk}}=1.6\times10^{-3}M_J$, validating our neglect of self gravity. On the other hand, the column density in the self-consistent accretion disks increase beyond $\sim30\,R_J$ reaching $\Sigma=9.6\times10^4$gcm$^{-2}$ for a vertical field, and $\Sigma=2.5\times10^5$gcm$^{-2}$ for an MRI field. Consequently, the disk masses are large, with $M_{\text{disk}}=0.5\,M_J$ for the vertical field, and $M_{\text{disk}}=0.64\,M_J$ for the MRI field. The disk mass increases as the inflow rate from the protoplanetary disks tapers, such that a factor 10 reduction in the inflow rate leads to an inward extension of the gravoturbulent region, and a disk mass $M_{\text{disk}}=0.42\,M_J$, independent of the field geometry. The centre-left panel of Fig. \[fig:models\] shows the aspect ratio for each model. The aspect ratio for the constant-$\alpha$ model ranges between $H/r=0.14$–$0.34$, with pressure dominating the scale height. Self-gravity is too weak to counteract the strong thermal pressure in the outer regions of the self-consistent accretion disks and so the disks are very thick, with the aspect ratio reaching a maximum of $H/r=0.63$, and $0.71$ for a vertical and MRI field, respectively. Our results agree with [@2013ApJ...767...63S] in that circumplanetary disks may be more aptly described as ‘slim’ (i.e., $H/r\lesssim1$) rather than ‘thin’. The centre-right panel of Fig. \[fig:models\] shows the radial profile for Toomre’s $Q$. Toomre’s $Q$ is large for the low mass constant-$\alpha$ disk, however, despite the high temperatures the self-consistent accretion disks reach $Q\sim1$ at the outer edge where the column density is highest. We fix $Q=1$ in the gravoturbulent region in the self-consistent accretion disk with MRI field. The bottom-left panel of Fig. \[fig:models\] shows the radial profile of the viscosity parameter, $\alpha$. The viscosity parameter is constant across the Canup & Ward and constant-$\alpha$ disks, and in the inner regions of the self-consistent accretion disks where magnetic coupling is good and $\alpha$ is saturates at its maximum value. Once the temperature drops below $\sim1000\,K$ thermal ionisation drops and with it the strength of magnetic coupling. Magnetic transport is less efficient with high diffusivity and so $\alpha$ is reduced, as per equation (\[eq:alphaSS02\]), reaching a minimum of $1.9\times10^{-7}$ for an MRI field, and $4.8\times10^{-7}$ for a vertical field. In the outer $\sim60\,R_J$ of the self-consistent MRI accretion disk, $\alpha$ increases radially to compensate for the decreasing column density. However, such a low required effective viscosity is potentially overwhelmed by other processes, such as stellar forcing or satellitesimal wakes which may contribute additional torque exceeding this level . Note that a property of this model is that temperature increases with decreasing $\alpha$. This result is counter intuitive given that viscosity, and hence dissipation, are directly proportional to $\alpha$. However, for a fixed $\dot{M}$, increasing $\alpha$ enhances the effectiveness of the turbulence and so reduces the required active column density \[see equation (\[eq:mdot\_alpha\_relation\])\]. The associated reduction in optical depth lowers the midplane temperature relative to the surface temperature. Consequently, if we increase $\alpha_{\text{sat}}$ to $10^{-2}$ which is appropriate for MRI with net magnetic flux, we find that the midplane temperature reaches at most $2100\,$K. We also find that the saturated magnetic transport region (i.e. where the diffusivities are below the coupling threshold) only reaches out to $6\,R_J$, whereas the gravoturbulent region extends in as far as $120\,R_J$. However, we also note that increasing $\alpha_{\text{sat}}$ requires a further reduction of the minimum value of $\alpha$ to $2.2\times10^{-8}$ (at the boundary of the marginally coupled and gravoturbulent regions). Opacity as a function of temperature is shown in the bottom–right panel of Fig. \[fig:models\], using the corresponding density profile \[i.e., $\kappa(\rho(r), T(r))$ vs $T(r)$\]. The opacity is complex and varies by four orders of magnitude throughout the disk. Despite differences in the temperature and density profiles, the opacity profile for the self-consistent accretion disks follow that of the constant-$\alpha$ disk. This is because the disks only deviate in the Grains opacity regime where the opacity is density independent (i.e., $a=0$). Ionisation ---------- ------------------------------------ ------------------------------------ {width="46.00000%"} {width="46.00000%"} ------------------------------------ ------------------------------------ Fig. \[fig:ionisation\_fraction\] shows the electron (solid curve), ion (dashed curve), and charge-weighted grain (dotted curve) number density fraction for the constant-$\alpha$ model (top-left panel), and self-consistent accretion disks with MRI field (centre-left panel) and vertical field (bottom-left panel). In the constant-$\alpha$ disk, the ionisation fraction is high within the inner disk. Close to the planet the disk is almost fully ionised by thermal ionisation of hydrogen and helium, and thermal ionisation continues out to $\sim30\,R_J$ where the temperature exceeds $\sim1000$K and potassium is thermally ionised. In the abundance of free electrons grains acquire a large negative charge, $Z_g\sim-660$, but with little effect on the total electron density. Beyond this distance, the disk is not hot enough for significant thermal ionisation and so the ionisation fraction drops sharply. Ionisation is primarily by radioactive decay beyond $60\,R_J$, and the ionisation fraction is low (i.e., $n_e/n\sim10^{-19}$). In these conditions grains are mostly neutral, but still remove a large proportion of free electrons, reducing the electron density by a factor of $\sim190$ relative to the ions. Thermal ionisation is strong over a larger portion of the self-consistent accretion disks, as the disk structure is reliant on a higher level of ionisation in the marginally magnetically coupled region. We rely on thermal ionisation to achieve magnetic coupling, as midplane ionisation from radioactive decay, cosmic rays and X-rays is too weak (see §\[sec:results\_coupling\]). Grain charging is important beyond $\sim40\,R_J$ for both field geometries, however it has a greater effect for the vertical field where the ionisation fraction is lower. All profiles are multivalued between $3\,R_J\le r\le5\,R_J$, in keeping with the temperature profiles. Depletion onto grains removes heavy elements from the gas phase, and consequently reduces the ionisation fraction between $3\,R_J\lesssim r \lesssim 60\,R_J$ in the constant-$\alpha$ disk. There is no depletion close to the planet where ionisation is from the non-depleted hydrogen and helium, and in the outer disk ionisation by radioactive decay is so weak that neutral metals are abundant (i.e., $n_i/n_n\ll x_{\text{metals}})$ and the reaction rate is not limited by depletion. In the intermediate region depletion reduces the ionisation fraction by up to the depletion factor, $10^{-\delta}=0.12$. The lowered electron density leads to a slight increase (up to $10\%$) in grain charge. Depletion at this level has no appreciable effect on the structure of the self-consistent accretion disks. Additional ionisation from MRI is ineffective for both the constant-$\alpha$ and fixed-temperature disks. Grain capture through vertical mixing rapidly removes ionisation in eddies produced in MRI active surface layers. If grains are absent, charges are removed by recombination quickly over a time-scale $\tau_R\approx4\Omega^{-1}$ at the outer edge. However, if grains are present, even at the level $f_{dg}\gtrsim10^{-11}$, grain charge capture is rapid. Thus, free charges are rapidly removed as they are mixed into the dead zone and so do not contribute to midplane ionisation. For ionisation produced through acceleration by MRI electric fields, we find that the electron energy is at most $\epsilon\approx5\times10^{-3}$eV in the constant-$\alpha$ disk, and lower still in higher density self-consistent accretion disks. This energy is orders of magnitude too low to ionise any atomic species. Thus, there is no appreciable contribution from self-sustaining MRI ionisation in circumplanetary disks. Self-sustaining ionisation is more successful in protoplanetary disks where the density is lower such that electrons are able to be accelerated over a longer mean free path. We have also calculated the charge number density fractions for the Canup & Ward $\alpha$ disk (top-right panel) and the MMJN (bottom-right panel) using the same method as given in §\[sec:thermal\_ionisation\]. In the Canup & Ward $\alpha$ disk thermal ionisation is high close to the planet with cosmic ray ionisation dominant beyond $20\,R_J$, similar to the constant-$\alpha$ disk. In the MMJN the ionisation fraction is very low ($n_e/n<10^{-16}$) due to both high surface density and low temperature. Magnetic field strength {#magnetic-field-strength} ----------------------- ![Radial dependence of the magnetic field strength, B, for the $\alpha$ model (solid curve), and fixed temperature model with MRI field (dotted curve), and vertical field (short-dashed curve), Canup & Ward $\alpha$ disk (dot-dashed curve), and MMJN (long-dashed curve).[]{data-label="fig:magnetic"}](fig4){width="46.00000%"} Fig. \[fig:magnetic\] shows the magnetic field strength for the constant-$\alpha$ model (solid curve), and self-consistent accretion disks with MRI field (dotted curve) and vertical field (dashed curve). The MRI field strength for the constant-$\alpha$ disk varies between $B=0.28$–$250\,$G, and follows $B\propto r^{-1.1}$ across most of the disk. The field strength for the self-consistent accretion disk with MRI field is almost identical to that of the constant-$\alpha$ disk, except for a small deviation at the outer edge where the temperature profiles diverge. The vertical field required for self-consistent accretion has a similar dependency, with $B\propto r^{-5/4}$, but it is $\sim$5 times weaker and decreases monotonically. All disk model fields are sub-equipartition and are consistent with the with the estimate of $B=10$–$50$G at $10R_J$ by . We have plotted the magnetic field strength required to drive accretion throughout the entire disk for the self-consistent accretion disk with MRI field, however beyond $200\,R_J$ accretion is powered by gravitoturbulence rather than magnetic fields. We have no information about the magnetic field in the gravoturbulent region. For comparison we have calculated the MRI magnetic field strength for the Canup & Ward $\alpha$ disk and the MMJN, which we also show in Fig. \[fig:magnetic\]. We calculate the field strength the Canup & Ward disk using equation (\[eq:alpha\_magnetic\_field\]) for their $\alpha=6.5\times0^{-3}$, and for the MMJN using equation (\[eq:BMRI\]) assuming an accretion rate of $\dot{M}=10^{-6}\,M_J/$year. Magnetic coupling {#sec:results_coupling} ----------------- Fig. \[fig:diffusivity\] shows the Ohmic (solid curve), Hall (dashed curve), and Ambipolar (dotted curve) magnetic diffusivities scaled by the coupling threshold for the constant-$\alpha$ disk (top panel), and self-consistent accretion disk with MRI field (centre panel) and vertical field (bottom panel). The coupling threshold $\eta\Omega/v_a^2=1$ is used for the constant-$\alpha$ disk and self-consistent accretion disk with MRI field whereas $\eta\Omega/c_s^2=1$ is used for the self-consistent accretion disk with vertical field. The threshold is shown as a dotted horizontal line, with strong magnetic coupling in regions where each of the Ohmic, Hall and Ambipolar diffusivities are below the coupling threshold. We find that all disks are dense enough that Ohmic diffusivity dominates over Hall and Ambipolar. The diffusivities follow the inverse of the ionisation fraction \[i.e., $\eta\propto n/n_e$, see equations (\[eq:ohmic\])–(\[eq:ambipolar\])\]. Within $30\,R_J$, the ionisation fraction is high and so the diffusivities are well below the coupling threshold, $\eta\Omega v_a^{-2}\ll1$ or $\eta\Omega c_s^{-2}\ll1$ . At $30\,R_J$ the diffusivities rise exponentially as thermal ionisation of potassium is suppressed by the low temperature. In the constant-$\alpha$ disk, ionisation from cosmic rays, X-rays and decaying radionuclides is too low for good magnetic coupling and so the majority of the disk, (i.e., $r>30\,R_J$), is uncoupled from the magnetic field. The magnetically coupled region is larger at higher inflow rates where the midplane temperature is higher (i.e., the disk is coupled within $90\,R_J$ for $\dot{M}=10^{-5}\,M_J/$year), however this also produces a higher disk scale height, (aspect ratio up to 0.79), violating the ‘thin-disk’ approximation. Diffusivity below the coupling threshold in the inner disk indicates that the evolution of the disk and magnetic field are locked together, however the bulk of the disk is uncoupled to the magnetic field and accretion cannot proceed in these regions. The boundary of the magnetically coupled region is controlled by the exponential rise in the diffusivity at the ionisation temperature of potassium. For instance, if a vertical field is used instead of an MRI field, the scaled diffusivity is reduced by a factor $\left(v_a/c_s\right)^2=4\alpha$ \[using the MRI field to evaluate $v_a$; see equation (\[eq:equipartition\_ratio\])\], but the steepness of the diffusivity profile at the coupling boundary means that there is no change in the magnetically-coupled boundary. Similarly, depletion of heavy elements onto grains increases the diffusivity between $3\,R_J\le r \le 60\,R_J$, but does not change the radius of the magnetically-coupled region. The diffusivity profile for the self-consistent accretion disk with MRI field follows the constant-$\alpha$ disk profiles out until $30\,R_J$, where Ohmic diffusivity reaches the coupling threshold. Here, the disk enters the marginally magnetic coupled region and the rise in the diffusivity is not as steep. Although magnetic coupling is only weak, as the diffusivities are above the coupling threshold, it is still enough to drive accretion at the level given by equation (\[eq:alphaSS02\]). This state of marginal coupling occurs out to $200\,R_J$, with Ohmic diffusivity up to $\sim10^4$ times greater than the coupling threshold. At the point where $Q=1$ gravitoturbulence becomes the dominant transport mechanism and the diffusivities resume their exponential rise. The coupling criterion for a vertical field is less stringent, and so the diffusivities are lower relative to the coupling threshold within $r\sim30\,R_J$. As with the self-consistent accretion disk with MRI field, the sharp rise in the diffusivity is reduced once the diffusivities reach the coupling threshold as the disk transitions to marginal magnetic coupling. However, in contrast, the disk never reaches $Q=1$ and so there is no transition to the gravoturbulent region. We have also calculated diffusivities for the Canup & Ward $\alpha$ disk (top-right panel) and the MMJN (bottom-right panel) for an MRI field. We show the absolute value of the Hall diffusivity for the Canup & Ward $\alpha$ disk as Hall diffusivity is negative beyond $r\sim70\,R_J$ (shown by a dotted curve when $\eta_H<0$). This occurs near the transition for ion re-coupling, and indicates that the Hall drift, between the field and neutrals, is in the opposite direction for a given field configuration. The diffusivities are above the coupling threshold for $r>10\,R_J$ for the Canup & Ward $\alpha$ disk and at all radii for the MMJN, preventing magnetically-driven accretion in these regions. ------------------------------------ ------------------------------------ {width="46.00000%"} {width="46.00000%"} ------------------------------------ ------------------------------------ Discussion {#sec:discussion} ========== In this paper we modelled steady-state accretion within a giant planet circumplanetary disk, and determined the effectiveness of magnetic fields and gravitoturbulence in driving accretion. We modelled the disk as a thin Shakura-Sunyaev $\alpha$ disk, heated by viscous transport and solved for the opacity simultaneously with the disk midplane structure using the @2009ApJ...694.1045Z opacity law, including the effects of self-gravity. Thermal ionisation dominates within $r\lesssim30\,R_J$ where the disk reaches the ionisation temperature of potassium ($T\sim10^3\,$K), but drops rapidly in cooler regions where ionisation is primarily by radioactive decay. The midplane is too dense for penetration of cosmic rays or stellar X-rays. We considered both an MRI field and a vertical field in driving accretion, and found that a field of order $10^{-2}$–$10\,$G is needed to account for the inferred accretion rate onto the young Jupiter. To quantify the strength of interaction between the magnetic field and disk we calculated Ohmic, Hall, and Ambipolar diffusivities which cause slippage of the field lines relative to the bulk motion of the disk, decoupling their evolution. In the standard constant-$\alpha$ disk, diffusivity is low enough for magnetic coupling in the inner region where potassium is thermally ionised. However, the remainder of the disk is too cool for thermal ionisation and so strong diffusivity prohibits magnetically-driven accretion throughout the bulk of the disk. The disk is gravitationally stable, with Toomre’s $Q\gg1$, and so there is no transport from gravitoturbulence either. This is inconsistent with the assumption of a constant-$\alpha$, and so we presented an alternate model in which $\alpha$ varies radially, ensuring that the accretion rate (taken to be uniform through the disk) is consistent with the level of magnetic coupling and gravitational instability. We achieved this by dividing the disk into three regions according to the mode of accretion: (i) the inner disk is hot enough for strong magnetic coupling through thermal ionisation and inflow is magnetically driven with $\alpha$ saturated at its maximum value; (ii) Beyond $30\,R_J$ the disk is too cool for sufficient thermal ionisation of potassium and diffusivity exceeds the coupling threshold. Accretion is still magnetically driven, however as the magnetic coupling is weak, it occurs at a reduced efficiency with $\alpha$ inversely proportional to the level of magnetic coupling [@2002ApJ...577..534S]; (iii) The disk is gravitationally unstable in the outer regions where $Q\sim1$, and so gravitoturbulence is produced and drives accretion. Accretion is self-regulated so that the disk maintains marginal stability with $Q=1$. We calculated the disk structure for accretion driven by either MRI or vertical fields, finding very similar disk structures. With $Q\sim1$ at the outer edge, the disks are massive with $M_{\text{disk}}=0.5\,M_J$. MHD analysis by @2011ApJ...743...53F and @2013arXiv1306.2276T argue against magnetically driven accretion through the midplane where the cosmic-ray and X-ray fluxes are too low. However, we find that midplane magnetic coupling relies primarily on thermal ionisation and so the disk temperature is crucial. @2011ApJ...743...53F use the surface temperature which is necessarily cooler than the midplane temperature, and so no thermal ionisation is expected. @2013arXiv1306.2276T considers both MMJN models and actively supplied accretion disk models, appropriate for a later, and so cooler, phase than we consider here. MMJN models are necessarily cold to match conditions recorded by the final, surviving generation of Jovian moons, however these are likely formed late after a succession of earlier generations were accreted by the planet [@2006Natur.441..834C]. Temperatures in actively accreting disks are controlled by the inflow rate which likely decreases as inflow from the protoplanetary disk tapers. @2013arXiv1306.2276T consider inflow rates that are lower than ours by a factor of 5–70, so these disks model a cooler stage and consequently thermal ionisation is limited to the inner $4\,R_J$ of their highest inflow disk. Additionally, we also consider accretion in regions which are only marginally coupled to the magnetic field. We find that while saturated magnetic transport (i.e. with strong magnetic coupling) is limited to the inner $30\,R_J$, magnetically driven accretion with marginal coupling can potentially occur across the entire disk. We have modelled steady-state accretion within the disk, with the assumption that the disk evolves toward or through this state during the proto-planet accretion phase. Numerical simulations indicate that accretion disks, including circumplanetary disks, rapidly evolve away from a self-gravitating state toward a quasi-steady state [@2011MNRAS.410..994F; @2013ApJ...767...63S], however there may be other time-dependent processes, such as short time-scale variability of inflow from the protoplanetary disk. Observations of accretion onto giant planets are needed to determine the accretion timescales, and how rapidly the accretion rate can change. The temperature profiles are multivalued in some regions of the disk, making the disks susceptible to viscous-thermal instability. This may lead to outbursts, undermining our steady-state assumption. This feature is only present when the inflow rate exceeds $\dot{M}=2\times10^{-8}\,M_J/$year, and so outbursting from the viscous-thermal instability will not occur at later time when the inflow rate has tapered off to below this value. While there is certainly the potential for outbursting at earlier times, our analysis centres on whether inflow driven by magnetic fields is plausible, rather than advocating a steady state solution. There may also be additional torques on the circumplanetary disk, from stellar forcing or spiral waves generated by satellitesimals , which we have not included. It is not clear what level of transport these processes produce during this phase of giant planet accretion and whether they can be incorporated as additional sources within the Shakura-Sunyaev $\alpha$ formalism. We can model minor variations on the inflow parameters, such as a reduction in the accretion rate which reproduces the necessary cooling and disk mass lowering as inflow from the protoplanetary disk tapers. However these results are uncertain as they require yet lower values of $\alpha$ in the self-consistent accretion disk which are likely overwhelmed by the additional torques mentioned above. Strong magnetic coupling near the surface of the planet will affect accretion onto the planet surface. The planetary magnetic field may channel the accretion flow onto the planet surface [@2011AJ....141...51L], effecting the spin evolution of the planet [@1996Icar..123..404T; @2011AJ....141...51L], and temperature of the planet. However magnetospheric accretion requires diffusivity in order for the inflow to transfer onto the planetary magnetic field from the disk field. Loading onto the planetary field lines is only expected to occur close to the surface, if at all (at $r\sim1$–$3\,R_J$; see ). However we find the diffusivity is very low at this distance, making loading of the gas onto the proto-planetary field lines from the disk field difficult. Magnetospheric accretion would require an additional source of diffusivity, such as electron momentum exchange with ion acoustic waves (e.g., see [@2006JGRA..111.1205P]), however it is not known how strong this effect is. Finally, the circumplanetary disk is the formation site for satellites. The composition of the present day satellite systems around Jupiter and Saturn record conditions in their circumplanetary disks at the time of their formation. In particular, the rock/ice compositional gradients through the satellite systems set the disk ice line ($T\approx250\,$K) at the the location of Ganymede, $r=15R_J$, and Rhea, $r=8.7\,R_S$, in the Jovian and Saturnian systems, respectively [@2003Icar..163..198M]. The location of the ice line is often incorporated or used as a measure of success in circumplanetary disk models (e.g., @1982Icar...52...14L [@2003Icar..163..198M; @2002AJ....124.3404C]), however no moons have been discovered beyond the Solar System and so it is not clear how typical the Jovian system is, nor to what degree these systems can vary [@2013arXiv1306.1530K]. It is not our aim to reproduce the conditions for moon formation, but rather we are focussed on modelling the early phase of the disk, in which the disk is hot and there is the significant inflow onto Jupiter. Consequently, the ice line in our constant-$\alpha$ disk is at $r=139\,R_J$, and the self-consistent accretion disks are too hot for ice. Several generations of satellites may have formed in these conditions, but the present day satellites likely form at a later stage when the disk has cooled as inflow into the circumplanetary disk tapers with the dispersal of the protoplanetary disk [@1989oeps.book..723C; @2006Natur.441..834C; @2010ApJ...714.1052S]. Our results support the two stage circumplanetary disk evolution proposed by @1989oeps.book..723C in which the disk is initially hot and turbulent, but evolves to the cool quiescent disk as recorded by the giant planet satellite systems. In summary, we have found that during the final gas accretion phase of a giant planet the circumplanetary disk is hot and steady-state accretion may be driven by a combination of magnetic fields and gravitoturbulence. Accretion maintains the disk at a high temperature so that there is thermal ionisation through most of the disk. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Yuri I. Fujii and Philippa K. Browning for valuable discussion and comments on the manuscript. We thank the anonymous referee for helpful comments which improved this manuscript. This work was supported in part by the Australian Research Council grant DP120101792. S.K. further acknowledges the support of an Australian Postgraduate Award and funding from the Macquarie University Postgraduate Research Fund scheme. This research has made use of NASA’s Astrophysical Data System. \[lastpage\] [^1]: E-mail: sarah.l.keith@mq.edu.au; mark.wardle@mq.edu.au [^2]: We have used the mean particle mass of molecular H/He gas in the conversion from pressure to density even though it is not strictly valid where hydrogen is ionised. Hydrogen is only ionised within the inner $5\,R_J$, at temperatures above $3000\,$K, and we find that correcting the mean particle mass (to $\mu=1.24$) leads to at most a 15% change in the temperature in this region. [^3]: is released in a boundary layer (thickness $\ll R_J$) above the planet surface where the disk angular velocity profile transitions sharply between keplerian and the planetary rotation rate [@1977MNRAS.178..195P]. This contributes an additional factor $\left(1-\sqrt{R_J/r}\right)$ to the right hand side to this viscosity-inflow relation. However, we find that this factor is only significant within $r<2\,R_J$, i.e., within the boundary layer. [^4]: For consistency we insert our equation (\[eq:alpha\_B\_relation\]) into equation (32) of @2012ApJ...760...56M, and account for self-gravity which leads to stricter criterion, independent of plasma $\beta$: $f_{\text{whb}}=5.4\times10^{-2}$ for $Q=0$ \[c.f., their equation (36)\]. [^5]: The profiles shown in Fig. \[fig:models\] are calculated using the full expression $\chi=1+\frac{3}{2}[r_c/r-\frac{1}{5}]^{-1}$ (given below equation 20 in @2002AJ....124.3404C), however we found $\chi=1$ was needed to reproduce the profiles in @2002AJ....124.3404C. For the parameter set used here, we find that the approximation leads to at most a $37\%$ increase in the surface density, and $27\%$ reduction in the temperature profile. The difference is greatest at $r=60R_J$, but decreases toward the inner and outer boundaries.

--- abstract: 'A prefix grammar is a context-free grammar whose nonterminals generate prefix-free languages. A prefix grammar $G$ is an ordinal grammar if the language $L(G)$ is well-ordered with respect to the lexicographic ordering. It is known that from a finite system of parametric fixed point equations over ordinals one can construct an ordinal grammar $G$ such that the lexicographic order of $G$ is isomorphic with the least solution of the system, if this solution is well-ordered. In this paper we show that given an ordinal grammar, one can compute (the Cantor normal form of) the order type of the lexicographic order of its language, yielding that least solutions of fixed point equation systems defining algebraic ordinals are effectively computable (and thus, their isomorphism problem is also decidable).' address: 'University of Szeged, Hungary' author: - Kitti Gelle - Szabolcs Iván bibliography: - 'biblio.bib' title: The ordinal generated by an ordinal grammar is computable --- Algebraic ordinals; Ordinal grammars; Parametric fixed-point equations over ordinals; Isomorphism of algebraic well-orderings Introduction ============ Least solutions of finite systems of fixed points equations occur frequently in computer science. Some very well-known instances of this are the regular and context-free languages, rational and algebraic power series, well-founded semantics of generalized logic programs, semantics of functional programs, just to name a few. A perhaps less-known instance is the notion of the algebraic linear orders of [@Bloom:2010:MTC:1655414.1655555]. A linear ordering is algebraic if it is (isomorphic to) the first component of the least solution of a finite system of fixed point equations of the sort $$F_i(x_0,\ldots,x_{n_i-1})=t_i,\quad i=1,\ldots,n,$$ where $n_1=0$ and each $t_i$ is an expression composed of the function variables $F_j$, $j=1,\ldots,n$, the variables $x_0,\ldots,x_{n_i-1}$ which range over linear orders, the constant $1$ and the sum operation $+$. As an example, consider the following system from [@DBLP:journals/fuin/BloomE10]: $$\begin{aligned} F_0 &= G(1)\\ G(x)&=x+G(F(x))\\ F(x)&=x+F(x)\end{aligned}$$ In this system, the function $F$ maps a linear order $x$ to $x+x+\ldots = x\times\omega$, the function $G$ maps a linear order $x$ to $x+G(x\times \omega)=x+x\times\omega+G(x\times\omega^2)+\ldots =x\times\omega^\omega$, thus the first component of the least solution of the system is $F_0=G(1)=\omega^\omega$. If the system in question is parameterless, that is, $n_i=0$ for each $i$, then the ordering which it defines is called a regular ordering. An ordinal is called algebraic (regular, respectively) if it is algebraic (regular, resp.) as a linear order. It is known [@ITA_1980__14_2_131_0; @BLOOM2001533; @10.1007/978-3-540-73859-6_1; @DBLP:journals/fuin/BloomE10; @10.1007/978-3-642-29344-3_25] that an ordinal is regular if and only if it is smaller than $\omega^\omega$ and is algebraic if and only if it is smaller than $\omega^{\omega^\omega}$. To prove the latter statement, the authors of [@DBLP:journals/fuin/BloomE10] applied a path first used by Courcelle [@ITA_1978__12_4_319_0]: every countable linear order is isomorphic to the frontier of some (possibly) infinite (say, binary) tree. Frontiers of infinite binary trees in turn correspond to prefix-free languages over the binary alphabet, equipped with the lexicographic ordering. Moreover, algebraic (regular, resp.) ordinals are exactly the lexicographic orderings of context-free (regular, resp.) prefix-free languages [@COURCELLE198395] (prefix-free being optional here as each language can be effectively transformed to a prefix-free order-isomorphic one for both the regular and the algebraic case). Thus, studying lexicographic orderings of prefix-free regular or context-free languages can give insight to regular or algebraic linear orders. The works [@BLOOM2001533; @6a0baa0d4e6744d38956d22057b410ce; @BLOOM200555; @10.1007/978-3-540-73859-6_1; @COURCELLE198395; @ITA_1980__14_2_131_0; @LOHREY201371; @ITA_1986__20_4_371_0] deal with regular linear orders this way, in particular [@LOHREY201371] shows that the isomorphism problem for regular linear orders is decidable in polynomial time The study of the context-free case was initiated in [@10.1007/978-3-540-73859-6_1], and further developed in [@DBLP:journals/fuin/BloomE10; @doi:10.1142/S0129054111008155; @ESIK2011107; @10.1007/978-3-642-22321-1_19; @10.1007/978-3-642-29344-3_25; @CARAYOL2013285; @KUSKE201446]. Highlighting the results from these works that are tightly connected to the current paper: the case of regular linear orders is well-understood, even their isomorphism problem (that is, whether two regular linear orders, given by two finite sets of fixed-point equations, are isomorphic) is decidable. For algebraic linear orders, there are negative results: it is already undecidable whether an algebraic linear ordering is dense, thus (as there are exactly four dense countable linear orders up to isomorphism) the isomorphism problem of algebraic linear orders is undecidable. On the other hand, deciding whether an algebraic linear order is scattered, or a well-order, is decidable. The frontier of decidability of the isomorphism problem of algebraic linear orderings is an interesting question: for the general case it is undecidable, while for the case of regular ordinals it is known to be decidable by [@LOHREY201371] and [@Khoussainov:2005:ALO:1094622.1094625]. In [@DBLP:journals/fuin/BloomE10], it was shown that a system of equations defining an algebraic ordering can be effectively transformed (in polynomial time) to a so-called prefix grammar $G$ (a context-free grammar whose nonterminals each generate a prefix-free language), such that the lexicographic order of the language generated by $G$ is isomorphic to the algebraic ordering in question. If the ordering is a well-ordering (i.e. the system defines an algebraic ordinal), then the grammar we get is called an ordinal grammar, that is, a prefix grammar generating a well-ordered language with respect to the lexicographic ordering. In this paper we show that given an ordinal grammar, the order type of the lexicographic ordering of the language it generates is computable (that is, we can effectively construct its Cantor normal form). Hence, applying the above transformation we get that the Cantor normal form of any algebraic ordinal is computable from its fixed-point system presentation, thus in particular, the isomorphism problem of algebraic ordinals is decidable. Notation ======== When $n\geq 0$ is an integer, $[n]$ denotes the set $\{1,\ldots,n\}$. (Thus, $[0]$ is another notation for the empty set $\emptyset$.) Linear orders, ordinals {#linear-orders-ordinals .unnumbered} ----------------------- In this paper we consider countable linear orderings. A good reference on the topic is [@rosenstein]. A linear ordering $(I,<)$ is a set $I$ equipped with a strict linear order: an irreflexive, transitive and trichotome relation $<$. When the order $<$ is clear from the context, we omit it. Set-theoretic properties of $I$ are lifted to $(I,<)$, thus we can say that a linear order is finite, countable etc. When $(I_1,<_1)$ and $(I_2,<_2)$ are linear orders, their (ordered) sum is $(I_1,<_1)+(I_2,<_2)=(I_1\uplus I_2,<)$ with $x<y$ if and only if either $x\in I_1$ and $y\in I_2$, or $x,y\in I_1$ and $x<_1y$, or $x,y\in I_2$ and $x<_2y$. A linear ordering $(I',<')$ is a subordering of $(I,<)$ if $I'\subseteq I$ and $<'$ is the restriction of $<$ onto $I'$. In order to ease notation, we usually use $<$ in these cases in place of $<'$ and so we will simply write $(I_1,<)+(I_2,<)=(I,<)$ or even $I_1+I_2=I$ in the case of sums. A linear ordering $I$ is called a *well-ordering* if there are no infinite descending chains $\ldots<x_2<x_1<x_0$ in $I$. Clearly, well-orderings are closed under (finite) sums and suborderings, and they are also closed under $\omega$-sums: if $I_1,I_2,\ldots$ are pairwise disjoint linear orderings, then their sum $I=I_1+I_2+\ldots$ is the ordering with underlying set $\bigcup_i I_i$ and order $x<y$ if and only if $x\in I_i$ and $y\in I_j$ for some $i<j$, or $x,y\in I_i$ for some $i$ and $x<_iy$, which is well-ordered if so is each $I_i$. Two linear orders $(I,<_i)$ and $(J,<_j)$ are called *isomorphic* if there is a bijection $h:I\to J$ with $x<_iy$ implying $h(x)<_jh(y)$. An *order type* is an isomorphism class of linear orderings. The order type of the linear order $I$ is denoted by $o(I)$. Clearly, if two orderings are isomorphic and one of them is a well-ordering, then so is the other one. The *ordinals* are the order types of well-orderings (for a concise introduction see e.g. the lecture notes of J. A. Stark [@jalex]). The order types of the finite ordered sets are identified with the nonnegative integers. The order type of the natural numbers themselves (whose set is $\mathbb{N}_0=\{0,1,\ldots\}$, equipped by their usual ordering) is denoted by $\omega$, while the order types of the integers and rational numbers are respectively denoted by $\zeta$ and $\eta$. Since if $o(I)=o(I')$ and $o(J)=o(J')$, then $o(I+J)=o(I'+J')$, the sum operation can be lifted to order types, even for $\omega$-sums. For example, $\omega+\omega$ is the order type of $\{0,1\}\times\mathbb{N}$, equipped with the lexicographic ordering $(b_1,n_1)<(b_2,n_2)$ if and only if either $b_1<b_2$ or ($b_1=b_2$ and $n_1<n_2$). Note that $1+\omega=\omega$ but $\omega+1\neq\omega$. The ordinals themselves are also equipped with a relation $<$ so that each set of ordinals is well-ordered by $<$, namely $o_1<o_2$ if $o_1\neq o_2$ and there are linear orderings $I$ and $J$ such that $o(I)=o_1$, $o(J)=o_2$ and $I$ is a subordering of $J$. With respect to this relation, every set $\Omega$ of ordinals have a least upper bound (a *supremum*) $\bigvee\Omega$ (which is also an ordinal), moreover, for each ordinal $\alpha$, the ordinals smaller than $\alpha$ form a set. Each ordinal $\alpha$ is either a *successor ordinal* in which case $\alpha=\beta+1$ for some smaller ordinal $\beta$, or a *limit ordinal* in which case $\alpha=\mathop\bigvee\limits_{\beta<\alpha}\beta$, the supremum of all the ordinals smaller than $\alpha$. These two cases are disjoint. For an example, $0=\bigvee\emptyset$ is a limit ordinal, and it is the smallest ordinal; $1$, $2$ and $42$ are successor ordinals, $\omega$ is a limit ordinal, $\omega+1$ is again a successor ordinal, $\omega+\omega$ is a limit ordinal and so on. Since every set of ordinals is well-ordered, and to each ordinal $\alpha$ the ordinals smaller than $\alpha$ form a set, the principle of *(well-founded) induction* is valid for ordinals: if $P$ is a property of ordinals, and - whenever $P$ holds for $\alpha$, then $P$ holds for $\alpha+1$ and - whenever $\alpha$ is a limit ordinal and $P$ holds for each ordinal $\beta<\alpha$, then $P$ holds for $\beta$, then $P$ holds for all the ordinals. (In practice we usually separate the case of $\alpha=0$ from the rest of the limit ordinals.) Over ordinals, the operations of (binary) product and exponentiation are defined via induction as follows: $$\begin{aligned} \alpha\times 0 &=0 & \alpha\times(\beta+1)&=\alpha\times\beta+\alpha& \alpha\times\beta^*&=\mathop\bigvee\limits_{\beta'<\beta^*}\left(\alpha\times\beta'\right)\\ \alpha^0 &=1 & \alpha^{\beta+1}&=\alpha^\beta\times\alpha&\alpha^{\beta^*}&=\mathop\bigvee\limits_{\beta'<\beta^*}\alpha^{\beta'}\end{aligned}$$ where the equations of the last column hold for limit ordinals $\beta^*$. Every ordinal $\alpha$ can be uniquely written as a finite sum $$\alpha=\omega^{\alpha_1}\times n_1+\omega^{\alpha_2}\times n_2+\ldots+\omega^{\alpha_k}\times n_k$$ where $k\geq 0$ and for each $1\leq i\leq k$, $n_i>0$ are integers, and $\alpha_1>\alpha_2>\ldots>\alpha_k$ are ordinals. The ordinal $\alpha_1$ in this form is called the *degree* of $\alpha$, denoted by $\deg(\alpha)$, and the sum itself is called the *Cantor normal form* of $\alpha$. The operations $+$ and $\times$ are associative, and the above operations satisfy the identities $$\begin{aligned} \alpha\times(\beta+\gamma)&=\alpha\times\beta+\alpha\times\gamma&\alpha^\beta\times\alpha^\gamma&=\alpha^{\beta+\gamma}&(\alpha^\beta)^\gamma&=\alpha^{\beta\times\gamma}\\ \deg(\alpha+\beta)&=\max\{\deg(\alpha),\deg(\beta)\}&\deg(\alpha\times\beta)&=\deg(\alpha)+\deg(\beta)&\deg(\alpha^\beta)&=\deg(\alpha)\times\beta,\end{aligned}$$ the last one being valid only when $\alpha\geq\omega$. From $\deg(\alpha+\beta)=\max\{\deg(\alpha),\deg(\beta)\}$ we get that if $o_1\leq o_2\leq \ldots$ are ordinals with $\deg(o_i)<\alpha$ for some ordinal $\alpha$, then $\deg(o_1+o_2+\ldots)\leq \alpha$ and equality holds if and only if $\bigvee\deg(o_i)=\alpha$ is a limit ordinal, in which case $o_1+o_2+\ldots=\omega^{\alpha}$. The following theorem from [@doi:10.1112/plms/s3-4.1.177] gives lower and upper bounds for the order type of the union of two well-ordered sets: \[thm-union\] Let $(I,<)$ be a countable well-ordered set and $I=A\cup B$. Let us write the order types of $A$ and $B$ as $$\begin{aligned} o(A) &= \omega^{\alpha_1}\times a_1+\ldots \omega^{\alpha_n}\times a_n\\ o(B) &= \omega^{\alpha_1}\times b_1+\ldots \omega^{\alpha_n}\times b_n \end{aligned}$$ for an integer $n\geq 0$, ordinals $\alpha_1>\alpha_2>\ldots> \alpha_n$ and integer coefficients $a_1,\ldots,a_n,b_1,\ldots,b_n\geq 0$ such that $\max\{a_i,b_i\}\geq 1$ for each $1\leq i\leq n$. Then $$\begin{aligned} o(I) &= \omega^{\alpha_1}\times c_1+\omega^{\alpha_2}\times c_2+\ldots +\omega^{\alpha_n}\times c_n \end{aligned}$$ for some integer coefficients $0\leq c_1,\ldots,c_n$ with $c_i\leq a_i+b_i$ for each $1\leq i\leq n$, and $c_1\geq\max\{a_1,b_1\}$. Observe that the Theorem can be applied as follows: if $o(A)<\omega^\alpha\times N$ and $o(B)<\omega^\beta\times M$, then $o(A\cup B)< \omega^{\max\{\alpha,\beta\}}\times(N+M-1)$: writing out the Cantor normal forms explicitly for $o(A)$ and $o(B)$ we would get the coefficients for $\omega^{\max\{\alpha,\beta\}}$ can be at most $N-1$ and $M-1$, respectively, making its coefficient in $o(I)$ to be at most $M+N-2$, thus (as the main term cannot be larger than $\omega^{\max\{\alpha,\beta\}}$ in either one of $o(A)$ and $o(B)$) we get $o(I)<\omega^{\max\{\alpha,\beta\}}\times(M+N-1)$. In particular, $\deg(o(A\cup B))=\max\{\deg(o(A)),\deg(o(B))\}$. Order types of context-free languages {#order-types-of-context-free-languages .unnumbered} ------------------------------------- For a nonempty finite set (an *alphabet*) $\Sigma$ of terminal symbols, also called *letters* equipped with a total ordering $<$, let $\Sigma^*$ denote the set of all finite words $a_1a_2\ldots a_n$, with $\varepsilon$ standing for the case $n=0$, the *empty word*, and let $\Sigma^\omega$ denote the set of all *$\omega$-word*s $a_1a_2\ldots$. The set of all finite and $\omega$-words is $\Sigma^{\leq\omega}=\Sigma^\omega\cup\Sigma^*$. When $u=a_1\ldots a_n$ is a finite word and $v=b_1b_2\ldots$ is either a finite or an $\omega$-word, then their product is the word $u\cdot v=a_1\ldots a_nb_1b_2\ldots$, also written $uv$. Also, when $u=a_1\ldots a_n$ is a finite word, then its *$\omega$-power* is the word $u^\omega=a_1\ldots a_na_1\ldots a_na_1\ldots$ which is $\varepsilon$ if $u=\varepsilon$ and is an $\omega$-word whenever $u$ is nonempty. Two (strict) partial orderings, the *strict ordering* $<_s$ and the *prefix ordering* $<_p$ are defined over $\Sigma^{\leq\omega}$ as follows: - $u<_sv$ if and only if $u=u_1au_2$ and $v=u_1bv_2$ for some words $u_1\in\Sigma^*$, $u_2,v_2\in\Sigma^{\leq\omega}$ and terminal symbols $a<b$ - $u<_pv$ if and only if $v=uw$ for some nonempty word $w\in\Sigma^{\leq\omega}$ (in particular, this implies $u\in\Sigma^*$). The union of these partial orderings, the *lexicographical ordering* $<_\ell~=~<_s\cup<_p$, simply written as $<$ when it is clear from the context, is a total ordering on $\Sigma^{\leq\omega}$, which is a complete lattice with respect to $<_\ell$. A *language* is an arbitrary set $L\subseteq\Sigma^*$ of *finite* words. The *supremum* of $L$, viewed as a subset of $\bigl(\Sigma^{\leq \omega},<_\ell\bigr)$ is denoted by $\bigvee L$ and is either a finite word $u\in L$, or an $\omega$-word. The *order type* $o(L)$ of $L$ is the order type of the linear ordering $(L,\leq_\ell)$. As an example, the order types of the languages $a^*$, $a^*\cup\{b\}$ and $b^*a^*$ are $\omega$, $\omega + 1$ and $\omega^2$, respectively. We say that $L$ is *well-ordered* if so is $(L,<_\ell)$. For example, the previous three languages are well-ordered but $a^*b$ is not (as it contains an infinite descending chain $\ldots<aab<ab<b$). When $K$ and $L$ are languages, then their product is $K\cdot L=\{uv:u\in K,v\in L\}$ and if $u\in\Sigma^*$, then the *left quotient of $L$ with respect to $u$* is $u^{-1}L=\{v\in\Sigma^*:uv\in L\}$, and of course, $K\cup L=\{u:u\in K\hbox{ or }u\in L\}$ is their *union*. We write $K<_\ell L$ if $u<_\ell v$ for each $u\in K$ and $v\in L$. Thus, if $K<_\ell L$, then viewing them as the linear orderings $(K,<_\ell)$ and $(L,<_\ell)$ we get their *sum* $K+L~=~(K\cup L,<_\ell)$. We put an emphasis here on the fact that taking the sum of two languages $K$ and $L$ is a *partial* operation, defined only if $K<_\ell L$. When $L$ is a language and $u$ is a (possibly infinite) word, then let $L^{<u}$ and $L^{\geq u}$ respectively denote the languages $\{v\in L:v<u\}$ and $\{v\in L:v\geq u\}$. Then clearly, $L=L^{<u}+L^{\geq u}$ for any $L$ and $u$. Note that $L^{<\varepsilon}=\emptyset$ and $L^{\geq\varepsilon}=L$, and also $L^{<a\cdot u}=L^{<a}+ a\bigl((a^{-1}L)^{<u}\bigr)$, $L^{\geq a\cdot u}=a\bigl((a^{-1}L)^{\geq u}\bigr) + L^{\geq b}$ for the least letter $b$ with $a<b$, if such a letter exists and $L^{\geq a\cdot u}=a\bigl((a^{-1}L)^{\geq u}\bigr)$ if $a$ is the last letter of the alphabet. Moreover, $(K\cup L)^{>u}=K^{>u}\cup L^{>u}$ and $(K\cup L)^{\geq u}=K^{\geq u}\cup L^{\geq u}$. A *context-free grammar* is a tuple $G=(N,\Sigma,P,S)$ with $N$ and $\Sigma$ being the disjoint alphabets of nonterminal and terminal symbols respectively, $S\in N$ is the *start symbol* and $P$ is a finite set of *productions* of the form $A\to \balpha$ with $A\in N$ being a nonterminal and $\balpha$ being a *sentential form*, i.e. $\balpha=X_1\ldots X_n$ for some $n\geq 0$ and $X_1,\ldots,X_n\in N\cup\Sigma$. If $\balpha=uX\bbeta$ for some $u\in\Sigma^*$, $X\in N$ and $\bbeta\in(N\cup\Sigma)^*$, and $X\to\boldsymbol{\gamma}$ is a production, then $\balpha$ can be rewritten to $u\boldsymbol{\gamma}\bbeta$, which is denoted by $\balpha\Rightarrow u\boldsymbol{\gamma}\bbeta$. The reflexive-transitive closure of the relation $\Rightarrow$ is denoted by $\Rightarrow^*$. For any set $\Delta$ of sentential forms, the *language generated by $\Delta$* is $L(\Delta)~=~\{u\in\Sigma^*:\balpha\Rightarrow^*u\hbox{ for some }\balpha\in\Delta\}$. For brevity, when $\Delta=\{\balpha_1,\ldots,\balpha_n\}$ is finite, we simply write $L(\balpha_1,\ldots,\balpha_n)$. Moreover, $o(\Delta)$ denotes $o(L(\Delta))$. The *language $L(G)$ generated by $G$* is $L(S)$. Languages generated by context-free grammars are called context-free languages. Two context-free grammars $G$ and $G'$ over the the same terminal alphabet are *equivalent* if $L(G)=L(G')$ and *order-equivalent* if $o(L(G))=o(L(G'))$. Any context-free grammar generating a nonempty language of nonempty words can be effectively transformed into a *Greibach normal form* in which the following all hold: - each production has the form $X\to aX_1\ldots X_n$ for some $a\in\Sigma$, - each nonterminal $X$ is *productive*, i.e., $L(X)\neq\emptyset$, and *accessible*, i.e., $S\Rightarrow^*uX\balpha$ for some $u\in\Sigma^*$ and $\balpha\in(N\cup\Sigma)^*$. Also, considering the grammar $G'=(N\cup\{S'\},\Sigma,P\cup\{S'\to aS\},S')$ for a fresh symbol $S'$, we get $L(G')=a\cdot L(G)$, and the order type of each $X\in N$ is the same in both cases, and of course $o(S)=o(S')$. Thus, to each grammar $G$ one can effectively construct another one $G'$ in Greibach normal form, with $o(L(G))=o(L(G'))$. Suppose $\balpha=aX_1\ldots X_n$ is a sentential form of a context-free grammar $G=(N,\Sigma,P,S)$ in Greibach normal form and $b$ is a terminal symbol. Then we define $\balpha^{<b}$, $\balpha^{\geq b}$ and $b^{-1}\balpha$ as the following finite sets of sentential forms: $$\begin{aligned} \balpha^{<b}&=\begin{cases} \{\balpha\}&\hbox{if }a<b\\ \emptyset&\hbox{otherwise} \end{cases} & \balpha^{\geq b}&=\begin{cases} \emptyset&\hbox{if }a<b\\ \{\balpha\}&\hbox{otherwise} \end{cases}\end{aligned}$$ $$\begin{aligned} b^{-1}\balpha&=\begin{cases} \{\varepsilon\}&\hbox{if }a=b\hbox{ and }n=0\\ \{X_1\ldots X_n\}&\hbox{if }a=b,~n>0\hbox{ and }X_1\in\Sigma\\ \{\boldsymbol{\delta} X_2\ldots X_n:X_1\to \boldsymbol{\delta}\in P\}&\hbox{if }a=b,~n>0\hbox{ and }X_1\in N\\ \emptyset&\hbox{otherwise} \end{cases}\end{aligned}$$ Then clearly, $L(\balpha^{<b})=L(\balpha)^{<b}$, $L(\balpha^{\geq b})=L(\balpha)^{\geq b}$ and $L(b^{-1}\balpha)=b^{-1}L(\balpha)$. Extending these definitions with $\varepsilon^{<b}=\{\varepsilon\}$, $\varepsilon^{\geq b}=b^{-1}\varepsilon=\emptyset$ and the recursion $$\begin{aligned} \balpha^{<b\cdot u}&=\balpha^{<b}\cup \{b\cdot(\boldsymbol{\gamma}^{<u}):\boldsymbol{\gamma}\in b^{-1}\balpha\}& \alpha^{\geq b\cdot u}&=\{b\cdot(\boldsymbol{\gamma}^{\geq u}):\boldsymbol{\gamma}\in b^{-1}\balpha\}\cup\balpha^{\geq c}\end{aligned}$$ where $c\in\Sigma$ is the first letter with $b<c$ if such a $c$ exists, otherwise $$\begin{aligned} \balpha^{\geq b\cdot u}&=\{b\cdot(\boldsymbol{\gamma}^{\geq u}):\boldsymbol{\gamma}\in b^{-1}\balpha\}\end{aligned}$$ and $(a\cdot u)^{-1}\balpha=\bigcup\bigl(u^{-1}\boldsymbol{\gamma}:\boldsymbol{\gamma}\in a^{-1}\balpha\bigr)$ we have $L(\balpha^{<u})=L(\balpha)^{<u}$, $L(\balpha^{\geq u})=L(\balpha)^{\geq u}$ and $L(u^{-1}\balpha)=u^{-1}L(\balpha)$ for any sentential form $\balpha$ not beginning with a nonterminal and word $u$, moreover, each member of any of these sets is still a sentential form not beginning with a nonterminal. Clearly, $\balpha^{<u}$, $\balpha^{\geq u}$ and $u^{-1}\balpha$ are all computable for any $u$ and $\balpha$. A context-free grammar $G=(N,\Sigma,P,S)$ is called an *ordinal grammar* if $o(X)$ is an ordinal and $L(X)$ is a *prefix-free language* (that is, there are no words $u,v\in L(X)$ with $u<_pv$) for each nonterminal $X\in N$. It is known [@DBLP:journals/fuin/BloomE10] that to each well-ordered context-free language $L$ there exists an ordinal grammar $G$ generating $L$. It is also known that for *regular* grammars (in which each production has the form $A\to uB$ or $A\to v$) generating a well-ordered language $L$, order equivalence is decidable [@DBLP:journals/ita/Thomas86], while for general context-free grammars, it is undecidable whether $o(L(G))=o(L(G'))$ for two grammars $G$ and $G'$: it is already undecidable whether $o(L(G))=\eta$ holds (or that whether $o(L(G))$ is dense) [@ESIK2011107]. In contrast, it is decidable whether $L(G)$ is well-ordered [@10.1007/978-3-642-22321-1_19]. It is unknown whether the order-equivalence problem is decidable for two grammars generating well-ordered languages. In this paper we show that it is decidable whether $o(L(G))=o(L(G'))$ for two *ordinal* grammars $G$ and $G'$. Thus, if there is an algorithm that constructs an ordinal grammar $G'$ for an input context-free grammar $G$ generating a well-ordered language (it is known that such an ordinal grammar $G'$ exists but the proof is nonconstructive), then the order-equivalence problem is decidable for well-ordered context-free languages. As any finite system $E$ of fixed point equations over variables taking ordinals as values can effectively by transformed into an ordinal grammar $G$ such that $o(L(G))$ coincides with the least fixed point of the first component of $E$ [@DBLP:journals/fuin/BloomE10], we also get as a byproduct that the Cantor normal form of an algebraic ordinal, given by a finite system of fixed point equations, is effectively computable. Thus, the isomorphism problem of algebraic ordinals is decidable. Ordinal grammars ================ In this section we recall some known properties of ordinal grammars and then we prove that the order type of the lexicographic ordering of a language, given by an ordinal grammar, is computable. It is known from [@DBLP:journals/fuin/BloomE10; @10.1007/978-3-642-29344-3_25] that the following are equivalent for an ordinal $\alpha$: 1. $\alpha<\omega^{\omega^\omega}$. 2. $\alpha=o(L(G))$ for a context-free grammar $G$. 3. $\alpha=o(L)$ for a deterministic context-free language $L$. 4. $\alpha=o(L(G))$ for an ordinal grammar $G$. If $G=(N,\Sigma,P,S)$ is a context-free grammar, we define the relation $\preceq$ on $N\cup\Sigma$ as follows: $Y\preceq X$ if and only if $X\Rightarrow^*\balpha Y\bbeta$ for some $\balpha,\bbeta\in(N\cup\Sigma)^*$. Clearly, $\preceq$ is reflexive and transitive (a preorder): $X\approx Y$ denotes that $X\preceq Y$ and $Y\preceq X$ holds. An equivalence class of $\approx$ is called a *component* of $G$. If $Y\preceq X$ and they do not belong to the same component, we write $Y\prec X$. As an extension, when $\balpha=X_1\ldots X_n$ is a sentential form with $X_i\prec X$ for each $i\in[n]$, we write $\balpha\prec X$. Productions of the form $X\to\balpha$ with $\balpha\prec X$ are called *escaping* productions, the others (when $X_i\approx X$ for some $i\in[n]$) are called *component* productions. A nonterminal $X$ is called *recursive* if $X\Rightarrow^+\balpha X\bbeta$ for some $\balpha,\bbeta\in(N\cup\Sigma)^*$. The following are known for ordinal grammars having only usable nonterminals: \[lem-product\] If $G$ is an ordinal grammar, then for any word $X_1\ldots X_n\in(\Sigma\cup N)^*$, $o(X_1\ldots X_n)=o(X_n)\times o(X_{n-1})\times\ldots\times o(X_1)$. We will frequently use the above Lemma in the following form: if $X\to X_1\ldots X_n$ is a production of the ordinal grammar $G$ (and thus $L(X_1\ldots X_n)\subseteq L(X)$), then $o(X_n)\times o(X_{n-1})\times\ldots\times o(X_1)\leq o(X)$. \[lem-u0\] To each recursive nonterminal $X$ there exists a nonempty word $u_X$ such that if $X\Rightarrow^+ uX\balpha$ for some $u\in\Sigma^*$ and $\balpha\in(N\cup\Sigma)^*$, then $u\in u_X^+$. Moreover, whenever $X\Rightarrow^*w$ for some word $w$, then $w<_s u_X^\omega$. \[lem-monotone\] If $Y\preceq X$ for the symbols $X,Y\in N\cup\Sigma$, then $o(Y)\leq o(X)$. So if $X\approx Y$, then $o(X)=o(Y)$. For the rest of the section, let $G=(N,\Sigma,P,S)$ be an ordinal grammar. Since it is decidable whether $L(G)$ is finite, and in that case its order type $o(G)=|L(G)|$ is computable, we assume from now on that $L(G)$ is infinite. Without loss of generality we can assume that $G$ is in *normal form*: - $G$ has only usable nonterminals: for each $X$, there are words $u,v,w\in\Sigma^*$ with $S\Rightarrow^*uXv$ and $X\Rightarrow^*w$. - $L(X)$ is infinite for each nonterminal $X$; - Each production in $P$ has the form $A\to a\balpha$ for some $A\in N$, $a\in\Sigma$ and $\balpha\in(N\cup\Sigma)^*$; - All nonterminals different from $S$ are recursive. To see that such a normal form is computable, consider the following sequence of transformations, starting from an ordinal grammar $G$: 1. Unusable nonterminals are eliminated applying the usual algorithm [@Hopcroft+Ullman/79/Introduction]. 2. If $L(A)$ is finite for some nonterminal $A$, then $A$ gets replaced by all the members of $L(A)$ on each right-hand side and gets erased from the set of nonterminals. The result of this transformation is still an ordinal grammar. 3. In particular, if $A\Rightarrow^*\varepsilon$, then by prefix-freeness of $L(A)$ we get that $L(A)=\{\varepsilon\}$, so after this step no $\varepsilon$-transitions remain. 4. Chain rules of the form $A\to B$ with $A,B\in N$ also get eliminated by the usual algorithm which still outputs an ordinal grammar as the generated languages do not change. 5. By Lemma \[lem-u0\], there are no left-recursive nonterminals, that is, no $A\in N$ with $A\Rightarrow^+A\balpha$ for some $\balpha\in(N\cup\Sigma)^*$. Hence, the relation $B<A$ if $A\Rightarrow^+B\balpha$ for some $\balpha\in(N\cup\Sigma)^*$ is a partial ordering. Thus, if we replace each rule of the form $A\to B\balpha$ by $A~\to~\bbeta_1\balpha~|~\bbeta_2\balpha~|~\ldots~|~\bbeta_k\balpha$ where $\bbeta_1,\ldots,\bbeta_k$ are all the alternatives of $B$, the process eventually terminates. 6. Finally, if $X\neq S$ is a nonrecursive nonterminal with $X~\to~\balpha_1~|~\ldots~|~\balpha_n$ being all the alternatives of $X$, let us erase $X$ from $N$ and replace $X$ by one of the $\balpha_i$’s in all possible ways in the right-hand sides of the productions. Clearly, this transformation does not change $L(Y)$ for any $X\neq Y$ and reduces the number of nonterminals in $G$. Applying this transformation for each nonrecursive nonterminal different from $S$ in some arbitrary order now results in an ordinal grammar in normal form. Clearly, for each $X$ it is decidable whether it is recursive, and if so, then an $u\in\Sigma^+$ can be computed for which $X\Rightarrow^+uX\balpha$ for some $\balpha\in(N\cup\Sigma)^*$. Thus, $u_X$ can be chosen as the (still computable) primitive root [@shyr1991free] of $u$. We can show also the following: \[lem-normal-form\] If $G=(N,\Sigma,P,S)$ is an ordinal grammar in normal form, then for each rule $X\to X_1\ldots X_n$ in $P$ one of the following holds: 1. either the production is an escaping one (clearly, for a nonrecursive nonterminal this is the only option), 2. or $X_i\approx X$ for a unique index $i\in[n]$, and $X_j\in\Sigma$ for each $j<i$. Assume that there is a production $X\to X_1\ldots X_n$ for which none of the conditions hold. This can happen in the following two cases: 1. If there are at least two indices $i<j$ with $X_i\approx X_j\approx X$, then by Lemma \[lem-product\] we get $\alpha\times o(X)\times \beta\times o(X)\times \gamma\leq o(X)$ for some nonzero ordinals $\alpha,\beta$ and $\gamma$, which is nonsense since if $G$ is in normal form, $L(X)$ is infinite, thus $o(X)>1$. 2. Similarly, assume there is a unique index $i\in[n]$ with $X_i\approx X$ (thus, $X_j\prec X$ for each $j\neq i$) and $X_j$ is a nonterminal for some $j<i$. Then again by Lemma \[lem-product\] we get $\alpha\times o(X_i)\times\beta\times o(X_j)\times\gamma\leq o(X)=o(X_i)$ for some nonzero ordinals $\alpha,\beta$ and $\gamma$. Since with $X_j$ being a nonterminal we have $o(X_j)>1$, this is again a contradiction. Operations on languages ----------------------- In this subsection we aim to show that whenever $\balpha\in(N\cup\Sigma)^*$ for some ordinal grammar $G=(N,\Sigma,P,S)$ in normal form, both the supremum $\bigvee L(\balpha)$ and whether $\bigvee L(\balpha)$ is a member of $L(\balpha)$ or not, are computable and also a technical decidability lemma which will be used in the proof of Theorem \[thm-computable-recursive-themingeszishere\]. Let $X$ be a recursive nonterminal. By Lemma \[lem-u0\], for each $X\Rightarrow^+w$ we have $w<_s u_X^\omega$, so $u_X^\omega$ is an upper bound of $L(X)$. It is also clear that if $X\Rightarrow^+u_X^tXv$, then $X\Rightarrow^+u_X^{t\cdot k}Xv^k$ for every $k\geq 0$. Hence for any integer $N>0$ there is a word $w\in L(X)$ (say, $w=u_X^{N\cdot t}w'v^N$ where $w'\in L(X)$ is an arbitrary fixed word) such that $u_X^N<_\ell w$, and as $\mathop\bigvee\limits_{N\geq 0}u_X^N=u_X^\omega$, we immediately get: \[lem-recursive-supremum\] Suppose $X$ is a recursive nonterminal. Then $\mathop\bigvee L(X)=u_X^\omega$. (Thus in particular, there is no largest element in $L(X)$, since $L(X)$ consists of finite words only.) It is obvious that for any $a\in\Sigma$ we have $\bigvee L(a)=a$ and $a\in L(a)$. For the case of nonrecursive nonterminals (that can be at most $S$) we need to handle the operations union and product. For union, we of course have $\bigvee (K\cup L)~=~\bigvee K\vee\bigvee L$ and this element $u$ belongs to $K\cup L$ if and only if $u=\bigvee K$ and $u\in K$, or $u=\bigvee L$ and $u\in L$ holds. For product, we state a useful property first: \[prop-strict\] If $L$ is prefix-free and $\bigvee L$ exists, then either $L<_s\bigvee L$, or $\bigvee L\in L$ holds. Assume neither of the two cases hold for the supremum of $L$. Then, since $\bigvee L\notin L$, we have $L<_\ell \bigvee L$. Thus, since $L\nless_s\bigvee L$, there is a word $u\in L$ with $u\nless_s\bigvee L$ and $u<_\ell \bigvee L$, hence $u<_p\bigvee L$. But since $L$ is prefix-free, there is no word $v\in L$ with $u<_pv$, thus – as there is no largest element in $L$ by $\bigvee L\notin L$ – there is a word $v\in L$ with $u<_s v$. But as $u<_p\bigvee L$, this yields $\bigvee L<_s v$, a contradiction since $v<_\ell\bigvee L$ has to hold. This proposition entails the following: \[cor-strict\] If $K$ and $L$ are nonempty prefix-free languages and both $\bigvee L$ and $\bigvee K$ exist, then $$\begin{aligned} \bigvee (KL) &= \begin{cases} \bigvee K&\hbox{ if }K<_s\bigvee K;\\ \bigvee K\cdot \bigvee L&\hbox{ otherwise}, \end{cases}\end{aligned}$$ and $\bigvee(KL)\in KL$ if and only if $K\in \bigvee K$ and $L\in\bigvee L$. If $K<_s\bigvee K$, then $K\Sigma^*<_s\bigvee K$, so $\bigvee K$ is an upper bound of $KL$ in that case. To see it’s the smallest one, assume $u<_\ell\bigvee K$. Since $\bigvee K$ is the supremum of $K$ with respect to the total ordering $<_\ell$, this means $u<_\ell v$ for some $v\in K$. But for this $v$ and an arbitrary $w\in L$ we still have $u<_\ell vw$, hence $u$ cannot be an upper bound of $KL$. Thus, $\bigvee K=\bigvee(KL)$. If $u=\bigvee K\in K$, then for any word $v\in K$ and $w\in L$ we have either $v<_s u$, in which case $vw<_s ux$ for any word $x\in\Sigma^{\leq\omega}$, or $v=u$, in which case $vw\leq_\ell u\bigvee L$ since $w\leq_\ell \bigvee L$. Thus, $\bigvee K\cdot\bigvee L$ is an upper bound of $\bigvee(KL)$. Again, if $v<_\ell u\bigvee L$ for some $v$, then either $v<_\ell u$ in which case $v<_\ell uw\in KL$ for any $w\in L$, thus $v$ cannot be the supremum of $KL$, or $u<_pv$ in which case $v=uw$ for some $w$ with $w<_\ell\bigvee L$. This in turn implies the existence of some $w'\in L$ with $w<_\ell w'$, thus $v=uw<_\ell uw'\in KL$, hence $v$ cannot be an upper bound of $KL$, showing the claim. The statement on membership is clear. \[cor-sup-computable\] For any ordinal grammar $G=(N,\Sigma,P,S)$ in normal form and $\balpha\in(N\cup\Sigma)^*$, the supremum $\bigvee L(\balpha)$ is computable and one of the following cases holds: - $\bigvee L(\balpha)=u$ for some finite $u\in\Sigma^*$, and $u\in L(\balpha)$; - $\bigvee L(\balpha)=uv^\omega$ for some finite $u\in\Sigma^*$ and $v\in\Sigma^+$, and (of course) $uv^\omega\notin L(\balpha)$. We already established $\bigvee L(X)=u_X^\omega$ when $X$ is a recursive nonterminal and that $\bigvee L(a)=a\in L(a)$ for terminals $a\in \Sigma$. Also, for any $\balpha=X_1X_2\ldots X_n\in (N\cup\Sigma)^+$ we can compute $\bigvee L(\balpha)$ with the recursion $$\begin{aligned} \bigvee L(X_1\ldots X_n) &=\begin{cases} \varepsilon&\hbox{if }n=0\\ \bigvee(X_1)&\hbox{if }n>0\hbox{ and }\bigvee X_1=uv^\omega\hbox{ for some }u\in\Sigma^*,v\in\Sigma^+\\ u\cdot\bigvee L(X_2\ldots X_n)&\hbox{if }n>0\hbox{ and }\bigvee X_1=u\in\Sigma^* \end{cases} \end{aligned}$$ using Corollary \[cor-strict\]. Then, if $X=S$ is a nonrecursive nonterminal and $X~\to~\balpha_1~|~\balpha_2~|\ldots|~\balpha_n$ are all the alternatives for $X$, then we have $\bigvee L(X)=\bigvee\limits_{i=1}^n L(\balpha_i)$, which yields an inductive proof for the only possible nonrecursive nonterminal $S$. Concluding the subsection, we show the following technical lemma: \[lem-transducer\] It is decidable for any context-free language $L\subseteq\Sigma^*$ and words $u,v$, whether there exists an integer $N\geq 0$ such that $uv^N\Sigma^*\cap L~=~\emptyset$. (If so, then $uv^M\Sigma^*\cap L=\emptyset$ for each $M\geq N$.) Let us define the following generalized sequential mappings $f,g:\Sigma^*\to a^*$: let $$\begin{aligned} f(x)&=\begin{cases} g(y)&\hbox{if }x=uy\\ \varepsilon&\hbox{otherwise,} \end{cases} & g(x)&=\begin{cases} a\cdot g(y)&\hbox{if }x=vy\\ \varepsilon&\hbox{otherwise.} \end{cases} \end{aligned}$$ We have that if $uv^N\Sigma^*\cap L$ is nonempty, then $f(L)$ contains some word of length at least $N$, and also, if $a^N\in f(L)$, then $uv^N\Sigma^*\cap L$ is nonempty. Thus, there is such an integer $N$ satisfying the condition of the lemma if and only if $f(L)$ is finite, which is decidable, since the class of context-free languages is effectively closed under generalized sequential mappings [@Ginsburg:1966:MTC:1102023]. The order type of recursive nonterminals ---------------------------------------- In this subsection we show that $o(X)$ is computable, whenever $X$ is a recursive nonterminal of an ordinal grammar $G=(N,\Sigma,P,S)$. Clearly, for each $a\in\Sigma$ we have $o(L(a))=1$. We will apply induction on the *height* of $X$, defined as the length of the longest chain $X_1\prec X_2\prec\ldots\prec X_n=X$ with each $X_i$ in $N\cup\Sigma$. (Thus, the height of the terminals is $0$, nonterminals have positive height.) Since $X$ is a recursive nonterminal, by Lemma \[lem-u0\] there is a (shortest, computable) nonempty word $u_X$ such that 1. $w<_su_X^\omega$ for each $w\in L(X)$; 2. whenever $X\Rightarrow^+uX\balpha$ for some $u\in\Sigma^*$ and $\balpha\in(N\cup\Sigma)^*$, then $u\in u_X^+$. This also implies that whenever $X$ and $Y$ are nonterminals belonging to the same component, then there is a unique word $u_{(X,Y)}<_pu_X$ such that $u_X^\omega = u_{(X,Y)}u_Y^\omega$. Moreover we have: \[prop-betakisebb\] If $Y\to\bbeta$ is an escaping production for $X\approx Y$, then $u_{(X,Y)}\cdot L(\bbeta)<_su_X^\omega$. In this case, $X\Rightarrow^+u_{(X,Y)}Y\balpha$ for some sentential form $\balpha$. Since $L(\bbeta)\subseteq L(Y)<_s u_Y^\omega$, we get $u_{(X,Y)}\cdot L(\bbeta)<_s u_{(X,Y)}u_Y^\omega=u_X^\omega$. Now by Lemma \[lem-normal-form\] we can deduce that any (leftmost) derivation from $X$ has the form $$\begin{aligned} \label{eq-leftmost} X&\Rightarrow~u_1X_1\balpha_1~\Rightarrow~u_1u_2X_2\balpha_2\balpha_1~\Rightarrow~\ldots\\&\Rightarrow~u_1u_2\ldots u_nX_n\balpha_n\ldots\balpha_2\balpha_1~\Rightarrow~u_1u_2\ldots u_n\bbeta\balpha_n\ldots\balpha_2\balpha_1 ~\Rightarrow^*~w\nonumber\end{aligned}$$ for some integer $n\geq 0$, nonempty words $u_1,\ldots,u_n\in\Sigma^+$ with $u_1\ldots u_n<_pu_X^\omega$, sentential forms $\balpha_1,\ldots,\balpha_n, \bbeta \in(N\cup\Sigma)^*$ with $\bbeta\prec X$, $X_i\approx X$ and $\balpha_i\prec X$ for each $i\in[n]$. By induction, $o(\bbeta)$ is computable (applying Lemma \[lem-product\]) for each possible $\bbeta\prec X_i$ with $X_i\approx X$ and production $X_i\to \bbeta$. Moreover, $o(\balpha)$ is also computable for each $\balpha\prec X_i$ with a production $X_i\to u_iX_{i+1}\balpha$, $X_{i+1}\approx X_i$ as there are only finitely many such productions. Let $v_1<_s v_2<_s\ldots<_sv_\ell$ be the complete enumeration of those words $v_i$ with $v_i<_s u_X$ having the form $v_i=ua$ with $u<_pu_X$. Observe that $L=L(X)$ is the disjoint union of languages of the form $u_X^Nv_i\Sigma^*~\cap~L(X)$, with $N\geq 0$ and $1\leq i\leq \ell$. Moreover, whenever $u\in u_X^Nv_i\Sigma^*$ and $v\in u_X^Mv_j\Sigma^*$, then $N<M$ or ($N=M$ and $i<j$) implies $u<_sv$. Thus, these languages form an $\omega$-sequence with respect to the lexicographic ordering and we can write $L$ as $$L~=~L_1+L_2+L_3+\ldots$$ We will construct an increasing sequence of ordinals $$o_1~\leq~o_2~\leq~o_3~\leq~\ldots$$ such that the following hold: - for each $i\geq 1$, there is a $j\geq 1$ with $o(L_i)\leq o_j$ and - for each $j\geq 1$, there is an $i\geq 1$ with $o_j\leq o(L_i)$. This implies $o(L)~=~o_1~+~o_2~+~o_3~+~\ldots$. Indeed: by the first condition we have $$o(L) ~=~o(L_1)~+~o(L_2)~+~\ldots\\ ~\leq~o_{f(1)}~+~o_{f(2)}~+~\ldots$$ for some indices $f(1)$, $f(2)$ and so on. Let us define for each $j$ the index $g(j)$ as follows: $g(1)=f(1)$ and for each $j>1$, let $g(j)=\max\{g(j-1)+1,f(j)\}$. Then we have $o(L)\leq o_{g(1)}+o_{g(2)}+\ldots$ and $g(1)<g(2)<\ldots$. Thus, $o(L)\leq o_1+o_2+\ldots$ holds (as the former order type is a sub-order type of the latter), the other direction being symmetric. Let us now consider one such language $L_t$. Then, $L_t$ is a finite union of languages of the form $$\begin{aligned} \label{eqn-theyhavethisform} u_1u_2\ldots u_nL'L(\balpha_n)\ldots L(\balpha_2)L(\balpha_1)\end{aligned}$$ where $u_1\ldots u_n<_p (u_X)^N$ for some $N$ depending only on $t$, moreover, applying Proposition \[prop-betakisebb\] we get that each such $L'$ has the form $\bigl((u_1\ldots u_n)^{-1}u_X^Nv_j\bigr)\Sigma^*~\cap~L(\bbeta)~=~u_{X'}^{M}v\Sigma^*~\cap~L(\bbeta)$, and for each $i\geq 0$ there is a production of the form $X_i\to u_iX_{i+1}\balpha_i$ (recall that due to the normal form each $u_i$ is nonempty) for some nonterminals $X_i\approx X$, $X_1=X$ and $X_{n+1}\to\bbeta$ with $\bbeta\prec X$. Clearly, for any fixed $N$ and $v_i$, there are only finitely many such choices. We do not have to explicitly compute the order type of each such $L_t$, due to the following lemma: \[lem-uniodeg\] Assume $o_1\leq o_2\leq \ldots$ is a sequence of ordinals and $K$, $L$ are languages with $\deg(o(L))$, $\deg(o(K))<\deg\bigl(\bigvee o_i\bigr)$. Then $o(K\cup L)<o_j$ for some index $j$. Without loss of generality, let $o(L)\leq o(K)$. By Theorem \[thm-union\] we have that $o(K\cup L)<\omega^{\deg(o(K))}\times T$ for some integer $T$. It suffices to show that for each integer $T>0$, there exists an $o_i$ with $o_i> \omega^{\deg(o(K))}\times T$. Assume to the contrary that each $o_i$ is at most $\omega^{\deg(o(K))}\times T'$ for some integer $T'$. But then, $\bigvee o_i\leq \omega^{\deg(o(K))}\times T'$ and thus $\deg(\bigvee o_i)\leq \deg(o(K))$, a contradiction. Equipped by our lemmas we are ready to prove the (technically most involved) main result of the subsection: \[thm-computable-recursive-themingeszishere\] Assume $G$ is an ordinal grammar in normal form and $X$ is a recursive nonterminal. Let $o_\balpha$ be the maximal order type of some $L(\balpha)$ for which a component production of the form $X'\to uX''\balpha$ exists in $G$ for some $X\approx X'$, and $o_\bbeta$ be the maximal order type of some $L(\bbeta)$ with $X'\to \bbeta$ being an escaping production of $G$ with $X'\approx X$. Then the order type of $L(X)$ is: 1. ${(o_\balpha)}^\omega$ if $o_\bbeta<{( o_\balpha)}^\omega$; 2. $o_\bbeta$ if $o_\bbeta=\omega^{\deg(o_\bbeta)}$ and for each escaping production $X'\to\bbeta$ with $\deg(o(L(\bbeta)))=\deg(o_\bbeta)$, the language $u_{X'}^N\Sigma^*\cap L(\bbeta)$ is nonempty for infinitely many integers $N\geq 0$; 3. $o_\bbeta\times\omega$, otherwise. So let $o_\balpha$ be the ordinal $\max\{o(L(\balpha)):~X'\to uX''\balpha\hbox{ is a production for some }X'\approx X''\approx X\}$. Since there are only finitely many such $\balpha$, and $\balpha\prec X$ holds for each of them, $o_\balpha$ is well-defined and computable by induction. Also, let $o_\bbeta$ be $\max\{o(L(\bbeta)):~X'\to \bbeta\hbox{ is a production for some }X'\approx X,\bbeta\prec X\}$. This ordinal $o_\bbeta$ is well-defined and computable as well. We also use the shorthands $\gamma=\deg(o_\balpha)$ and $\delta=\deg(o_\bbeta)$. These ordinals are also computable (as an ordinal “being computable” means in our context that the Cantor normal form of the ordinal is computable). Now we apply a case analysis, based on $\delta$ and $\gamma$. We note next to these (sub, subsub)cases to which case of the theorem they correspond. Case 1: $\delta<\gamma\times\omega$ {#case-1-deltagammatimesomega .unnumbered} ----------------------------------- This case corresponds to Case $1$ of the theorem. We claim that in this case $o(X)={(o_\balpha)}^\omega$. To see this, it suffices to show for each integer $N\geq 0$ that ${(o_\balpha)}^N<o(X)$ and that there is an $L_i$ with $o(L_i)<{(o_\balpha)}^N$. For ${(o_\balpha)}^N<o(X)$, let $X'\to uX''\balpha$ be a component production with $o(L(\balpha))=o_\balpha$ and let $u_0,v_0,u_1,v_1\in\Sigma^*$ be so that $X''\Rightarrow^* u_1X'v_1$ and $X\Rightarrow^* u_0X'v_0$. Finally, let $w\in L(X')$. Then we have $$X\Rightarrow^*u_0(uu_1)^Nw(v_1\balpha)^Nv_0.$$ Since by Lemma \[lem-product\] the order type of the language generated by this sentential form is at least ${(o_\balpha)}^N$, and this language is a subset of $L(X)$, this direction is proved. For the other direction, note that $\deg({(o_\balpha)}^\omega)=\gamma\times\omega$. Thus, since each $L_i$ is a finite union of languages of the form \[eqn-theyhavethisform\], in which $L'\subseteq L(\bbeta)$ for some $\bbeta$, by Lemma \[lem-uniodeg\] it suffices to show that $$\deg(o(u_1\ldots u_nL(\bbeta)L(\balpha_n)L(\balpha_{n-1})\ldots L(\balpha_1)))<\gamma\times\omega.$$ But, as each $o(\balpha_i)$ is at most $o_\balpha$ and $o(\bbeta)\leq o_\bbeta$, we get that this sentential form has the order type at most ${(o_\balpha)}^n\times o_\bbeta$. We have that $\deg({(o_\balpha)}^n\times o_\bbeta)=\gamma\times n+\delta$ which is smaller than $\gamma\times\omega$ if so is $\delta$ and the claim is proved. Case 2: $\gamma\times\omega\leq \delta$ {#case-2-gammatimesomegaleq-delta .unnumbered} --------------------------------------- Observe that this case applies if and only if $\deg(\gamma)<\deg(\delta)$ and that this cannot happen within Case $1$ of the theorem. We split the analysis of this case to several subcases. For each escaping production $X'\to\bbeta$ with $\deg(o(\bbeta))=\delta$, we decide whether there exists an $N\geq 0$ such that $u_{X'}^N\Sigma^*\cap L(\bbeta)~=~\emptyset$. By Lemma \[lem-transducer\], this is decidable. ### Subcase 2.1: $\gamma\times\omega\leq\delta$ and there exists a $\bbeta$ such that $u_{X'}^N\Sigma^*\cap L(\bbeta)~=~\emptyset$ for some $N$ {#subcase-2.1-gammatimesomegaleqdelta-and-there-exists-a-bbeta-such-that-u_xnsigmacap-lbbetaemptyset-for-some-n .unnumbered} This subcase rules out Case $2$ of the theorem by the condition $u_{X'}^N\Sigma^*\cap L(\bbeta)~=~\emptyset$, so this subcase falls under Case $3$ of the theorem, and we claim $o(X)=o_\bbeta\times\omega$ in this subcase. In this subcase, $L(\bbeta)$ is a finite union of languages of the form $K_{N,v}~=~u_{X'}^Nv\Sigma^*~\cap~L(\bbeta)$ for some word $v=v'a<_su_{X'}$ with $v'<_p u_{X'}$ (see Figure \[fig-tree\]). Thus, there is one $K_{N,v}$ among these languages with $\deg(o(K_{N,v}))=\delta$ (since the degree of this finite union is $\delta$). Such a language is a subset of a factor $L'$ of a language of the form (\[eqn-theyhavethisform\]), moreover, such an $L'$ occurs as a factor in infinitely many languages $L_i$: if $X'\Rightarrow^+ u_{X'}^tX'\boldsymbol{\alpha}$, and $K_{N,v}$ is a subset of one of the languages $L'$ belonging to $L_i$, then it also belongs to the same factor $L'$ of $L_{i+t}$. Hence, we have the lower bound $\omega^\delta\times\omega=\omega^{\delta+1}={o_\bbeta}\times\omega\leq o(X)$. To see that this is an upper bound as well, it suffices to show that each language of the form (\[eqn-theyhavethisform\]) has an order type less than $o_\bbeta\times\omega$, that is, has a degree at most $\delta$. Again, similarly to Case 1 we get that the order type of such a language is upperbounded by ${(o_\balpha)}^n\times o_\bbeta$ whose degree is $\gamma\times n+\delta$ which is $\delta$ since the degree of $\gamma$ is smaller than the degree of $\delta$. (In this case it can happen that $o_\balpha<\omega$ but for finite powers, $\deg(\alpha^n)=\deg(\alpha)\times n$ still holds.) Thus, in this subcase the order type of $L(X)$ is $o_\bbeta\times\omega$. (0,10) node\[anchor=south\][$X$]{} – (3,7) node\[anchor=north\] – (-3,7) node\[anchor=south\] – cycle; (-3,7) – (0,7) node \[black,midway,yshift=-0.6cm\] [$u^M_Xu_{X,X'}$]{}; (0,7) – (3,7) node \[black,midway,yshift=-0.6cm\] [$\balpha_m\ldots\balpha_1$]{}; (0,7) node\[anchor=south\][$X'$]{} – (1.5,4) node\[anchor=north\] – (-1.5,4) node\[anchor=south\] – cycle; (-0.5,6) – node\[anchor=north\][$\bbeta$]{} (0.5,6); (-1.5,4) – (1,4) node \[black,midway,yshift=-0.6cm\] [$(u_{X'}^N)v$]{}; ### Subcase 2.2: $\gamma\times\omega\leq\delta$ and for all $\bbeta$ and $N$, $u_{X'}^N\Sigma^*\cap L(\bbeta)~\neq~\emptyset$ {#subcase-2.2-gammatimesomegaleqdelta-and-for-all-bbeta-and-n-u_xnsigmacap-lbbetaneqemptyset .unnumbered} In this subcase, the order type of each such $\bbeta$ can be written as an infinite sum of nonempty ordinals $o_\bbeta=o_{\bbeta_1}+o_{\bbeta_2}+\ldots$, $L(\bbeta)$ being the ordered disjoint union of the nonempty languages $K_{N,v}$. Now again, we have two subsubcases: either $o_\bbeta=\omega^\delta$ (this subsubcase corresponds to Case $2$ of the theorem) or $o_\bbeta>\omega^\delta$ (which in turn falls under Case $3$ of the theorem as well). [**If $o_\bbeta=\omega^\delta$**]{}, then the degree of each such $o_{\bbeta_i}$ is strictly smaller than $\delta$. In this case, each language of the form (\[eqn-theyhavethisform\]) has an order type at most ${(o_\balpha)}^n\times o$ for some $o$ with $\deg(o)=\delta'<\delta$, the degree of which ordinal is $\gamma\times n+\delta'$. Since $\deg(\gamma\times n)<\deg(\delta)$, we have $\gamma\times n+\delta'<\gamma\times n+\delta=\delta$, thus each such language $L_i$ has a degree still strictly smaller than $\delta$. Thus, $o_\bbeta=\omega^\delta$ is an upper bound for $o(X)$ in this case. Since $o(\bbeta)$ occurs as a subordering in $o(X)$, we also have $o_\bbeta\leq o(X)$, thus $o(X)=o_\bbeta$ in this subsubcase. [**If $o_\bbeta>\omega^\delta$**]{}, then there exists an $o_{\bbeta_i}$ with degree $\delta$. Proceeding with the argument exactly as in Subcase 2.1, we get that $o(L)=o_\bbeta\times\omega$ in this subsubcase. Thus in particular, as each condition is decidable if the order types $o(\bbeta)$ and $o(\balpha)$ are computable, which are, applying the induction hypothesis, we get decidability: \[thm-recursive\] Assume $G$ is an ordinal grammar in normal form and $X$ is a recursive nonterminal. Then $o(X)$ is computable. The order type of nonrecursive nonterminals ------------------------------------------- Recall that if $G$ is an ordinal grammar in normal form, then its only nonrecursive nonterminal can be its starting symbol $S$. Thus, if $\balpha_1,\ldots,\balpha_n$ are all the alternatives of $S$, then $L(G)=\mathop\bigcup\limits_{i=1}^nL(\balpha_i)$ and all the $\balpha_i$s consist of terminal symbols and recursive nonterminals, whose order type is already known to be computable. Hence we only have to show that the following problem is computable: - [**Input:**]{} An ordinal grammar $G=(N,\Sigma,P,S)$ (in normal form), and a finite set $\{\balpha_1,\ldots,\balpha_n\}$ of sentential forms such that for each symbol $X$ occurring in the set, $o(X)$ is known. - [**Output:**]{} The order type of $L=\mathop\bigcup\limits_{i=1}^nL(\balpha_i)$. We claim that the following algorithm $A$ solves this problem: ``` {mathescape="true" style="myScalastyle"} function $A(\{\balpha_1,\ldots,\balpha_n\})$ if( $n$ == $0$ ) return $0$ $\Right$ := $\{\balpha_1,\ldots,\balpha_n\}$ $\Left$ := $\emptyset$ $u$ := $\varepsilon$ while( true ) { $w$ := $\max\{\bigvee L(\balpha) :\balpha\in\mathrm{Right} \}$ $\Right_1$ := $\{\balpha\in\Right:~\bigvee L(\balpha)<w\}$ $\Right_2$ := $\{\balpha\in\Right:~\bigvee L(\balpha)=w\}$ $o$ := $\max\{o(L(\balpha)):\balpha\in\mathrm{Right}_2\}$ if( $o=\omega^\gamma$ for some $\gamma$ ) Let $w'$ be a finite prefix of $w$ such that for each $\balpha\in\Right_1$, $L(\balpha)<w'$ already holds. return $A(\mathrm{Left})+A\Bigl(\Right_1\cup\bigl\{(\balpha^{<{w'}}):\balpha\in\Right_2\bigr\}\Bigr)~+~\omega^\gamma$ Let $a$ be the largest letter of $\Sigma$ such that there exists some $a\balpha\in\mathrm{Right}$ $\mathrm{Left}$ := $\mathrm{Left}\cup\{u\cdot\balpha:\balpha\in\mathrm{Right},~\mathrm{First}(\balpha)\neq a\}$ $\mathrm{Right}$ := $a^{-1}\mathrm{Right}$ $u$ := $u\cdot a$ $\mathrm{Right}$ := $\{\boldsymbol{\delta}\balpha':\exists X\to\boldsymbol{\delta}\in P,X\balpha'\in\mathrm{Right}\}~\cup~\{\balpha:\alpha\in\mathrm{Right},~\mathrm{First}(\balpha)\notin N\}$. } ``` In the above algorithm, for a sentential form $\balpha=X\cdot\balpha'$, $\mathrm{First}(\balpha)=X$ and $\mathrm{First}(\varepsilon)=\varepsilon$. We use induction on $o(L)$ to show that the algorithm always terminates, and it does so with the right answer. Since $G$ is in normal form, we can restrict the proof to those cases when each $\balpha_i$ is either $\varepsilon$ or starts with a terminal symbol. If this order type is $0$, then (since each nonterminal is productive as $G$ is in normal form) $n=0$ has to hold, in which case the algorithm indeed returns $0$. Now assume $o(L)>0$, thus $n>0$. For the sake of convenience, let $L(\mathrm{Left})$ stand for the language $\bigcup_{\bbeta\in\mathrm{Left}}L(\bbeta)$ and similarly for $L(\Right)$. We claim that the following invariants are preserved in the loop of the algorithm: $$\begin{aligned} L(\Left)&< u& &\hbox{and}& L&=L(\Left)~\cup~u\cdot L(\Right).\end{aligned}$$ Also, $\Right\neq\emptyset$ and after each execution of Line 7, $u\cdot w=\bigvee L$ . Upon entering the loop, $\Left=\emptyset$ and from $u=\varepsilon$ we have $u\cdot L(\Right)=L(\Right)=L$. Within the loop, if $L=L(\Left)\cup u\cdot L(\Right)$ and $L(\Left)< u$ before executing Line 7, then $\bigvee L~=~\bigvee \bigl(u\cdot L(\Right)\bigr)~=~u\cdot \bigvee L(\Right)~=~u\cdot\max\{\bigvee L(\balpha):\balpha\in\Right\}$, thus indeed, $u\cdot w=\bigvee L$. Now assuming $L(\Left)< u$ holds when we start an iteration of the loop, we have to see that $L(\Left)\cup u\cdot\bigl(\bigcup L(\balpha):\balpha\in\Right,\First(\balpha)\neq a\bigr)<u\cdot a$ for the letter $a$ chosen during Line 14. The part $L(\Left)<u<u\cdot a$ is clear. The latter part is equivalent to $L(\balpha)<a$ holds for each $\balpha\in\Right$ with $\First(\balpha)\neq a$, which holds since if such an $\balpha$ begins with a terminal symbol $b$ then by the choice of $a$ we have $b<a$, and if $\balpha=\varepsilon$, then also $\varepsilon<a$, showing preservation of the property $L(\Left)<u$. It is also clear that the operation in Line 16 can’t make $\Right$ empty by the choice of $a$ (also, since $\Right$ is nonempty and by assumption, each $\balpha\in\Right$ begins with a terminal symbol, such a letter $a$ always exists: if $\Right=\{\varepsilon\}$, then $o=1=\omega^0$ and the algorithm terminates at Line 13). Assuming $L=L(\Left)~\cup~u\cdot L(\Right)$ when starting an iteration, after executing Line 17 we have to show that $L=L(\Left)\cup\{u\cdot L(\balpha):\balpha\in\Right,\First(\balpha)\neq a\}~\cup~u\cdot a\cdot L(a^{-1}\Right)$ for the original values of $\Left$ and $\Right$ to see preservance of this property. But this clearly holds for arbitrary set of sentential forms $\Left$ and $\Right$, thus this property is again a loop invariant. After executing Line 16, it may happen that $\Right$ contains some sentential form(s) starting with a nonterminal; executing Line 18 does not change $L(\Right)$ but restores the property of $\Right$ that each $\balpha\in\Right$ begins with a terminal symbol (or $\balpha=\varepsilon$). Now by the first two properties we have $o(L)~=~o(L(\Left))+o(u\cdot L(\Right))~=~o(L(\Left))+o(L(\Right))$. We show that this is exactly the ordinal we return in Line 13, should the condition of Line 11 hold. Consider the sets $\Right_1$ and $\Right_2$ of sentential forms. By the definition of $w$, $\Right_2$ is nonempty and $\Right=\Right_1\uplus\Right_2$. By the choice of $w'$, we have that $L(\Right_1)<w'$ and of course $L(\Right_2)=L({\Right_2}^{<w'})+L({\Right_2}^{\geq w'})$, thus $$L~=L(\Left)+\Bigl(L(\Right_1)\cup L({\Right_2}^{<w'})\Bigr)+L({\Right_2}^{\geq w'}).$$ Observe that such a $w'$ is computable as $w$ is a computable word (possibly having the form $xy^\omega$ for some finite words $x,y$ by Corollary \[cor-sup-computable\]), so its prefixes can be enumerated and for each prefix $w_0$, the emptiness of the context-free language $L(\balpha^{\geq w_0})$ can be decided for each $\balpha\in\Right_1$; as for these strings $\balpha$ we have $\bigvee L(\balpha)<w$, there is a finite prefix $w_0$ of $w$ with $L(\balpha)$ being already smaller than $w_0$. Thus, even the shortest such prefix $w'$ of $w$ can be computed. Since $L({\Right_2}^{\geq w'})$ is nonempty (as $w'<w=\bigvee L(\Right_2)$) we get that $o(L(\Left))$ and $o\Bigl(L(\Right_1)\cup L({\Right_2}^{<w'})\Bigr)$ are both strictly smaller than $o(L)$, thus applying the induction hypothesis we get that the algorithm terminates with a correct answer in Line 13 if $o(L({\Right_2}^{\geq w'}))=\omega^\gamma$. Since each nonempty suffix of $\omega^\gamma$ is itself, and $\omega^\gamma$ is the order type of at least one $L(\balpha)$ with $\balpha\in\Right_2$ by the choice of $o$, we have $o(\balpha^{\geq w'})=\omega^\gamma$, thus $\omega^\gamma\leq o(L({\Right_2}^{\geq w'}))$. For the lower bound, note that $L({\Right_2}^{\geq w'})$ is a finite union of languages $L_i$ such that for each $i$, $\bigvee L_i=w$ and $o(L_i)\leq \omega^\gamma$. If $\gamma=0$, then all these languages are singletons containing the word $w$ and the claim holds. Otherwise, none of the languages $L_i$ have a largest element and so for any word $v\in L({\Right_2}^{\geq w'})$ we have $o({L_i}^{< v})<\omega^\gamma$ (by that $v<\bigvee L({\Right_2}^{\geq w'})=\bigvee L_i=w$ and so ${L_i}^{\geq v}$ is nonempty) and so $L({\Right_2}^{<v})$ is a finite union of languages, each having an order type strictly less than $\omega^\gamma$, thus the union itself also has an order type less than $\omega^\gamma$. So each prefix of $o(L(\Right_2))$ is less than $\omega^\gamma$ which makes $o(L(\Right_2))\leq \omega^\gamma$ and the claim holds. Thus, if the algorithm makes a recursive call in Line 13, then it returns with a correct answer. We still have to show that the algorithm eventually terminates. To see this, observe that $u\cdot w=\bigvee L$ holds after each iteration of the loop and $u$ gets longer by one letter in each iteration. Hence, if the algorithm does not terminate, then the supremum of the values of the variable $u$ is $\bigvee L$. Since $o(L)\neq 0$, say $o(L)=\omega^{\gamma_1}\times n_1+\ldots +\omega^{\gamma_k}\times n_k$ for some integer $k>0$, integer coefficients $n_i>0$ and ordinals $\gamma_1>\gamma_2>\ldots>\gamma_k$, so there exists some word $x\in L$ with $o(L^{\geq x})=\omega^{\gamma_k}$. Clearly, after some finite number (say, $|x|$) of iterations we have $x<u$, this makes $o(L^{\geq u})=\omega^{\gamma_k}$, and by $u\cdot L(\Right)$ being a nonempty suffix of $L^{\geq u}$, we get that $o(L(\Right))=\omega^{\gamma_k}$: as $L(\Right_2)\subseteq L(\Right)$ is a finite union of languages, we have $o(L(\balpha))\leq \omega^{\gamma_k}$ for each $\balpha\in\Right_2$ and equality holds for at least one of them. Hence, the loop terminates in at most $|x|$ steps, finishing the proof of termination as well. Theorem \[thm-recursive\], in conjunction with the correctness of the above algorithm yields the main result of the paper: \[thm-main\] Given an ordinal grammar $G$, one can compute the order type $o(G)$ in Cantor normal form. Applying the construction of [@DBLP:journals/fuin/BloomE10], we get the following corollary: \[cor-main\] The Cantor normal form of an algebraic ordinal, given by a finite system of fixed point equations, is effectively computable. Thus, the isomorphism problem of algebraic ordinals is decidable. Conclusion and acknowledgement ============================== We have shown that the isomorphism problem of algebraic ordinals is decidable, studying the order types of well-ordered context-free languages, given by an ordinal grammar. It is an interesting question whether the proof can be lifted to scattered linear orders: in many cases, scattered linear orders behave almost as well as well-orders. Also, it would be interesting to analyze the runtime of our algorithm: we only know that by well-founded induction the computation eventually terminates. The authors wish to thank Prof. Zoltán Fülöp for the discussion on generalized sequential mappings on context-free languages.

--- abstract: | The results in [@O2] (see [@O1] for the quasistatics regime) consider the Helmholtz equation with fixed frequency $k$ and, in particular imply that, for $k$ outside a discrete set of resonant frequencies and given a source region $D_a\subset \RR^d$ ($d=\overline{2,3}$) and $u_0$, a solution of the homogeneous scalar Helmholtz equation in a set containing the control region $D_c\subset \RR^d$, there exists an infinite class of boundary data on $\partial D_a$ so that the radiating solution to the corresponding exterior scalar Helmholtz problem in $\RR^d\setminus D_a$ will closely approximate $u_0$ in $D_c$. Moreover, it will have vanishingly small values beyond a certain large enough “far-field" radius $R$ (see Figure \[fig:mainsetup\] for a geometric description). In this paper we study the minimal energy solution of the above problem (e.g. the solution obtained by using Tikhonov regularization with the Morozov discrepancy principle) and perform a detailed sensitivity analysis. In this regard we discuss the stability of the the minimal energy solution with respect to measurement errors as well as the feasibility of the active scheme (power budget and accuracy) depending on: the mutual distances between the antenna, control region and far field radius $R$, value of regularization parameter, frequency, location of the source. author: - Mark Hubenthal - Daniel Onofrei bibliography: - 'controlpaper.bib' title: Sensitivity analysis for active control of the Helmholtz equation --- Introduction ============ During recent years, there has been a growing interest in the development of feasible strategies for the control of acoustic and electromagnetic fields with one possible application being the construction of robust schemes for sonar or radar cloaking. One main approach controls fields in the regions of interest by changing the material properties of the medium in certain surrounding regions ([@Chan3; @Chan2; @Cummer; @Green2; @Green1; @Gunther-pr; @Pendry] and references therein). Several alternative techniques are proposed in the literature (other than transformation optics strategies) such as: plasmonic designs (see [@Alu] and references therein), strategies based on anomalous resonance phenomena (see [@Mil1; @Mil3; @Mil2]), conformal mapping techniques (see [@Ulf2; @Ulf1]), and complementary media strategies (see [@Chan]). In the applied community, active designs for the manipulation of fields appear to have occurred initially in the context of low-frequency acoustics (or active noise cancellation). Especially notable are the pioneering works of Lueg [@Lueg] (feed-forward control of sound) and Olson & May [@Olson-May] (feedback control of sound). The reviews [@Elliot; @Fuller; @Tsynkov; @T1; @Peake; @Peterson], provide detailed accounts of past and recent developments in acoustic active control. In the context of cloaking, the [**[*interior*]{}**]{} strategy proposed in [@Miller] employs a continuous active layer on the boundary of the control region while the [**[*exterior*]{}**]{} scheme discussed in [@OMV4; @OMV1; @OMV2; @OMV3] (see also [@CTchan-num]), uses a discrete number of active sources located in the exterior of the control region to manipulate the fields. The active exterior strategy for 2D quasistatics cloaking was introduced in [@OMV1], and, based on *a priori* information about the incoming field, the authors constructively described how one can create an almost zero field control region with very small effect in the far field. However, the proposed strategy did not work for control regions close to the active source. It “cloaked" large objects only when they are far enough from the source region (see [@OMV4]) and was not adaptable to three space dimensions. The finite frequency case was studied in the last section of [@OMV1] and in [@OMV3] (see also [@OMV4] for a recent review) where three (or four in 3D) active sources were needed to create a zero field region in the interior of their convex hull, while creating a very small scattering effect in the far field. The broadband character of the proposed scheme was numerically observed in [@OMV2]. All the above results were obtained assuming large amplitude and highly oscillatory currents on the active source regions. In this regard, in [@Norris] (see also [@Devaney0; @Miller]) the authors presented theoretical and numerical evidence that increasing the number of sources will decrease the power needed on each source and thus increase the feasibility of the scheme. Experimental designs and testing of active cloaking schemes in various regimes are reported in [@Du; @Ma; @Eleftheriades1; @Eleftheriades2]. In a recent development in [@O1], a general analytical approach based on the theory of boundary layer potentials is proposed for the active control problem in the quasi-static regime. By using the same integral equation approach, in [@O2] we extended the results presented in [@O1] to the active control problem for the exterior scalar Helmholtz equation. In particular, we characterized an infinite class of boundary functions on the source boundary $\partial D_a$ so that we achieve the desired manipulation effects in several mutually disjoint exterior regions. The method is novel in the sense that instead of using microstructures, exterior active sources modeled with the help of the above boundary controls are employed for the desired control effects. Such exterior active sources can represent velocity potential, pressure or currents. In the current paper we study the active control problem in the context of cloaking, where one antenna $D_a$ protects a given control region $D_c$ from far field interrogation on $\partial B_{R}(\mathbf{0})$, with $R \gg 1$ (see Figure \[fig:mainsetup\]). We make use of the results in [@O2] and present a detailed sensitivity and feasibility study for the minimal norm solution of the problem. The paper is organized as follows: In Section \[sec:background\] we recall the general result obtained in [@O2] in the context of exterior active cloaking. In Section \[sec:stability\] we present an $L^2$ conditional stability result for the minimal norm solution with respect to measurement errors of the incoming field. In Section \[sec:numerics\] we present the numerical details of the Tikhonov regularization algorithm with the Morozov discrepancy principle for the computation of the minimal norm solution of the exterior active cloaking problem in two dimensions. We will numerically observe the fact that the scheme requires large antenna powers in the far field and we will provide numerical support for our theoretical stability results. An important part of this section will be focused on the sensitivity analysis, where we will study: the dependence of the control results as a function of mutual distances between the antenna, control region and far field region; and the broadband character of our scheme in the near field region. Finally, in Section \[sec:conclusions\] we highlight the main results of the paper and discuss current and future challenges and extensions of our research. Background {#sec:background} ========== In this section we will recall the main result regarding the active exterior control problem for the Helmholtz equation obtained in [@O2]. We will focus only on the case where one active external source (antenna) $D_a$ protects a control region $D_c$ from an interrogating far field and maintains an overall small signature beyond a disk of large enough radius $R$. The general setup for this question will be as follows. Let $B_{R} \subset \mathbb{R}^{d}$ be the ball of radius $R > 0$. We assume $\B0\in D_{a} \subset B_{R}$ is the region inside a single antenna with sufficiently smooth boundary $\partial D_{a}$. We also let $D_{c} \Subset B_{R}$ be the control region, which is assumed to satisfy $\overline{D_{c}} \cap \overline{D_{a}} = \emptyset$ (see Figure \[fig:mainsetup\]). The numerical simulations in the current work are performed for the two dimensional case but the methods are adaptable to the three dimensional setting as well. Consider the function space $$\Xi = L^{2}(\partial D_{c}) \times L^{2}(\partial B_{R}),$$ endowed with the scalar product $$(\phi,\psi)_{\Xi} = \int_{\partial D_{c}}\phi_{1}(\mathbf{y})\overline{\psi}_{1}(\mathbf{y})\,dS_{\mathbf{y}} + \int_{\partial B_{R}} \phi_{2}(\mathbf{y})\overline{\psi}_{2}(\mathbf{y})\,dS_{\mathbf{y}},$$ which is a Hilbert space. For the remainder of the paper we will assume that every $L^2$ space of complex valued functions will be endowed with the usual inner product. As in [@O2] consider $K: L^{2}(\partial D_{a}) \to \Xi$, the double layer potential operator restricted to $\partial D_{c}$ and $\partial B_{R}$, respectively, defined by $$\label{EQ:K} K\phi(\Bx,\Bz) = (K_{1}\phi(\Bx), K_{2}\phi(\Bz)), \quad \phi \in L^{2}(\partial D_{a}),$$ where $$\begin{aligned} K_1:L^{2}(\partial D_{a})\rightarrow L^{2}(\partial D_{c}), \, K_1\phi(\Bx) & = \int_{\partial D_{a}}\phi(\By)\frac{\partial \Phi(\Bx,\By)}{\partial {{\mbox{\boldmath ${\nu}$}}}_{\By}}ds_{\By}, \mbox{ for }\Bx\in \partial D_c, \notag\\ K_2:L^{2}(\partial D_{a})\rightarrow L^{2}(\partial B_R), \, K_2\phi(\Bz) & = \int_{\partial D_{a}}\phi(\By)\frac{\partial \Phi(\Bz,\By)}{\partial {{\mbox{\boldmath ${\nu}$}}}_{\By}}ds_{\By}, \mbox{ for }\Bz\in \partial B_R(\B0). \label{8}\end{aligned}$$ Here $\Phi(\Bx, \By)$ represents the fundamental solution of the relevant Helmholtz operator, i.e., $$\label{9} \Phi(\Bx,\By)=\vspace{0.15cm}\left\{\begin{array}{ll} \vspace{0.15cm}\displaystyle\frac{e^{ik|\Bx-\By|}}{4\pi|\Bx-\By|}, \mbox{ for } d=3\vspace{0.15cm}\\ \frac{i}{4}H_0^{(1)}(k|\Bx-\By|) , \mbox{ for } d=2\end{array}\right.$$ with $H_{0}^{(1)} = J_{0} + iY_{0}$ representing the Hankel function of first type. Note that in the integrals are to be understood as singular integrals defined through an operator extension from $C(\partial D_a)$. We will also consider $k$ such that $$\begin{aligned} \label{ass-lambda-0} && \mathbf{1)} \, -k^2 \mbox{ is not a Neumann eigenvalue for the Laplace operator in $D_a$ or $B_R(\B0)$},\nonumber\\ && \mathbf{2)} \, -k^2 \mbox{ is not a Dirichlet eigenvalue for the Laplace operator in $D_c$}.\end{aligned}$$ As in [@O2] we introduce the adjoint operator $K^{*}: \Xi \to L^{2}(\partial D_{a})$, which can be shown to satisfy $$K^{*}\psi(\mathbf{x}) = \int_{\partial D_{c}} \psi_{1}(\mathbf{y}) \overline{\frac{\partial \Phi(\mathbf{y},\mathbf{x})}{\partial \nu_{\mathbf{x}}}}\,dS_{\mathbf{y}} + \int_{\partial B_{R}} \psi_{2}(\mathbf{y}) \overline{\frac{\partial \Phi(\mathbf{y},\mathbf{x})}{\partial \nu_{\mathbf{x}}}}\, dS_{\mathbf{y}}, \quad \mathbf{x} \in \partial D_{a}. \label{eq:dlpotentialadjoint}$$ This paper proposes a sensitivity study for the following problem: Let $V\Subset D_c$ and $R'>R$. For a fixed wave number $k > 0$ and fixed $0 < \mu \ll 1$, find a function $h \in C(\partial D_{a})$ such that there exists $u \in C^{2}(\RR^{n} \setminus \overline{D_{a}}) \cap C^{1}(\RR^{n} \setminus D_{a})$ solving $$\left\{ \begin{array}{rl} (\Delta + k^{2}) u(\mathbf{x}) & = 0 \quad \mathbf{x} \in \RR^{n} \setminus \overline{D_{a}}\\ u & = h \quad \textrm{ on $\partial D_{a}$}\\ \|u - f_{1}\|_{C(\overline{V})} & = \mathcal{O}(\mu)\mbox { and }\|u\|_{C(\RR^{n} \setminus B_{R'}(\mathbf{0}))} = \mathcal{O}(\mu), \end{array} \right. \label{eq:inverseproblem}$$ where $f_{1}$ is a solution of the Helmholtz equation in a neighborhood of the control region $D_{c}$. In fact, by using the operator $K$ and regularity arguments it is shown in [@O2] that a class of solutions for problem can be obtained by considering the following problem: for a fixed wave number $k>0$ satisfying conditions , a given function $f=(f_1,0) \in \Xi$ and $\mu > 0$, find a density function $\phi \in C(\partial D_{a})$ such that $$\|K\phi - f\|_{\Xi} \leq \mu. \label{eq:controlinequality}$$ Problem is in fact a Fredholm integral equation of the first kind, and it was studied in a very general setting in [@O2]. There the authors proved that the bounded and compact operator $K$ is also one-to-one and has a dense (but not closed) range, thus proving the existence of a class of solutions for (and thus for ). However, the fact that $K$ is compact and that its range is not closed also implies that problem is ill-posed. By using regularization, one can approximate a solution to problem with an arbitrary level of accuracy $\mu \ll 1$. There are several methods known in the literature, but we will use in this paper the Tikhonov regularization method [@reg1; @reg2]. This method, when applied to the operator $K:L^2(\partial D_a)\rightarrow \Xi$, proposes a solution $\phi_{\alpha}\in C(\partial D_{a})$ of the form $$\label{Tikhonov} \phi_{\alpha}=(\alpha I+K^*K)^{-1}K^*f, \textrm{ for } 0<\alpha \ll 1,$$ where $\alpha$ is a suitably chosen regularization parameter. It is known that $\Vert K\phi_\alpha-f\Vert_\Xi\rightarrow 0$ as $\alpha\rightarrow 0$, (see [@Kirsch-Book96], Theorem 2.16), but the optimal choice of $\alpha$ is an essential step in designing a feasible method (e.g., finding a minimal norm solution), and there are various modalities to do this. In this paper we will use the Morozov discrepancy principle associated to the following weighted residual: $$\label{disc-fun} E(\phi,h)= \displaystyle\frac{1}{\|h_{1}\|_{L^{2}(\partial D_{c})}^{2}}\| K_1\phi - h_{1}\|_{L^{2}(\partial D_{c})}^{2} + \frac{1}{2\pi R} \|K_2\phi \|_{L^{2}(\partial B_{R})}^{2},$$ for every given $h=(h_1,0)\in \Xi$. The reasoning behind using the weighted residual discrepancy functional defined at \[disc-fun\] is as follows. Due to the asymptotic behavior of $\frac{\partial \Phi(\mathbf{x},\, \mathbf{y})}{\partial \nu_{\mathbf{y}}} = \mathcal{O}(|\mathbf{x}-\mathbf{y}|^{-1/2})$ as $|\mathbf{x}-\mathbf{y}| \to \infty$, we have that given a fixed density $\phi$, $\|K\phi\|_{L^{2}(\partial B_{R})} = \mathcal{O}(1)$ as $R \to \infty$. In other words, using the space $L^{2}(\partial B_{R})$ with the standard surface measure is not really suited to the decay properties of double layer potential solutions, because the decay of the normal derivative $\partial_{\nu}\Phi$ is too weak. Similarly, we use the relative norm $$\label{eq:F1} \frac{\|K_1\phi - h_{1}\|_{L^{2}(\partial D_{c})}}{\|h_{1}\|_{L^{2}(\partial D_{c})}}$$ on $\partial D_{c}$ because this is a useful quantity for determining how good the control is, regardless of the norm of $h_{1}$. Thus our procedure for finding an approximate solution for problem is to first make use of the Tikhonov regularization for the operator $K:L^2(\partial D_a)\rightarrow \Xi$ as described in to obtain $\phi_\alpha$ and then apply the Morozov’s discrepancy principle for the unique choice of $\alpha$ ([@Kress-Book99]), i.e. such that $$\label{Morozov0} E(\phi_{\alpha}, f)=\delta^2,$$ with $\displaystyle\delta^2\leq \mu^2\min\left\{\frac{1}{2\|f_{1}\|^2_{L^{2}(\partial D_{c})}}, \frac{1}{4\pi R}\right\}$. In what follows, we will account for noise and measurement errors and will consider with $f=(f_1,0)\in \Xi$ replaced by $f_\epsilon=(f_{\epsilon,1}, f_{\epsilon,2})\in \Xi$, given by $$\label{random-f} f_{\epsilon} = (f_{1} + \epsilon s, \, 0)\in \Xi,$$ where $s\in L^{2}(\partial D_{c})$ is a random perturbation with $\|s\|_{L^{2}(\partial D_{c})} \leq 2 \|f_{1}\|_{L^{2}(\partial D_{c})}$ and $f_1$ is a solution of the Helmholtz equation in a neighborhood of the control region $D_{c}$. We mention that in the numerical experiments of Section \[sec:numerics\], $f_{1}$ denotes the $k$ frequency component of the far field of a far field observer. Note that this assumption about the interrogating signal ensures that $f_1$ is a solution of the Helmholtz equation in $B_R$. In the noisy case (i.e. when $f$ is replaced by $f_\epsilon$) equation becomes $$\label{Morozov} E(\phi_{\alpha}, f_\epsilon)=\delta^2,$$ where $\phi_{\alpha}=(\alpha I+K^*K)^{-1}K^*f_\epsilon$ is the Tikhonov regularization solution. From the definition of $E$ and classical results, [@Kirsch-Book96; @Kress-Book99], it follows that admits at least a solution $\alpha$. Moreover, as we will discuss in Section \[sec:stability\], motivated by numerical evidence, we formulate the hypothesis that there exists $\epsilon_0>0$ such that for each $\epsilon\in (0,\epsilon_0)$, problem has a unique solution $\alpha(\epsilon)$ which uniquely defines a differentiable function $\epsilon\rightsquigarrow\alpha(\epsilon)$. We will study the minimal norm solution uniquely determined by , discuss its stability for given noisy data in $\Xi$ and, in the case of data corresponding to a point source, analyze its sensitivity with respect to parameters such as: mutual distances between $D_a$, $D_c$ and $B_R(\B0)$; wave number $k$; and the location of the point source. Stability estimate for the Tikhonov regularization {#sec:stability} ================================================== In this section we present analytical and numerical arguments which indicate the stability of the minimum norm solution $\phi_{\alpha}$ with respect to noise level $\epsilon$ for a given fixed discrepancy level $\delta$. Next, we present below Lemma \[lemma:phif1constant\] which will provide bounds for $\|f_{1}\|_{L^{2}(\partial D_{C})}$ and $\alpha$ in terms of the operatorial norm of $K_{1}^{*}$. Let $0<\delta<\frac{1}{\sqrt{2}}$ and $z=(z_1,0) \in \Xi'$ with $z_1\neq 0$. Consider the Tikhonov regularization solution $\phi_{\alpha}=(\alpha I + K^{*}K)^{-1}K^*z \in C(\partial D_a)$, with $\alpha$ such that $\|K\phi_{\alpha} - z\|_{\Xi'} \leq \delta$. Then we have $$\begin{aligned} \|z_{1}\|_{L^2(\partial D_c)} & \leq 4\|K_1^*\|_{\mathcal O}\|\phi_{\alpha}\|_{L^{2}(\partial D_{a})},\label{lb-phi}\\ \alpha & \leq 4\delta \Vert K_1^*\Vert^2_{\mathcal O},\label{ub-alpha}\end{aligned}$$ where $K_1^*$ is the adjoint operator for $K_1$ defined by and $\|\cdot\|_{\mathcal O}$ denotes the operatorial norm. \[lemma:phif1constant\] We will start with the proof of . Note that since $E(\phi_{\alpha}, z)=\delta^{2}$, we have $$\label{5.17} \|K_{1}\phi_{\alpha} - z_{1}\|_{L^{2}(\partial D_{c})}^{2} \leq \delta^{2}\|z_{1}\|_{L^2(\partial D_c)}^{2}.$$ From we obtain $$\|K_{1}\phi_{\alpha}\|_{L^2(\partial D_c)}^{2} - 2 \, {\textrm{Re}}(K_{1}\phi_{\alpha}, \, z_{1})_{L^2(\partial D_c)} + \|z_{1}\|_{L^2(\partial D_c)}^{2} \leq \delta^{2}\|z_{1}\|_{L^2(\partial D_c)}^{2},$$ and this implies $$\|z_{1}\|_{L^2(\partial D_c)}^{2}(1 -\delta^{2}) \leq 2 \, {\textrm{Re}}(\phi_{\alpha}, \, K_{1}^{*}z_{1})_{L^2(\partial D_a)} \leq 2 \|\phi_{\alpha}\|_{L^2(\partial D_a)}\|K_{1}^{*} z_{1}\|_{L^2(\partial D_a)}. \label{pas1}$$ Then gives $$\begin{aligned} \|\phi_{\alpha}\|_{L^2(\partial D_a)} & \geq \frac{\|z_{1}\|_{L^2(\partial D_c)}^{2}(1 - \delta^{2})}{2 \|K_{1}^{*}z_{1}\|_{L^2(\partial D_a)}} = {\left ( \frac{1 - \delta^{2}}{2}\right )}{\left ( \frac{\|z_{1}\|_{L^2(\partial D_c)}}{\|K_{1}^{*}z_{1}\|_{L^2(\partial D_a)}} \right )}\|z_{1}\|_{L^2(\partial D_c)}\notag\\ & \geq \frac{1 - \delta^{2}}{2} {\left ( \frac{\|z_{1}\|_{L^2(\partial D_c)}}{\|K_{1}^{*}\|_{\mathcal O}}\right )}\notag\\ &\geq \frac{\|z_1\|_{L^2(\partial D_c)}}{4\|K_{1}^{*}\|_{\mathcal O}}.\label{margine}\end{aligned}$$ Next we proceed towards proving . From the definition of $\phi_{\alpha}$ we have $$\begin{aligned} \alpha \phi_{\alpha} + K^{*}K\phi_{\alpha}&=& K^*z=K_1^*z_1,\nonumber\\ \alpha \phi_{\alpha}+K_2^{*}K_2\phi_{\alpha} &=& K_1^*(z_1-K_1\phi_{\alpha}).\label{5.18}\end{aligned}$$ Here we have used from and that $$\begin{aligned} \label{adjunct-rel} K^*\psi=K_1^*\psi_1+K^*_2\psi_2, \mbox{ for all }\psi\in\Xi,\nonumber\\ K^*Kv=K_1^*K_1v+K_2^*K_2v, \mbox{ for all }v\in L^2(\partial D_a).\end{aligned}$$ Multiplying with $\phi_{\alpha}$ in the sense of the usual scalar product in $L^2(\partial D_a)$, we obtain $$\label{rel11} \alpha \|\phi_{\alpha}\|^2_{L^2(\partial D_a)} +\|K_2\phi_{\alpha}\|^2_{L^2(\partial D_a)} = (K_1^*(z_1-K_1\phi_{\alpha}),\phi_{\alpha})_{L^2(\partial D_a)}.$$ Using , and in we then have $$\alpha \|\phi_{\alpha}\|_{L^2(\partial D_a)} \leq \delta\|K_1^*\|_{\mathcal O}\|z_1\|_{L^2(\partial D_c)}\Longrightarrow \alpha\leq 4\delta\|K_1^*\|^2_{\mathcal O} \label{alfa}.$$ Next, before presenting the main stability result of this section, i.e., Proposition \[theorem:stabilityestimate\] below, we must understand the conditions on $\epsilon > 0$ under which admits a unique solution $\alpha(\epsilon)$ with the property that the resulting function $\epsilon\rightsquigarrow \alpha(\epsilon)$ is differentiable. For this, we consider the function $g:(0,\infty)\times(0,\infty)\rightarrow (0,\infty)$ defined by $$g(\alpha, \epsilon) = E(\phi_\alpha, f_\epsilon)\label{eq:galphaeps},$$ where $f_{\epsilon} \in \Xi$ was introduced in , and $\phi_\alpha$ is the Tikhonov regularization solution introduced in . With this notation, can be rewritten as $$\label{Morozov-g} g(\alpha, \epsilon) = \delta^2,$$ where $\delta$ is the desired fixed discrepancy level. By using classical results (e.g., [@Kirsch-Book96; @Kress-Book99]) it can be observed that for every $\epsilon$, admits at least one solution in $(0,\infty)$ and that $g$ defined by is differentiable with respect to positive $\alpha$ and $\epsilon$, respectively. In fact, it follows from classical arguments that a maximum value of $\alpha$ for a given $\epsilon$ exists. This solution of corresponds to the $L^2$ minimal energy solution and we will further refer to it as the Morozov solution. For the remainder of the paper, unless otherwise specified, $C$ will denote a generic constant which depends only on the operator $K$, $d_c=diam(D_c)$ and $d=dist(\partial D_c,\partial D_a)$. The next Proposition states a central stability result concerning the Morozov solution of . We have, Let $0<\delta$ be as above, and $f_\epsilon$ and $f_1$ as defined in . For every $\epsilon \geq 0$ consider $\phi_{\alpha_{\epsilon}}=(\alpha_\epsilon I + K^{*}K)^{-1}K^{*}f_{\epsilon}\in C(\partial D_a)$ with $\alpha_\epsilon=\alpha(\epsilon)$ the Morozov solution of . Then we have, $$\displaystyle\frac{\| \phi_{\alpha_\epsilon} - \phi_{\alpha_0}\|_{L^{2}(\partial D_{a})}}{\|\phi_{\alpha_{\epsilon}}\|_{L^{2}(\partial D_{a})}} \leq \displaystyle\frac{ \left|\displaystyle\frac{\alpha_{\epsilon}}{\alpha_0} - 1\right| + \sqrt{ \left|\displaystyle\frac{\alpha_{\epsilon}}{\alpha_0} -1 \right|^{2} + 16\displaystyle \frac{\epsilon{\left ( 2\delta + C\delta\epsilon +C\epsilon\right )}}{\alpha_0}\|K^*_{1}\|_{\mathcal O}}}{2}.$$\[theorem:stabilityestimate\] Fix $\epsilon > 0$ and let $f=f_\epsilon\mbox{ for } \epsilon=0$. Let us recall that $\alpha(\epsilon)$ is uniquely implicitly defined by the equation $E((\alpha_{\epsilon}I+K^*K)^{-1}K^*f_{\epsilon}, f_{\epsilon})= \delta^2$ and by Lemma \[lemma-diff-alpha\] it will be differentiable in some interval $(0,\epsilon_0)$ for all wavenumbers $k$. Next consider $$\begin{aligned} \alpha_{\epsilon}\phi_{\alpha_{\epsilon}} + K^{*}K\phi_{\alpha_{\epsilon}} & = K^{*}f_{\epsilon},\\ \alpha_{0} \phi_{\alpha_{0}} + K^{*}K\phi_{\alpha_0} & = K^{*}f.\end{aligned}$$ Subtracting, we obtain $$\begin{aligned} \alpha_{0}\phi_{\alpha_0}-\alpha_{\epsilon}\phi_{\alpha_{\epsilon}} + K^{*}K(\phi_{\alpha_0}-\phi_{\alpha_{\epsilon}}) & = K^{*}(f - f_{\epsilon}),\notag\\ \alpha_0(\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}}) + (\alpha_0 - \alpha_{\epsilon})\phi_{\alpha_{{\epsilon}}} + K^{*}K(\phi_{\alpha_0}-\phi_{\alpha_{\epsilon}}) & = K^{*}(f - f_{\epsilon}).\label{3.1.1}\end{aligned}$$ Integrating both sides of against $\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}}$ yields $$\begin{aligned} & \displaystyle\alpha_0 \|\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}}\|_{L^2(\partial D_a)}^{2} + (\alpha_0 - \alpha_{{\epsilon}})(\phi_{\alpha_{{\epsilon}}}, \, \phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}})_{L^2(\partial D_a)} + \|K(\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}})\|_\Xi^{2}\notag\\ & \quad\quad= \displaystyle(K(\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}}), \, f - f_{\epsilon})_{\Xi}= \displaystyle(K_{1}(\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}}), \, f_{1} - f_{\epsilon, 1})_{L^2(\partial D_c)}.\label{3.1.2}\end{aligned}$$ where, we have used and the fact that $f_2=f_{\epsilon,2}=0$ in the last equality above. Thus, $$\begin{aligned} \label{ecuatie-1} \!\!\!\!\!\!\!\!\!\alpha_0 \|\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}}\|_{L^2(\partial D_a)}^{2} \leq & |\alpha_0 - \alpha_{{\epsilon}}|\|\phi_{\alpha_{{\epsilon}}}\|_{L^2(\partial D_a)}\|\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}}\|_{L^2(\partial D_a)} \nonumber\\ + & \|f_{1} - f_{\epsilon, 1}\|_{L^2(\partial D_c)}\|K_{1}(\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}})\|_{L^2(\partial D_c)}\end{aligned}$$ Observe that $$\begin{aligned} \|K_{1}(\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}})\|_{L^2(\partial D_c)} & \leq \|K_{1} \phi_{\alpha_0} - f_{ 1}\|_{L^2(\partial D_c)} + \|K_{1} \phi_{\alpha_{{\epsilon}}} - f_{{\epsilon},1}\|_{L^2(\partial D_c)} + \|f_{1} - f_{\epsilon, 1}\|_{L^2(\partial D_c)}\nonumber\\ &\leq {\delta}\|f_{1}\|_{L^2(\partial D_c)}+ {\delta}\|f_{{\epsilon},1}\|_{L^2(\partial D_c)} + C\epsilon \|f_{1}\|_{L^2(\partial D_c)}\nonumber\\ & = {\left ( 2\delta + C\delta\epsilon +C\epsilon\right )}\|f_{1}\|_{L^2(\partial D_c)}. \label{ecuatie-2}\end{aligned}$$ where $f_{\epsilon,1}=f_{1} + \epsilon s$ with $\|s\|_{L^{2}(\partial D_{c})} \leq C \|f_{1}\|_{L^{2}(\partial D_{c})}$, and we have used the definition of $\phi_{\alpha_{\epsilon}}$ and in the inequalities above. Using in we obtain $$\begin{aligned} \alpha_0 \|\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}}\|_{L^2(\partial D_a)}^{2} & \leq |\alpha_0 - \alpha_{{\epsilon}}| \|\phi_{\alpha_{{\epsilon}}}\|_{L^2(\partial D_a)}\|\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}}\|_{L^2(\partial D_a)} \notag\\ & + \epsilon{\left ( 2\delta + C\delta\epsilon +C\epsilon\right )}\|f_{1}\|^2_{L^2(\partial D_c)}. \label{ecuatie-3}\end{aligned}$$ If we define $A := \frac{\|\phi_{\alpha_{\epsilon}} - \phi_{\alpha_{0}} \|_{L^{2}(\partial D_{a})}}{\|\phi_{\alpha_{{\epsilon}}}\|_{L^{2}(\partial D_{a})}}$, inequality implies that $$\begin{aligned} \alpha_0 A^{2} & \leq |\alpha_0 - \alpha_{{\epsilon}}| A + \frac{\epsilon {\left ( 2\delta + C\delta\epsilon +C\epsilon\right )} \|f_{1}\|_{L^{2}(\partial D_{c})}^{2}}{\|\phi_{\alpha_{{\epsilon}}}\|_{L^{2}(\partial D_{a})}^{2}} \notag\\ & \leq |\alpha_0 - \alpha_{{\epsilon}}| A + 16\epsilon {\left ( 2\delta + C\delta\epsilon +C\epsilon\right )}\|K^*_1\|^2_{\mathcal O}, \label{quadratic}\end{aligned}$$ where we have used of Lemma \[lemma:phif1constant\] in the last inequality above. Next, consider $$h(A) := A^{2} - \left|\displaystyle\frac{\alpha_{{\epsilon}}}{\alpha_0} - 1 \right| A - 16\|K^*_1\|^2_{\mathcal O}\displaystyle\frac{\epsilon {\left ( 2\delta + C\delta\epsilon +C\epsilon\right )}}{\alpha_0}.$$ Then, from we have $h(A) \leq 0$ and this implies that $$A \leq \displaystyle\frac{ \left|\displaystyle\frac{\alpha_{{\epsilon}}}{\alpha_0} - 1 \right| + \sqrt{\left|\displaystyle\frac{\alpha_{{\epsilon}}}{\alpha_0} - 1 \right|^{2} + 64\|K^*_1\|^2_{\mathcal O}\displaystyle\frac{\epsilon {\left ( 2\delta + C\delta\epsilon +C\epsilon\right )}}{\alpha_0}}}{2}$$ which completes the proof. Regarding the monotonic character of $g$, we note that, as suggested by the numerics, $g$ is not in general *globally* monotonic with respect to $\alpha$ as can be seen in Figure \[fig:nonincreasing\_F\], which considers an antenna of radius $a = 0.01$, region of control characterized in polar coordinates by $r_{1} = 0.011$, $r_{2} = 0.015$, $\theta \in [3\pi/4, 5\pi/4]$, wave number $k=10$, $f_\epsilon$ given by with $f_1=\frac{1}{4}H^{(1)}_0(k|\Bx-\Bx_0|)$ with $\mathbf{x}_{0} = [10000,0]^{T}$, and noise level $\epsilon=0.005$. But on the other hand, for the same geometry and functional settings as in Figure \[fig:nonincreasing\_F\], we observe in Figure \[fig:plot\_g\_dg\_alpha\] that for each $\epsilon<0.015$, $g(\alpha, \epsilon) = E(\phi_{\alpha_{\epsilon}}, \, f_{\epsilon})$ is strictly increasing with respect to $\alpha$ in the interval $(10^{-4}, 1)$. Moreover, for every $\epsilon<0.015$ the Morozov solution $\alpha(\epsilon)$ is the unique solution of in $(10^{-4}, 1)$. In fact, for the same geometry and functional settings as in Figure \[fig:plot\_g\_dg\_alpha\] and for $k \in [1,100]$ and $\epsilon = 0.005$, Table \[tab:pktable\] summarizes the values of $p_k>0$ for which $g(\alpha, \epsilon)$ is locally strictly monotonic with respect to $\alpha$ in an interval $(10^{-p_k}, 1)$, as well as the value of the Morozov solution for each $k$ (also see Figure \[fig:pk\_plot\]). Together with Figure \[fig:plot\_g\_dg\_alpha\] where $k=10$ and $\epsilon$ is varied in the interval $[0,0.015]$, this suggests that the Morozov solution $\alpha(\epsilon)$ of satisfies $\alpha(\epsilon)\in(10^{-p_k}, 1)$ at least for $\epsilon < 0.015$. This in turn together with the strict monotonicity $g$ implies the existence of a unique solution $\alpha(\epsilon)$ for . Then, uniqueness together with the fact that $\frac{\partial}{\partial \alpha}g(\alpha, \epsilon)\neq 0$ in $ (10^{-p_k}, 1)\times (0,0.015)$ (for $k=10$ shown in Figure \[fig:plot\_g\_dg\_alpha\]) implies the differentiability of the Morozov solution $\alpha(\epsilon)$ by using the implicit function theorem. --------------- ---------------- ------------------ ------ ---------------- ------------------ $k$ $-p_{k}$ Morozov $\alpha$ $k$ $-p_{k}$ Morozov $\alpha$ \[0.5ex\] 1.0 -5.74057337341 0.0021397 51.0 -5.63387601498 0.023987 6.0 -5.15371857022 0.0022445 56.0 -5.5538513243 0.028226 11.0 -4.61487654411 0.0027985 61.0 -5.58052339557 0.032734 16.0 -4.43348001737 0.0037643 66.0 -7.93866063316 0.038053 21.0 -7.22374205154 0.0052228 71.0 -7.78393971408 0.043891 26.0 -6.73292218326 0.0072707 76.0 -7.60786606025 0.048884 31.0 -6.41280555768 0.0098052 81.0 -7.41580451387 0.05425 36.0 -6.18339417827 0.012607 86.0 -7.2077344143 0.060871 41.0 -5.97530913764 0.015782 91.0 -7.00500135956 0.066982 46.0 -5.78858575492 0.01971 96.0 -6.90364624001 0.07278 --------------- ---------------- ------------------ ------ ---------------- ------------------ : Table of values $-p_{k}$ such that $g(\alpha, \epsilon)$ is increasing with respect to $\alpha$ for $\alpha \geq 10^{-p_{k}}$. $\epsilon$ is fixed at $0.005$ in this case.[]{data-label="tab:pktable"} ![Threshold value $p_{k}$ for which $F(\alpha)$ is increasing when $\alpha > 10^{-p_{k}}$. Also shows the value of $\alpha$ for which $F(\alpha) = \delta^{2}$ with the same setting as in Figure \[fig:plot\_g\_dg\_alpha\], where $\delta = 0.02$ and the noise level $\epsilon = 0.005$.[]{data-label="fig:pk_plot"}](pk_plot){width="\textwidth"} For simplicity of notation, in what follows we will write sometimes $\alpha$ instead of $\alpha_\epsilon=\alpha(\epsilon)$ and we will use $\alpha'$ and $f_{\epsilon,i}'$ to denote $\frac{d\alpha}{d\epsilon}$ and $\frac{df_{\epsilon,i}}{d\epsilon}$ respectively. Motivated by the above numerics, we formulate the following more general hypothesis: \[hypothesis-I\] Assume the same geometrical setup as in Section \[sec:background\] and let $f_\epsilon, f_1$ be as in . Then there exists $p_0>0$ and $\epsilon_0>0$ such that $\frac{\partial}{\partial \alpha}g(\alpha, \epsilon)\neq 0$ in $ (10^{-p_0}, 1)\times (0,\epsilon_0)$ for all wave numbers $k$, and the Morozov solution $\alpha(\epsilon)$ is the unique solution of in $(10^{-p_0}, 1)$. For example, as shown in Table \[tab:pktable\] and Figure \[fig:pk\_plot\], for $f_\epsilon, f_1$ as in with $s=\widehat{\nu}\Vert f_1\Vert_{ L^2(\partial D_c)}$ where $\widehat{\nu}\in L^2(\partial D_c)$ is a random perturbation with $\Vert \widehat{\nu}\Vert_{ L^2(\partial D_c)}=1$, and for the same geometry and data as in Figure \[fig:plot\_g\_dg\_alpha\], we have that Hypothesis \[hypothesis-I\] is satisfied for $p_0=10^{-4}$ and $\epsilon_0=0.015$ for all $k=\overline{1,100}$. Thus, whenever Hypothesis \[hypothesis-I\] is satisfied, the definition of $\alpha(\epsilon)$ and the implicit function theorem imply: \[lemma-diff-alpha\] There exists $\epsilon_0>0$ such that for every $\epsilon\in(0,\epsilon_0)$, the function $\alpha:(0,\epsilon_0)\rightarrow(0,\infty)$, where for each $\epsilon\in(0,\epsilon_0)$, $\alpha(\epsilon)$ represents the Morozov solution of , will be differentiable for all wave numbers $k$. The next Lemma is a technical result needed in the stability estimate obtained in Corollary \[corolar-1\]. \[teorema-int\] Let $f_{1}$ be a solution of the Helmholtz equation in a neighborhood of $D_c$ satisfying the following source type condition: $$\label{source-cond} \displaystyle\left\Vert K_1\psi_0-\frac{f_1}{\Vert f_1\Vert_{L^{2}(\partial D_{c})}}\right\Vert_{L^{2}(\partial D_{c})}\!\!\!\!\leq C\delta \mbox{ for some }\psi_0\in L^2(\partial D_a) \mbox{ with } \Vert\psi_0\Vert_{L^2(\partial D_a)}\leq C \delta.$$ Assume $R$ (radius of $B_R(\B0)$) is such that, $$\label{R-bound} \Vert f_1\Vert_{L^{2}(\partial D_{c})} \leq \sqrt{\pi R}.$$ Consider $s_\epsilon=f_{1} + \frac{\epsilon}{2} \widehat{\nu}\|f_{1}\|_{L^{2}(\partial D_{c})}$ where $\widehat{\nu}\in L^2(\partial D_c)$ is a random perturbation with $\|\widehat{\nu}\| \leq 1$. Assume the same functional framework as in Proposition \[theorem:stabilityestimate\] and that Hypothesis \[hypothesis-I\] holds true in the case when $f_\epsilon$ is given by $$\label{hyp-1} f_{\epsilon} = (f_{\epsilon,1}, f_{\epsilon,2})= (f_{1} + \epsilon s_\epsilon, \, 0).$$ Then, there exists $\epsilon_0>0$ such that the Morozov solution of equation $\alpha=\alpha(\epsilon)$ satisfies $$\alpha|\alpha'| \leq C\frac{\delta^2}{\sqrt{\alpha}}, \mbox{ for all } \epsilon<\epsilon_0.$$ Define the weights $$\begin{aligned} w_{1} & := \frac{1}{\|f_{\epsilon, 1}\|_{L^{2}(\partial D_{c})}^{2}}\\ w_{2} & := \frac{1}{2 \pi R}.\end{aligned}$$ and denote $T_{\alpha} := (K^{*}K + \alpha I)^{-1}$, $R_{\alpha} := T_{\alpha}K^{*}$. Then using the Einstein summation convention, we may write $$E(\phi_{\alpha}, f_{\epsilon})= w_{i}\|K_{i}\phi_{\alpha} - f_{\epsilon,i} \|_{L^{2}(W_{i})}^{2} = w^{i} {\left ( K_{i}\phi_{\alpha} - f_{\epsilon,i}, \, K_{i}\phi_{\alpha} - f_{\epsilon,i}\right )}_{L^{2}(W_{i})},$$ where $W_{1} = \partial D_{c}$, $W_{2} = \partial B_{R}$. Next, as in Lemma \[lemma-diff-alpha\], we observe that Hypothesis \[hypothesis-I\] together with the implicit function theorem imply the uniqueness and differentiability of $\alpha(\epsilon)$, on the interval $\epsilon \in (0, \epsilon_0)$ for some $\epsilon_0>0$, where $\alpha(\epsilon)$ is uniquely and implicitly defined by the equation $E((\alpha_{\epsilon}I+K^*K)^{-1}K^*f_{\epsilon}, \, f_{\epsilon})= \delta^2$. Differentiating the equation $E((\alpha_{\epsilon}I+K^*K)^{-1}K^*f_{\epsilon}, \, f_{\epsilon})= \delta^2$ with respect to $\epsilon$ and noting that $\delta$ is fixed, we obtain $$\begin{aligned} 0 = \partial_{\epsilon}E(\phi_{\alpha}, f_{\epsilon})& = 2w_{i}\,{\textrm{Re}}{ {\left ( K_{i}\phi_{\alpha}' - f_{\epsilon,i}', \, K_{i}\phi_{\alpha} - f_{\epsilon, i}\right )}}_{L^{2}(W_{i})} \notag\\ & \quad - 2(w_{1})^{2}\,{\textrm{Re}}{{\left ( f_{\epsilon,1}', f_{\epsilon, 1}\right )}}_{L^{2}(\partial D_{c})}\|K_{1}\phi_{\alpha} - f_{\epsilon,1}\|_{L^{2}(\partial D_{c})}^{2}\label{eq:diffnorm1}.\end{aligned}$$ Next, from $(K^{*}K + \alpha I)\phi_{\alpha} = K^{*}f_{\epsilon}$ we observe that $$\phi_{\alpha}' = R_{\alpha}f_{\epsilon}' - \alpha' T_{\alpha}\phi_{\alpha}. \label{eq:phialphadiff}$$ Thus, we may write $$K_{i}\phi_{\alpha}' - f_{\epsilon, i}' = -\alpha' K_{i} T_{\alpha}\phi_{\alpha} + K_{i}R_{\alpha}f_{\epsilon}' - f_{\epsilon, i}'.\label{eq:sub1}$$ By using (\[eq:sub1\]) and we obtain that $$\begin{aligned} \label{ec-11} \!\!\!\!\!\!\!\!\! 2{\left ( K_{i}\phi_{\alpha}' - f_{\epsilon,i}', \, K_{i}\phi_{\alpha}\right )}_{L^{2}(W_{i})} &= -2\alpha' {\left ( T_{\alpha}\phi_{\alpha}, \, K_{i}^{*}K_{i}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})}\notag\\ & + 2{\left ( K_{i}R_{\alpha}f_{\epsilon}' - f_{\epsilon, i}', \, K_{i}\phi_{\alpha}\right )}_{L^{2}(W_{i})},\end{aligned}$$ and $$\begin{aligned} \label{ec-12} \!\!\!\!\!\!\!\!\!-2{\left ( f_{\epsilon,i}, \, K_{i}\phi_{\alpha}' - f_{\epsilon, i}'\right )}_{L^{2}(W_{i})} & = 2\alpha'{\left ( f_{\epsilon,i}, \, K_{i}T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(W_{i})} - 2{\left ( f_{\epsilon,i}, \, K_{i}R_{\alpha}f_{\epsilon}' - f_{\epsilon, i}'\right )}_{L^{2}(W_{i})}\notag\\ & = 2\alpha'{\left ( K_{i}^{*}f_{\epsilon,i}, \, T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} - 2{\left ( f_{\epsilon,i}, \, K_{i}R_{\alpha}f_{\epsilon}' - f_{\epsilon, i}'\right )}_{L^{2}(W_{i})}.\end{aligned}$$ Let $P,Q$ be defined by $$\begin{aligned} P & = 2w_{i}\left[ {\left ( K_{i}\phi_{\alpha} - f_{\epsilon, i}, \, K_{i}R_{\alpha}f_{\epsilon}' - f_{\epsilon,i}'\right )}_{L^{2}(W_{i})} + \alpha' {\left ( K_{i}^{*}f_{\epsilon,i} - K_{i}^{*}K_{i}\phi_{\alpha}, \, T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} \right]\label{P}\\ Q & = 2(w_{1})^{2}\,{\left ( f_{\epsilon,1}', f_{\epsilon, 1}\right )}_{L^{2}(\partial D_{c})}\|K_{1}\phi_{\alpha} - f_{\epsilon,1}\|^{2}.\label{Q}\end{aligned}$$ Then from , used in we obtain $$\label{ec-15'} 0=\partial_{\epsilon}E(\phi_{\alpha}, f_{\epsilon})= {\textrm{Re}}(P) - {\textrm{Re}}(Q).$$ We focus first on $P$ introduced in . In this regard, let us define $$\label{P1} L_{i} := {\left ( K_{i}\phi_{\alpha} - f_{\epsilon, i}, \, K_{i}R_{\alpha}f_{\epsilon}' - f_{\epsilon, i}'\right )}_{L^{2}(W_{i})}.$$ Observe that implies $$f_\epsilon'=(f'_{\epsilon,1}, f'_{\epsilon,2})=(f_1+\epsilon\widehat{\nu}\Vert f_1\Vert_{L^2(\partial D_c)},0)\label{P2}.$$ First note that by using classical arguments based on the singular value decomposition for $K:L^2(\partial D_a)\rightarrow\Xi$, one can adapt the results in [@Kirsch-Book96] (Theorem 2.7) and obtain, $$\label{regl-1} \|KR_{\alpha}Kz - Kz \|_{\Xi} \leq C\sqrt{\alpha} \Vert z\Vert_{L^2(\partial D_a)}, \mbox{ for every } z\in L^2(\partial D_a).$$ Let $\displaystyle f=\left(\frac{f_1}{\Vert f_1\Vert_{L^2(\partial D_c)}},0\right)$ and $\displaystyle v=\frac{f_\epsilon'}{\Vert f_1\Vert_{L^2(\partial D_c)}}-f$. By using the definition of $E$ and $\Xi$, , , , , , Cauchy’s inequality and the triangle inequality in , we obtain $$\begin{aligned} \label{P4} 2|w_iL_i| &\leq& C\delta (\sqrt{w_1}\|K_1R_{\alpha}f_{\epsilon}' - f_{\epsilon,1}' \|_{L^2(\partial D_c)}+\sqrt{w_2}\|K_2R_{\alpha}f_{\epsilon}' - f_{\epsilon,2}' \|_{L^2(\partial B_R(\B0))})\nonumber\\ &\leq&C\delta (\|K_1R_{\alpha}(v+f) - (v_1+f_1) \|_{L^2(\partial D_c)}+\|K_2R_{\alpha}(v+f) - (v_2+f_2) \|_{L^2(\partial B_R(\B0))})\nonumber\\ &\leq& C\delta \|KR_{\alpha}(v+f) - (v+f) \|_{\Xi}\nonumber\\ &\leq& C\delta ( \|KR_{\alpha}f - f\|_{\Xi}\!+\!\|KR_{\alpha}v\! - \!v\|_{\Xi})\nonumber\\ &\leq& C\delta(\|KR_{\alpha}K\psi_0 - K\psi_0 \|_{\Xi}+\|KR_{\alpha}(K\psi_0 - f) \|_{\Xi}\!+\!\|K\psi_0 - f\|_{\Xi})+C\frac{\delta\epsilon}{\sqrt{\alpha}}\nonumber\\ &\leq& C\delta^2\sqrt{\alpha} +C\frac{\delta^2}{\sqrt{\alpha}}+C\delta^2+C\frac{\delta\epsilon}{\sqrt{\alpha}}\nonumber\\ &\leq& C\frac{\delta^2}{\sqrt{\alpha}},\end{aligned}$$ where Einstein summation convention was used and where, in the second inequality above we make use of to obtain $\sqrt{\frac{w_2}{w_1}}\leq 1$ and $\sqrt{w_1}\Vert f_1\Vert_{L^2(\partial D_c)}<C$ for small enough $\epsilon$, and in the fourth and fifth inequalities above we have used that $||R_\alpha||_{{\cal O}}\leq\frac{1}{2\sqrt{\alpha}}$ (e.g. see [@Kirsch-Book96]), and respectively, that $\epsilon < \delta$ and $\psi_0$ satisfies the source condition . Expanding $P$ defined in and using the fact that $f_{\epsilon,2} = 0$ and $\phi_{\alpha} = T_{\alpha}K^{*}f_{\epsilon} = T_{\alpha}K_{1}^{*}f_{\epsilon,1}$, we obtain $$\begin{aligned} \label{ec-13} P & = \frac{2\alpha'}{\|f_{\epsilon,1}\|_{L^{2}(\partial D_{c})}^{2}} {\left ( K_{1}^{*}f_{\epsilon,1},\, T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} - \frac{2\alpha'}{\|f_{\epsilon,1}\|_{L^{2}(\partial D_{c})}^{2}}{\left ( K_{1}^{*}K_{1}\phi_{\alpha}, \, T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})}\notag\\ & \quad - \frac{2 \alpha'}{2\pi R} {\left ( K_{2}^{*}K_{2}\phi_{\alpha},\, T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} + 2w_{i}L_{i}\notag\\ & = \frac{2\alpha'}{\|f_{\epsilon,1}\|^{2}} {\left ( T_{\alpha}^{-1}\phi_{\alpha},\,T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} - \frac{2\alpha'}{\|f_{\epsilon,1}\|^{2}}{\left ( K_{1}^{*}K_{1}\phi_{\alpha}, T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})}\notag\\ & \quad - \frac{2 \alpha'}{2\pi R} {\left ( K_{2}^{*}K_{2}\phi_{\alpha},\, T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} + 2w_{i}L_{i}\notag\\ & = \frac{2\alpha'}{\|f_{\epsilon,1}\|^{2}} {\left ( (\alpha I + K_{2}^{*}K_{2})\phi_{\alpha},\,T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} - \frac{2 \alpha'}{2\pi R} {\left ( K_{2}^{*}K_{2}\phi_{\alpha},\, T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} + 2w_{i}L_{i}\notag\\ & = \frac{2\alpha \alpha'}{\|f_{\epsilon,1}\|^{2}} {\left ( \phi_{\alpha},\,T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} + \frac{\alpha'}{\|f_{\epsilon,1}\|^{2}}B{\left ( K_{2}^{*}K_{2}\phi_{\alpha}, \, T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} + 2w_{i}L_{i},\end{aligned}$$ where $B = 2 - \displaystyle\frac{\|f_{\epsilon,1}\|_{L^{2}(\partial D_{c})}^{2}}{\pi R}$ and we have used that $T_\alpha^{-1}=\alpha I+K_1^*K_1+K_2^*K_2$ in the third equality above. Observe that implies $B\geq 0$. Introduce the notation $\widetilde{K}_{2} := \sqrt{B}K_{2}$, and denote $\displaystyle v_{\alpha} := \frac{\phi_{\alpha}}{\|f_{\epsilon,1}\|_{L^{2}(\partial D_{c})}}$. Then becomes $$\begin{aligned} \label{ec-14} P & = 2 \alpha \alpha' {\left ( v_{\alpha}, \, T_{\alpha}v_{\alpha}\right )}_{L^{2}(\partial D_{a})} + \alpha' {\left ( \widetilde{K}_{2}^{*}\widetilde{K}_{2}v_{\alpha}, \, T_{\alpha}v_{\alpha}\right )}_{L^{2}(\partial D_{a})} + 2w_{i}L_{i}\notag\\ & = \alpha \alpha' {\left ( v_{\alpha}, \, T_{\alpha}v_{\alpha}\right )}_{L^{2}(\partial D_{a})} + \alpha' {\left ( (\alpha I + \widetilde{K}_{2}^{*}\widetilde{K}_{2})v_{\alpha}, \, T_{\alpha}v_{\alpha}\right )}_{L^{2}(\partial D_{a})} + 2w_{i}L_{i}.\end{aligned}$$ From [@kato] (see Section V.3.10), for any self-adjoint linear operator $A:H\rightarrow H$, where $H$ is a given Hilbert space (real or complex), we have that: $$\label{Kato} 0<\gamma = \inf_{\lambda \in \mathrm{Sp}(A)}\lambda \Longrightarrow (Ax, x)_H \geq \gamma(x,x)_H,$$ where $(\cdot,\cdot)$ in denotes the usual Hilbert product and where $\mathrm{Sp}(A)$ denotes the real spectrum of the operator $A$. Then, by using for the operator $T_\alpha$ we obtain $$\begin{aligned} \label{Kato-1} {\left ( v_{\alpha}, \, T_{\alpha}v_{\alpha} \right )}_{L^{2}(\partial D_{a})} &\geq \frac{1}{\alpha + \mu_{1}^{2}}\|v_{\alpha}\|_{L^{2}(\partial D_{a})}^{2}\geq \frac{1}{1 + \mu_{1}^{2}}\|v_{\alpha}\|_{L^{2}(\partial D_{a})}^{2}\notag\\ &\geq C \|v_{\alpha}\|_{L^{2}(\partial D_{a})}^{2},\end{aligned}$$ where we have used that $\displaystyle \frac{1}{\alpha+\mu_1^2} = \inf_{\lambda \in \mathrm{Sp}(T_\alpha)}\lambda$ with $\mu_1$ denoting the largest singular value of $K$. Next consider $\displaystyle D_{\alpha} := \alpha I + \widetilde{K}_{2}^{*}\widetilde{K}_{2}$. Then, because $D_\alpha$ and $T_\alpha$ are linear, bounded, self-adjoint, invertible and positive definite operators, we have that $D_\alpha T_{\alpha}$ will also be linear, bounded, self-adjoint, invertible and have strictly positive eigenvalues. Indeed, for any eigenvalue-eigenvector pair $(x,\lambda)$ of $D_\alpha T\alpha$ we have $$D_\alpha T_\alpha x=\lambda x\Rightarrow T_\alpha x=\lambda D_\alpha^{-1}x\Rightarrow \lambda=\frac{( T_\alpha x,x)}{( D^{-1}_\alpha x,x)}\geq 0.$$ Observing that $0\notin \mathrm{Sp}(D_\alpha T_\alpha)$ we have the claim, and the positive definiteness of $D_\alpha T_\alpha$ follows. Thus we have $$\label{ec-15} {\left ( (\alpha I + \widetilde{K}_{2}^{*}\widetilde{K}_{2})v_{\alpha}, \, T_{\alpha}v_{\alpha}\right )}_{L^{2}(\partial D_{a})} = {\left ( D_{\alpha}v_{\alpha},\, T_{\alpha}v_{\alpha}\right )}_{L^{2}(\partial D_{a})}\geq 0$$ From , and used in , we obtain $$\begin{aligned} \label{ec-17} |2w_{i}L_{i}| + |Q| & \geq |\alpha'| \left| \alpha {\left ( v_{\alpha}, \, T_{\alpha} v_{\alpha}\right )}_{L^{2}(\partial D_{a})} + {\left ( D_{\alpha}v_{\alpha}, \, T_{\alpha}v_{\alpha}\right )}_{L^{2}(\partial D_{a})} \right|\notag\\ & \geq |\alpha'| C\alpha\|v_{\alpha}\|_{L^{2}(\partial D_{a})}^{2} \notag\\ & \geq C \alpha |\alpha'| \|v_{\alpha}\|_{L^{2}(\partial D_{a})}^{2}.\end{aligned}$$ From and elementary algebraic manipulations we obtain, $$\label{ec-18} |Q| \leq \frac{2 \|f_{1}\|_{L^{2}(\partial D_{c})}}{\|f_{\epsilon, 1}\|_{L^{2}(\partial D_{c})}^{4}} \|f_{\epsilon, 1}\|_{L^{2}(\partial D_{c})}\cdot \delta^{2} \|f_{\epsilon,1}\|_{L^{2}(\partial D_{c})}^{2} = \frac{2 \|f_{1}\|_{L^{2}(\partial D_{c})} \delta^{2}}{\|f_{\epsilon, 1}\|_{L^{2}(\partial D_{c})}} \leq C\delta^{2}$$ Recalling that $\displaystyle v_{\alpha} := \frac{\phi_{\alpha}}{\|f_{\epsilon,1}\|_{L^{2}(\partial D_{c})}}$, Lemma \[lemma:phif1constant\] implies $$\label{ec-19} \|v_{\alpha}\|_{L^{2}(\partial D_{a})} \geq C.$$ Then from , , and used in we finally obtain the statement of the Lemma: $$\label{ecuatie-4} \alpha |\alpha'| \leq C \delta^{2}+ C\frac{\delta^2}{\sqrt{\alpha}}\leq C\frac{\delta^2}{\sqrt{\alpha}}, \mbox{ for } \epsilon<\epsilon_0.$$ Note that $\|s_\epsilon\|_{L^{2}(\partial D_{c})}\leq (1+\frac{\epsilon}{2})\|f_{1}\|_{L^{2}(\partial D_{c})}$ and thus $f_\epsilon$ as defined above satisfies . The next result is a simple consequence of Proposition \[theorem:stabilityestimate\] and Lemma \[teorema-int\]. We have, \[corolar-1\] Assume the same notation and the same framework as in Proposition \[theorem:stabilityestimate\]. Assume also that for $d_c=diam(D_c)$ small enough there exists $d=dist(\partial D_c,\partial D_a)$ small enough so that $\alpha(0)=\alpha_0\approx\delta$. Then we have $$\displaystyle\frac{\| \phi_{\alpha_\epsilon} - \phi_{\alpha_0}\|_{L^{2}(\partial D_{a})}}{\|\phi_{\alpha_{\epsilon}}\|_{L^{2}(\partial D_{a})}} \leq C\sqrt{\epsilon}.$$ From Lemma \[teorema-int\] we obtain that $$\label{estimare} |\alpha_\epsilon'| \alpha_\epsilon^{\frac{3}{2}}\leq C\delta^2.$$ Estimate and Cauchy’s theorem implies that $$\label{ecuatie-5} \left|\frac{\alpha_\epsilon^{\frac{5}{2}}}{\alpha_0^{\frac{5}{2}}}-1\right|= \frac{5}{2}\epsilon\left|\alpha'_{\epsilon_*}\alpha_{\epsilon_*}^{\frac{3}{2}}\alpha_0^{-\frac{5}{2}}\right|\leq C\epsilon\delta^2\alpha_0^{-\frac{5}{2}}\leq C\sqrt{\epsilon},$$ where $\epsilon_*\in(0,\epsilon)$ and we used that $\epsilon<\delta$. Next, simple algebraic manipulation and imply $$\left|\frac{\alpha_\epsilon}{\alpha_0}-1\right|\leq \left|\frac{\alpha_\epsilon^5}{\alpha_0^5}-1\right|\leq C\sqrt{\epsilon}\left(\frac{\alpha_\epsilon^{\frac{5}{2}}}{\alpha_0^{\frac{5}{2}}}+1\right)\leq C\sqrt{\epsilon}(2+ C\sqrt{\epsilon})\leq C\sqrt{\epsilon}.$$ This together with Proposition \[theorem:stabilityestimate\] imply the statement of the Corollary. We note that all the above results can be adapted to three dimensions. The proof follows exactly the same steps by considering the natural extension of the definition of the discrepancy function $E$ in three dimensions. The assumption made in Corollary \[corolar-1\] that $\alpha_0\approx\delta$ for $d_c=diam(D_c)$ and $dist(\partial D_c,\partial D_a)=d$ small enough is based on of Lemma \[lemma:phif1constant\] and on the numerical results presented in Section \[sec:numerics\]. In particular, for the same settings as in Figure \[fig:nonincreasing\_F\], Figure \[fig:besselFS\_alpha0\_vary\_k\_mu\] shows that given small $d_c$ for small enough $d$, we have roughly that $10^{-2.5} \leq \alpha_{0} \leq 10^{-1}$. Since $\delta = 2\cdot10^{-2}$ we have in this case that $\frac{1}{2\sqrt{5}} \delta \lesssim \alpha_{0} \lesssim 5 \delta$. \[constant\] As suggested by our numerical results in Section \[sec:numerics\] we beleive that the constants (denoted by $C$) in Proposition \[theorem:stabilityestimate\], Lemma \[teorema-int\] and Corollary \[corolar-1\] will only have small values for $d_c=diam(D_c)$ and $dist(\partial D_c,\partial D_a)=d$ small enough. Numerics {#sec:numerics} ======== In this section we proceed with the numerical study of the minimal norm solution for obtained through Tikhonov regularization with the Morozov discrepancy principle for the choice of the regularization parameter in two dimensions. First we focus on the general setup of our numerical approach, and then in Section \[subsec:parametersetup\] we discuss more specifically the parameters used in our numerical examples. In Sections \[subsec:nearfieldstability\] and \[subsec:sphericalsource\] we present numerical data which demonstrates how stably $\phi$ depends on $f$ and various control statistics for a spherical point source. All figures generally display their respective parameters in an offset legend. For all of the numerical computations done in this section, we discretize the integral operator $K$ via the method of moment collocation. We refer to ([@Kress-Book99], §17.4) for more details on the method. First we choose an approximate basis of functions for $L^{2}(\partial D_{a})$. To do this we suppose the domain $D_{a}$ can be parametrized in polar coordinates by points $$(s(\tau)\cos{\tau}, \, s(\tau)\sin{\tau})), \quad \tau \in [0,2\pi],$$ where $s:\mathbb{R} \to \mathbb{R}_{+}$ is a $2\pi$-periodic smooth function. Using these coordinates, any function $\phi$ defined on $\partial D_{a}$ can be realized via the pullback as a function of $\tau$: $$\phi(s(\tau)\cos{\tau}, \, s(\tau)\sin{\tau}).$$ For convenience, let us use the notation $\widehat{\tau} = (\cos{\tau},\sin{\tau})$ and $\widehat{\tau}^{\perp} = (-\sin{\tau}, \cos{\tau})$. Now let $n_{a} \in \mathbb{N}$ and let $\tau_{j} = \frac{2\pi j}{n_{a}}$, $0 \leq j \leq n_{a}-1$ be $n_{a}$ equally spaced points on the interval $[0,2\pi)$. We then use the exponential basis functions $\{e^{il\tau}\}_{l=0}^{n_{a}-1}$ for $L^{2}([0,2\pi])$ and approximate a given $\phi \in L^{2}(\partial D_{a})$ via interpolation at the points $\{\tau_{j}\}_{j=0}^{n_{a}-1} \subset [0,2\pi]$. Note that $$\begin{aligned} \int_{\partial D_{a}}\phi(\mathbf{y}) \frac{\partial \Phi}{\partial \nu_{\mathbf{y}}}(\mathbf{x}, \, \mathbf{y})\,dS_{\mathbf{y}} & = \int_{0}^{2\pi}\phi(s(\tau)\cos{\tau}, \, s(\tau)\sin{\tau})\frac{\partial \Phi}{\partial \nu_{y}}(\mathbf{x}, (s(\tau)\cos{\tau}, \, s(\tau)\sin{\tau})) \notag\\ & \quad \cdot \sqrt{s(\tau)^{2} + s'(\tau)^{2}}\,d\theta.\end{aligned}$$ Furthermore, since $\left( s'(\tau)\cos{\tau} - s(\tau)\sin{\tau}, \, s(\tau)\cos{\tau} + s'(\tau)\sin{\tau}\right)$ is a tangent vector to $\partial D_{a}$, we have that $$\begin{aligned} \nu(\mathbf{y}) = \nu(\tau) & = \frac{(s(\tau)\cos{\tau} + s'(\tau)\sin{\tau}, \, s(\tau)\sin{\tau} - s'(\tau)\cos{\tau})}{\sqrt{ s(\tau)^{2} + s'(\tau)^{2}}}\\ & = \frac{s(\tau)\widehat{\tau} - s'(\tau)\widehat{\tau}^{\perp}}{\sqrt{ s(\tau)^{2} + s'(\tau)^{2}}}\end{aligned}$$ is the unit outward normal vector to $\partial D_{a}$. It is then straightforward to compute in the case of the Helmholtz equation in 2D that $$\begin{aligned} & \quad \frac{\partial \Phi}{\partial \nu_{\mathbf{y}}}(\mathbf{x}, \, (s(\tau)\cos{\tau}, \, s(\tau)\sin{\tau}))\\ & = \nabla_{y}\Phi(\mathbf{x}, (s(\tau)\cos{\tau}, \, s(\tau)\sin{\tau})) \cdot \nu(\tau)\\ & = \frac{ik}{4}H_{0}^{(1)'}(k|\mathbf{x} - s(\tau)\widehat{\tau}|)\frac{s(\tau)\widehat{\tau} - \mathbf{x}}{\sqrt{s(\tau)^{2} + |\mathbf{x}|^{2} - 2s(\tau)\mathbf{x}\cdot \widehat{\tau}}} \cdot \frac{s(\tau)\widehat{\tau} - s'(\tau)\widehat{\tau}^{\perp}}{\sqrt{s(\tau)^{2} + s'(\tau)^{2}}}.\end{aligned}$$ Let $n_{a} \in \mathbb{N}$ be the number of sample points on the antenna, $\partial D_{a}$, and let $n_{c} \in \mathbb{N}$ be the total number of sample points on $\partial D_{c}$. Also let $n_{R}$ be the total number of sample points on $\partial B_{R}$. We write the $2 \times (n_{c} + n_{R})$ matrix of points $$\mathbf{X} := [x_{1}, \ldots, x_{n_{c}+n_{R}}],$$ where each $x_{j}$ is a $2$-vector, $\{x_{j}\}_{j=1}^{n_{c}} \subset \partial D_{c}$ and $\{x_{j}\}_{j=n_{c}+1}^{n_{c} + n_{R}} \subset \partial B_{R}$. Approximations of all the integrals involved are then computed using a standard left endpoint sum with the appropriate quadrature weights. All the numerical examples presented herein take $D_{c}$ to be an annular sector parametrized by $r \in [r_{1}, r_{2}]$ and $\theta \in [\theta_{1}, \theta_{2}]$. See Figure \[fig:numericssetup\] for details. For each $1 \leq j \leq n_{c} + n_{R}$ and each $0 \leq l \leq n_{a}-1$, we compute $K[e^{il\tau}](x_{j})$ via the approximation $$\frac{2\pi}{n_{a}}\sum_{m=0}^{n_{a}-1} \frac{\partial \Phi( x_{j}, [s(\tau_{m})\cos{\left ( \tau_{m}\right )}, \, s(\tau_{m})\sin{\tau_{m} }]^{T})}{\partial \nu_{\mathbf{y}}} e^{il \tau_{m}}\sqrt{s(\tau_{m})^{2} + s'(\tau_{m})^{2}}.$$ If we fix $j$ and vary $l$, we see that the above sum is equivalent to computing the discrete Fourier transform of the $n_{a}$-vector $$\mathbf{v}_{j} := \left[\frac{\partial \Phi( x_{j}, [s(\tau_{m})\cos{\tau_{m}}, \, s(\tau_{m})\sin{\tau_{m} }]^{T})}{\partial \nu_{\mathbf{y}}}\sqrt{s(\tau_{m})^{2} + s'(\tau_{m})^{2}}\right]_{m=0}^{n_{a}-1}, \label{eq:vj}$$ which can be computed efficiently using the Fast Fourier Transform algorithm. In particular, for the vector $\mathbf{v}_{j}$ in $$[K\{e^{il\tau}\}(x_{j})]_{l=0}^{n_{a}-1} \approx 2\pi \mathsf{FFT}(\mathbf{v}_{j}),$$ where $\mathsf{FFT}$ is defined on $n_{a}$-vectors $\mathbf{w} = [w_{1},\ldots, w_{n_{a}}]^{T} \in \mathbb{C}^{n_{a}}$ by $$\mathsf{FFT}(\mathbf{w}) = \left[ \frac{1}{n_{a}}\sum_{j=1}^{n_{a}}w_{j}e^{\frac{2\pi i (j-1)(l-1)}{n_{a}}} \right]_{l=1}^{n_{a}}. \label{eq:fft}$$ So the matrix representation of $K$ is then the $n_{a} \times (n_{c} + n_{R})$ matrix $$A := 2\pi [\mathsf{FFT}(\mathbf{v}_{1}), \cdots, \mathsf{FFT}(\mathbf{v}_{n_{c}+n_{R}})].$$ Now, in order to approximately solve the ill-posed problem $K\phi = (f_{1}, f_{2})$, we attempt to solve the linear system $$\begin{aligned} K_{1}\phi(x_{j}) & = f_{1}(x_{j}), \quad 1 \leq j \leq n_{c}\\ K_{2}\phi(x_{j}) & = f_{2}(x_{j}), \quad n_c + 1 \leq j \leq n_c + n_{R}.\end{aligned}$$ Since $A$ is computed with respect to the functions $e^{il\theta}$, we first consider the approximate coefficients of $\phi$ with respect to the finite basis $\{e^{il\tau}\}_{l=0}^{n_{a}-1}$, given by $$c_{l} := \frac{1}{n_{a}}\sum_{m=0}^{n_{a}-1}e^{-il\tau_{m}}\phi(s(\tau_{m})\cos(\tau_{m}), \, s(\tau_{m})\sin(\tau_{m}))\,d\tau \approx \frac{1}{2\pi}\int_{0}^{2\pi} e^{-il\tau}\phi(s(\tau)\cos(\tau), \, s(\tau)\sin(\tau))\,d\tau.$$ Let $$h = [c_{0}, c_{1}, \ldots, c_{n_{a} - 1}]^{T} \in \mathbb{C}^{n_{a}}.$$ We then numerically compute the Tikhonov regularized solution $$h_{\alpha} := (A^{*}A + \alpha I)^{-1}A^{*}f,$$ with $\alpha > 0$. The solution vector $h_{\alpha}$ yields the approximate coefficients $c_{l}$ of the desired density $\phi$ with respect to the functions $\{e^{il\tau}\}_{l=0}^{n_{a}-1}$. We obtain the density $\phi_{\alpha}$ corresponding to $h_{\alpha}$ sampled at the angles $\tau_{m}$ on $\partial D_{a}$ by the formula $$\phi_{\alpha}(\tau_{m}) := \sum_{l=0}^{n_{a}-1}[h_{\alpha}]_{l}e^{il\tau_{m}}.$$ After computing the residual $K\phi - f$ (e.g. for $\phi = \phi_{\alpha}$), we will then need to compute $$E(\phi_\alpha, f) = \frac{1}{\|f_{1}\|_{L^{2}(\partial D_{c})}^{2}}\| K_{1}\phi - f_{1}\|_{L^{2}(\partial D_{c})}^{2} + \frac{1}{2\pi R} \|K_{2}\phi - f_{2}\|_{L^{2}(\partial B_{R})}^{2}.$$ Recall that the discrepancy function $F(\alpha)$ was defined by $$F(\alpha) = E(\phi_\alpha, f) - \delta^{2}, \label{eq:F}$$ where $\delta > 0$ is a fixed error parameter. As discussed in Section \[sec:stability\], the mapping $$\alpha \mapsto E(\phi_\alpha, f)$$ is not globally increasing, as can be numerically demonstrated. However, for certain feasible regions of $(\alpha, \epsilon)$, $F$ is increasing. And in this case, there is a unique $\alpha_{\delta}$ such that $F(\alpha_{\delta}) = 0$. To find $\alpha_{\delta}$, we use Newton’s method combined with an initial coarse line search to identify a good starting point. First note that if we split the matrix $A$ into two blocks $A_{near}$ ($n_{c}$ by $n_{a}$) and $A_{far}$ ($n_{R}$ by $n_{a}$) so that $$A = \left[ \begin{array}{c} A_{near}\\ A_{far} \end{array}\right],$$ then $[A\phi]_{1} = A_{near}\phi$, $[A\phi]_{2} = A_{far}\phi$, and $A^{*}A = A_{near}^{*}A_{near} + A_{far}^{*}A_{far}$. In the discretized setting, instead of (\[eq:F1\]) we take $$F(\alpha) =\frac{1}{\|f_{1}\|^{2}} \|A_{near}h_{\alpha} - f_{1}\|_{L^{2}(\partial D_{c})}^{2} + \frac{1}{2\pi R}\|A_{far}h_{\alpha} - f_{2}\|_{L^{2}(\partial B_{R})}^{2} - \delta^{2}$$ with $$h_{\alpha} = (A^{*}A + \alpha I)^{-1}A^{*}f = (A_{near}^{*}A_{near} + A_{far}^{*}A_{far} + \alpha I)^{-1}{\left ( A_{near}^{*}f_{1} + A_{far}^{*}f_{2}\right )}.$$ Then in the same spirit as that presented in [@CoKr-Book98] for Tikhonov regularization with respect to the standard $L^2$ norm, we compute $$\begin{aligned} F'(\alpha) & = \frac{-2\alpha}{\|f_{1}\|_{L^{2}(\partial D_{c})}^{2}}{\textrm{Re}}{\left ( \frac{\partial h_{\alpha}}{\partial \alpha}, \, h_{\alpha}\right )} \notag\\ & \quad + {\left ( \frac{1}{\pi R} - \frac{2}{\|f_{1}\|_{L^{2}(\partial D_{c})}^{2}}\right )}{\textrm{Re}}{\left ( \frac{\partial h_{\alpha}}{\partial \alpha}, \, A_{far}^{*}A_{far}h_{\alpha} - A_{far}^{*}f_{2}\right )} \label{eq:dF1}\\ \frac{\partial h_{\alpha}}{\partial \alpha} & = -(A^{*}A + \alpha I)^{-1}h_{\alpha}, \label{eq:dF2}\end{aligned}$$ where $( \cdot, \cdot )$ denotes the $L^{2}$ inner product on $\partial D_{a}$. The function $f_{1}$ defined on $\partial D_{c}$ could be, for example, the trace of a plane wave, or of the fundamental solution to the Helmholtz equation based at some fixed point $\mathbf{x}_{0}$, i.e. a point source. For this paper, we focus on the case where $f_{1}$ is a point source and where $f_{2} \equiv 0$ on $\partial B_{R}$. A spherical point source is represented as $$\frac{i}{4}H_{0}^{(1)}(k|\mathbf{x} - \mathbf{x}_{0}|), \label{eq:sphericalsource}$$ where $\mathbf{x}_{0}$ is the source point (typically outside of $B_{R}$). For such an $f_{1}$, there are some quantities in which we will be interested so as to determine the effectiveness of a given density $\phi$ in solving the problem $K\phi =f$. These are: the relative error of $K\phi$ on $\partial D_{c}$; the $L^{2}$ average of $K\phi$ on $\partial B_{R}$; the relative and absolute stability of $\phi$ when applying a small perturbation to $f_{1}$; the norm of $\phi$ on $\partial D_{a}$. In other words, we will measure $$\frac{\|K_{1} \phi - f_{1}\|_{L^{2}(\partial D_{c})}}{\|f_{1}\|_{L^{2}(\partial D_{c})}}, \quad \frac{1}{\sqrt{2\pi R}} \|K_{2}\phi\|_{L^{2}(\partial B_{R})}, \label{eq:computevar1}$$ $$\frac{\|\phi_{\alpha_{\epsilon}}- \phi_{\alpha_0}\|_{L^{2}(\partial D_{a})}}{\|\phi_{\alpha_0}\|_{L^{2}(\partial D_{a})}}, \quad \|\phi_{\alpha_\epsilon} - \phi_{\alpha_0}\|_{L^{2}(\partial D_{a})}, \label{eq:computevar2}$$ and $$\|\phi\|_{L^{2}(\partial D_{a})}, \label{eq:phinorm}$$ where $\phi_{\alpha_\epsilon}$ is the Tikhonov regularization solution to $K\phi = (f_{1,\epsilon},0)$ with $\|f_{1} - f_{1,\epsilon}\|_{L^{2}(\partial D_{c})} = \epsilon \|f_{1}\|_{L^{2}(\partial D_{c})}$, and $\phi_{\alpha_0}$ is the solution with unperturbed $f_{1}$. The Morozov solution $\alpha_0$ and $\alpha_\epsilon$ are computed via Newton’s Method using and such that $$\begin{aligned} E(\phi_{\alpha_0}, f) & = \delta^2 \notag\\ E(\phi_{\alpha_\epsilon}, f_{\epsilon}) & = \delta^2.\end{aligned}$$ See also the discussion following . Recall from (\[random-f\]), that when adding noise to the data $(f_{1}, 0)$, we choose a random perturbation $\eta \in L^{2}(\partial D_{c})$ and set $$f_{1,\epsilon} = f_{1} + \epsilon \widehat{\eta} \|f_{1}\|_{L^{2}(\partial D_{c})}, \label{eq:noiseterm}$$ where $\epsilon > 0$ represents the relative percentage of noise added. In the discrete case, the noise is chosen to be a complex $n_c$-vector $\nu$ whose real and imaginary components are pseudorandom numbers (we used uniformly distributed noise, but any distribution would yield similar results) on the interval $(-1,1)$. Furthermore, for reproducibility, whenever generating $\nu$ using a pseudorandom number generator, we always reset the seed to the same value. Parameters Used for Numerical Experiments\[subsec:parametersetup\] ------------------------------------------------------------------ Here we describe some of the parameters used for the various numerical experiments presented. In Section \[subsec:sphericalsource\] we always assume that $\partial D_{a}$ is a circle with radius given by $a = 0.01$, and that $\partial D_{c}$ is a sector of an annulus with $\theta_{1} = 3\pi/4$ and $\theta_{2} = 5\pi/4$. We also take $R = 10$ in all computational examples. We remark that we always restrict the distance from $D_{c}$ to $D_{a}$ to be no smaller than $10^{-3}$ due to the numerical limitations of our approach. This is due to the fact that $K\phi$ is a singular integral when evaluating at points very near to $\partial D_{a}$. Therefore, at points on $\partial D_{c}$ that are near $\partial D_{a}$ the limitations of machine precision become more and more apparent. Numerically, we observed that our direct approach starts to break down near $d = \mathrm{dist}(\partial D_{c}, \partial D_{a}) = 10^{-4}$. However, we stress that one could most likely obtain high accuracy in computing $K\phi$ for $d \leq 10^{-4}$ by using the Nyström method as discussed in [@Kress-Book99]. For the collocation method, we use $n_{a} = 256$ sample points on $\partial D_{a}$, and $n_{\mathrm{arc}_{1}} = 256$ points on the inner arc of $\partial D_{c}$, with the remaining points chosen so as to keep the quadrature weights approximately constant. Thus for a very thin region, $n_{c} \approx 512$. We also take $n_{R} = 256$ (number of sample points on $\partial B_{R}$). Note that increasing $n_{c}$ or $n_{R}$ will put more emphasis on matching $f$ on $\partial D_{c}$ or $\partial B_{R}$, respectively. The discrepancy parameter $\delta$ used for Tikhonov regularization will typically be fixed at $0.02$. The key variables under consideration are $d = r_{1} - a$, $k$, and $\epsilon$ (perturbation parameter for adding noise to $f_{1}$). All of the plots presented in the following sections involve varying two of the aforementioned parameters and plotting different quantities of interest, as stated in (\[eq:computevar1\])-(\[eq:phinorm\]). When evaluating the relative change in $\phi$ given a perturbation of $f_{1}$, denoted by $f_{\epsilon, 1}$, we remark that for the parameter choices we used, a $0.5\%$ change ($\epsilon = 0.005$) in $f_{1}$ yielded a roughly $5\%$ change in $\phi$. However, one must keep in mind that this depends quite a lot on the parameters used. In particular, setting the discrepancy $\delta = 0.02$ in all the simulations had an important effect on the numerical results. If we had used $\delta = 0.05$ instead (which leads to approximately a $5 \%$ mismatch on the region $\partial D_{c}$), then the relative change in $\phi$ given $\epsilon = 0.005$ would be noticeably smaller. So ultimately there is a tradeoff between $\mu$, $R$, $k$, $\delta$, and $\epsilon$ which is not entirely trivial, but this still can be examined experimentally as we have done. Near field stability\[subsec:nearfieldstability\] ------------------------------------------------- We present below Figure \[fig:singularvalues\], which shows how the first $50$ singular values of the operator $K = (K_{1}, K_{2})$ vary with $d$. It is clear that for $d$ small (i.e. for control in the nearfield of the antenna), the rate of decay of the singular values of $K$ is considerably slower than for larger $d$. This in turn provides some experimental evidence for the fact that nearfield control seems to be more feasible in terms of stable dependence of the solution $\phi$ on $f$. We also show Figure \[fig:singularvaluesdiff\], which shows the behavior of the first and sixth singular value of $K$ with respect to $d$ and $k$. ![Plot of first $50$ singular values of $K:L^{2}(\partial D_{a}) \to \Xi$ for $\partial D_{a}$ a circular antenna of radius $a = 0.01$ and $\partial D_{c}$ an annular region of varying distance from $\partial D_{a}$.[]{data-label="fig:singularvalues"}](50_singular_values){width="\textwidth"} ![Plot of first singular value of $K:L^{2}(\partial D_{a}) \to \Xi$ as well as the relative difference of the first and sixth singular values with respect to $d$ and $k$. Again, $\partial D_{a}$ is a circular antenna of radius $a = 0.01$ and $\partial D_{c}$ an annular region.[]{data-label="fig:singularvaluesdiff"}](singular_values_diff){width="\textwidth"} Control for a Spherical Point Source\[subsec:sphericalsource\] -------------------------------------------------------------- We now consider the case that $$f_{1}(\mathbf{x}) = \frac{i}{4}H_{0}^{(1)}(k|\mathbf{x} - \mathbf{x}_{0}|),$$ where $\mathbf{x}_{0}$ is the source point. In all examples presented in this section, we have $R = 10$ unless otherwise noted, and $\mathbf{x}_{0} = [20, 0]^{T}$ or $\mathbf{x}_{0} = [10000, 0]^{T}$ (to approximate a source at infinity). First we observe how the frequency $k$ and distance $d$ from $\partial D_{c}$ to $\partial D_{a}$ affects the various control criteria. In figures \[fig:besselFS\_epsilon\_0005\_delta\_002\_vary\_k\_01\_to\_100\_mu\_0001\_to\_003\_data\] and \[fig:besselNS\_epsilon\_0005\_delta\_002\_vary\_k\_01\_to\_100\_mu\_0001\_to\_003\_data\] we vary $k$ from $0.1$ to $100$ and $d$ from $0.001$ to $0.003$ with $a = 0.01$. With the error discrepancy set at $\delta = 0.02$, we have in both figures that relative error on $\partial D_{c}$ is very close to $2\%$ for all data points. Moreover, with $0.5\%$ noise added to $f$, roughly a $5\%$ change in $\phi$ is observed at all frequencies when $d$ is near its lower limit. A bit more sensitivity is observed for frequencies $k < 20$ when $d$ increases beyond $0.01$. Interestingly, for $k > 20$ the optimal parameter $\alpha$ is larger and corresponding power budget smaller in order to achieve discrepancy $\delta$. In figures \[fig:besselFS\_epsilon\_0005\_delta\_002\_vary\_k\_01\_to\_100\_mu\_0001\_to\_003\_data\] and \[fig:besselNS\_epsilon\_0005\_delta\_002\_vary\_k\_01\_to\_100\_mu\_0001\_to\_003\_data\] it is clear that for smaller $d$ values the sensitivity of $\phi$ to $0.5\%$ noise added to $f$ is close to $5\%$. As $d$ increases, sensitivity of $\phi$ to noise increases as expected. Having $\mathbf{x}_{0}$ nearer or farther from $\partial B_{R}$ does not have a very significant effect on the overall shape of each subplot. Figures \[fig:besselFS\_vary\_mu\_epsilon\] and \[fig:besselNS\_vary\_mu\_epsilon\] show how the quantities of interest change with $d$ and the noise factor $\epsilon$, both with $k = 10$. The reason for choosing $k=10$ instead of, e.g., $k=1$ is that from figure \[fig:besselFS\_epsilon\_0005\_delta\_002\_vary\_k\_01\_to\_100\_mu\_0001\_to\_003\_data\] we see a slightly higher sensitivity of $\phi$ to noise for approximately $1 < k < 20$ when $d$ starts to increase. So the goal was to capture the worst case scenario for the control stability. For smaller values of $d$ we see as before that a roughly $0.5\%$ change in $f_{1}$ yields about a $5\%$ change in $\phi$. Moreover, the dependence on $\epsilon$ for fixed $d$ is superlinear, consistent with the illposedness of the problem. Interestingly, sensitivity of $\phi$ at $d \approx 0.015$ is better than at nearby values, but of course such a value depends on the other parameters of the problem setup. Finally, we consider Figure \[fig:besselFS\_vary\_k\_R\], which shows the dependence on $R$ and $k$ for a source at $\mathbf{x}_{0} = [10000,\, 0]^{T}$. Overall, one can see that $R$ can be decreased to around $R = 3$ at any frequency between $0.1$ and $100$ and still achieve the same approximate level of sensitivity for $\phi$ as in the previous plots with $R = 10$. Conclusions and Future Work {#sec:conclusions} =========================== In this paper we studied the feasibility of the active control scheme for the scalar Helmholtz equation. In the $L^2$ setting, we presented analytic conditional stability results as well as detailed numerical sensitivity studies for the minimal energy solution. We provided several analytic and numerical arguments for the scheme’s feasibility and broadband character in the near field when the interrogating field is a far field of a far field observer. We focused our discussion in this paper only on the case of an interrogating far field point source (i.e. similar to a plane wave with corresponding decay) because we believe that this situation is relevant in usual radar or sonar detection problems. In contrast, the case of an interrogating plane wave corresponds to a different problem, where the observer is close to the source and control region and thus the interrogating signal does not have sufficient decay. In fact, we have numerically studied the case when the interrogating field is a plane wave without decay or a given uniform field. We observed the scheme does not behave well for the uniform field and that although the stability and accuracy of the near field scheme are essentially independent of the plane wave direction, the overall performance of the scheme is not as good when compared to the case of an interrogating signal coming from a far field observer presented above. In fact, for the same settings as in Figure \[fig:nonincreasing\_F\] when comparing the case of an interrogating far field point source with an interrogating plane wave, we obtained $5\%$ versus $8\%$ stability error and power budget levels of $\approx 10^{-1}$ versus $\approx 10$. We conclude that the scheme performance depends not only on the location of the control region with respect to the source region but also on the amplitude and oscillatory pattern of the incoming field. Currently we are considering a more localized basis for $L^2(\partial D_a)$ (e.g. delta function basis, or splines) in order to better observe the field characteristics in the control region $D_c$ and around the antenna $D_a$. We also plan to study the active control scheme for linear arrays and for large elongated antennas. Then, as a next step in our research efforts, we will work on the extension of the current numerical sensitivity study to three dimensions and full Maxwell system and on the study of near field control with planar and conformal arrays.

--- abstract: 'The extremal coefficient function (ECF) of a max-stable process $X$ on some index set $T$ assigns to each finite subset $A \subset T$ the effective number of independent random variables among the collection $\{X_t\}_{t \in A}$. We introduce the class of Tawn–Molchanov processes that is in a 1:1 correspondence with the class of ECFs, thus also proving a complete characterization of the ECF in terms of negative definiteness. The corresponding Tawn–Molchanov process turns out to be exceptional among all max-stable processes sharing the same ECF in that its dependency set is maximal [w.r.t. ]{}inclusion. This entails sharp lower bounds for the finite dimensional distributions of arbitrary max-stable processes in terms of its ECF. A spectral representation of the Tawn–Molchanov process and stochastic continuity are discussed. We also show how to build new valid ECFs from given ECFs by means of Bernstein functions.' address: 'Institute of Mathematics, University of Mannheim, 68131 Mannheim, Germany.\' author: - - title: 'An exceptional max-stable process fully parameterized by its extremal coefficients' --- Introduction {#sect:intro} ============ Besides the class of [square integrable processes]{}, the class of temporal or spatial *max-stable processes* is of common interest in stochastics and statistics, [cf. ]{}[@dehaan_84; @ginehahnvatan_90; @wangstoev_10; @blanchetdavison_11; @buishandetalii_08; @naveauetalii_09], for example. In spite of considerable differences between these two classes, for example, the non-existence of the first moments in case of max-stable processes with *unit Fréchet marginals*, connections between the two classes have been made for instance, by the *extremal Gaussian process* [@schlather_02] and the *Brown–Resnick process* [@kabluchkoetalii_09] that are parameterized by a [correlation function]{} and a [variogram]{}, respectively. Naturally, extremal dependence measures such as the *extremal coefficients* [@smith_90; @schlathertawn_02], the *(upper) tail dependence coefficients* [@beirlantetalii_03; @davismikosch_09; @falk_05; @colesheffernantawn_99] or other special cases of the *extremogram* [@davismikosch_09] are appropriate summary statistics for max-stable processes. In this article, we capture the full set of extremal coefficients of a max-stable process $X=\{X_t\}_{t \in T}$ on some space $T$ in the so-called *extremal coefficient function (ECF)* $\theta$, which assigns to each finite subset $A$ of $T$ the effective number of independent variables among the collection $\{X_t\}_{t \in A}$. We introduce a subclass of max-stable processes that is parameterized by the ECF, and thus reveal some analogies to *Gaussian processes* and *positive definite functions* as follows: Among (zero mean) square integrable processes, the subclass of *Gaussian processes* takes a unique role, since it is in a 1–1 correspondence with the set of *covariance functions*, which are precisely the *positive definite functions*. This fact can be proven by means of Kolmogorov’s extension theorem and is illustrated in the following graph: ----------------------  ---------------------- In case $T$ is a metric space, the Gaussian process $Z^*(C)$ is continuous in the mean square sense (and then also stochastically continuous) if and only if the covariance function $C$ is continuous if and only if $C$ is continuous on the diagonal ([cf. ]{}[@scheuerer_10], Theorem 5.3.3). Well-known operations on the set of positive definite functions $C$, and hence on the corresponding Gaussian processes $Z^*(C)$, include convex combinations and pointwise limits. Moreover, *Bernstein functions* play an important role for the construction of positive definite functions. In our case, the crucial role of zero mean Gaussian processes is taken by the class of *Tawn–Molchanov processes (TM processes)*, which are in fact the spatial generalization of the multivariate *max-linear* model of [@schlathertawn_02]. Using Kolmogorov’s extension theorem, we shall see that each ECF $\theta$ (of some max-stable process) uniquely determines a TM process $X^*(\theta)$ having the same ECF (Theorem \[thm:ECF\_ND\]). Alongside, we generalize a multivariate result [@molchanov_08], Corollary 1, to the spatial setting, proving that the ECFs coincide with the functions $\theta$ on ${\mathcal{F}}(T)$ (the *set of finite subsets* of $T$) that are normalized to $\theta (\varnothing )=0$ and $\theta(\{t\})=1$ for $t \in T$ and that are *negative definite* (or equivalently *completely alternating*) in a sense to be explained below ([cf. ]{}Definition \[def:ND\_CA\]). This can be illustrated in analogy to the above sketch: ----------------------  ---------------------- Having identified the ECF $\theta$ as a negative definite quantity allows for several immediate consequences: First, we obtain an *integral representation* of $\theta$ as a mixture of maps $A \mapsto\mathbh{1}_{A \cap Q \neq\varnothing}$ (Corollary \[cor:intrep\]) and derive a *spectral representation* for the corresponding TM process $X^*(\theta)$ (Theorem \[thm:spectralrep\]). Second, we consider operations on ECFs that allow to build new ECFs from given ones. We find that ECFs allow for convex combinations and pointwise limits (Corollaries \[cor:Theta\_convex\] and \[cor:Theta\_compact\]) and that the class of *Bernstein functions* operates on ECFs (Corollary \[cor:opBernstein\]). We also recover the “triangle inequalities” for $\theta$ from [@cooleyetalii_06], Proposition 4, and see that the inequalities therein correspond to three specific choices of a Bernstein function, whereas we may plug in arbitrary Bernstein functions to obtain further “triangle inequalities” (Corollary \[cor:triangleineq\]). For $T$ being a metric space, we discuss *stochastic continuity*: The TM process $X^*(\theta)$ is stochastically continuous if and only if $\theta$ is continuous ([cf. ]{}Definition \[def:ECF\_cts\]) if and only if the bivariate map $(s,t) \mapsto\theta(\{s,t\})$ is continuous if and only if the bivariate map $(s,t) \mapsto\theta(\{s,t\})$ is continuous on the diagonal (Theorem \[thm:ECprocess\_cty\]). Finally, we address the exceptional role of the TM processes among simple max-stable processes. To this end, Molchanov’s *dependency set* ${{\mathcal K}}$ [@molchanov_08] is transferred to max-stable processes $X$. It comprises the finite dimensional distributions (f.d.d.) of $X$ (Lemma \[lemma:DepSetprops\]). Now, let ${{\mathcal K}}^*(\theta)$ denote the dependency set of the process $X^*(\theta)$. Then we identify ${{\mathcal K}}^*(\theta)$ as intersection of halfspaces that are directly given by the ECF $\theta$ (Theorem \[thm:starDepSet\]). It turns out that ${{\mathcal K}}^*(\theta)$ is exceptional among the dependency sets ${{\mathcal K}}$ of all max-stable processes sharing the same ECF $\theta$, since ${{\mathcal K}}^*(\theta)$ is maximal [w.r.t. ]{}inclusion as illustrated in Figure \[fig:ballDepSet\]. Since inclusion of dependency sets corresponds to stochastic ordering, this observation leads to sharp inequalities for the [f.d.d. ]{}of max-stable processes in terms of its ECF $\theta$ (Corollary \[cor:fddinequalities\]). The text is structured as follows. After the introductory Section \[sect:foundations\], the characterization of ECFs and the existence of TM processes is established in Section \[sect:TMprocess\]. Section \[sect:consequences\] collects several immediate consequences and related results, while Section \[sect:depset\] exhibits the exceptional role of TM processes. Sections \[sect:consequences\] and \[sect:depset\] can be read independently. ![Examples of dependency sets in a trivariate setting: a “typical” dependency set ${{\mathcal K}}$ (left) and a dependency set ${{\mathcal K}}^*$ stemming from a TM process (right). It is shown that ${{\mathcal K}}\subset{{\mathcal K}}^*$ (middle). For further details, see the introduction, Example \[example:ball\], Lemma \[lemma:DepSet\] and Theorem \[thm:starDepSet\].[]{data-label="fig:ballDepSet"}](567f01.eps) Foundations and definitions {#sect:foundations} =========================== Notation for max-stable processes and ECFs ------------------------------------------ A stochastic process $X=\{X_t\}_{t \in T}$ on an arbitrary index set $T$ is said to be *max-stable* if for each $n \in{\mathbb{N}}$ and independent copies $X^{(1)},\ldots,X^{(n)}$ of $X$ the process of the maxima $\{\bigvee_{i=1}^n X^{(i)}\}_{t \in T}$ has the same law as $\{ a_n(t) X_t + b_n(t)\}_{t \in T}$ for suitable norming functions $a_n(t)>0$ and $b_n(t)$ on $T$. Without loss of generality, we shall deal with max-stable processes that have *standard Fr[é]{}chet marginals*, that is, ${\mathbb{P}}(X_t \leq x)=\mathrm{e}^{-1/x}$ for $t \in T$ and $x \geq0$, and set $a_n(t)=n$ and $b_n(t)=0$. Such processes are called *simple* max-stable processes. It has been shown ([cf. ]{}[@kabluchko_09; @dehaan_84; @stoev_08]) that (simple) max-stable processes $X=\{X_t\}_{t \in T}$ allow for a *spectral representation* $(\Omega,{{\mathcal A}},\nu,V)$: there exists a (sufficiently rich) measure space $(\Omega,{{\mathcal A}},\nu )$ and measurable functions $V_t:\Omega\rightarrow{\mathbb{R}}_+$ (with $\int_{\Omega} V_t(\omega) \nu({\mathrm{d}}\omega) = 1$ for each $t \in T$), such that the law of $X=\{X_t\}_{t \in T}$ equals the law of $$\label{eqn:spectralrepresentation} \biggl\{ \bigvee_{(U,\omega)\in\Pi} U V_t(\omega) \biggr\}_{t \in T}.$$ Here $\Pi$ denotes a Poisson point process on ${\mathbb{R}}_+ \times\Omega$ with intensity $u^{-2}\, {\mathrm{d}}u \times\nu({\mathrm{d}}\omega)$. The functions $\{ V_t\}_{t \in T}$ are called *spectral functions* and the measure $\nu$ is called *spectral measure*. In order to describe the *finite dimensional distributions* (f.d.d.) of $X$, we shall fix some convenient notation first: Let $M \subset T$ be some non-empty finite subset of $T$. By ${\mathbb{R}}^M$ ([resp. ]{}$[0,\infty]^M$) we denote the space of real-valued ([resp. ]{}$[0,\infty ]$-valued) functions on $M$. Elements of these spaces are denoted by $x=(x_t)_{t \in M}$ where $x_t=x(t)$. The standard scalar product is given through $\langle x,y \rangle= \sum_{t \in M} x_t y_t$. For a subset $L \subset M$, we write ${\mathbf{1}}_L$ for the indicator function of $L$ in ${\mathbb{R}}^M$ ([resp. ]{}$[0,\infty]^M$), such that $\{{\mathbf{1}}_{\{t\}}\}_{t \in M}$ forms an orthonormal basis of ${\mathbb{R}}^M$. In this sense, the origin of ${\mathbb{R}}^M$ equals ${\mathbf{1}}_{\varnothing }$ being zero everywhere on $M$. Using this notation, we emphasize the fact that a multivariate distribution of a stochastic process is not any $|M|$-variate distribution, but it is bound to certain points (forming the set $M$) in the space $T$. Finally, we consider some *reference norm* $\| \cdot\|$ on ${\mathbb{R}}^M$ (not necessarily the one from the scalar product) and denote the positive unit sphere $S_M:=\{ a \in[0,\infty)^M {\dvtx }\| a \| = 1 \}$. In terms of a spectral representation $(\Omega,{{\mathcal A}},\nu,V)$, the [f.d.d. ]{}of $X$ are given through $$\label{eqn:fddspectralrep} -\log{\mathbb{P}}(X_t\leq x_t, t \in M) = \int _{\Omega} \biggl(\bigvee_{t \in M} \frac{V_{t}(\omega)}{x_t} \biggr) \nu({\mathrm{d}}\omega)$$ for $x \in[0,\infty)^M\setminus\{{\mathbf{1}}_\varnothing\}$. Alternatively, the [f.d.d. ]{}of $X$ for a finite subset $\varnothing\neq M \subset T$ may be described by means of one of the following three quantities that are all equivalent to the knowledge of the f.d.d.: - the *(finite dimensional) spectral measure* $H_M$ ([cf. ]{}[@dehaanresnick_77; @resnick_08]), that is, the Radon measure on $S_M$ such that for $x \in[0,\infty)^M\setminus\{{\mathbf{1}}_\varnothing\}$ $$\label{eqn:fddspectralmeasure} - \log{\mathbb{P}}(X_{t} \leq x_t, t \in M) = \int _{S_M} \biggl(\bigvee_{t \in M} \frac{a_t}{x_t} \biggr) H_M ({\mathrm{d}}a),$$ - the *stable tail dependence function* $\ell_M$ ([cf. ]{}[@beirlantetalii_03]), that is, the function on $[0,\infty)^M$ defined through $$\label{eqn:fddstabletaildepfn} \ell_M(x) := - \log{\mathbb{P}}(X_{t} \leq1/x_t, t \in M) = \int_{S_M} \biggl( \bigvee _{t \in M} a_t x_t \biggr) H_M ({\mathrm{d}}a),$$ - the *(finite dimensional) dependency set* ${{\mathcal K}}_M$ ([cf. ]{}[@molchanov_08]), that is, the largest compact convex set ${{\mathcal K}}_M \subset[0,\infty)^M$ satisfying $$\label{eqn:DepSetdefn} \ell_M(x)=\sup \bigl\{ \langle x,y \rangle{\dvtx }y \in {{\mathcal K}}_M \bigr\}\quad\quad \forall x \in[0,\infty)^M.$$ In order to be a valid finite dimensional spectral measure of a simple max-stable random vector $\{X_t\}_{t \in M}$, the only constraint that a Radon measure $H_M$ on $S_M$ has to satisfy is that $$\int_{S_M} a_t H_M({\mathrm{d}}a)=1$$ for each $t \in M$. This ensures standard Fr[é]{}chet marginals. Given a simple max-stable process $X$ on $T$, we may assign to a non-empty finite subset $A \subset T$ the *extremal coefficient* $\theta(A)$ ([cf. ]{}[@smith_90; @schlathertawn_02]), that is $$\label{eqn:ECFintro} \theta(A):= \lim_{x \to\infty}\frac{\log{\mathbb{P}}(\bigvee_{t \in A} X_t \leq x )}{\log{\mathbb{P}}(X_t \leq x)} = \int _{S_M} \biggl(\bigvee_{t \in A} {a_t} \biggr) H_M ({\mathrm{d}}a) = \ell_M ( {\mathbf{1}}_A ).$$ Indeed, the expression $\log{\mathbb{P}}(\bigvee_{t \in A} X_t \leq x)/\log{\mathbb{P}}(X_t \leq x)$ does not depend on $x$ and equals the right-hand side (r.h.s.) for $A \subset M$. Observe that $\theta(A)$ takes values in the interval $[1,|A|]$, where the value $1$ corresponds to full dependence of the random variables $\{X_t\}_{t \in A}$ and the value $|A|$ (number of elements in $A$) corresponds to independence. Roughly speaking, the extremal coefficient $\theta(A)$ detects the effective number of independent variables among the random variables $\{ X_t\}_{t \in A}$. It is coherent to set $\theta(\varnothing):=0$ to obtain a function $\theta$ on ${\mathcal{F}}(T)$, the *set of finite subsets of $T$*. We call the function $$\theta\dvtx {\mathcal{F}}(T) \rightarrow[0,\infty)$$ *extremal coefficient function (ECF)* of the process $X$. The set of all ECFs of simple max-stable processes on a set $T$ will be denoted by $$\label{eqn:Theta_defn} \Theta(T)= \bigl\{\theta\dvtx {\mathcal{F}}(T) \rightarrow[0, \infty) {\dvtx }\theta \mbox{ is an ECF of a simple max-stable process on } T \bigr\}.$$ The simplest ECFs are the functions $\theta(A)=|A|$ corresponding to a process of independent random variables, and the indicator function $\theta(A) = \mathbh{1}_{A \neq\varnothing}$ corresponding to a process of identical random variables. Rather sophisticated examples of ECFs can be obtained using spectral representations $(\Omega,{{\mathcal A}},\nu,V)$ of processes $X$. In these terms the ECF $\theta$ of a process $X$ is given by $$\label{eqn:ECFspectralrep} \theta(A)=\int_{\Omega} \biggl(\bigvee _{t \in A} {V_t(\omega)} \biggr) \nu ({\mathrm{d}}\omega)$$ for $A \in{\mathcal{F}}(T) \setminus\{\varnothing\}$ and $\theta(\varnothing)=0$. Consider the simple max-stable stationary process $X$ on ${\mathbb{R}}^d$ that is given through the following spectral representation $(\Omega ,{{\mathcal A}},\nu,V)$: - $(\Omega,{{\mathcal A}},\nu)=({\mathbb{F}}\times{\mathbb{R}}^d, {\mathcal F}\otimes {{\mathcal B}}({\mathbb{R}}^d), \mu\otimes{\mathrm{d}}z)$, where $({\mathbb{R}}^d,{{\mathcal B}}({\mathbb{R}}^d),{\mathrm{d}}z)$ denotes the Lebesgue-measure on the Borel-$\sigma$-algebra of ${\mathbb{R}}^d$ and where $({\mathbb{F}},{\mathcal F},\mu)$ denotes a measure space of non-negative measurable functions on ${\mathbb{R}}^d$ with $\int_{{\mathbb{F}}} (\int_{{\mathbb{R}}^d} f(z)\, {\mathrm{d}}z ) \mu({\mathrm{d}}f) = 1$, - $V_t((f,z))=f(t-z)$ for $t \in{\mathbb{R}}^d$, then we call $X$ a *Mixed Moving Maxima process (M3 process)* ([cf. ]{}also [@kabluchkostoev_12; @schlather_02; @stoev_08; @stoevtaqqu_06]). Because of (\[eqn:ECFspectralrep\]) the ECF $\theta$ of a Mixed Moving Maxima process $X$ can be computed as $$\theta(A) =\int_{{\mathbb{F}}} \int_{{\mathbb{R}}^d} \biggl( \bigvee_{t \in A} f(t-z) \biggr) \,{\mathrm{d}}z \mu ({\mathrm{d}}f)$$ for $A \in{\mathcal{F}}({\mathbb{R}}^d) \setminus\{\varnothing\}$ and $\theta (\varnothing)=0$. In case $\mu$ is a point mass at an indicator function $f$ the bivariate coefficient $\theta(\{s,t\})$ will be given by $\theta(\{ s,t\} )=2-f*\check f(s-t)$, where $f*\check f$ means the convolution of $f$ with $\check f$ and $\check f(t)=f(-t)$. \[example:BR\] Consider the simple max-stable stationary process $X$ on ${\mathbb{R}}^d$ that is given through the following spectral representation $(\Omega ,{{\mathcal A}},\nu,V)$: - $(\Omega,{{\mathcal A}},\nu)$ denotes the probability space of a Gaussian process $W$ on ${\mathbb{R}}^d$ with stationary increments and variogram $\gamma(z)={\mathbb{E}}(W_z-W_o)^2$ for $z \in{\mathbb{R}}^d$. - $V_t(\omega)=\exp ({W_t(\omega)-\sigma^2(t)/2} )$ for $t \in{\mathbb{R}}^d$, where $\sigma^2(t)$ denotes the variance of $W_t$, then we call $X$ a *Brown–Resnick process* ([cf. ]{}[@kabluchkoetalii_09]). Because of (\[eqn:ECFspectralrep\]) the ECF $\theta$ of a Brown–Resnick process $X$ is $$\theta(A)= {\mathbb{E}}\exp \biggl(\bigvee_{t \in A} W_t- \sigma^2(t)/2 \biggr)$$ for $A \in{\mathcal{F}}({\mathbb{R}}^d) \setminus\{\varnothing\}$ and $\theta (\varnothing)=0$. Since the [f.d.d. ]{}of $X$ only depend on the variogram $\gamma$, the extremal coefficient $\theta(A)$ will also depend only on the values $\{ \gamma(s-t)\}_{s,t \in A}$. In particular, we have $\theta(\{s,t\} )=1+{\operatorname{erf}}(\sqrt{\gamma(s-t)/8})$ for the bivariate coefficient $\theta(\{ s,t\})$, where ${\operatorname{erf}}(x)=2/\sqrt{\uppi} \int_{0}^x \mathrm{e}^{-t^2} \,\mathrm{d}t$ denotes the error function ([cf. ]{}[@kabluchkoetalii_09]). In case the variogram equals $\gamma(z)=\lambda\| z \|^2$ for some $\lambda> 0$, explicit expressions for multivariate coefficients of higher orders up to $d+1$ can be found in [@gentonetal_11]. A consistent max-linear model ----------------------------- A multivariate simple max-stable distribution is called *max-linear* (or *spectrally discrete*) if it arises as the distribution of a random vector $X$ of the following form $$X_i = \bigvee_{j=1}^q a_{ij} Z_j,\quad\quad i=1,\ldots,p,$$ where $Z=\{Z_j\}^q_{j=1}$ is a vector of [i.i.d. ]{}unit Fr[é]{}chet random variables and where $\{a_{ij}\}_{p \times q}$ is a matrix of non-negative entries with $\sum_{j=1}^q a_{ij}=1$ for each row $i=1,\ldots,p$. This is equivalent to requiring the spectral measure $H_M$ from (\[eqn:fddspectralmeasure\]) for $M=\{1,\ldots,\ldots,p\}$ to be the following *discrete* measure on $S_M$ $$H_M=\sum_{j=1}^q \| a_j \| \delta_{a_j/\| a_j\|},$$ where $a_j$ denote the column vectors of the matrix $\{a_{ij}\}_{p \times q}$. Conversely, any discrete spectral measure of a simple max-stable random vector gives rise to such a matrix. Surely, the ECF of such a random vector $X=\{X_i\}_{i \in M}$ is $$\label{eqn:ECFmaxlinear} \theta(A)=\sum_{j=1}^q \bigvee_{i \in A}a_{ij}$$ for $\varnothing\neq A \subset M$ and $\theta(\varnothing)=0$ ([cf. ]{}(\[eqn:ECFintro\])). In [@schlathertawn_02], the authors introduce a max-linear model for $X^*=\{X^*_i\}_{i \in M}$ where the column index $j$ ranges over all non-empty subsets $L$ of $M$ and where non-negative coefficients $\tau_L$ are given for each column $\varnothing\neq L \subset M$, more precisely $$X^*_i = \bigvee_{\varnothing\neq L \subset M} a_{i,L} Z_L ,\quad\quad i \in M, \mbox{ with } a_{i,L} = \tau_L \mathbh{1}_{i \in L},$$ which is equivalent to $$\label{eqn:taumodel} X^*_i = \bigvee_{i \in L \subset M} \tau_{L} Z_L,\quad\quad i \in M.$$ The model (\[eqn:taumodel\]) is simple if and only if $\sum_{\varnothing\neq L \subset M} a_{iL} = \sum_{L \subset M {\dvtx }i \in L} \tau_L=1$ for each $i \in M$. It follows from (\[eqn:ECFmaxlinear\]) that the ECF of model (\[eqn:taumodel\]) is $$\theta(A)=\sum_{L \subset M {\dvtx }A \cap L \neq\varnothing} \tau_L$$ for $\varnothing\neq A \subset M$ and $\theta(\varnothing)=0$. Now, the interesting aspect of this model (\[eqn:taumodel\]) with given coefficients $\tau_L$ is that such models are in 1–1 correspondence with ECFs $\theta$ on the finite set $M$ ([cf. ]{}[@schlathertawn_02], Theorems 3 and 4). Alongside, this leads to a set of inequalities which fully characterizes the set of ECFs $\Theta(M)$ for finite sets $M$ ([cf. ]{}[@schlathertawn_02], Corollary 5). An alternative proof for these inequalities is offered in [@molchanov_08], Corollary 1, and it is noticed therein that they are equivalent to a property called complete alternation (see below). As we seek a spatial generalization of these results, let us consider a max-stable processes $X^*=\{X^*_t\}_{t \in T}$ on an arbitrary index set $T$, whose [f.d.d. ]{}for a finite set $M$ are precisely of the above form (\[eqn:taumodel\]), where the coefficients $\tau_L$ now additionally depend on $M$. That means we set the spectral measure $H^*_M$ of the random vector $\{X^*_t\}_{t \in M}$ $$\label{eqn:starspectralmeasure} H^*_M := \sum_{\varnothing\neq L \subset M} \tau^M_L \| {\mathbf{1}}_L \| \delta_{{\mathbf{1}}_L/\| {\mathbf{1}}_L \|},$$ such that the [f.d.d. ]{}of the process $X^*$ are given by ([cf. ]{}(\[eqn:fddspectralmeasure\])) $$\label{eqn:starfdd} -\log{\mathbb{P}}\bigl(X^*_t\leq x_t, t \in M \bigr) =\sum_{\varnothing\neq L \subset M} \tau^M_L \bigvee_{t \in L} \frac{1}{x_t}.$$ Here $M$ ranges over all non-empty finite subsets of $T$, which we express as $M \in{\mathcal{F}}(T)\setminus\{\varnothing\}$. Figure \[fig:spectralmeasure\] illustrates this spectral measure for a trivariate distribution where $M=\{1,2,3\}$ in case the reference norm is the maximum norm. ![Spectral measure representation of $\{X^*_{t}\}_{t \in M}$ for $M = \{ 1,2,3\}$ if we choose the reference norm on ${\mathbb{R}}^M$ to be the maximum norm. In this case, the spectral measure simplifies to a sum of weighted point masses on the vertices of a cube: $H^*_M = \sum_{\varnothing\neq L \subset M} \tau^M_L \delta_{{\mathbf{1}}_L}$.[]{data-label="fig:spectralmeasure"}](567f02.eps) \[lemma:tauprocess\] Let $T$ be an arbitrary set and let coefficients $\tau^M_L$ be given for $M \in{\mathcal{F}}(T)\setminus\{\varnothing\}$ and $L \in{\mathcal{F}}(M)\setminus\{\varnothing\}$, such that (i) $\tau^M_L \geq0$ for all $M \in{\mathcal{F}}(T)\setminus\{\varnothing\}$ and $L \in{\mathcal{F}}(M)\setminus\{ \varnothing\}$, (ii) $\tau^M_L=\tau^{M \cup\{t\}}_L+\tau^{M \cup\{ t\}}_{L \cup\{t\}}$ for all $M \in{\mathcal{F}}(T)\setminus\{\varnothing\}$ and $L \in{\mathcal{F}}(M)\setminus\{\varnothing\}$ and $t \in T\setminus M$, (iii) $\tau^{\{t\}}_{\{t\}}=1$ for all $t \in T$. Then the spectral measures $\{H^*_M\}_{M \in{\mathcal{F}}(T) \setminus\{ \varnothing\}}$ from (\[eqn:starspectralmeasure\]) define a simple max-stable process $X^*=\{X^*_t\}_{t \in T}$ on $T$ with [f.d.d. ]{}as in (\[eqn:starfdd\]). Condition (i) ensures that each spectral measure $H^*_M$ defines a max-stable distribution with Fr[é]{}chet marginals. Subsequently, condition (ii) ensures consistency of these distributions (i.e., the conditions for Kolmogorov’s extension theorem are satisfied). Hence the spectral measures $H^*_M$ define a max-stable process $X^*$ on $T$. Finally, condition (iii) ensures that the process $X^*$ has standard Fr[é]{}chet marginals. Condition (ii) is equivalent to $$\label{eqn:tau4} \tau^A_K = \sum _{ J \subset M \setminus A} \tau^M_{K \cup J} \quad\quad \forall M \in {\mathcal{F}}(T)\setminus\{\varnothing\}, \varnothing\neq K \subset A \subset M.$$ The TM process and negative definiteness of ECFs {#sect:TMprocess} ================================================ For the following characterization of the set of ECFs $\Theta(T)$, we use the fact that ${\mathcal{F}}(T)$, the set of finite subsets of $T$, forms a semigroup with respect to the union operation $\cup$ and with neutral element the empty set $\varnothing$. The following notation is adopted from [@molchanov_05] and [@bcr_84]. For a function $f\dvtx {\mathcal{F}}(T) \rightarrow{\mathbb{R}}$ and elements $K,L \in{\mathcal{F}}(T)$, we set $$\begin{aligned} (\Delta_{K}f ) (L):= f(L)-f(L\cup K).\end{aligned}$$ \[def:ND\_CA\] A function $\psi\dvtx {\mathcal{F}}(T) \rightarrow{\mathbb{R}}$ is called *negative definite (in the semigroup sense)* on ${\mathcal{F}}(T)$ if for all $n \geq2$, $\{K_1,\ldots, K_n\} \subset{\mathcal{F}}(T)$ and $\{ a_1,\ldots,a_n\} \subset{\mathbb{R}}$ with $\sum_{j=1}^n a_j=0$ $$\sum_{j=1}^n \sum _{k=1}^n a_j a_k \psi(K_j \cup K_k) \leq0.$$ A function $\psi\dvtx {\mathcal{F}}(T) \rightarrow{\mathbb{R}}$ is called *completely alternating* on ${\mathcal{F}}(T)$ if for all $n \geq1$, $\{K_1,\ldots, K_n\} \subset{\mathcal{F}}(T)$ and $K \in{\mathcal{F}}(T)$ $$(\Delta_{K_1}\Delta_{K_2} \cdots\Delta_{K_n} \psi ) (K) = \sum_{I \subset\{1, \ldots, n\}} (-1)^{|I|} \psi \biggl(K \cup \bigcup_{i \in I} K_i \biggr) \leq0.$$ Because the semigroup $({\mathcal{F}}(T),\cup,\varnothing)$ is idempotent, these two terms coincide. That means $\psi\dvtx {\mathcal{F}}(T) \rightarrow {\mathbb{R}}$ is *completely alternating* if and only if $\psi$ is *negative definite (in the semigroup sense)*, [cf. ]{}[@bcr_84], 4.4.16. \[example:cap\_ND\] An important example of a negative definite (completely alternating) function on ${\mathcal{F}}(T)$ is the *capacity functional* $C\dvtx {\mathcal{F}}(T) \rightarrow{\mathbb{R}}$ of a binary process $Y=\{Y_t\}_{t \in T}$ with values in $\{0,1\}$, which is given by $C(\varnothing) = 0$ and $$C(A) = {\mathbb{P}}(\exists t \in A \mbox{ such that } Y_t = 1).$$ Now, we are in position to characterize the set $\Theta(T)$ of possible ECFs on ${\mathcal{F}}(T)$ and to define a corresponding max-linear process $X^*$. \[thm:ECF\_ND\] (a) The function $\theta\dvtx {\mathcal{F}}(T) \rightarrow{\mathbb{R}}$ is the ECF of a simple max-stable process on $T$ if and only if the following conditions are satisfied: (i) $\theta$ is negative definite, (ii) $\theta(\varnothing) = 0$, (iii) $\theta(\{t\}) = 1$ for all $t \in T$. (b) If these conditions are satisfied, the following choice of coefficients $$\begin{aligned} &&\tau^M_L := - \Delta_{\{t_1\}} \cdots \Delta_{\{t_l\}} \theta(M \setminus L) = \sum_{I \subset L} (-1)^{|I|+1} \theta \bigl((M \setminus L) \cup I \bigr) \\ &&\quad \forall M \in{\mathcal{F}}(T)\setminus\{\varnothing\}, \varnothing \neq L = \{t_1,\ldots,t_l\} \subset M\end{aligned}$$ for model (\[eqn:starspectralmeasure\]) defines a simple max-stable process $X^*$ on $T$ which realizes $\theta$ as its own ECF $\theta^*$. Referring to the previous work in [@colestawn_96; @molchanov_08; @schlathertawn_02], we will call the simple max-stable process $X^*$ from Theorem \[thm:ECF\_ND\] *Tawn–Molchanov process (TM process)* henceforth. [Proof of Theorem \[thm:ECF\_ND\]]{} If $\theta$ is an ECF of a simple max-stable process $X$ on $T$, then necessarily $\theta(\varnothing) = 0$ and $\theta(\{t\}) = 1$ for all $t \in T$ ([cf. ]{}(\[eqn:ECFintro\])). Further, it is an application of l’H[ô]{}pitals rule that for $A \subset{\mathcal{F}}(T)\setminus\{ \varnothing\}$ $$\begin{aligned} \label{eqn:lhopital} \nonumber \theta(A)&=& \lim_{x \to\infty} \frac{-\log{\mathbb{P}}(\bigvee_{t \in A} X_t \leq x )}{-\log{\mathbb{P}}(X_t \leq x)} = \lim_{x \to\infty} \frac{1-{\mathbb{P}}(\bigvee_{t \in A} X_t \leq x )}{1-{\mathbb{P}}(X_t \leq x )} \\[-8pt] \\[-8pt] &=& \lim_{x \to\infty} \frac{{\mathbb{P}}(\exists t \in A \mbox{ such that } X_t \geq x )}{{\mathbb{P}}(X_t \geq x )} = \lim_{x \to\infty} \frac{C^{(x)}(A)}{p^{(x)}}, \nonumber\end{aligned}$$ where $C^{(x)}$ denotes the capacity functional for the binary process $Y_t=\mathbh{1}_{X_t \geq x}$ and $p^{(x)} = {\mathbb{E}}Y_t = 1-\mathrm{e}^{-1/x}$. Since negative definiteness respects scaling and pointwise limits, negative definiteness of $\theta$ follows from Example \[example:cap\_ND\]. This shows the necessity of (i)–(iii). Now, let $\theta\dvtx {\mathcal{F}}(T)\rightarrow{\mathbb{R}}$ be a function satisfying conditions (i)–(iii) and let the coefficients $\tau^M_L$ be given as above. We need to check that they fulfill the (in)equalities from Lemma \[lemma:tauprocess\]. Indeed we have: - The inequalities $\tau^M_L = - \Delta_{\{t_1\}} \cdots\Delta_{\{t_l\}} \theta(M \setminus L) \geq0$ follow directly from the complete alternation of $\theta$ that is equivalent to (i). - From the definition of $\Delta_{\{t\}}$ we observe $$\begin{aligned} \tau^{M \cup\{t\}}_{L \cup\{t\}} &=& - \Delta_{\{t\}} \Delta_{\{t_1\}} \cdots\Delta_{\{t_l\}}\theta \bigl( \bigl(M \cup\{t\} \bigr) \setminus \bigl(L \cup\{t\} \bigr) \bigr) \\ &=& - \Delta_{\{t_1\}} \cdots\Delta_{\{t_l\}}\theta (M \setminus L ) + \Delta_{\{t_1\}} \cdots\Delta_{\{t_l\}}\theta \bigl(M \cup \{ t\} \setminus L \bigr) \\ &=& \tau^M_L -\tau^{M \cup\{t\}}_L.\end{aligned}$$ - For $t \in T$, we have $\tau^{\{t\}}_{\{t\}} = \theta(\{t\}) = 1$ because of (iii). Thus, the coefficients $\tau^M_L$ define a simple max-stable process $X^*$ on $T$ as given by model (\[eqn:starspectralmeasure\]). Finally, we compute the ECF $\theta^*$ of $X^*$ and see that it coincides with $\theta$: For the empty set, we have $\theta ^*(\varnothing )=0=\theta(\varnothing)$ because of (ii); otherwise we compute for $A \subset{\mathcal{F}}(T)\setminus\{\varnothing\}$ that $$\begin{aligned} \theta^*(A) & \stackrel{{\scriptsize{(\ref{eqn:ECFintro}),(\ref {eqn:starspectralmeasure})}}} {=}& \sum_{\varnothing\neq L \subset A} \tau^A_L = \sum _{\varnothing\neq L \subset A} \sum_{I \subset L} (-1)^{|I|+1} \theta \bigl((A \setminus L) \cup I \bigr) \\ &=& \sum_{\varnothing\neq K \subset A} \theta(K) \mathop{\sum _{\varnothing \neq L \subset A}}_{ A \setminus L \subset K} (-1)^{|K \cap L|+1} = \sum _{\varnothing\neq K \subset A} \theta(K) \bigl(- (-\mathbh {1}_{K=A} ) \bigr) = \theta(A).\end{aligned}$$ This shows sufficiency of (i)–(iii) and part (b). Theorem \[thm:ECF\_ND\] is in analogy to the following standard result for Gaussian processes (as illustrated in the sketches in the ): (a) A function $C\dvtx T \times T \rightarrow{\mathbb{R}}$ is a covariance function if and only if it is positive definite. (b) If $C\dvtx T \times T \rightarrow{\mathbb{R}}$ is positive definite, we may choose a (zero mean) Gaussian process which realizes $C$ as its own covariance function. Both statements are intrinsically tied together. When proving them by means of Kolmogorov’s extension theorem, one proceeds in the same manner as we did for Theorem \[thm:ECF\_ND\]. The necessity of positive definiteness of covariance functions is easily derived even for the bigger class of square-integrable processes, whilst sufficiency can be established by showing that Gaussian processes can realize any positive definite function as covariance function. [In some points (such as continuity relations), this analogy will be deepened. Other aspects (such as the exceptional role of dependency sets in Section \[sect:depset\]) seem unsuitable for a direct comparison.]{} In order to incorporate stationarity [w.r.t. ]{}some group $G$ acting on $T$ (for example, ${\mathbb{R}}^d$ acting on ${\mathbb{R}}^d$ by translation), we just have to add the following condition (iv) $\theta(gA)=\theta(A)$ for all $A \in{\mathcal{F}}(T)\setminus\{ \varnothing\}$ and for all $g \in G$. Then the process $X^*$ will be stationary [w.r.t. ]{}this group action. Instead of requiring the max-stable processes in Theorem \[thm:ECF\_ND\] to have *standard* Fr[é]{}chet marginals everywhere, we can admit a different scale at different locations, that is, ${\mathbb{P}}(X_t \leq x)=\exp(-s_t/x)$ for a positive scaling parameter $s_t$ for $t \in T$. In that case Theorem \[thm:ECF\_ND\] holds true without condition (iii) and the word “simple”. To make sense of the ECF as in (\[eqn:ECFintro\]) in this case, either use a reference point $t \in T$ or set $\log{\mathbb{P}}(X_t\leq x)=-1/x$ in the denominator. Beware of that the ECF $\theta$ cannot be interpreted as the number of independent variables anymore in this case. In [@schlathertawn_02], the last issue of the proof is derived for finite sets $T$ by a Moebius inversion. The relation to the proof therein becomes more transparent if we compute $\theta^*(A)$ for $A \subset M$ from the coefficients $ \{\tau^M_L \}_{\varnothing \neq L \subset M}$ for arbitrary $M \supset A$ instead of $M=A$: $$\label{eqn:thetaA.from.tauML} \theta^*(A) \stackrel{{\scriptsize{(\ref{eqn:ECFintro}),(\ref {eqn:starspectralmeasure})}}} {=} \sum_{\varnothing\neq K \subset A}\tau^A_K \stackrel{{\scriptsize{(\ref{eqn:tau4})}}} {=} \sum_{\varnothing\neq K \subset A} \sum_{J \subset M \setminus A} \tau^M_{K \cup J} = \sum _{L \subset M {\dvtx }L \cap A \neq\varnothing} \tau^M_L.$$ Direct consequences of Theorem 8 {#sect:consequences} ================================ Here, we collect some direct consequences of the above Theorem \[thm:ECF\_ND\]. Therefore, note that the first part of Theorem \[thm:ECF\_ND\] can also be expressed as ([cf. ]{}(\[eqn:Theta\_defn\])) $$\label{eqn:Theta_ND} \Theta(T)= \bigl\{\theta\dvtx {\mathcal{F}}(T) \rightarrow[0,\infty) {\dvtx }\theta\mbox{ is negative definite, } \theta(\varnothing)=0, \theta \bigl(\{t\} \bigr)=1 \mbox{ for } t \in T \bigr\}.$$ Convexity and compactness ------------------------- \[cor:Theta\_convex\] The set of ECFs $\Theta(T)$ is convex. This can be seen directly from (\[eqn:Theta\_ND\]) since all involved properties are compatible with convex combinations. As a constructive argument, use the fact that the ECF of the max-combination $\alpha X \vee(1-\alpha) Y$ of two independent simple max-stable processes $X$ and $Y$ on $T$ is the convex combination of their ECFs for $\alpha\in(0,1)$. \[cor:Theta\_compact\] The set of ECFs $\Theta(T)$ is compact [w.r.t. ]{}the topology of pointwise convergence. The topology of pointwise convergence on ${\mathbb{R}}^{{\mathcal{F}}(T)}$ is the product topology. Since $\theta(\varnothing)=0$ and $\theta (A)\in [1,|A|]$ for $\theta\in\Theta(T)$ and $A \in{\mathcal{F}}(T)\setminus\{ \varnothing\}$, the set $\Theta(T)$ is a subset of the product space $$\{0\} \times\prod_{A \in{\mathcal{F}}(T)\setminus\{\varnothing\}} \bigl[1,|A|\bigr],$$ which is compact by Tychonoff’s theorem. Moreover, since elements of $\Theta(T)$ are completely characterized by finite dimensional equalities and inequalities involving $\leq$ only (stemming from (\[eqn:Theta\_ND\])), the set $\Theta(T)$ is closed. Hence, $\Theta(T)$ is compact. Note that even though we say “the topology of pointwise convergence”, the “points” meant here are indeed elements of ${\mathcal{F}}(T)$, that is, finite subsets of $T$. In particular it follows from the compactness of $\Theta(T)$ that $\Theta(T)$ is sequentially closed. That means if $(\theta_n)_{n \in{\mathbb{N}}}$ is a sequence of ECFs such that $\theta_n(A)$ converges to some value $f(A)$ for each $A \in{\mathcal{F}}(T)$, then $f$ is an ECF. Spectral representation of the TM process ----------------------------------------- Another consequence of Theorem \[thm:ECF\_ND\] is that ECFs allow for an *integral representation* as a mixture of functions $A \mapsto \mathbh{1}_{A \cap Q \neq\varnothing}$, where $Q$ is from the power set of $T$. To be more precise, let us denote the power set of $T$ by ${\mathcal{P}}(T)$ and consider the topology on ${\mathcal{P}}(T)$ that is generated by the maps $Q \mapsto\mathbh{1}_{A \cap Q \neq\varnothing}$ for $A \in{\mathcal{F}}(T)$ or equivalently (since ${\mathcal{F}}(T)$ is generated by the singletons $\{\{t\}\}_{t \in T}$) the topology on ${\mathcal{P}}(T)$ that is generated by the maps $Q \mapsto\mathbh{1}_{t \in Q}$ for $t \in T$. Identifying ${\mathcal{P}}(T)$ with $\{0,1\}^T$, this space is also known as *Cantor cube*. As in [@bcr_84], Definition 2.1.1, a measure $\mu$ on the Borel-$\sigma$-algebra of ${\mathcal{P}}(T)$ [w.r.t. ]{}this topology will be called *Radon measure* if $\mu$ is finite on compact sets and $\mu$ is inner regular. \[cor:intrep\] Let $\theta\in\Theta(T)$ be an ECF. Then $\theta$ uniquely determines a positive Radon measure $\mu$ on ${\mathcal{P}}(T)\setminus\{\varnothing \}$ such that $$\theta(A) = \mu \bigl( \bigl\{Q \in{\mathcal{P}}(T) \setminus\{\varnothing\} {\dvtx }A \cap Q \neq\varnothing \bigr\} \bigr) = \int_{{\mathcal{P}}(T) \setminus\{\varnothing \}} \mathbh{1}_{A \cap Q \neq\varnothing} \mu({\mathrm{d}}Q),$$ where $\theta(\{t\})=1$ for $t \in T$. The function $\theta$ is bounded if and only if $\mu({\mathcal{P}}(T) \setminus\{\varnothing\}) < \infty$. Since $\theta$ is negative definite (Theorem \[thm:ECF\_ND\]) and ${\mathcal{F}}(T)$ is idempotent, we may apply [@bcr_84], Proposition 4.4.17. It says that $\theta$ uniquely determines a positive Radon measure $\widetilde\mu$ on $\widehat{{\mathcal{F}}(T)}\setminus\{1\}$, where $\widehat{{\mathcal{F}}(T)}$ denotes the dual semigroup of ${\mathcal{F}}(T)$ ([cf. ]{}[@bcr_84], 4.2.1 and 4.4.16), such that $\theta(A) = \widetilde\mu(\{\rho\in \widehat{{\mathcal{F}}(T)} \setminus\{1\} \mid\rho(A)=0 \})$. The function $\theta$ is bounded if and only if $\widetilde\mu(\widehat{{\mathcal{F}}(T)} \setminus\{1\}) < \infty$. Now, it can be easily seen that semicharacters on ${\mathcal{F}}(T)$ are in a 1–1 correspondence with subsets of $T$ via $\widehat{{\mathcal{F}}(T)} \ni \rho\rightarrow\{ t \in T {\dvtx }\rho(\{t\})=0 \} \in{\mathcal{P}}(T)$ and ${\mathcal{P}}(T) \ni Q \rightarrow\mathbh{1}_{(\cdot) \cap Q = \varnothing} \in\widehat{{\mathcal{F}}(T)}$. Here the constant function $1$ corresponds to the empty set. Moreover, the topology considered on $\widehat{{\mathcal{F}}(T)}$ is the topology of pointwise convergence. Transported to ${\mathcal{P}}(T)$ this is the topology generated by the maps $Q \mapsto\mathbh{1}_{A \cap Q \neq\varnothing}$ for $A \in{\mathcal{F}}(T)$. Let $\mu$ denote the Radon measure $\widetilde\mu$ transported to ${\mathcal{P}}(T)\setminus\{\varnothing\}$. Then the corollary follows. \[remark:tauFourier\] In case $T=M$ is finite, we have that ${\mathcal{P}}(M)={\mathcal{F}}(M)$ carries the discrete topology and $$\theta(A) = \mu \bigl( \bigl\{Q \in{\mathcal{F}}(M) \setminus\{\varnothing\} {\dvtx }A \cap Q \neq \varnothing \bigr\} \bigr) = \sum_{Q \in{\mathcal{F}}(M) \setminus\{\varnothing\}} \mu \bigl( \{Q\} \bigr) \mathbh {1}_{A \cap Q \neq\varnothing}.$$ A comparison with (\[eqn:thetaA.from.tauML\]) reveals that $\mu(\{ Q\} )=\tau^M_Q$. In this sense, the coefficients $\tau^M_Q$ of the max-linear model (\[eqn:starspectralmeasure\]) can be interpreted as finite dimensional “Fourier coefficients” of the negative definite function $\theta$. The integral representation of the ECF $\theta$ also yields a spectral representation for the corresponding TM process $X^*$. \[thm:spectralrep\] The TM process $X^*=\{X^*_t\}_{t \in T}$ with ECF $\theta$ has the following spectral representation $(\Omega,{{\mathcal A}},\nu,V)$ ([cf. ]{}(\[eqn:spectralrepresentation\])): - $(\Omega,{{\mathcal A}},\nu)$ is the measure space $({\mathcal{P}}(T),{{\mathcal B}}({\mathcal{P}}(T)),\mu)$ from Corollary \[cor:intrep\], - $V_t(Q)=\mathbh{1}_{t \in Q}$. We need to check that the [f.d.d. ]{}of $X^*$ satisfy (\[eqn:fddspectralrep\]). The [f.d.d. ]{}of $X^*$ are given by ([cf. ]{}(\[eqn:starfdd\])) $$\begin{aligned} -\log{\mathbb{P}}\bigl(X^*_t\leq x_t, t \in M \bigr) =\sum _{\varnothing\neq L \subset M} \tau^M_L \bigvee _{t \in L} \frac{1}{x_t},\end{aligned}$$ where the coefficients $\tau^M_L$ can be computed from the ECF $\theta$ as in Theorem \[thm:ECF\_ND\](b) and $\theta$ satisfies the integral representation from Corollary \[cor:intrep\], that is, $$\begin{aligned} \tau^M_L &=& \sum_{I \subset L} (-1)^{|I|+1} \theta \bigl((M \setminus L) \cup I \bigr) \\ &=& \sum _{I \subset L} (-1)^{|I|+1} \int_{{\mathcal{P}}(T) \setminus\{ \varnothing\}} \mathbh{1}_{((M \setminus L) \cup I) \cap Q \neq \varnothing } \mu({\mathrm{d}}Q).\end{aligned}$$ Using the identity $$\begin{aligned} &&\sum_{I \subset L} (-1)^{|I|+1} \mathbh{1}_{((M \setminus L) \cup I) \cap Q \neq\varnothing} \\[-0.5pt] &&\quad= \sum_{I \subset L} (-1)^{|I|+1} ( \mathbh{1}_{(M \setminus L) \cap Q \neq\varnothing} + \mathbh{1}_{I \cap Q \neq\varnothing} - \mathbh{1}_{(M \setminus L) \cap Q \neq\varnothing} \mathbh{1}_{I \cap Q \neq\varnothing} ) \\[-0.5pt] &&\quad= 0 \cdot\mathbh{1}_{(M \setminus L) \cap Q \neq\varnothing} + (1-\mathbh{1}_{(M \setminus L) \cap Q \neq\varnothing} ) \sum_{I \subset L} (-1)^{|I|+1}\mathbh{1}_{I \cap Q \neq\varnothing} \\[-0.5pt] &&\quad= \mathbh{1}_{(M \setminus L) \cap Q = \varnothing} \mathbh{1}_{L \subset Q} = \mathbh{1}_{L=M \cap Q},\end{aligned}$$ we obtain that $$\tau^M_L = \int_{{\mathcal{P}}(T) \setminus\{\varnothing\}} \mathbh{1}_{L=M \cap Q} \mu({\mathrm{d}}Q).$$ It follows that the [f.d.d. ]{}of $X^*$ satisfy $$\begin{aligned} -\log{\mathbb{P}}\bigl(X^*_t\leq x_t, t \in M \bigr) &=& \int _{{\mathcal{P}}(T) \setminus\{\varnothing\}} \sum_{\varnothing\neq L \subset M} \mathbh{1}_{L=M \cap Q} \bigvee_{t \in L} \frac{1}{x_t} \mu ({\mathrm{d}}Q) \\[-0.5pt] &=& \int_{{\mathcal{P}}(T) \setminus\{\varnothing\}} \bigvee_{t \in M} \frac {\mathbh{1}_{t \in Q}}{x_t} \mu({\mathrm{d}}Q) = \int_{\Omega} \biggl(\bigvee _{t \in M} \frac{V_{t}(\omega )}{x_t} \biggr) \nu({\mathrm{d}}\omega)\end{aligned}$$ as desired. This finishes the proof. Triangle inequalities and operation of Bernstein functions ---------------------------------------------------------- In [@cooleyetalii_06], Proposition 4, it is shown that an ECF $\theta$ on ${\mathcal{F}}(T)$ satisfies the following bivariate inequalities for $r,s,t\in T$: $$\begin{aligned} \theta \bigl(\{s,t\} \bigr) & \leq&\theta \bigl(\{s,r\} \bigr)\theta \bigl(\{r,t \} \bigr), \\[-0.5pt] \theta \bigl(\{s,t\} \bigr)^\alpha& \leq&\theta \bigl(\{s,r\} \bigr)^\alpha+ \theta \bigl(\{r,t\} \bigr)^\alpha- 1,\quad\quad 0 < \alpha \leq1, \\[-0.5pt] \theta \bigl(\{s,t\} \bigr)^\alpha& \geq&\theta \bigl(\{s,r\} \bigr)^\alpha+ \theta \bigl(\{r,t\} \bigr)^\alpha- 1,\quad\quad \alpha\leq0.\end{aligned}$$ These inequalities have in common, that they are in fact triangle inequalities of the form $$\label{eqn:BernsteinTriangle} g\circ\eta \bigl(\{s,t\} \bigr) \leq g\circ\eta \bigl(\{s,r\} \bigr) + g\circ\eta \bigl(\{r,t\} \bigr),$$ if we rewrite them in terms of $\eta:= \theta- 1$ and $$\begin{aligned} g(x)&=&\log(1+x), \\[-0.5pt] g(x)&=&(1+x)^\tau-1,\quad\quad 0 < \alpha\leq1, \\[-0.5pt] g(x)&=&1-(1+x)^\tau, \quad\quad\alpha\leq0.\end{aligned}$$ These functions $g$ have in common that they are in fact *Bernstein functions*. A function $g\dvtx [0,\infty) \rightarrow[0, \infty)$ is called a *Bernstein function* if one of the following equivalent conditions is satisfied ([cf. ]{}[@bcr_84], 4.4.3 and page 141) (i) The function $g$ is of the form $$g(r)= c + br + \int_0^\infty \bigl(1- \mathrm{e}^{-\lambda r } \bigr) \nu({\mathrm{d}}\lambda),$$ where $c,b \geq0$ and $\nu$ is a positive Radon measure on $(0,\infty )$ with $\int_0^\infty\frac{\lambda}{1+\lambda} \nu({\mathrm{d}}\lambda) < \infty$. (ii) The function $g$ is continuous and $g \in C^\infty((0,\infty))$ with $g \geq0$ and $(-1)^n g^{(n+1)} \geq0$ for all $n\geq0$. (Here, $g^{(n)}$ denotes the $n$th derivative of $g$.) (iii) The function $g$ is continuous, $g \geq0$ and $g$ is negative definite as a function on the semigroup $([0,\infty),+,0)$. For a comprehensive treatise on Bernstein functions including a table of examples, see [@schillingetalii_10]. Bernstein functions play already an important role in the construction of advanced Gaussian processes by generating novel covariance functions from given ones, [cf. ]{}[@zastavnyiporcu_11] and [@porcuschilling_11]. Here, we see that they are equally useful for generating new ECFs from given ECFs and correspondingly new Tawn–Molchanov processes from given ones. \[cor:opBernstein\] Let $T$ be a set and $\theta\in\Theta(T)$ an ECF. Let $g$ be a Bernstein function which is not constant. Then the function on ${\mathcal{F}}(T)$ $$A \mapsto \frac{g(\theta(A))-g(0)}{g(1)-g(0)}$$ is again an ECF in $\Theta(T)$. The result is immediate from Theorem \[thm:ECF\_ND\], since Bernstein functions operate on negative definite kernels ([cf. ]{}[@bcr_84], 3.2.9 and 4.4.3). For instance, if $\theta$ is an ECF, then also $\log(1+\theta)/\log(2)$ or $((\theta+a)^q-a^q)/((1+a)^q-a^q)$ are ECFs for $0<q<1$ and $a \geq0$. Finally, we show that (\[eqn:BernsteinTriangle\]) holds true for arbitrary Bernstein functions. In fact, the result of [@cooleyetalii_06], Proposition 4, can be generalized to the following extent as a corollary to Theorem \[thm:ECF\_ND\]. \[cor:triangleineq\] Let $\theta\in\Theta(T)$ be an ECF. Set $\eta:=\theta-1$ and let $g$ be a Bernstein function. Then we have for $A,B,C \in{\mathcal{F}}(T)\setminus \{ \varnothing\}$ that $$g\circ\eta(A \cup B) \leq g\circ\eta(C) + g\circ\eta(A \cup B) \leq g\circ\eta(A \cup C) + g\circ\eta(C \cup B).$$ Since $\theta$ is an ECF, it is negative definite ([cf. ]{}Theorem \[thm:ECF\_ND\]). Subtracting $1$ does not change this property. Notice further that $\theta$ takes values in $\{0\} \cup[1,\infty)$, where the value $0$ is only attained for the empty set $\varnothing$ (the neutral element of ${\mathcal{F}}(T)$). Thus, the function $\eta:=\theta-1: {\mathcal{F}}(T) \setminus\{\varnothing\} \rightarrow{\mathbb{R}}$ is negative definite and takes values only in $[0,\infty)$. Applying a Bernstein function $g$ does not change this property ([cf. ]{}[@bcr_84], 3.2.9 and 4.4.3). By [@bcr_84], 8.2.7, this also means that $f:=g \circ\eta: {\mathcal{F}}(T)\setminus\{\varnothing\} \rightarrow{\mathbb{R}}$ is negative definite on ${\mathcal{F}}(T)\setminus\{\varnothing\}$. Since we have also $f \geq0$ on ${\mathcal{F}}(T)\setminus\{\varnothing\}$, we may derive for $A,B,C \in{\mathcal{F}}(T) \setminus\{\varnothing\}$ $$\begin{aligned} &&f(C)+f(A \cup B)-f(A \cup C)-f(C \cup B) \\ &&\quad= \bigl(f(C)-f(A \cup C)-f(C \cup B)+f(A \cup B \cup C) \bigr)+ \bigl(f(A \cup B)-f(A \cup B \cup C) \bigr) \\ &&\quad=\Delta_{A}\Delta_{B}f(C) + \Delta_{C}f(A \cup B) \leq0\end{aligned}$$ as desired. This finishes the proof. Stochastic continuity --------------------- In this section, we require $T$ to be a metric space. We need to define the notion of continuity that we will use in connection with ECFs $\theta\dvtx {\mathcal{F}}(T) \rightarrow[0,\infty)$. Therefore, let $f\dvtx {\mathcal{F}}(T) \rightarrow{\mathbb{R}}$ be a function on the finite subsets of $T$. Then $f$ induces a family of functions $\{f^{(m)}\}_{m \geq0}$ where $f^{(m)}\dvtx T^m \rightarrow{\mathbb{R}}$ is given by $$f^{(m)}(t_1,\ldots,t_m)=f \bigl( \{t_1,\ldots,t_m\} \bigr).$$ \[def:ECF\_cts\] Let $f\dvtx {\mathcal{F}}(T) \rightarrow{\mathbb{R}}$ be a function on the finite subsets of a metric space $T$. We say that $f$ is *continuous* if all induced functions $f^{(m)}\dvtx T^m \rightarrow{\mathbb{R}}$ are continuous for all $m \geq0$, where $T^m$ is endowed with the product topology. \[lemma:ECF\_cty\] Let $X=\{X_t\}_{t \in T}$ be a simple max-stable process with ECF $\theta$. Then the following implication holds: $$X \mbox{ is stochastically continuous} \quad\Longrightarrow \quad\theta\mbox{ is continuous.}$$ Stochastic continuity of $X$ means that for any $\varepsilon> 0$, for any $t \in T$ and sequence $t^{(n)} \rightarrow t$ we have ${\mathbb{P}}(| X_{t^{(n)}} - X_t | > \varepsilon) \rightarrow0$. From this, we can easily derive that for any $\varepsilon>0$, any $m \in {\mathbb{N}}$, any $(t_1,\ldots,t_m) \in T^m$ and a sequence $(t^{(n)}_1,\ldots ,t^{(n)}_m) \rightarrow(t_1,\ldots,t_m)$, also ${\mathbb{P}}(\| (X_{t^{(n)}_i} - X_{t_i})_{i=1}^m \| > \varepsilon) \rightarrow0$ for any reference norm $\| \cdot\|$ on ${\mathbb{R}}^m$. The latter implies the corresponding convergence in distribution: $F_{(t^{(n)}_1,\ldots,t^{(n)}_m)} \rightarrow F_{(t_1,\ldots,t_m)}$. Since $\log F_{(t_1, \ldots, t_m)}: [0,\infty)^m \rightarrow{\mathbb{R}}$ is monotone and homogeneous, we have that for $x>0$ the point $(x,\ldots,x) \in(0,\infty)^m$ is a continuity point of $F_{(t_1, \ldots, t_m)}$ ([cf. ]{}[@resnick_08], page 277). Thus, the induced function $\theta^{(m)}$ on $T^m$ is continuous, since $\theta^{(m)}(t_1,\ldots,t_m)= - x \log F_{(t_1, \ldots, t_m)} (x, \ldots, x)$. Hence, $\theta$ is continuous. Second, we prove the following upper bound that shows that stochastic continuity of the TM process $X^*$ is indeed controlled by the bivariate extremal coefficients. \[lemma:ctyestimate\] Let $X^*=\{X^*_t\}_{t \in T}$ be the TM process with ECF $\theta$. Set $\eta:=\theta-1$. Then we have for any $\varepsilon> 0$ $${\mathbb{P}}\bigl( \bigl|X^*_s - X^*_t \bigr| > \varepsilon \bigr) \leq2 \biggl( 1 - \exp \biggl( - \frac {\eta(\{s,t\})}{\varepsilon} \biggr) \biggr) \leq \frac {2}{\varepsilon} \eta \bigl(\{s,t\} \bigr).$$ Let $\varepsilon> 0$. We will prove the statement for $2 \varepsilon$ instead of $\varepsilon$. Therefore, consider the following disjoint events on a corresponding probability space $(\Omega, {\mathcal A}, {\mathbb{P}})$ for $k=0,1,2, \ldots$ $$\begin{aligned} A_k := \bigl\{\omega\in\Omega{\dvtx }\bigl(X^*_s( \omega),X^*_t(\omega) \bigr) \in(k \varepsilon, (k+2) \varepsilon]^2 \setminus\bigl((k+1)\varepsilon, (k+2)\varepsilon\bigr]^2 \bigr\}.\end{aligned}$$ The disjoint union $\bigcup_{k=0}^{\infty} A_k$ is a subset of $\{ \omega\in\Omega{\dvtx }|X^*_s(\omega) - X^*_t(\omega)| \leq 2\varepsilon\}$ and so $${\mathbb{P}}\bigl( |X^*_s - X^*_t | \leq2\varepsilon \bigr) \geq {\mathbb{P}}\Biggl(\bigcup_{k=0}^{\infty} A_k \Biggr) = \sum_{k=0}^\infty {\mathbb{P}}(A_k) = \lim_{n \to \infty} \sum _{k=0}^n {\mathbb{P}}(A_k).$$ From (\[eqn:starfdd\]) and Theorem \[thm:ECF\_ND\], we see that the bivariate distribution of the process $X^*$ is given by $$\label{eqn:starbivariate} - \log{\mathbb{P}}\bigl(X^*_s \leq x, X^*_t \leq y \bigr) = \frac{\eta(\{s,t\})}{x \vee y}+\frac{1}{x \wedge y}.$$ For further calculations, we abbreviate for $p,q \in{\mathbb{N}}\cup\{0\}$ $$B(p,q):={\mathbb{P}}\bigl(X^*_s \leq p \cdot\varepsilon, X^*_t \leq q \cdot \varepsilon \bigr).\vadjust{\goodbreak}$$ Note that $B(p,q)=B(q,p)$ and $B(p,0)=0$. With this notation, we rearrange $$\sum_{k=0}^n {\mathbb{P}}(A_k) = -B(n+1,n+1) + 2 \sum_{k=0}^n \bigl[ B(k+2,k+1)- B(k+2,k) \bigr].$$ For the second summand, we have ([cf. ]{}(\[eqn:starbivariate\])) $$\begin{aligned} && \sum_{k=0}^n \bigl[ B(k+2,k+1)- B(k+2,k) \bigr] \\[-1pt] &&\quad\stackrel{{\scriptsize{(\ref{eqn:starbivariate})}}} {=} \sum _{k=0}^n \biggl[ \exp \biggl(-\frac{1}{\varepsilon} \biggl[\frac{\eta(\{s,t\})}{k+2} + \frac {1}{k+1} \biggr] \biggr) - \exp \biggl(- \frac{1}{\varepsilon} \biggl[ \frac {\eta(\{s,t\})}{k+2} + \frac{1}{k} \biggr] \biggr) \biggr] \\[-1pt] &&\quad= \sum_{k=0}^n \exp \biggl(- \frac{1}{\varepsilon} \biggl[\frac{\eta (\{s,t\} )}{k+2} \biggr] \biggr) \biggl[ \exp \biggl(- \frac {1}{(k+1)\varepsilon } \biggr) - \exp \biggl(-\frac{1}{k \varepsilon} \biggr) \biggr] \\[-1pt] &&\quad \geq\sum_{k=0}^n \exp \biggl(- \frac{\eta(\{s,t\})}{2 \varepsilon } \biggr) \biggl[ \exp \biggl(-\frac{1}{(k+1)\varepsilon} \biggr) - \exp \biggl(-\frac{1}{k \varepsilon} \biggr) \biggr] \\[-1pt] &&\quad= \exp \biggl(-\frac{\eta(\{s,t\})}{2 \varepsilon} \biggr) \exp \biggl(-\frac{1}{(n+1) \varepsilon} \biggr).\end{aligned}$$ Finally, $$\begin{aligned} &&{\mathbb{P}}\bigl( \bigl|X^*_s - X^*_t \bigr| > 2\varepsilon \bigr) \\[-1pt] &&\quad= 1 - {\mathbb{P}}\bigl( \bigl|X^*_s - X^*_t \bigr| \leq 2 \varepsilon \bigr) \leq1 - \lim_{n \to\infty} \sum _{k=0}^n {\mathbb{P}}(A_k)\vadjust{\goodbreak} \\ &&\quad= 1 + \lim_{n \to\infty} B(n+1,n+1) - 2 \lim _{n \to\infty} \sum_{k=0}^n \bigl[ B(k+2,k+1)- B(k+2,k) \bigr] \\ &&\quad\leq1 + \lim_{n \to\infty} \exp \biggl(-\frac{\eta(\{s,t\} )+1}{(n+1)\varepsilon} \biggr) - 2 \lim_{n \to\infty} \biggl(\exp \biggl(-\frac{\eta(\{s,t\})}{2 \varepsilon} \biggr) \exp \biggl(-\frac{1}{(n+1) \varepsilon} \biggr) \biggr) \\ &&\quad= 2 - 2 \exp \biggl(-\frac{\eta(\{s,t\})}{2 \varepsilon} \biggr) \leq \frac{2}{2\varepsilon} \eta \bigl(\{s,t\} \bigr).\end{aligned}$$ This finishes the proof. \[thm:ECprocess\_cty\] Let $X^*=\{X^*_t\}_{t \in T}$ be the TM process with ECF $\theta$. Then the following statements are equivalent: (i) $X^*$ is stochastically continuous. (ii) $\theta$ is continuous. (iii) The bivariate map $(s,t) \mapsto\theta(\{s,t\})$ is continuous. (iv) The bivariate map $(s,t) \mapsto\theta(\{s,t\})$ is continuous on the diagonal. The implication $\mathrm{(i)} \Rightarrow\mathrm{(ii)}$ follows from Lemma \[lemma:ECF\_cty\]. Clearly, continuity of $\theta$ implies continuity of the induced function $\theta^{(2)}(s,t):=\theta(\{s,t\})$, which implies continuity of $\theta^{(2)}$ on the diagonal. This shows the implications $\mathrm{(ii)} \Rightarrow\mathrm{(iii)}$ and $\mathrm{(iii)} \Rightarrow\mathrm{(iv)}$. Finally, the implication $\mathrm{(iv)} \Rightarrow\mathrm{(i)}$ follows from Lemma \[lemma:ctyestimate\], since $\eta(\{t,t\})=\theta(\{t\})-1=0$. Dependency sets – the special role of TM processes {#sect:depset} ================================================== In this section, we show that the TM process $X^*$ with ECF $\theta$ is exceptional among all max-stable processes sharing the same ECF $\theta $ as $X^*$ in the sense that its dependency set ${{\mathcal K}}^*$ (to be introduced below) is maximal [w.r.t. ]{}inclusion. Therefore, recall that for a finite non-empty subset $M \subset T$ the dependency set ${{\mathcal K}}_M$ of $\{X_t\}_{t \in M}$ is the largest compact convex set ${{\mathcal K}}_M \subset[0,\infty)^M$ satisfying ([cf. ]{}(\[eqn:DepSetdefn\])) $$\ell_M(x)=\sup \bigl\{ \langle x,y \rangle{\dvtx }y \in {{\mathcal K}}_M \bigr\} \quad\quad\forall x \in[0,\infty)^M.$$ The closed convex set ${{\mathcal K}}_M$ may also be described as the following intersection of half spaces ([cf. ]{}[@schneider_93], Section 1.7): $$\label{eqn:DepSetdirect} {{\mathcal K}}_M= \bigcap_{x \in S_M} \bigl\{ y \in[0,\infty)^M {\dvtx }\langle x,y \rangle\leq \ell_M(x) \bigr\}.$$ The simplest examples for dependency sets ${{\mathcal K}}_M$ are the unit cube $[0,1]^M$ corresponding to a collection of independent random variables $\{X_t\}_{t \in M}$ and the cross-polytope $D^M:=\{x \in[0,\infty)^M {\dvtx }\sum_{t \in M} x_t \leq1\}$ corresponding to identical random variables $\{X_t\}_{t \in M}$. Any dependency set ${{\mathcal K}}_M$ satisfies $$D^M \subset{{\mathcal K}}_M \subset[0,1]^M.$$ \[example:DepSetBR\] The [f.d.d. ]{}of a Brown–Resnick process ([cf. ]{}Example \[example:BR\]) are the multivariate Hüsler–Reiss distributions ([cf. ]{}[@hueslerreiss_89]). In the bivariate case, when $M=\{1,2\}$ consists of two points only, the distribution function of a Hüsler–Reiss distributed random vector $(X_1,X_2)$, standardized to unit Fr[é]{}chet marginals, is $$-\log{\mathbb{P}}_{\gamma}(X_1 \leq x_1, X_2 \leq x_2) = \frac{1}{x_1} \Phi \biggl(\frac{\sqrt{\gamma}}{2} + \frac{\log (x_2/x_1 )}{\sqrt {\gamma }} \biggr) + \frac{1}{x_2} \Phi \biggl(\frac{\sqrt{\gamma}}{2} + \frac{\log (x_1/x_2 )}{\sqrt{\gamma}} \biggr)$$ for $x_1,x_2 \geq0$. Here $\Phi$ denotes the distribution function of the standard normal distribution and the parameter $\gamma$ is the value of the variogram between the two points ([cf. ]{}Example \[example:BR\]). Figure \[fig:DepSetBR\] illustrates, how the corresponding dependency sets range between full dependence ($\gamma =0$) and independence ($\gamma=\infty$). ![Nested dependency sets ${{\mathcal K}}^{(\gamma)}_{M}$ of the bivariate Brown–Resnick ([resp. ]{}Hüsler–Reiss) distribution where $M=\{1,2\}$ ([cf. ]{}Example \[example:DepSetBR\]). The dependency sets grow as the parameter $\gamma$ increases. They range between full dependence ($\gamma=0$) and independence ($\gamma=\infty$).[]{data-label="fig:DepSetBR"}](567f03.eps) In order to define a single dependency set for a simple max-stable process comprising all multivariate dependency sets, we write $${\operatorname{pr}}_M\dvtx [0,\infty)^T \rightarrow[0, \infty)^M,\quad\quad (x_t)_{t \in T} \mapsto(x_t)_{t \in M}$$ for the natural projection. Let $X$ be a simple max-stable process $X=\{X_t\}_{t \in T}$ and denote for finite $M\in{\mathcal{F}}(T)\setminus\{\varnothing\}$ the multivariate dependency set of the random vectors $\{X_t\}_{t \in M}$ by ${{\mathcal K}}_M$. Then we define the *dependency set* ${{\mathcal K}}\subset[0,\infty)^T$ of $X$ as $${{\mathcal K}}:= \bigcap_{M \in{\mathcal{F}}(T)\setminus\{\varnothing\}} {\operatorname{pr}}_M^{-1} ({{\mathcal K}}_M ).$$ Analogously to (\[eqn:DepSetdefn\]), the dependency set ${{\mathcal K}}$ may be characterized as follows. \[lemma:DepSetprops\] The dependency set ${{\mathcal K}}$ of a simple max-stable process $X=\{X_t\}_{t \in T}$ is the largest compact convex set ${{\mathcal K}}\subset[0,\infty)^T$ satisfying $$\label{eqn:DepSetprops} \ell_M(x)=\sup \biggl\{ \sum _{t \in M} x_ty_t {\dvtx }y \in{{\mathcal K}}\biggr\}\quad\quad \forall x \in[0,\infty)^M \forall \varnothing\neq M \in {\mathcal{F}}(T),$$ where $\ell_M$ is the stable tail dependence function of $\{X_t\}_{t \in M}$. Convexity of ${{\mathcal K}}$ follows from the convexity of each ${{\mathcal K}}_M$ and from the linearity of the projections ${\operatorname{pr}}_M$ for $M\in{\mathcal{F}}(T)\setminus\{\varnothing\}$. Since ${{\mathcal K}}_{\{t\}}=[0,1]$ is the unit interval for each $t \in T$, the set ${{\mathcal K}}$ is contained in the compact space $[0,1]^T$. Moreover, ${{\mathcal K}}$ is closed as the intersection of closed sets, hence ${{\mathcal K}}$ is compact. Next, we prove that ${{\mathcal K}}_M={\operatorname{pr}}_M({{\mathcal K}})$. By definition of ${{\mathcal K}}$ it is clear that ${\operatorname{pr}}_M({{\mathcal K}}) \subset{{\mathcal K}}_M$ for $M \in{\mathcal{F}}(T)\setminus\{\varnothing\}$. To prove the reverse inclusion, let $y_M$ be an element of ${{\mathcal K}}_M$ and set $V(y_M):= {\operatorname{pr}}_M^{-1}(\{y_M\})\cap{{\mathcal K}}= {\operatorname{pr}}_M^{-1}(\{y_M\})\cap{{\mathcal K}}\cap[0,1]^T$. We need to show that $V(y_M) \neq\varnothing$. Denoting $V(y_M,A):={\operatorname{pr}}_M^{-1}(\{y_M\}) \cap{\operatorname{pr}}_A^{-1} ({{\mathcal K}}_A ) \cap[0,1]^T$, we see that $$\begin{aligned} V(y_M) =\bigcap_{A \in{\mathcal{F}}(T)\setminus\{\varnothing\}} V(y_M,A).\end{aligned}$$ Note that each $V(y_M,A)$ is a closed subset of the compact Hausdorff space $[0,1]^T$. Therefore, it suffices to verify the finite intersection property for the system of sets $\{V(y_M,A)\}_{A \in {\mathcal{F}}(T)\setminus\{\varnothing\}}$ in order to show that $V(y_M)\neq \varnothing$. But this follows from the consistency of the finite dimensional dependency sets $\{{{\mathcal K}}_A\}_{A \in{\mathcal{F}}(T)\setminus \{ \varnothing\}}$ as follows: As [@molchanov_08], Section 7, Proposition 8, essentially says, we have that if $A$ and $B$ are non-empty finite subsets of $T$ with $A \subset B$, then ${{\mathcal K}}_A$ is the projection of ${{\mathcal K}}_B$ onto the respective coordinate space. In particular, ${\operatorname{pr}}_B^{-1}({{\mathcal K}}_B) \subset{\operatorname{pr}}_A^{-1}({{\mathcal K}}_A)$ and ${\operatorname{pr}}_A^{-1}(\{y_A\}) \cap{\operatorname{pr}}_B^{-1}({{\mathcal K}}_B) \cap[0,1]^T \neq \varnothing$ for $y_A \in{{\mathcal K}}_A$. Now, let $A_1,\ldots,A_k$ be non-empty finite subsets of $T$. Then $$\begin{aligned} \varnothing &\neq&{\operatorname{pr}}_M^{-1} \bigl(\{y_M\} \bigr) \cap{\operatorname{pr}}_{M \cup\bigcup_{i=1}^k A_i}^{-1} ({{\mathcal K}}_{M \cup\bigcup_{i=1}^k A_i} ) \cap[0,1]^T \\ & \subset&{\operatorname{pr}}_M^{-1} \bigl(\{y_M\} \bigr) \cap \bigcap_{i=1}^k {\operatorname{pr}}^{-1}_{A_i} ({{\mathcal K}}_{A_i} ) \cap[0,1]^T = \bigcap _{i=1}^k V(y_M,A_i),\end{aligned}$$ as desired and we have shown that ${{\mathcal K}}_M \subset{\operatorname{pr}}_M({{\mathcal K}})$. Both inclusions give ${{\mathcal K}}_M = {\operatorname{pr}}_M({{\mathcal K}})$. By definition, we have $\ell_M(x)=\sup \{ \langle x,y \rangle {\dvtx }y \in{{\mathcal K}}_M \}$ for $x \in[0,\infty)^M$. Thus, (\[eqn:DepSetprops\]) follows from ${{\mathcal K}}_M={\operatorname{pr}}_M({{\mathcal K}})$. Finally, let ${{\mathcal L}}\subset[0,\infty)^T$ be also convex compact and satisfying (\[eqn:DepSetprops\]) with ${{\mathcal K}}$ replaced by ${{\mathcal L}}$. Then it follows immediately that ${\operatorname{pr}}_M({{\mathcal L}})={{\mathcal K}}_M$ for any non-empty finite subset $M \subset T$. We conclude that ${{\mathcal L}}\subset{{\mathcal K}}$ by definition of ${{\mathcal K}}$. This finishes the proof. In particular, the ECF $\theta$ of a simple max-stable process $X=\{ X_t\}_{t \in T}$ can be expressed in terms of the dependency set ${{\mathcal K}}$ of $X$ as $$\label{eqn:ECFfromDepSet} \theta(A)=\sup \biggl\{\sum_{t \in A} x_t {\dvtx }x \in{{\mathcal K}}\biggr\}.$$ In order to make statements about the dependency sets ${{\mathcal K}}$ of processes $X=\{X_t\}_{t \in T}$ in terms of the ECF $\theta$, we introduce the following notation: For any non-empty finite subsets $A$ of $T$, we set the halfspace $${{\mathcal H}}_A(\theta):= \biggl\{ x \in[0,\infty)^T {\dvtx }\sum _{t \in A} x_t \leq\theta(A) \biggr\} \phantom{.}$$ that is bounded by the hyperplane $${{\mathcal E}}_A(\theta):= \biggl\{ x \in[0,\infty)^T {\dvtx }\sum _{t \in A} x_t = \theta(A) \biggr\}.$$ \[lemma:DepSet\] Let ${{\mathcal K}}$ be the dependency set of a simple max-stable process $X=\{ X_t\}_{t \in T}$ with ECF $\theta$. Then the following inclusion holds $${{\mathcal K}}\subset\bigcap_{A \in{\mathcal{F}}(T)\setminus\{\varnothing\}} {{\mathcal H}}_A( \theta).$$ On the other hand for each $A \in{\mathcal{F}}(T)\setminus\{\varnothing\}$ there is at least one point $\mathbf{x^A}$ in the intersection $$\mathbf{x^A} \in{{\mathcal K}}\cap{{\mathcal E}}_A(\theta).$$ Let $A \in{\mathcal{F}}(T)\setminus\{\varnothing\}$ and $x \in{{\mathcal K}}$. Then the assumption $\sum_{t \in A} x_t > \theta(A)$ contradicts $\theta (A)=\sup\{\sum_{t \in A} x_t {\dvtx }x \in{{\mathcal K}}\} > \theta(A)$ ([cf. ]{}(\[eqn:ECFfromDepSet\])). So $\sum_{t \in A} x_t \leq\theta(A)$. This proves the inclusion. Second, since ${{\mathcal K}}$ is compact and the map $[0,\infty)^T \ni x \rightarrow\sum_{t \in A} x_t$ is continuous, we know that it attains its supremum at some $\mathbf{x^A} \in{{\mathcal K}}$. \[example:ball\] We give a simple multivariate example for Lemma \[lemma:DepSet\] (as illustrated in Figure \[fig:ballDepSet\] in the introduction for the trivariate case): The Euclidean norm $\ell_M(x)=\| x \|_2$ is a stable tail dependence function on $[0,\infty)^M$ ([cf. ]{}[@molchanov_08], Example 2) and defines a simple max-stable distribution ([cf. ]{}(\[eqn:fddstabletaildepfn\])) with ECF $\theta(A)=\sqrt{|A|}$ for $A \subset M$, such that $$\begin{aligned} {{\mathcal H}}_A(\theta)&=& \bigl\{ x \in[0,\infty)^M {\dvtx }\langle x , {\mathbf{1}}_A\rangle \leq\sqrt{|A|} \bigr\}, \\ {{\mathcal E}}_A(\theta)&=& \bigl\{ x \in[0,\infty)^M {\dvtx }\langle x , {\mathbf{1}}_A\rangle= \sqrt{|A|} \bigr\}\end{aligned}$$ for $\varnothing\neq A \subset M$. It can be easily seen that for $x \in[0,\infty)^M\setminus\{{\mathbf{1}}_\varnothing\}$ $$\ell_M(x)=\| x \|_2 = \bigl\langle x, x/\| x \|_2 \bigr\rangle= \sup \bigl\{ \langle x,y \rangle{\dvtx }y \in B^{+} \bigr\},$$ where $B^{+}:=\{y \in[0,\infty)^M {\dvtx }\| y \|_2 \leq1\}$ denotes the positive part of the (Euclidean) unit ball. So, the dependency set ${{\mathcal K}}$ is clearly $B^{+}$ in this case. Now, the planes ${{\mathcal E}}_A(\theta)$ are tangent to the boundary of $B^{+}$ with common points $\mathbf{x^A}={\mathbf{1}}_A/\sqrt{|A|}$ for $\varnothing\neq A \subset M$, which makes it easy to see that Lemma \[lemma:DepSet\] holds true in this example. Figure \[fig:ballDepSet\] shows the dependency set ${{\mathcal K}}=B^+$ (left) and the intersection of halfspaces bounded by the planes ${{\mathcal E}}_A(\theta)$ (right). In the middle it is illustrated that this intersection contains $B^+$ and the points $\mathbf{x^A}$ are marked. The following theorem shows that the inclusion from Lemma \[lemma:DepSet\] is sharp and attained by TM processes. Figure \[fig:starDepSet\] illustrates the dependency set of a trivariate distribution of a TM process. ![Dependency set ${{\mathcal K}}^*$ of the random vector $\{X^*_t\}_{t \in M}$ for $M=\{1,2,3\}$. The dependency set ${{\mathcal K}}^*$ is bounded by the hyperplanes ${{\mathcal E}}_A(\theta)$ that are given by the equations $\sum_{t \in A} x_t = \theta(A)$, where $\theta$ denotes the ECF of $X^*$. The coefficients $\tau^L_{\{t\}}$ for $L \in {\mathcal{F}}(M)\setminus\{\varnothing\}$ and $t \in L$ turn up as lengths of the resulting polytope ${{\mathcal K}}^*$ ([cf. ]{}Theorem \[thm:ECF\_ND\] (b) and Theorem \[thm:starDepSet\]).[]{data-label="fig:starDepSet"}](567f04.eps) \[thm:starDepSet\] Let ${{\mathcal K}}^*$ be the dependency set of the TM process $X^*=\{X^*_t\} _{t \in T}$ with ECF $\theta$. Then $${{\mathcal K}}^* = \bigcap_{A \in{\mathcal{F}}(T)\setminus\{\varnothing\}} {{\mathcal H}}_A( \theta).$$ First, we prove the theorem in the case, when $T=M$ is finite and ${{\mathcal K}}^*={{\mathcal K}}^*_M$: Therefore, write $${{\mathcal L}}_M := \bigcap_{\varnothing\neq A \subset M} {{\mathcal H}}_A(\theta) = \bigl\{ x \in[0,\infty)^M {\dvtx }\langle x , {\mathbf{1}}_A\rangle \leq\theta(A) \mbox{ for all } \varnothing\neq A \subset M \bigr\}.$$ The inclusion ${{\mathcal K}}^*_M \subset{{\mathcal L}}_M$ is proven in Lemma \[lemma:DepSet\]. So, it remains to show the other inclusion ${{\mathcal L}}_M \subset{{\mathcal K}}^*_M$. Due to (\[eqn:DepSetdirect\]), we have that $${{\mathcal K}}^*_M= \bigcap_{x \in S_M} \bigl\{ y \in[0,\infty)^M {\dvtx }\langle x,y \rangle\leq\ell^*_M(x) \bigr\},$$ where $$\ell^*_M(x) = \sum_{\varnothing\neq L \subset M} \tau^M_L \bigvee_{t \in L} x_{t}$$ is the stable tail dependence function of $\{X^*_t\}_{t \in M}$, here expressed in terms of the coefficients $\tau^M_L$ from Theorem \[thm:ECF\_ND\] (b) ([cf. ]{}(\[eqn:starfdd\])). Thus, it suffices to show the following implication in order to prove ${{\mathcal L}}_M \subset{{\mathcal K}}^*_M$: $$x \in S_M \quad\mbox{and}\quad y \in{{\mathcal L}}_M \quad\Longrightarrow \quad\langle x,y \rangle\leq\ell^*_M(x).$$ We now prove this implication: Without loss of generality, we may label the elements of $M=\{t_1, \ldots, t_m\}$ such that $x_{t_1} \geq x_{t_2} \geq\cdots\geq x_{t_m}$. Then we may write $x=(x_t)_{t \in M} \in S_M \subset[0,\infty)^M$ as $$x = \underbrace{x_{t_m}}_{\geq0} {\mathbf{1}}_M + \underbrace {(x_{t_{n-1}}-x_{t_m})}_{\geq0} {\mathbf{1}}_{M \setminus\{t_m\}} + \cdots+ \underbrace{(x_{t_{2}}-x_{t_3})}_{\geq0} {\mathbf{1}}_{\{t_1,t_2\}} + \underbrace{(x_{t_{1}}-x_{t_2})}_{\geq0} {\mathbf{1}}_{\{t_1\}}.$$ Taking the scalar product with $y \in{{\mathcal L}}_M$, we conclude $$\begin{aligned} \label{eqn:ellcoincide} \langle x,y \rangle & \leq& x_{t_m} \theta(M) + (x_{t_{n-1}}-x_{t_m}) \theta \bigl(M \setminus\{ t_m \} \bigr) \nonumber \\ &&{} + \cdots+ (x_{t_{2}}-x_{t_3}) \theta \bigl( \{t_1,t_2\} \bigr) + (x_{t_{1}}-x_{t_2}) \theta \bigl(\{t_1\} \bigr) \\ & =& x_{t_m} \bigl(\theta(M)-\theta \bigl(M \setminus\{ t_m\} \bigr) \bigr) + \cdots+ x_{t_{2}} \bigl(\theta \bigl( \{t_1,t_2\} \bigr)-\theta \bigl(\{t_1\} \bigr) \bigr) + x_{t_{1}} \theta \bigl(\{t_1\} \bigr). \nonumber\end{aligned}$$ On the other hand the stable tail dependence function $\ell^*_M$ is by this ordering of the components of $x$ given as $$\ell^*_M(x) = \sum_{\varnothing\neq L \subset M} \tau^M_L \bigvee_{t \in L} x_{t} = \sum_{i=1}^m x_{t_i} \biggl( \sum_{L \subset M {\dvtx }t_1,\ldots ,t_{i-1} \notin L, t_{i} \in L} \tau^M_L \biggr).$$ From (\[eqn:thetaA.from.tauML\]), we see that this expression coincides with the [r.h.s. ]{}of $(\ref{eqn:ellcoincide})$. Thus, we have our desired inequality $\langle x,y \rangle\leq\ell^*_M(x)$. This finishes the proof in the case, when $T=M$ is finite. Otherwise, the definition of the dependency set ${{\mathcal K}}^*$ and the result for finite $M$ give $${{\mathcal K}}^* = \bigcap_{M \in{\mathcal{F}}(T)\setminus\{\varnothing\}} {\operatorname{pr}}_M^{-1} \bigl({{\mathcal K}}^*_M \bigr) = \bigcap_{M \in{\mathcal{F}}(T)\setminus\{\varnothing\}} \bigcap_{\varnothing\neq A \subset M} {\operatorname{pr}}_M^{-1} \bigl( {{\mathcal H}}^M_A(\theta) \bigr),$$ where ${{\mathcal H}}^M_A(\theta)= \{ x \in[0,\infty)^M {\dvtx }\sum_{t \in A} x_t \leq\theta(A) \}$. Since ${\operatorname{pr}}_M^{-1} ( {{\mathcal H}}^M_A(\theta) ) = {{\mathcal H}}_A(\theta)$ for $\varnothing\neq A \subset M$, the claim follows. So, if we fix the ECF $\theta$ of a simple max-stable process on $T$, then the TM process yields a maximal dependency set ${{\mathcal K}}^*$ [w.r.t. ]{}inclusion, that is $$\label{eqn:DepSetinclusion} {{\mathcal K}}^* = \mathop{\bigcup_{{{\mathcal K}}\mbox{ {\scriptsize{dependency set}}}}}_{ \mbox{{\scriptsize{with the same ECF as}} } {{\mathcal K}}^*} {{\mathcal K}}.$$ Now, inclusion of dependency sets corresponds to stochastic ordering in the following sense ([cf. ]{}[@molchanov_08], page 242): If ${{\mathcal K}}'$ and ${{\mathcal K}}''$ denote the dependency sets of the simple max-stable processes $X'$ and $X''$ respectively, then ${{\mathcal K}}' \subset{{\mathcal K}}''$ implies $${\mathbb{P}}\bigl(X_{t}' \leq x_t, t \in M \bigr) \geq{\mathbb{P}}\bigl(X_{t}'' \leq x_t, t \in M \bigr)\quad\quad \forall x \in[0,\infty)^M$$ for all $M \in{\mathcal{F}}(T)\setminus\{\varnothing\}$. This leads to the following sharp inequality. \[cor:fddinequalities\] Let $X=\{X_t\}_{t \in T}$ be a simple max-stable process with ECF $\theta$. Let $M$ be a non-empty finite subset of $T$. Then $$\label{eqn:DepSet_inequality} {\mathbb{P}}(X_{t} \leq x_t, t \in M) \geq \exp \biggl(-\sum_{\varnothing\neq L \subset M} \tau_{L}^{M} \bigvee_{t \in L}\frac{1}{x_t} \biggr) \quad\quad \forall x \in[0, \infty)^M,$$ where the coefficients $\tau_{L}^{M}$ depend only on $\theta$ and can be computed as in Theorem \[thm:ECF\_ND\]. Equality holds for the TM process $X^*$. Let us abbreviate $\eta_A:=\theta(A)-1$. In the bivariate case, the inequality (\[eqn:DepSet\_inequality\]) reads as $$\begin{aligned} {\mathbb{P}}(X_{s} \leq x_s, X_{t} \leq x_t) & \geq&\exp \biggl(- \biggl[\frac{\eta_{st}}{x_s \vee x_t} + \frac{1}{x_s \wedge x_t} \biggr] \biggr) \\ &=& \exp \biggl(- \frac{\eta_{st}+1}{x_s \wedge x_t} \biggr) \exp \biggl(\eta _{st} \biggl{\vert}\frac{1}{x_s} - \frac{1}{x_t} \biggr{\vert}\biggr) .\end{aligned}$$ Indeed this inequality is much better then the trivial inequality ${\mathbb{P}}(X_{s} \leq x_s, X_{t} \leq x_t) \geq{\mathbb{P}}(X_{s} \leq x_s \wedge x_t , X_{t} \leq x_s \wedge x_t)$, which can be written in the above terms as $${\mathbb{P}}(X_{s} \leq x_s, X_{t} \leq x_t) \geq \exp \biggl(- \frac{\eta_{st}+1}{x_s \wedge x_t} \biggr).$$ Further note that $\eta_{st}=\theta(\{s,t\})-1$ can be interpreted as a normalized madogram: $$\eta_{st}\stackrel{\scriptsize{(\ref{eqn:lhopital})}} {=}\lim _{x \to\infty} \frac{{\mathbb{P}}(X_s \geq x \mbox{ or } X_t \geq x)}{{\mathbb{P}}(X_t \geq x)}-1 = \lim_{x \to \infty} \frac{{\mathbb{E}}|\mathbh{1}_{X_s \geq x} - \mathbh{1}_{X_t \geq x}|}{2 {\mathbb{E}}\mathbh{1}_{X_t \geq x}}.$$ If we additionally take into account that ([cf. ]{}[@schlathertawn_02], inequality (13)) $$\eta_{rs} \vee\eta_{st} \vee\eta_{rt} \vee( \eta_{rs}+\eta _{st}+\eta_{rt}-1) \leq \eta_{rst} \leq(\eta_{rs}+\eta_{st}) \wedge( \eta_{st}+\eta_{rt}) \wedge (\eta _{rt}+ \eta_{rs}),$$ we obtain from (\[eqn:DepSet\_inequality\]) the following (sharp) inequality for the trivariate distribution of a simple max-stable random vector $(X_r,X_s,X_t)$ from bivariate quantities: $$\begin{aligned} && {\mathbb{P}}(X_{r} \leq x_r, X_{s} \leq x_s, X_{t} \leq x_t) \\ && \quad\geq\exp \biggl(- \biggl[ \frac{1-\eta_{rs} \vee\eta_{st} \vee\eta_{rt}}{x_r \wedge x_s \wedge x_t} + (a_{rst} \wedge1) \biggl(\frac{1}{x_r \wedge x_s} + \frac{1}{x_s \wedge x_t} + \frac{1}{x_r \wedge x_t} \biggr) \\ &&\quad\quad{} - \biggl( \frac{\eta_{rs}}{x_r \wedge x_s} + \frac {\eta _{st}}{x_s \wedge x_t} + \frac{\eta_{rt}}{x_r \wedge x_t} \biggr) + a_{rst} \biggl(\frac{1}{x_r}+ \frac{1}{x_s}+ \frac{1}{x_t} \biggr) - \biggl(\frac{\eta_{st}}{x_r}+ \frac{\eta_{rt}}{x_s}+ \frac{\eta _{rs}}{x_t} \biggr) \biggr] \biggr),\end{aligned}$$ where $a_{rst}:=(\eta_{rs}+\eta_{st}) \wedge(\eta_{rs}+\eta_{rt}) \wedge(\eta_{st}+\eta_{rt})$. Thus, if one can handle the ECF of a max-stable process, sharp lower bounds for its [f.d.d. ]{}are available. However, beware that higher variate cases of these inequalities will be numerically unstable. It is an open problem and it would be interesting to know whether there exist also minimal dependency sets in the sense of (\[eqn:DepSetinclusion\]) and if they would help to better understand the classification of all dependency structures. In view of Lemma \[lemma:DepSet\] and Theorem \[thm:starDepSet\] a very naive idea would be to take one point from each of the sets ${{\mathcal K}}^* \cap{{\mathcal E}}_A$ where $A \in{\mathcal{F}}(T)\setminus\{\varnothing\}$ and then to take the convex hull with 0 included. However, this fails to be a dependency set in dimensions $|T| \geq3$, since it is not even a zonoid, which would be necessary ([cf. ]{}[@molchanov_08]). Acknowledgements {#acknowledgements .unnumbered} ================ We would like to thank Ilya Molchanov for an inspiring discussion and Zakhar Kabluchko for pointing us to the Cantor cube. We are grateful to two unknown referees for their valuable hints and comments that helped to significantly improve the paper. Financial support for K. Strokorb by the German Research Foundation DFG through the Research Training Group 1023 and for M. Schlather by Volkswagen Stiftung within the “WEX-MOP” project is gratefully acknowledged. [32]{} , , (). . . : . , (). . . : . (). . . , (). . . , (). . . (). . . , (). . In . . : . (). . . (). . . (). . . (). . . , (). . . , (). . . (). . . (). . . , (). . . (). . (). . . : . (). . . , , (). . . (). . . (). . . : . (). . , (). . . : . (). . . (). . . (). . . : . (). . (). . . (). . . (). . . (). . .

--- author: - 'S. D. P. Vitenti' - 'M. Penna-Lima' title: A general reconstruction of the recent expansion history of the universe --- Introduction {#sec:introduction} ============ Many indications of the accelerated expansion of the universe come from distance measurements, such as the distance modulus of type Ia supernovae (SNe Ia) [@Riess1998; @Perlmutter1999]. In the last two decades, several models have been proposed in order to explain this phenomenon and, in general, they can be classified into dynamic and kinematic models. Assuming the general relativity, the first is described by adding a fluid, Dark Energy (DE), in which several propositions provide different DE equation of state (EoS) (for a review, see [@Joyce2015] and references therein). Other common dynamic approach is to modify the geometric setting of the gravitational theory instead of the energy-momentum tensor, such as the high-dimensional models [@Dvali2000] and $f(R)$ theories [@Sotiriou2010; @Nojiri2011]. These approaches are labeled as dynamic in the sense that there are differential equations of motion for the metric, whose modifications consist in altering the source term or the equation of motion itself. In the context of kinematic models, the expansion history of the universe can be probed without assuming any theory of gravitation nor its matter content, and one only needs to define the space-time metric to study it. Considering the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, the recent expansion of the universe is described in terms of the scale factor and its $n$-order derivatives with respect to time, such as the Hubble, deceleration and jerk functions [@Turner2002c; @Visser2004; @Rapetti2007; @Zhai2013], as well as in terms of the luminosity distance [@Daly2003; @Benitez-Herrera2012] (and references therein).[^1] Since the only unknown in this metric is the scale factor (and possibly the spatial curvature), once one of the kinematic functions is determined, the others can be found by integrating and/or differentiating it. Therefore, all kinematic functions are related. Which function will be chosen to be reconstructed depends on the questions one wants to answer.[^2] For example, one can model the luminosity distance by a linear piecewise function and obtain a statistically sound and unbiased fit using observational data. However, this study will not contribute at all to the understanding of the recent accelerated expansion since, in this case, the deceleration function is assumed zero in the entire redshift interval.[^3] Keeping the above idea in mind, the reconstruction of a kinematic function has been addressed using different methods. To understand some of these different methodologies, we divide the problem in two parts: (i) the theory underlying the observable quantities and (ii) the relation between the observables and the data, including the data probability distribution. Regarding (i), the kinematic model provides a perfect description of the observables, not taking into account any noises nor errors in the measurements. For the sake of argument, suppose that (ii) is not part of the problem, i.e., the data is perfectly known. In this case, we could use a parametric function and adjust its parameters, such that the observable (hopefully) matches the data points, or we could use the data points to determine the observable function using, for example, interpolation. In this sense, we say that the analysis is model-dependent when a given parametric function is chosen a priori and model-independent when we use the data to determine it. There are also two main procedures to treat (ii). We can assume which is the probability distribution of the data and, consequently, the only problem left is to determine the observable curve, which can be done in a model-dependent or independent way, as discussed above. In statistics texts, this is described as a parametric method. On the other hand, we can follow an even more conservative path and not impose a given probability distribution for the data. This way, known as non-parametric, also uses the data to reconstruct their own probability distribution. In the model-dependent parametric approach, one assumes a priori a specific functional form of kinematic quantities, such as the deceleration function $q(z)$, and a probability distribution of the data [@Riess2004; @Shapiro2006; @Avgoustidis2009; @Nair2012]. A feature of this strategy is that its results have potentially smaller error bars when compared to the others. After all, one is introducing a reasonable set of assumptions which can lead to biased results. A natural improvement to this is to apply a model-independent approach, where one tries to reconstruct the curve when still using the assumed distribution for the data. Among these approaches is the Principal Component Analyses (PCA), in which the kinematic function is described in terms of a set of basis functions and the data is used to determine which subset of this basis is better constrained. Then, the function is reconstructed by using this subset [@Huterer2003; @Shapiro2006; @Clarkson2010; @Ishida2011; @Ruiz2012; @Benitez-Herrera2013]. Another possibility is to use smoothing methods [@Shafieloo2006; @Shafieloo2012]. In this model-independent non-parametric case, only mild assumptions are made about the data and, usually, no assumption is made about the model. This allows a direct translation of the data into a kinematic curve. Still in this context, we also have the Gaussian Process (GP), in which one chooses to model directly the probability distribution of the kinematic function itself [@Holsclaw2010; @Seikel2012]. For a more complete list of non-parametric methods see [@Montiel2014] and references therein. Recovering both the probability distribution of the data and a reconstructed kinematic function require a large amount of data and, in practice, the current observational cosmology did not seem to have reached this level yet. This is evinced by the results obtained so far in the literature [@Ruiz2012; @Shafieloo2012; @Seikel2012; @Montiel2014]. Regarding the data, there is a good perspective to increasingly improve their probabilistic descriptions, since different error sources, such as the systematic ones, are being included in their modeling (e.g., [@Conley2011; @Betoule2014] ). This presents an additional challenge to the non-parametric methods, as they must incorporate all the error sources in their reconstruction. Even in a model-independent and non-parametric approach, the estimated curves are not free from assumptions. Each method has some internal choices of parameters. Currently in the literature, these parameters are obtained using the observational data. However, as we usually have only one set of data, doing so will calibrate the method for this one particular realization of the data. In this case, there is no way to know if this calibration provides the best balance between bias and variance. This difficulty can be circumvented using different realizations of the **same** data set. For a given calibration, i.e., for a given choice of the internal parameters, the method is applied to a large number of simulations obtaining the bias and variance for this calibration. Then, repeating this process for different calibrations one can find the best suited one for the chosen data set. In other words, the internal parameters in these reconstructions must not be related to one particular realization of the data, but to their probability distribution. This idea can be extended to the study of the statistical properties of the data. For example, in [@Montiel2014], among other results, the authors apply a bootstrap-like procedure to calibrate the smoothing parameter applied to the data. This kind of analysis can provide a insightful information about the statistical properties of the data when little is known about their relationships. In this work, we use the current available observational data for small redshifts ($z \lesssim 2.3$) and their likelihoods, namely, the Sloan Digital Sky Survey-II and Supernova Legacy Survey 3 years (SDSS-II/SNLS3) combined Joint Light-curve Analysis (JLA) SNe Ia sample [@Betoule2014], baryon acoustic oscillation (BAO) data [@Beutler2011; @Padmanabhan2012; @Kazin2014; @Ross2014] and $H(z)$ measurements [@Stern2010; @Riess2011; @Moresco2012; @Busca2013]. Currently, there is not enough data to perform a full model-independent and non-parametric reconstruction of the recent evolution of the universe. Therefore, we use the usual likelihood for these data, but, to be conservative, we reconstruct $q(z)$ along with some astrophysical parameters of SN Ia, the drag scale (present in the BAO likelihood) and the Hubble parameter $H_0$. Besides the above data, there is also a wealth of data concerning the large scale structure connected to the perturbations around a FLRW metric, such as the temperature fluctuations of the cosmic microwave background (CMB) [@Hinshaw2013; @Planck2015]. Since we assume no dynamic model, we would have to propose a kinematic one for the perturbations. Such model is not feasible as it would require a set of functions of both time and space. In principle, one could also directly use derived observables, as, e.g., the CMB distance priors [@Komatsu2011], to fit the background model. However, these parameters are obtained in the context of a specific model, e.g., $\Lambda$CDM. Thus this model would be indirectly reintroduced in the results. For this reason, we choose not to use these data and focus on the distance-like measurements. We reconstruct $q(z)$ using a model-independent parametric approach, which consists in describing $q(z)$ by a cubic spline. The choice of the deceleration function comes from the fact that one can use such function to directly test the energy conditions [@Lima2008; @Lima2008a]. Likewise, the deceleration function is related to the underlying dynamics of the metric, since it is a simple combination of first and second derivatives of the scale factor. Therefore, their knowledge is necessary to constrain models which chooses a different dynamic for the gravitation sector (Alves et al. to be submitted). In addition, the spline method allows us to vary the complexity of the functional form as a function of the knots number, for example. Here we introduce a novel method to continuously vary the complexity of the reconstructed function. As a result of this analysis, we also obtain the reconstruction of those SN Ia parameters and the drag scale. The paper is organized as follows. In section \[sec:def\_q\] we review basic concepts, such as $q(z)$, and the few assumptions made here. In section \[sec:review\], we discuss some approaches used in the literature and some of their respective drawbacks. In section \[sec:reconst\] we present our reconstruction method and the tools to vary the function complexity and to select the best one. Following, section \[sec:data\], we specify the observational data sets used in our study and their respective likelihood functions. We then perform a Monte Carlo analysis in different scenarios calibrating the method (section \[sec:meth\_valid\]). Finally, we use this calibration to reconstruct the deceleration function using the observational data, in section \[sec:real\_reconst\], and we summarize our conclusions in section \[sec:conclusions\]. Deceleration function $q(z)$ {#sec:def_q} ============================ Measurements of the CMB [@Hinshaw2013; @Planck2015] show that the universe is nearly homogeneous and isotropic at large scales. Therefore, we assume that the universe follows the cosmological principle, restricting the metric to the FLRW metric, $$\label{eq:rw_metric} ds^2 = -c^2\,dt^2 + a^2 (t) \left [ dr^2 + S_k^2(r) (d\theta^2 + \sin^2 \theta d\phi^2) \right ]\;,$$ where $S_k(r)=(r\,$, $\sin(r)$, $\sinh(r))$ for flat, spherical and hyperbolic spatial section ($k=0, 1, -1$), respectively, $c$ is the speed of light and $a(t)$ is the cosmological scale factor. In this case, the expansion history of the universe can be defined knowing $a(t)$ and $k$. In practice, we do not measure $a(t)$ directly, but related quantities such as the distances to astronomical objects. Considering a null trajectory of photons emitted by a galaxy traveling along the radial direction to us, we have that $$\label{eq:com_dist1} r = c \int_{t_e}^{t_0} \frac{{\mathrm{d}}t^\prime}{a(t^\prime)},$$ where $t_e$ and $t_0$ are the emitted and observed times, respectively. Expanding the scale factor to second order around $t_0$, gives $$a(t) = a_0 + H_0 (t - t_0) - \frac{q_0 H_0^2}{2} (t - t_0)^2,$$ where $a_0$ is the scale factor today and $H_0$ and $q_0$ are, respectively, the Hubble and deceleration functions at $t_0$, $$H_0 = \left(\frac{ \dot{a}}{a} \right)_{t_0} \quad \text{and} \quad q_0 \equiv - \left(\frac{\ddot{a}a}{\dot{a}^2}\right)_{t_0}.$$ Rewriting the comoving distance $D_c$ and the deceleration function in terms of the redshift, $1 + z = a_0/a$, we obtain $$\label{eq:com_dist} D_c(z) = a_0 r = \frac{c}{H_0} \int_0^z \frac{{\mathrm{d}}z^\prime}{E(z^\prime)}$$ and $$\label{eq:qz} q(z) = \frac{(1 + z)}{H(z)}\frac{\mathrm{d}H(z)}{\mathrm{d}z} - 1,$$ whose integral solution is $$\label{eq:Ez} E(z) = \exp{\int_0^z \frac{1+q(z^\prime)}{1+z^\prime}{\mathrm{d}}z^\prime},$$ and where $E(z)$ is the normalized Hubble function, $E(z) = H(z)/H_0$. Equations  – evince that we shall reconstruct $q(z)$ in order to access local information about the accelerated/decelerated phase [@Lima2008]. Besides, assuming that $q(z)$ is continuous guarantees that $E(z)$ and $D_c(z)$ are also continuous functions and at least once and twice differentiable, respectively, although the opposite is not true. Review of other approaches {#sec:review} ========================== In this section we briefly present some of the most used methods in reconstructing the expansion history of the universe. We discuss some intrinsic issues of these methodologies, which motivated us to develop a novel approach (presented in section \[sec:reconst\]). Parametric models {#sec:PM} ----------------- In general the Taylor series approaches have two related problems. The first is the convergence radius of the Taylor expansion itself, which can only be estimated since the real scale factor is naturally unknown. On the other hand, as we are fitting the coefficients of such expansion, the functional form for the scale factor (or distance) can be interpreted as a simple polynomial interpolation. In this sense, changing the time parameter can be useful [@Cattoeen2008] and provide a better polynomial interpolation. However, this leads us to the second problem, the Runge’s phenomenon. That is, after a given order, higher order polynomials provide worse and worse approximations. Therefore, when using the Taylor expansion one should stay on a small convergence radius, which would restrain the analysis to a very small but unknown redshift or use a polynomial interpolation keeping in mind its caveats. More generally, the problem of finding a good kinematic description of the expansion history can be addressed using a parametric method. In this context, one assumes a priori a specific functional form of a kinematic function, like the polynomial form discussed above, then proceeds by fitting its parameters using observational data. For example, in Refs. [@Riess2004; @Shapiro2006; @Avgoustidis2009; @Nair2012] they fit different functional forms of the deceleration function $q(z)$. The drawback of this method is that the choice of a functional form introduces a form-bias in the estimates if the functional form is different from the true one. As we do not know it, the result of such fit can be misleading since, even if the parameters’ error bars are small, their form-biases can still be large. A more conservative approach is to use flexible functional forms. However, this translates in using many parameters and, consequently, obtaining larger error bars. In this way, there is a natural trade-off between variance and form-bias which should be evaluated to determine the optimal reconstruction. An additional, less discussed, difficulty is the estimator-bias. The functional forms are usually fitted using a Least-Squares (LS) or Maximum-Likelihood (ML) approach. As it is well known, both approaches can provide biased estimators for the parameters (ML estimators are usually asymptotically unbiased). This means that, even if we knew the correct functional form, the fact that we have only a finite number of observations can lead to estimator-biases.[^4] Heuristically, when fitting a functional form with $n$ parameters using $m$ data points we will have $m/n$ observations per point. Hence, if the estimators are only asymptotically unbiased, then the higher the number of parameters higher the estimator-biases. Thus, the final trade-off must consider variance, form-bias and estimator-bias. Principal Component Analysis {#sec:PCA} ---------------------------- Another popular methodology is the Principal Component Analysis (PCA) [@Huterer2003; @Shapiro2006; @Clarkson2010; @Ishida2011; @Ruiz2012]. In this approach the kinematic function is described by a flexible parametrization (e.g., in [@Huterer2003] they express the DE EoS as a constant piecewise function within $N=50$ bins). From the estimated covariance matrix of these parameters, the eigenvectors and eigenvalues are deduced. Say, for example, that we choose the first 5 eigenvectors, whose eigenvalues correspond to the smallest variance terms. This means that the original division in 50 bins is being described by a 5-dimensional parametrization. The eigenvectors, whose eigenvalues correspond to the smallest variance terms, provide the parametrization which is better constrained by the data. The appeal of this method is that it provides a straightforward way to determine the curves better constrained by the data. The choice of how many eigenvectors should be used can be answered by looking the variance-bias trade-off [@Huterer2003]. On the other hand, the method as described above is subject to a potential drawback. The relationship between different kinematic functions are given in terms of their integral in time, for example, the relation between the deceleration and the Hubble functions is given by Eq.  while the distance is another integral of the Hubble function \[Eq. \]. Note that, to calculate the distance to a given SNIa at redshift $z_i$, the deceleration function is integrated twice between $[0, z_i]$. Therefore, all bins in this interval contribute to the final value of the distance. This leads to the following problem, the first bins contribute to the value of the distance for almost all supernovae, while the final bins affect only few objects. Besides, since the initial bins always contribute to the value of the distance at higher redshifts, they will be naturally correlated to them, any modification in their value will have to be compensated by the other. These facts have the following consequence, the best constrained modes will be strongly connected to the first bins while the worst will be related mostly to the last bins. Therefore, when one chooses to use only a few (better constrained) eigenvectors, the final parametrization will provide almost no power in the last bins. This is natural since it is equivalent to choose the coefficients of the last eigenvectors to be fixed at zero. This problem can be seen in [@Huterer2003], where the issue persists even when the authors consider a forecast with 3000 SNe Ia uniformly distributed in redshift. This problem was also noted in [@Ishida2011]. In the latter, they realized that the first modes have this deficiency and proposed the addition of a new parameter to circumvent this problem. Similarly to what is commonly used in the PCA approach, the authors in [@Crittenden2012; @Zhao2012] reconstructed the EoS function describing it as a set of bins (discontinuous piecewise constant function of the scale factor). Instead of limiting the estimated curve variance by using a small set of eigenvectors (principal components), they introduced the correlated prior, in an approach similar to the GP, which treats the reconstructed curve as a set of random variables emerging from a multidimensional Gaussian distribution. This prior introduces a correlation between the bins controlled by the correlation length and prior strength. Once the prior is calibrated, the problem described in the above paragraph can be systematically resolved. The discussion above elucidates an important characteristic of the PCA, it is dependent on the initial description of the function. In many works the simple step function (or bins) were used. If another basis functions were used, with different behavior at large redshifts, we would end up with a similar problem, i.e., the large redshift behavior of the best constrained basis function would dominate at large redshifts, providing biased estimates at these points. We conclude that, in the specific context of constraining a function by its integral, the PCA approach has this potential problem. Smoothing methods {#sec:SM} ----------------- As we mentioned in section \[sec:introduction\], these approaches are model-independent and non-parametric. Therefore, they suit the study cases where the data probability distribution is unknown. However, as pointed out by Montiel et al. [@Montiel2014], these methods find difficulties due to the limited amount of observational data or even due to some features of the methods themselves, such as the size of the smoothing parameter and the assumptions on priors or fiducial models. Besides, since no assumption is made about the data distribution, one cannot use resampling to perform a self-validation, but only bootstrap like procedures, e.g., jackknife, which usually requires large samples. Finally, the fact that this method makes such minimal assumptions is not necessarily useful. As one can only reconstruct the direct variable associated to the observable, i.e., cosmological distances and $H(z)$, any inference about the derived kinematic quantities is limited since it is highly dependent on the smoothing technique as, e.g., smoothing linear splines has always zero second derivative, and top-hat moving average filters are non-continuous. Therefore, any analysis about kinematic quantities, different of the reconstructed one, requires new assumptions [@Daly2004; @Shafieloo2012]. Gaussian Process {#sec:GP} ---------------- In this approach, instead of modeling a kinematic function, one chooses to model a probability distribution for the curve as a Gaussian probability distribution. This assumption dictates the data probability distribution by relating both the curve and observable probability distributions. In this sense, this approach unifies the two aspects of the model, the data distribution and the curve reconstruction. All the assumptions are comprised in the mean curve and the two-point covariance, which define the Gaussian distribution of the curve. The drawback is similar to the parametric procedure. In general, one has to assume a function to describe the mean of the GP and a two-point function to describe the variance. If the considered mean function differs from the true one, it will impose a bias in the reconstruction. See, for example, references [@Holsclaw2010; @Seikel2012], where they assume a constant mean function. Since the GP determines the observable statistical distribution, it is also necessary to include the data distribution to calculate the joint probability distribution of both curve and data. In this case, one has a model-independent but parametric method in the sense that one is assuming a particular distribution for the data. The GP validation also has to be performed for a number of realizations of the data, since the indirect curve determination through a Gaussian distribution can lead to bias in an unpredictable way. Reconstruction of $q(z)$ {#sec:reconst} ======================== Broadly speaking each strategy described in section \[sec:review\] has a better suited application. As a rule of thumb, to use less hypothesis it is necessary to have more data. Therefore, if the amount of data is limited, the generality of GP and soothing methods, for example, are restricted, leading to underdetermined problems. Besides that, including natural hypothesis can also be considerably difficult in those non-parametric studies, e.g., after the determination of the cosmological distances through a smoothing method it is necessary to add new assumptions to describe the Hubble function. Therefore, in this work we adopted a model-independent parametric approach. Nonetheless, to be conservative we reconstruct the kinematic curve along with all the phenomenological parameters related to the modeling of each data set. Doing so, we minimize the assumptions on the data distribution bypassing any bias which could result from it. In a model-independent technique we need to use a set of functions to perform the reconstruction. To avoid the problems described above on the PCA approach, we use a cubic spline to reconstruct our observable, as described in section \[sec:spline\]. One advantage of the cubic spline is that it is continuous and twice differentiable on every knot, thus, all the parameters (the value of the function on each knot) are related through these conditions.[^5] Another point on the model-independent approach is how to deal with the bias. In the PCA approach one uses the particular set of data to determine which basis functions form the minimal set. Depending on the data distribution, this tends to favor the intervals where the sample is denser, which usually creates biases on the other regions. To avoid this issue, we impose a global penalization (see section \[sec:penalty\]) on the curvature of the curve. Applying the same penalization in each interval, we evade the localization problem described before. Piecewise deceleration function {#sec:spline} ------------------------------- In this work, we avoid making arbitrary choices of the $q(z)$ form, and, consequently, *a priori* restricting it to specific functional forms, by approximating $q(z)$ by a piecewise third-order polynomial function, i.e., a cubic spline. The first step to build an estimator of $q(z)$ \[denoted as $\hat{q}(z)$\] is to specify the redshift interval (domain $\mathrm{D}$) in which the function is defined. This interval is $\mathrm{D} = [z_{min}, z_{max}]$, where $z_{min}$ ans $z_{max}$ are the minimum and maximum redshifts of the used data. The next step is to choose the partition of the domain $\mathrm{D}$ into $n$ sub-intervals in which we define $\hat{q}(z)$ as a cubic polynomial function, namely, $$\hat{q}(z) = \left\{ \begin{array} {llllll} p_0(z) = a_0(z-z_0)^3 + b_0(z-z_0)^2 + c_0(z-z_0) + d_0 \quad & {z \in [z_0,z_1)}\\ p_1(z) = a_1(z-z_1)^3 + b_1(z-z_1)^2 + c_1(z-z_1) + d_1 & {z \in [z_1,z_2)}\\ \qquad\qquad\qquad\qquad \vdots & \quad \vdots\\ p_{n-1}(z) = a_{n-1}(z-z_{n-1})^3 + b_{n-1}(z-z_{n-1})^2 + c_{n-1}(z-z_{n-1}) + d_{n-1} & z \in [z_{n-1}, z_n], \end{array} \right.$$ where $z_0 = z_{min}$, $z_n = z_{max}$ and $p_i$ is the cubic polynomial defined in the $i$-th sub-interval. Note that each polynomial $p_i(z)$ in the segment $[z_i,z_{i+1})$ depends on 4 parameters ($a_i$, $b_i$, $c_i$ and $d_i$) and, consequently, we would need to estimate $4n$ parameters to define $\hat{q}(z)$ in the whole domain $\mathrm{D}$. However, imposing the following continuity conditions, $$\begin{aligned} p_{i+1}(z_{i+1}) &=& p_i(z_{i+1}), \\ p^{\prime}_{i+1}(z_{i+1}) &=& p^{\prime}_i(z_{i+1}), \\ \quad p^{\prime\prime}_{i+1}(z_{i+1}) &=& p^{\prime\prime}_i(z_{i+1}),\end{aligned}$$ on the $n-1$ internal knots $i \in (1,n-1)$, where ${}^\prime$ denotes the derivative with respect to $z$, and the two not-a-knot boundary conditions $$\hat{q}^{\prime\prime\prime}_{0}(z_{1}) = \hat{q}^{\prime\prime\prime}_1(z_{1}) \quad \text{and} \quad \hat{q}^{\prime\prime\prime}_{n-2}(z_{n-1}) = \hat{q}^{\prime\prime\prime}_{n-1}(z_{n-1}),$$ we end up with only $n + 1$ parameters to determine $\vec{Q} = \{d_0, ..., d_{n}\}$, i.e., the values of $\hat{q}(z)$ at each knot. As we have a one to one relation between the function in each knot $\hat{q}_i \equiv \hat{q}(z_i)$ and the parameter $d_i$ of each polynomial, from now on we rename the set as $\vec{Q} = \{\hat{q}_i\}$ for the sake of notational simplicity. Thus, using this cubic spline approximation, we fit the vector $\vec{Q}$ in order to obtain the estimates of the deceleration function (see description in section \[sec:meth\_valid\]). Any interpolation method introduces an error source limiting the set of functions able to be reconstructed. The interpolation error for a cubic spline has an upper bound proportional to both the largest distance between nearby knots to the fourth power and the fourth derivative of the function with respect to $z$ (for details see [@Boor2001]). At first sight, larger the number of knots, smaller the interpolation error. But in practice, the number of knots is limited by the increasing number of parameters. Besides, there is also the overfitting that rises when fitting the parameters using data points. As a rule of thumb, one should choose $n$ and, consequently, the intervals between knots, such that the estimated function is expected to be well approximated by a cubic polynomial in these intervals. One can test the choice of $n = n_1$ applying the reconstruction for another one, e.g., $n = n_1 + 1$, and probing the results for any significant improvements on the fit. Notwithstanding the interpolation error, another important source of uncertainty comes from the statistical errors (bias/overfitting), as we will discuss in the next section. Finally, the use of cubic splines represents a large advantage in comparison to the step functions frequently used in the literature. For a constant piecewise function, the interpolation error is bounded by the first derivative times the largest distance between nearby knots. As a result, the number of knots (and, consequently, parameters) necessary to reconstruct functions with the same interpolation error bound is much larger in a binned approach. Function complexity {#sec:penalty} ------------------- Assuming a cubic spline to approximate $q(z)$, we are able to address both model-dependent and model-independent parametric methods by varying the number of knots. The simplest case, $n = 4$, is equivalent to consider that $q(z)$ is a third-order polynomial. On the other hand, we approach a model-independent case increasing $n$. The complexity of the function $\hat{q}(z)$ is, in principle, parameterized by the number of knots, as the number of knots goes to infinity any interpolation error drops to zero. Nonetheless, the choice of the domain partition is rather arbitrary and, at first, one would have to test different options in order to achieve, for example, a “model-independent limit” trying to minimize the over-fitting error. Another difficulty inherent of this approach is that the number of knots is a discrete variable and, as such, it is difficult to include it as another parameter in the analysis. Instead of varying the number of knots by adding/removing actual knots to the function representation, we can fix the number of knots in some large value and penalize independent values of the parameters. For example, given that the parameters are just the values of the function at each knot, a penalty factor as a increasing function of $\vert\hat{q}_i - \hat{q}_{i+1}\vert$ will correlate all parameters $\hat{q}_i$. The correlation will be proportional to the weight of the penalization in the analysis, higher the weight more correlated are the parameters. In the strong correlation limit, all parameters would be equal, i.e., $\hat{q}_i = \hat{q}_{i+1}$. In this last case, even with a large number of knots, the effective number of degrees of freedom would be one. In short, varying the weight of the penalization, we can vary the effective number of degrees of freedom, circumventing the difficulties described above.[^6] In practice, we include a set of penalty factors $P_i(\sigma_i)$ in our estimator. The initial likelihood is $L(\vec{D}, \vec{\theta})$, where the vector $\vec{D}$ represents the data set and the vector $\vec{\theta}$ all the parameters, including the spline parameters $\hat{q}_i$ and other parameters as described in sections \[sec:mc\_sneia\] and \[sec:mc\_alldata\]. To obtain the parameter estimators, we add to the likelihood $L(\vec{D}, \vec{\theta})$ the penalization $P_i(\sigma_i)$ defining the penalized likelihood $$\label{eq:likel_penalty} -2 \ln\left[L_P (\vec{D}, \vec{\theta})\right] \equiv -2 \ln\left[L\left(\vec{D}, \vec{\theta}\right)\right] + \sum_{i = 2}^{n-1} P_i(\sigma_i),$$ where the penalty factor is given by $$\label{eq:penalty_fac} P_i(\sigma_i) = \left(\frac{{\bar{\hat{q}}_i - \hat{q}_i}}{\sigma_i}\right)^2,\qquad \bar{\hat{q}}_i = \frac{(\hat{q}_{i-1} + \hat{q}_{i+1})}{2}, \qquad \sigma_i = \sigma_{abs} + \bar{\hat{q}}_i{\sigma_{\text{rel}}},$$ and we use $\sigma_{abs} = 10^{-5}$.[^7] The penalization factor is schematically illustrated in figure \[fig:basis\] showing the positions of $\hat{q}_1$ and $\bar{\hat{q}}_1$. We control the complexity of $\hat{q}(z)$ by varying the value of the relative error ${\sigma_{\text{rel}}}$. For example, we are able to recover a high complexity function, in particular, a full $n+1$ knots spline for large ${\sigma_{\text{rel}}}$, and a straight line in the entire redshift interval when ${\sigma_{\text{rel}}}$ goes to zero. The former has many coefficients and can tend to fit the data noise, i.e., it is over-fitting dominated. The second naturally sharpen the constraints on $\{\hat{q}_i\}$, but they can be biased if the assumed functional form significantly differs from the true one. It is worth mentioning that this penalty factor allows us to explore a wide range of functional forms, since its simplest case is a linear function. Meanwhile, without using the penalty factor, the simplest model would be a third-order polynomial. Finally, we emphasize that, in principle, one could use a large set of knots while constraining the allowed shape with the penalty factor. The restriction will be practical, the computational cost increases with the number of knots. Therefore, one should find the best balance between computational cost and flexibility of the method. ![\[fig:basis\] A descriptive example of the penalization on a cubic spline. For each three knots, the penalization is proportional to the distance between the straight line connecting the first and third knots, and the interpolation function.](penal_example.pdf) Bias-variance trade-off {#sec:bv_trade} ----------------------- Giving the penalty function which allows us to explore different complexity forms of $\hat{q}(z)$, we now have to introduce a criterion to determine the best ${\sigma_{\text{rel}}}$ value. We want to find the scenario where the combined error due to the biases (estimator- and form-bias) and the over-fitting error is minimized. For this, we decompose the error into bias and variance components as described below [@Bishop1996; @Keijzer2000; @Wasserman2001]. We create a controlled environment introducing a fidicual deceleration function $q^{{\text{fid}}}(z)$, which is determined by a given set of values for the parameters $\vec{\theta}^{{\text{fid}}}$. The idea is to use this function to generate a new data set $\vec{D}$, which is possible since we know the likelihood of the data $L(\vec{D},\vec{\theta})$. We define the ML estimators using the penalized likelihood, then, given a simulated sample $\vec{D}^{(l)}$, we obtain the estimates $\tilde{\vec{\theta}}(\vec{D}^{(l)})$ computing $$\frac{\partial L_P}{\partial \vec{\theta}}\left(\vec{D}^{(l)}, \tilde{\vec{\theta}}\right) = 0.$$ This provides an implicit definition of the function $\vec{\theta}^{(l)} \equiv \tilde{\vec{\theta}}(\vec{D}^{(l)})$. In principle, we could calculate the bias in the estimator integrating the function $\vec{\theta}^{(l)}$, i.e., $$\left\langle\tilde{\vec{\theta}}\right\rangle = \int{\mathrm{d}}\vec{D} \, \vec{\theta}^{(l)}L_P\left(\vec{D},\vec{\theta}^{{\text{fid}}}\right).$$ However, such integration is computationally unfeasible. It is a $N$ dimensional integration, where $N$ is the number of data points, and the function $\vec{\theta}^{(l)}$ is usually only determined numerically by maximizing the penalized likelihood. Instead, we use the Monte Carlo (MC) approach to deal with such integrals.[^8] Since we know the probability distribution of the data, we can create a new realization of the data (mock catalog) $\vec{D}^{(l)}$ by resampling,[^9] i.e., using a (pseudo)random number generator to create a new data set. Given a large enough number of realizations $m$, we can approximate the expected value of a function of the data as $$\left\langle f\left(\vec{D}\right) \right\rangle \simeq \frac{1}{m} \sum_{l=1}^{m} f\left(\vec{D}^{(l)}\right).$$ Using these tools, we introduce the mean squared error (MSE) of approximating a fiducial function $q^{{\text{fid}}}(z)$ by $\hat{q}(z; {\sigma_{\text{rel}}})$. For a fixed ${\sigma_{\text{rel}}}$, it is $$\text{MSE} = \left\langle\left[\hat{q}^{(l)}(z; {\sigma_{\text{rel}}}) - q^{{\text{fid}}}(z)\right]^2\right\rangle \simeq \frac{1}{m} \sum_{l=1}^{m} \left[\hat{q}^{(l)}(z; {\sigma_{\text{rel}}}) - q^{{\text{fid}}}(z)\right]^2.$$ Given the estimate of the expected value, $$\label{eq:expec_val} \langle \hat{q}(z, {\sigma_{\text{rel}}}) \rangle \simeq \frac{1}{m} \sum_{l=1}^{m} \hat{q}^{(l)}(z; {\sigma_{\text{rel}}}),$$ we have that (omitting ${\sigma_{\text{rel}}}$ for simplicity) $$\begin{aligned} \left[\hat{q}^{(l)}(z; {\sigma_{\text{rel}}}) - q^{{\text{fid}}}(z)\right]^2 &= \left[\hat{q}^{(l)}(z) - \langle \hat{q}(z) \rangle + \langle \hat{q}(z) \rangle - q^{{\text{fid}}}(z)\right]^2 \nonumber \\ &= \left[\hat{q}^{(l)}(z) - \langle \hat{q}(z) \rangle \right]^2 + \left[\langle \hat{q}(z) \rangle - q^{{\text{fid}}}(z)\right]^2 \nonumber \\ & + 2 \left[\hat{q}^{(l)}(z) - \langle \hat{q}(z) \rangle\right]\left[\langle \hat{q}(z) \rangle - q^{{\text{fid}}}(z)\right],\end{aligned}$$ and, therefore, $$\label{eq:mse_1} \text{MSE} \simeq \left(\langle \hat{q}(z) \rangle - q^{{\text{fid}}}(z)\right)^2 + \frac{1}{m} \sum_{l=1}^{m} \left(\hat{q}^{(l)}(z) - \langle \hat{q}(z) \rangle \right)^2,$$ since $$\frac{1}{m} \sum_{l=1}^{m} \left(\hat{q}^{(l)}(z) - \langle \hat{q}(z) \rangle\right)\left(\langle \hat{q}(z) \rangle - q^{{\text{fid}}}(z)\right) \simeq 0.$$ The first term of eq.  is the squared bias $b_{\hat{q}}(z)^2$ and the second is the variance. Since this variance estimator is biased, in this work, we evaluate the bias-variance trade-off computing $$\begin{aligned} \label{eq:mse_2} \text{MSE} &\simeq \left(\langle \hat{q}(z) \rangle - q^{{\text{fid}}}(z)\right)^2 + \frac{1}{m -1} \sum_{l=1}^{m} \left(\hat{q}^{(l)}(z) - \langle \hat{q}(z) \rangle\right)^2 \nonumber \\ &\equiv b_{\hat{q}}(z; {\sigma_{\text{rel}}})^2 + \text{Var}\left(\hat{q}(z; {\sigma_{\text{rel}}})\right).\end{aligned}$$ Both the squared bias and the variance have the same weight in the above expression. Nonetheless, when applying the reconstruction for real data, we usually do not have access to an estimate of the bias. Therefore, minimizing MSE can potentially lead us to a methodology with a large bias (as we will see in sections \[sec:mc\_sneia\] and \[sec:mc\_alldata\]). To avoid this, in what follows we will minimize the MSE satisfying the constraint $$\label{eq:m_b} \frac{b_{\hat{q}}(z; {\sigma_{\text{rel}}})}{\sigma(\hat{q}(z; {\sigma_{\text{rel}}}))} \leq m_b, \qquad \sigma(\hat{q}(z; {\sigma_{\text{rel}}})) \equiv \sqrt{\text{Var}(\hat{q}(z; {\sigma_{\text{rel}}}))},$$ where $m_b$ controls the maximum ratio between bias and variance. The variance of the reconstructed curve $\hat{q}(z; {\sigma_{\text{rel}}})$ can be written in terms of the covariance of the spline parameters $\text{Cov}\left(\hat{q}_i, \hat{q}_j\right)$. In turn this covariance can be estimated using the unbiased covariance estimator $$\label{eq:cov_q} \text{Cov}\left(\hat{q}_i, \hat{q}_j\right) = \frac{1}{m - 1} \sum_{l = 1}^{m} \left(\hat{q}_{i}^{(l)} - \langle \hat{q}_i \rangle\right) \left(\hat{q}_{j}^{(l)} - \langle \hat{q}_j\rangle \right),$$ where $\hat{q}_i^{(l)}$ is the best-fitting value of the $i$-th spline parameter using the $l$-th mock catalog. Observational data {#sec:data} ================== As $q(z)$ is not a direct observable, we need to use other quantities to access $\{\hat{q}_i\}$. In this section, we present the samples of type Ia SNe, BAO and $H(z)$ measurements, and also their respective likelihood functions that we utilize to recover the deceleration function $q(z)$. Type Ia supernova data ---------------------- We use the JLA sample [@Betoule2014] of 740 SNe Ia, whose likelihood is $$\label{eq:lik_snia} -2\ln(L_{SNIa}) = \Delta{}\vec{m}^T\mathsf{C}_{SNIa}^{-1}\Delta{}\vec{m},$$ where the data covariance is a combination of the systematic and statistical errors $\mathsf{C}_{SNIa} = \mathsf{C}_{sys} + \mathsf{C}_{stat}(\alpha,\beta)$, and $$\begin{aligned} \label{eq:snia_mb} \Delta{}m_i &=& m_{Bi} - m_{Bi}^{\text{th}}\\ &=& m_{Bi} -5\log_{10}(\mathcal{D}_L(z^{\text{hel}}_i, z^{{\text{cmb}}}_i)) + \alpha X_i - \beta \mathcal{C}_i - M_{h_i} + 5\log_{10}(c/H_0) - 25. \nonumber\end{aligned}$$ $m_{Bi}$ is the rest-frame peak B-band magnitude of the $i$-th SN Ia, and $z^{\text{hel}}_i$ and $z^{\text{cmb}}_i$ are its heliocentric and CMB frame redshits, respectively. The SN Ia astrophysical model contains four parameters $(\alpha, \beta, M_1, M_2)$, where the first two are related to the stretch-luminosity and colour-luminosity, respectively, and $M_1$ and $M_2$ are absolute magnitudes. The luminosity distance [@Davis2011] is $$\begin{aligned} \label{eq:dis_SN} D_L(z^{\text{hel}}, z^{\text{cmb}}) &= \frac{c}{H_0} \mathcal{D}_L(z^{\text{hel}}, z^{\text{cmb}}) \nonumber \\ &= (1 + z^{\text{hel}}) D_M(z^{\text{cmb}}),\end{aligned}$$ where the transverse comoving distance in a flat spatial sections universe is $D_M(z) = D_c(z)$. Baryon acoustic oscillation --------------------------- The peak position of the angular correlation function of the matter density can be measured by the distance ratio $D_V (z) / r_s(z_d)$ (see [@Thepsuriya2014; @Aubourg2014] and references therein). The volume-averaged-distance for perturbations along and orthogonal to the line of sight is defined as [@Eisenstein2005] $$\label{eq:bao_dv} D_V(z) \equiv \left[ D_M(z)^2 \frac{cz}{H(z)}\right]^{1/3}.$$ The sound horizon $r_s(z_d)$ at the drag redshift $z_d$ (i.e., epoch at which baryons were released from photons) is $$\label{eq:bao_rd} r_d \equiv r_s(z_d) = \frac{1}{H_0}\int_{z_d}^\infty dz \frac{c_s(z)}{E(z)},$$ where $c_s(z)$ is the sound wave speed in the photon-baryon fluid. In this work we use 6 BAO data points as described in table \[tab:bao\] in appendix \[app:sampling\]. The first is measured by Beutler et al. [@Beutler2011] using galaxies from the 6dF Galaxy Survey. Padmanabhan et al. [@Padmanabhan2012] reported an improved data obtained with the reconstruction method [@Eisenstein2007] using Luminous Red Galaxy sample from SDSS Data Release 7 (DR7). Kazin et al. [@Kazin2014] give three points computing the power spectrum and correlation function of galaxies from the WiggleZ Survey in three correlated redshift bins. The last data is obtained by Ross et al. [@Ross2014] which used galaxies from SDSS DR7 with $z < 0.2$. The BAO likelihood is $$\label{eq:lik_bao} -2 \ln L_{\text{BAO}} = \left(\vec{b}^{\text{th}} - \vec{b} \right)^T\mathsf{C}_{\text{BAO}}^{-1}\left( \vec{b}^{\text{th}} -\vec{b} \right) - 2 \ln L_{\text{Ross}},$$ where $\vec{b}^{\text{th}}$ is the observable vector calculated using the theoretical model, i.e., the components are given by $b^{\text{th}}_i = D_V(z_i)/r_d$ calculated at each redshift (second column of table \[tab:bao\]). The vector $\vec{b}$ represent the observed version of these quantities and its components are provided by the third column of table \[tab:bao\]. The matrix $\mathsf{C}^{-1}_{\text{BAO}}$ is the inverse covariance matrix appearing in the BAO likelihood (see table \[tab:bao\]). Finally, Ross et al. [@Ross2014] pointed out that their data should be used considering their estimate of the likelihood distribution \[which we called $L_{\text{Ross}}$ in eq. \], since, in this case, the Gaussian distribution is not a good approximation. References [@Beutler2011] and [@Padmanabhan2012] used the Eisenstein & Hu [@Eisenstein1998] (EH98) fitting function to compute $r_d^{{\text{fid}}}$ while [@Kazin2014; @Ross2014] used CAMB [@Lewis2000]. As mentioned in Ref. [@Kazin2014], the difference between $r^{{\text{fid}}}_{d,\text{EH98}}$ and $r^{{\text{fid}}}_{d, \text{CAMB}}$ is of order of $3\%$. So due to the current error magnitude of these data, this difference is relevant and, hence, we have to re-scale the data such that all measurements refer to the same method. In particular, we multiply Beutler and Padamanabhan’s data by $r_{d, \text{EH98}}^{{\text{fid}}} / r_{d, \text{CAMB}}^{{\text{fid}}} = 1.027$ and 1.025, respectively. The BAO observable depends on the kinematic model through $D_V(z)$ and also requires the $r_d$ value. However, to calculate $r_d$ theoretically, we would need to extend the kinematic model to high redshifts and to compute the decoupling redshift $z_d$. We avoid this making $r_d$ a free parameter in the analysis. Therefore, throughout this work we fit $r_d$ along with the other parameters. Hubble function --------------- We work with 21 measurements of $H(z)$: 11 are provided by Stern et al. [@Stern2010] in the redshift range $0.1 \leq z \leq 1.75$ obtained from the spectra of red-enveloped galaxies; Riess et al. [@Riess2011] estimated the Hubble constant $H_0$ using optical and infrared observations of over 600 Cepheid variables; other 8 measurements are comprised within $0.1791 \leq z \leq 1.037$ as presented in Moresco et al. [@Moresco2012]. These last were obtained from the differential spectroscopic evolution of early-type galaxies with respect to $z$. In particular, we use their $H(z)$ values computed assuming the BC03 model for the differential evolution. The last datum is an estimate of $H(z)r_d / (1+z)$ at $z = 2.3$, and it is derived from the transmitted flux fraction in the Ly$\alpha$ forest of over 48,000 quasars combined with CMB observations as showed by Busca et al. [@Busca2013]. Assuming that $H(z)$ follows a Gaussian distribution and given that the error of each measurement is independent, we have that the likelihood is $$\label{eq:lik_hz} -2 \ln L_{H} = \sum_{i = 1}^{20} \frac{\left(H(z_i) - H^{\text{obs}}_i\right)^2}{\sigma_i^2} + \frac{\left.\left[\frac{H(z)r_d}{1+z} - H_{r}^{\text{obs}}\right]^2\right\vert_{z=z_{21}}}{\sigma_{21}^2},$$ where $H_i^{\text{obs}}$ and $\sigma_i$ are the data points and their respective errors, $H(z) = H_0 E(z)$ and $E(z)$ is given by eq. . The observed values of the $20$ measures of the Hubble function are depicted in table \[tab:Hz\] in appendix \[app:sampling\]. The last observable $H^{\text{obs}}_r$ is described in the footnote of the same table. Methodology and Validation {#sec:meth_valid} ========================== The observational data set represent just one realization of their underlying data probability distributions. So by construction, we cannot access the bias of an estimated function, e.g., $\hat{q}(z)$, by fitting it using this data set. As we discussed in section \[sec:bv\_trade\], the $\hat{q}(z)$ bias is inferred knowing both the expected value and the true value of $q(z)$ \[see eq. \]. This last is the missing piece to compute the bias-variance trade-off. However, one can indirectly infer the bias without knowing the underlying $q(z)$.[^10] First, one fits the model obtaining the best fit for the real data. Then, using this best fit as the fiducial model, several mock catalogs are generated and, following the description in section \[sec:bv\_trade\], one calculates the bias and, consequently, the bias-variance trade-off. The problem with this procedure is that one must choose the parameters of the reconstruction method to compute the best fit. In particular, we have to fix the number of knots and the complexity of the function $\hat{q}(z)$, through the penalization parameter ${\sigma_{\text{rel}}}$. Thus, if the initial best fit for a given ${\sigma_{\text{rel}}}$ is already significantly biased, so will be the bias-variance trade-off analysis. In other words, performing a MC analysis of the bias-variance trade-off around the best fit does not take into account the variance of the estimated curve. Therefore, a better approach consists in using not only the best fit curve but a set of curves inside some statistical significance, i.e., curves whose parameters are inside some confidence interval of the best fit. This means that the procedure should be capable of reconstructing not only the best fit curve but also every other curves inside some significance level. There is a high computational cost to study the bias-variance trade-off for a given fiducial curve. It is necessary to resample from the model $m$ times and to find the best fit for each realization. The whole calculation must be performed for different values of ${\sigma_{\text{rel}}}$ until the best bias-variance trade-off is attained. In this work, we address the problem by performing the MC analyses for three different fiducial curves and seven ${\sigma_{\text{rel}}}$ values (see sections \[sec:mc\_sneia\] and \[sec:mc\_alldata\]). The functional forms of these fiducial models were purposely defined to have quite different features as shown in figure \[fig:fiducial\_models\]. Given these distinct scenarios, we can explore the capability and efficiency of the proposed method in reconstructing $q(z)$ and also verify any dependence on the underlying model. Essentially, we want to find the ${\sigma_{\text{rel}}}$ value which best reconstructs all the three fiducial models. We carry out this study considering: (i) only the SN Ia data and (ii) jointly the SN Ia, BAO and $H(z)$ measurements, in sections \[sec:mc\_sneia\] and \[sec:mc\_alldata\], respectively. Therefore, each realization is a (pseudo)randomly-generated catalog of SNe Ia $\{m_{B, a}, X_a, \mathcal{C}_a\}$ or random samples of all three observables, i.e., $\{\{m_{B,a}, X_a, \mathcal{C}_a\}, \{(D_V/r_d)_b\}, \{H_c\}\}$, where $a = 1,...,740$, $b = 1,...,6$ and $c = 1,...,21$. The methodology and algorithms to generate these samples are described in appendix \[app:sampling\]. We use the Monte Carlo object of the *Numerical Cosmology* Library (NumCosmo) [@DiasPintoVitenti2014], called . This object proceeds as describe in algorithm \[algo:NcmFitMC\]. The code generates a minimum of $\mathsf{prerun}$ mock samples and the catalog of the best fit for each sample. In the list below, we summarize the steps to obtain $\hat{q}(z)$ using the MC method: 1. Define the redshift domain $\mathrm{D} = [z_{min}, z_{max}]$, which is determined by the real data sample as described in appendix \[app:sampling\]. 2. Define the fiducial model $q^{\text{fid}}(z)$, where $z \in \mathrm{D}$. 3. Choose a ${\sigma_{\text{rel}}}$ value and the number of knots $n$, which will be the same for all ${\sigma_{\text{rel}}}$ values to be tested. 4. Run the MC algorithm for $m = \mathsf{prerun}$ minimum realizations. 5. Finally, compute the MSE. These steps are repeated for different values of ${\sigma_{\text{rel}}}$ and their respective results are compared in order to determine the best scenario which minimizes the MSE \[eq. \] for a given $m_b$ \[eq. \]. $l = 0$\ Monte Carlo analyses: SNe Ia {#sec:mc_sneia} ---------------------------- We carry out the first analysis using the SNe Ia data, such that the realizations are generated using the covariance matrix and redshifts of the JLA sample (see appendix \[app:sampling\]). Thus, $\mathrm{D} = [0.0, 1.3]$ which we divide in 7 equally spaced intervals, i.e., $n + 1 = 8$ knots.[^11] The first and second fiducial models, denoted as $q^{\text{fid1}}(z)$ and $q^{\text{fid2}}(z)$, are defined by the blue and green curves, respectively, in figure \[fig:fiducial\_models\]. The third one, $q^{\text{fid3}}(z)$, is the $\Lambda$CDM model (red curve), in which the cold dark matter and baryon density parameters are $\Omega_c = 0.3$ and $\Omega_b = 0.05$ and the DE EoS is $w = -1.0$. In all scenarios we assume a flat universe ($\Omega_k = 0$).[^12] Finally, these three fiducial models are completely defined by fixing the SN Ia astrophysical parameters, namely $\alpha^{\text{fid}}= 0.141$, $\beta^{\text{fid}}= 3.101$, $M_1^{\text{fid}}= -19.05$ and $M_2^{\text{fid}}= -19.12$. These values correspond to the best-fit obtained in [@Betoule2014] considering $\Lambda$CDM and SNe Ia data (including both systematic and statistical errors).[^13] ![The three $q^{\text{fid}}(z)$ fiducial models for which we study the reconstruction method via cubic spline. []{data-label="fig:fiducial_models"}](fiducial_models_new_z2.pdf) ![The top part of each panel shows the reconstructed curve of $q(z)$ using 8 knots, MC approach and sampling from SNe Ia data. The colored (blue, red and green) lines and shaded regions are the mean function $\langle\hat{q}(z)\rangle$ and their $1\sigma$ error bar, respectively, obtained for a given ${\sigma_{\text{rel}}}$ and $q^{\text{fid}}(z)$. The black lines correspond to $q^{{\text{fid}}1}$ (upper panel), $q^{{\text{fid}}2}$ (middle) and $\Lambda$CDM (lower). The bottom part of each panel shows the bias (dashed lines) and its $1\sigma$ error bar of the mean curve.[]{data-label="fig:qz_qs8"}](qz_plot_fiduc1_s8_jla.pdf "fig:") ![The top part of each panel shows the reconstructed curve of $q(z)$ using 8 knots, MC approach and sampling from SNe Ia data. The colored (blue, red and green) lines and shaded regions are the mean function $\langle\hat{q}(z)\rangle$ and their $1\sigma$ error bar, respectively, obtained for a given ${\sigma_{\text{rel}}}$ and $q^{\text{fid}}(z)$. The black lines correspond to $q^{{\text{fid}}1}$ (upper panel), $q^{{\text{fid}}2}$ (middle) and $\Lambda$CDM (lower). The bottom part of each panel shows the bias (dashed lines) and its $1\sigma$ error bar of the mean curve.[]{data-label="fig:qz_qs8"}](qz_plot_fiduc2_s8_jla.pdf "fig:") ![The top part of each panel shows the reconstructed curve of $q(z)$ using 8 knots, MC approach and sampling from SNe Ia data. The colored (blue, red and green) lines and shaded regions are the mean function $\langle\hat{q}(z)\rangle$ and their $1\sigma$ error bar, respectively, obtained for a given ${\sigma_{\text{rel}}}$ and $q^{\text{fid}}(z)$. The black lines correspond to $q^{{\text{fid}}1}$ (upper panel), $q^{{\text{fid}}2}$ (middle) and $\Lambda$CDM (lower). The bottom part of each panel shows the bias (dashed lines) and its $1\sigma$ error bar of the mean curve.[]{data-label="fig:qz_qs8"}](qz_plot_xcdm_s8_jla.pdf "fig:") In order to obtain the features of the reconstruction procedure as a function of the penalty factor (through ${\sigma_{\text{rel}}}$) and $q^{\text{fid}}(z)$, we perform the MC analyses considering 7 different ${\sigma_{\text{rel}}}$ values, ${\sigma_{\text{rel}}}= \{5\%, 15\%, 30\%, 50\%, 75\%, 100\%, 150\% \},$ for each fiducial model. Independently of ${\sigma_{\text{rel}}}$ and $q^{\text{fid}}(z)$, we standardize our analyses fixing the number of realizations to $m= 42000$, with which we obtain small $\mathsf{lre} < 1\%$ (algorithm \[algo:NcmFitMC\]) in all cases. Then, for each mock catalog, we minimize the function $$\label{eq:f_snia} -2 \ln (L_{\text{SNIa},P}) = -2 \ln (L_{\text{SNIa}}) + \sum_{i = 2}^{n-1} P_i({\sigma_{\text{rel}}}),$$ with respect to the free parameters $$\label{eq:set_p1} \vec{\theta} \doteq \{\hat{q}_0, \, \hat{q}_1, \, \hat{q}_2, \, \hat{q}_3, \, \hat{q}_4, \, \hat{q}_5, \, \hat{q}_6, \, \hat{q}_7, \, \alpha, \, \beta, \, M_1, \, M_2\},$$ where the right-side terms of eq.  are given in eqs.  and , respectively. Therefore, the expected values $\langle \hat{q}(z; {\sigma_{\text{rel}}}) \rangle$ and the error bars $\sigma(\hat{q}(z; {\sigma_{\text{rel}}})) \equiv \sqrt{\text{Var}(\hat{q}(z; {\sigma_{\text{rel}}}))}$ of the reconstructed function $\hat{q}(z; {\sigma_{\text{rel}}})$ are estimated by computing eqs.  and . Figure \[fig:qz\_qs8\] displays the reconstructed curves (colored solid lines) for ${\sigma_{\text{rel}}}= 5\%, 50\%$ and $150\%$, along with their respective fiducial models (black lines). As we discussed in section \[sec:penalty\], a small ${\sigma_{\text{rel}}}$ implies in a big constraint on the function complexity. Indeed, we see that ${\sigma_{\text{rel}}}= 5\%$ imposes $\hat{q}(z; 5\%)$ to be a linear function independently of the underlying fiducial model (upper-left part of the three panels). Naturally, if the true model differs from a linear function, the result will be strongly biased and, consequently, the estimated curve will not be capable to recover the true form. In particular, $\hat{q}(z; 5\%)$ is outside the $1\sigma$ error bar in a large fraction of the redshift interval for $q^{{\text{fid}}1}(z)$ and $q^{{\text{fid}}2}(z)$. The bottom-left part of the three panels in figure \[fig:qz\_qs8\] show the bias (colored dashed lines) as a function of the redshift and its $1\sigma$ error bar. Overall, until $z \lesssim 0.6$, the bias $b(z; {\sigma_{\text{rel}}})$ decreases as the $\hat{q}(z; {\sigma_{\text{rel}}})$ function complexity (i.e., ${\sigma_{\text{rel}}}$) increases, such that the reconstructed function approximates better and better the fiducial curve. On the other hand, due to the small amount of data at higher redshifts, mainly for $z \gtrsim 1.0$, the standard deviation $\sigma(\hat{q}(z; {\sigma_{\text{rel}}}))$ greatly increases, as evinced in figure \[fig:qz\_qs8\] by the colored shaded areas. As a direct result, we have that the constraints on the highest parameters $\hat{q}_i$ are degenerated causing an increment on $b(z; 150\%)$, for $z \gtrsim 0.6$, in comparison to $b(z; 50\%)$ for all three fiducial models, as shown in figure \[fig:qz\_qs8\]. ![The top part of each panel shows the MSE of the reconstructed $\hat{q}(z)$ for 7 different ${\sigma_{\text{rel}}}$ values $\in [5\%, 150\%]$, and obtained using 8 knots, MC approach and sampling from SNe Ia data. The upper, middle and lower panels refer to $q^{{\text{fid}}1}(z)$, $q^{{\text{fid}}2}(z)$ and $\Lambda$CDM models, respectively. The MSE decomposition into variance (dotted lines) and squared bias (dashed lines) is displayed in the bottom part of each panel.[]{data-label="fig:mse_qs8"}](fiduc1_bias_var_qs8.pdf "fig:") ![The top part of each panel shows the MSE of the reconstructed $\hat{q}(z)$ for 7 different ${\sigma_{\text{rel}}}$ values $\in [5\%, 150\%]$, and obtained using 8 knots, MC approach and sampling from SNe Ia data. The upper, middle and lower panels refer to $q^{{\text{fid}}1}(z)$, $q^{{\text{fid}}2}(z)$ and $\Lambda$CDM models, respectively. The MSE decomposition into variance (dotted lines) and squared bias (dashed lines) is displayed in the bottom part of each panel.[]{data-label="fig:mse_qs8"}](fiduc2_bias_var_qs8.pdf "fig:") ![The top part of each panel shows the MSE of the reconstructed $\hat{q}(z)$ for 7 different ${\sigma_{\text{rel}}}$ values $\in [5\%, 150\%]$, and obtained using 8 knots, MC approach and sampling from SNe Ia data. The upper, middle and lower panels refer to $q^{{\text{fid}}1}(z)$, $q^{{\text{fid}}2}(z)$ and $\Lambda$CDM models, respectively. The MSE decomposition into variance (dotted lines) and squared bias (dashed lines) is displayed in the bottom part of each panel.[]{data-label="fig:mse_qs8"}](xcdm_bias_var_qs8.pdf "fig:") In order to define which ${\sigma_{\text{rel}}}$ provides the best reconstructed curve, we compute the MSE \[eq. \] as a function of $z$ of all 21 reconstructed curves. Figure \[fig:mse\_qs8\] shows the MSE (solid lines) for each ${\sigma_{\text{rel}}}$ and $q^{{\text{fid}}1}(z)$ (upper panel), $q^{{\text{fid}}2}(z)$ (middle) and $\Lambda$CDM (lower) models. For these three fiducial models, $\hat{q}(z; 5\%)$ is the function with the smallest MSE for any $m_b$. However, these are also the most biased reconstructed curves and, even through visual inspection (figure \[fig:qz\_qs8\]), they do not give satisfactory reconstructions to their respective true $q^{\text{fid}}(z)$. Nevertheless, if we have only analyzed the fiducial curve closer to the best fit one (fiducial 3), we would wrongly conclude that ${\sigma_{\text{rel}}}= 5\%$ provides the smallest MSE with a insignificant bias (see the last panel in figure \[fig:mse\_qs8\]). More importantly, when performing the analysis with real data, we will be able to estimate the error bars but not the bias. We can note in figure \[fig:mse\_qs8\] that the MSE for ${\sigma_{\text{rel}}}= 5\%$ has about the same contribution from bias and variance. Thus, if we choose the smallest MSE (${\sigma_{\text{rel}}}= 5\%$) disregarding $m_b$, the estimated error in the real data analysis would provide only half of the total uncertainty in the reconstruction. In a conservative approach one would estimate the variance and then double by hand the error bars.[^14] Notwithstanding, inspecting the ${\sigma_{\text{rel}}}= 5\%$ reconstructions in figure \[fig:qz\_qs8\], we note that this reconstruction looses all information about the shape of the curve. So even correcting the error bar for $\hat{q}(z)$ by doubling the computed error, for example, the form of the curve will always be close to a straight line. The objective of this work is to reconstruct the form of the kinematic curve. We select the best reconstructed function (smallest MSE) requiring the bias to be at most $10\%$ of the total error, i.e., $m_b = 0.1$. Making this imposition, the estimated variance for the fit using real data will provide a good approximation of the uncertainty of the reconstruction. Therefore, taking into account the three fiducial models, we find that the best bias-variance trade-off as a function of $z$ is achieved for ${\sigma_{\text{rel}}}= 30\%$. It is worth noting that the MC results for $\alpha$, $\beta$, $M_1$ and $M_2$ are unbiased (bias smaller than $0.1\%$) for all 21 cases that we studied. Monte Carlo analyses: SNe Ia + BAO + $H(z)$ {#sec:mc_alldata} ------------------------------------------- In this section, we perform the MC analyses combining the SNe Ia, $H(z)$ and BAO data. In this case, we equally divide the redshift interval, $\mathrm{D} = [0.0, 2.3]$, using $n + 1 = 12$ knots. The fiducial models and parameters are those defined in section \[sec:mc\_sneia\] and, additionally, $r_d^{\text{fid}}= 103.5 \, h^{-1}\text{Mpc}$ and $H_0^{\text{fid}}= 73.0 \, \text{km} \, \text{s}^{-1} \text{Mpc}^{-1}$. Similarly, the MC study is done for the same ${\sigma_{\text{rel}}}$ set and the three fiducial models. In view of the increased amount of data, the present $q(z)$ reconstruction is better constrained and $m= 20000$ mock catalogs (for each MC analysis) are sufficient to provide $\langle \hat{q}(z; {\sigma_{\text{rel}}}) \rangle$ with small $\mathsf{lre} < 1\%$ (see algorithm \[algo:NcmFitMC\]). Since the SN Ia, BAO and $H(z)$ data sets are independent, the joint likelihood is $$-2\ln(L_{\text{SBH},P}) = -2 \left( \ln L_{\text{SNIa}} + \ln L_{\text{BAO}} + \ln L_{H} \right) + \sum_{i = 2}^{n-1} P_i({\sigma_{\text{rel}}}),$$ where $-2 \ln L_{SNIa}$, $-2 \ln L_{BAO}$ and $-2 \ln L_{H(z)}$ are given by equations , and , respectively. Then, for each realization, we compute the best-fitting values of the following 18 parameters, $$\label{eq:set_p2} \vec{\theta} \doteq \{\hat{q}_0, \, \hat{q}_1, \, \hat{q}_2, \, \hat{q}_3, \, \hat{q}_4, \, \hat{q}_5, \, \hat{q}_6, \, \hat{q}_7, \, \hat{q}_8, \, \hat{q}_9, \, \hat{q}_{10}, \, \hat{q}_{11}, \, \alpha, \, \beta, \, M_1, \, M_2, H_0, r_d\},$$ and, with them, we calculate the expected value of each parameter estimator, $\langle \vec{\theta} \rangle$, and their covariance matrix. ![The top part of each panel shows the reconstructed curve of $q(z)$ using 12 knots, MC approach and sampling from SNe Ia, BAO and $H(z)$ data. The colored (blue, red and green) lines and shadow regions are the mean function $\langle\hat{q}(z)\rangle$ and their $1\sigma$ error bar, respectively, obtained for a given ${\sigma_{\text{rel}}}$ and $q^{\text{fid}}(z)$. The black lines correspond to $q^{{\text{fid}}1}$ (upper panel), $q^{{\text{fid}}2}$ (middle) and $\Lambda$CDM (lower). The bottom part of each panel shows the bias (dashed lines) and its $1\sigma$ error bar of the mean curve.[]{data-label="fig:qz_qs12"}](qz_plot_fiduc1_s12_jla.pdf "fig:") ![The top part of each panel shows the reconstructed curve of $q(z)$ using 12 knots, MC approach and sampling from SNe Ia, BAO and $H(z)$ data. The colored (blue, red and green) lines and shadow regions are the mean function $\langle\hat{q}(z)\rangle$ and their $1\sigma$ error bar, respectively, obtained for a given ${\sigma_{\text{rel}}}$ and $q^{\text{fid}}(z)$. The black lines correspond to $q^{{\text{fid}}1}$ (upper panel), $q^{{\text{fid}}2}$ (middle) and $\Lambda$CDM (lower). The bottom part of each panel shows the bias (dashed lines) and its $1\sigma$ error bar of the mean curve.[]{data-label="fig:qz_qs12"}](qz_plot_fiduc2_s12_jla.pdf "fig:") ![The top part of each panel shows the reconstructed curve of $q(z)$ using 12 knots, MC approach and sampling from SNe Ia, BAO and $H(z)$ data. The colored (blue, red and green) lines and shadow regions are the mean function $\langle\hat{q}(z)\rangle$ and their $1\sigma$ error bar, respectively, obtained for a given ${\sigma_{\text{rel}}}$ and $q^{\text{fid}}(z)$. The black lines correspond to $q^{{\text{fid}}1}$ (upper panel), $q^{{\text{fid}}2}$ (middle) and $\Lambda$CDM (lower). The bottom part of each panel shows the bias (dashed lines) and its $1\sigma$ error bar of the mean curve.[]{data-label="fig:qz_qs12"}](qz_plot_xcdm_s12_jla.pdf "fig:") ![The top part of each panel shows the MSE of the reconstructed $\hat{q}(z)$ for 7 different ${\sigma_{\text{rel}}}$ values $\in [5\%, 150\%]$, and obtained using 12 knots, MC approach and sampling from SNe Ia, BAO and $H(z)$ data. The upper, middle and lower panels refer to $q^{{\text{fid}}1}(z)$, $q^{{\text{fid}}2}(z)$ and $\Lambda$CDM models, respectively. The MSE decomposition into variance (dotted lines) and squared bias (dashed lines) is displayed in the bottom part of each panel.[]{data-label="fig:mse_qs12"}](fiduc1_bias_var_qs12.pdf "fig:") ![The top part of each panel shows the MSE of the reconstructed $\hat{q}(z)$ for 7 different ${\sigma_{\text{rel}}}$ values $\in [5\%, 150\%]$, and obtained using 12 knots, MC approach and sampling from SNe Ia, BAO and $H(z)$ data. The upper, middle and lower panels refer to $q^{{\text{fid}}1}(z)$, $q^{{\text{fid}}2}(z)$ and $\Lambda$CDM models, respectively. The MSE decomposition into variance (dotted lines) and squared bias (dashed lines) is displayed in the bottom part of each panel.[]{data-label="fig:mse_qs12"}](fiduc2_bias_var_qs12.pdf "fig:") ![The top part of each panel shows the MSE of the reconstructed $\hat{q}(z)$ for 7 different ${\sigma_{\text{rel}}}$ values $\in [5\%, 150\%]$, and obtained using 12 knots, MC approach and sampling from SNe Ia, BAO and $H(z)$ data. The upper, middle and lower panels refer to $q^{{\text{fid}}1}(z)$, $q^{{\text{fid}}2}(z)$ and $\Lambda$CDM models, respectively. The MSE decomposition into variance (dotted lines) and squared bias (dashed lines) is displayed in the bottom part of each panel.[]{data-label="fig:mse_qs12"}](xcdm_bias_var_qs12.pdf "fig:") Analogously to the results presented in section \[sec:mc\_sneia\], in figure \[fig:qz\_qs12\] we show the reconstructed curves, $\langle \hat{q} (z; {\sigma_{\text{rel}}}) \rangle$ (colored solid lines), the biases (dashed lines) and their respective $1\sigma$ error bars (shaded areas) for ${\sigma_{\text{rel}}}= 5\%, 50\%$ and $150\%$ and the three fiducial models. As expected, combining SN Ia, BAO and $H(z)$ data decreases $\sigma(\hat{q}(z; {\sigma_{\text{rel}}}))$ and we note, in the three upper-left panels of figure \[fig:qz\_qs12\], that these improved constraints no longer restrict $\langle \hat{q}(z; 5\%) \rangle$ to be a linear function.[^15] Besides, as displayed in figure \[fig:mse\_qs12\], it also leads to a stronger dependency of $b(z; {\sigma_{\text{rel}}})$ on the fiducial model. For example, the squared bias function of $q^{{\text{fid}}2}(z)$ (dashed lines on the middle panel of figure \[fig:mse\_qs12\]) presents a large variation with respect to ${\sigma_{\text{rel}}}$, similar to what we obtained considering only SN Ia data. On the other hand, the bias function of the $\Lambda$CDM fiducial model is nearly invariant with respect to ${\sigma_{\text{rel}}}$ (lower panel of figure \[fig:mse\_qs12\]). This highlights that, giving a sufficient amount of data, if the true model has low complexity form then, naturally, a satisfactory reconstruction will rapidly be reached even for small ${\sigma_{\text{rel}}}$. In particular, taking into account the bias-variance trade-off and $m_b = 0.1$, the best reconstructed curve for the $\Lambda$CDM fiducial model is obtained for ${\sigma_{\text{rel}}}= 15\%$. As it was shown in figures \[fig:mse\_qs8\] and \[fig:mse\_qs12\], the MSE is dominated by the variance of the fitted parameters and, consequently, by the measurement errors of the observable data sets. In this way, we note that the MSE is independent of the fiducial model, depending only on the penalty factor, i.e., ${\sigma_{\text{rel}}}$. This aspect points out the fact that an statistical inference with relative small error bars can be completely misleading. For example, the well-constrained function $\hat{q}(z; 5\%)$, obtained for $q^{{\text{fid}}2}(z)$, is not at all a good reconstruction of the true model, diverging with more than $1\sigma$ in a large fraction of the redshift interval. Ultimately, following the discussion on section \[sec:PM\], the nature of the biases for ${\sigma_{\text{rel}}}= 5\%$ and $150\%$ can be classified as form-bias and estimator-bias, respectively. As in section \[sec:mc\_sneia\], we expect that a good reconstruction is one which does not present a significant bias. Therefore, applying the same requirement $m_b = 0.1$, we obtain that ${\sigma_{\text{rel}}}= 30\%$ provides the best balance between bias and variance, given the current data sets, to reconstruct the $q(z)$ curve. In summary, the MC outcomes (sections \[sec:mc\_sneia\] and \[sec:mc\_alldata\]) show that our method is efficient in reconstructing $q(z)$ and, given the current errors of the observational data, ${\sigma_{\text{rel}}}= 30\%$ is a safe and conservative choice to recover $q(z)$ imposing minimal assumptions and guaranteeing that the reconstructed curve will not be bias dominated. Results {#sec:real_reconst} ======= Defined the best estimator, $\hat{q}(z; 30\%)$, we now obtain the deceleration function given the real JLA, BAO and $H(z)$ samples (see section \[sec:data\]). As before, we reconstruct $q(z)$ (i) in the redshift interval $\mathrm{D} = [0, 1.3]$ using JLA SN Ia data and (ii) in $\mathrm{D} = [0, 2.3]$ combining those three observable data. The likelihood function is now interpreted as the posterior distribution $$\mathcal{P}(\vec{\theta} | \vec{D}) = L_P(\vec{D}| \vec{\theta}),$$ since we are assuming flat priors on all parameters. We then use the NumCosmo algorithm ,[^16] which implements an ensemble sampler with affine invariance for Markov Chain Monte Carlo analysis, to compute the mean function $\overline{q}(z)$ given the JLA sample and its $68.27\%$, $95.45\%$ and $99.73\%$ confidence intervals (CI). We ran 100 chains, computing $8\times 10^5$ points in the 12-dimensional parametric space, shown in eq. , attaining a multivariate potential scale reduction factor (MPSRF) equal to $1.016$.[^17] The best-fitting values and the error bars of the SNe Ia parameters are $\alpha =0.141 \pm 0.007$, $\beta = 3.108 \pm 0.081$, $M_1 = -19.05 \pm 0.03$ and $M_2 = -19.12 \pm 0.03$. ![The model-independent reconstructed $q(z)$ (blue solid line) and its $68.23\%$, $95.45\%$ and $99.73\%$ confidence intervals (blue shaded areas), for ${\sigma_{\text{rel}}}= 30\%$, using SNe Ia (left panel) and SNe Ia + BAO + $H(z)$ data (right panel). The red line and contours are the $q(z)$ mean and its CI’s obtained, assuming XCDM model, fitting $\Omega_c$ (with $w = -1$) (left panel) and $(H_0, \Omega_c, w)$ (right panel) along with the SNe Ia nuisance parameters. The black lines correspond to the Planck+BAO+JLA+$H_0$ best-fit assuming $\Lambda$CDM model.[]{data-label="fig:real_qz"}](qz_plot_real_data_s8_planckbf.pdf "fig:") ![The model-independent reconstructed $q(z)$ (blue solid line) and its $68.23\%$, $95.45\%$ and $99.73\%$ confidence intervals (blue shaded areas), for ${\sigma_{\text{rel}}}= 30\%$, using SNe Ia (left panel) and SNe Ia + BAO + $H(z)$ data (right panel). The red line and contours are the $q(z)$ mean and its CI’s obtained, assuming XCDM model, fitting $\Omega_c$ (with $w = -1$) (left panel) and $(H_0, \Omega_c, w)$ (right panel) along with the SNe Ia nuisance parameters. The black lines correspond to the Planck+BAO+JLA+$H_0$ best-fit assuming $\Lambda$CDM model.[]{data-label="fig:real_qz"}](qz_plot_real_data_s12_planck.pdf "fig:") The mean (blue line) and CI’s (blue shaded areas) are displayed in the left panel of figure \[fig:real\_qz\], where we note that the ESMCMC $68.27\%$ CI is in agreement to the respective MC error bar. Naturally, as in the MC study, the difficulty in constraining $q(z)$ with minimal assumptions is that the result is highly degenerated. So, besides the fact that $\overline{q}(z) < 0$ in the entire redshift, this is not statistically significant. The main results of this $q(z)$ reconstruction, using only SNe Ia data, are: the indication of a transition redshift at $z_{\tiny{T}} \simeq 0.53$ with $68.27\%$ significance level, and the evidence of an accelerated expansion phase $\geq 99.73\%$ within the redshift interval $\sim [0.04, 0.19]$. To compare this result with the flat $\Lambda$CDM model, i.e., assuming GR and DE EoS given by $w = -1.0$, we carry out the ESMCMC in the 5-dimensional parametric space $$\vec{\theta} \doteq (\Omega_c, \alpha, \beta, M_1, M_2).$$ For this, we fixed the other cosmological parameters to the JLA best-fit. We obtain the following best-fitting values and standard deviations: $\Omega_c = 0.244 \pm 0.034$, $\alpha = 0.141 \pm 0.007$, $\beta = 3.103 \pm 0.081$, $M_1 = -19.05 \pm 0.023$ and $M_2 = -19.12 \pm 0.026$. The mean $\overline{q}^{\text{XCDM}} (z)$ (red line) and the CI’s (red shaded areas) are consistent with our model-independent reconstruction overall redshift range, as shown in the left panel of figure \[fig:real\_qz\]. In both analyses, the SN Ia nuisance parameters $\alpha$ and $\beta$ are weakly (anti-) correlated ($\lesssim 0.1$) to $\{\hat{q}_i\}$ $(i = 0,..., 7)$ and $\Omega_c$. $M_1$ and $M_2$ are also weakly correlated to most parameters excepted for $q_0$ and $\Omega_c$, in which there are moderate correlations $\sim 0.56 - 0.66$. At last, we also plot the Planck best-fitting curve (black line), $\overline{q}^{\text{Planck}}(z)$, which we computed using the Planck+BAO+JLA+$H_0$ best-fitting parameters obtained assuming $\Lambda$CDM [@Planck2015], namely, $H_0 = 67.74$, $\Omega_c = 0.259$ and $\Omega_b =0.049$. We note that $\overline{q}^{\text{Planck}}(z)$ is inside the $68.27\%$ CI in the entire redshift interval. We follow the same procedure to compute $\overline{q}(z)$ and their CI in the redshift interval $[0.0, 2.3]$ given the JLA, BAO and $H(z)$ data. In this case, $8\times 10^5$ points are calculated in the 18-dimensional parametric space \[eq. \], getting $\text{MPSRF} = 1.033$. In this case, the best-fitting and standard deviation values of the non-$q_i$ parameters are $H_0 = 71.68 \pm 1.69 \, \text{km} \, \text{s}^{-1} \text{Mpc}^{-1}$, $r_d = 101.15 \pm 1.8 \, h^{-1}\text{Mpc}$, $\alpha = 0.141 \pm 0.007$, $\beta = 3.111 \pm 0.081$, $M_1 = -19.01 \pm 0.05$ and $M_2 = -19.07 \pm 0.05$. We note in the right panel of figure \[fig:real\_qz\] an expressive improvement on the constraints due to the combined data, regarding that we are assuming only FLRW metric and flat space, and given the high dimension of the parametric space. In this case, we obtain $z_{\tiny{T}} \simeq 0.58$ with $68.27\%$ CI and we can attest with significance $\geq 99.73\%$ that the universe is accelerating within $0.02 \lesssim z \lesssim 0.22$. At last, we also have an indication of a decelerated phase in the redshift interval $1.84 \lesssim z \lesssim 2.13$ with $68.27\%$. We compare the model-independent $q(z)$ reconstruction with the XCDM model, i.e., constant EoS for the dark energy component $w = constant$, performing the ESMCMC on the 7-dimensional parametric space $$\vec{\theta} \doteq (\Omega_c, H_0, w, \alpha, \beta, M_1, M_2).$$ Their best-fitting values and error bars are $\Omega_c = 0.269 \pm 0.017$, $H_0 = 71.04 \pm 1.59 \, \text{km} \, \text{s}^{-1} \text{Mpc}^{-1}$, $w = -1.11 \pm 0.07$, $\alpha = 0.142 \pm 0.007$, $\beta = 3.108 \pm 0.081$, $M_1 = -19.03 \pm 0.050$ and $M_2 = -19.10 \pm 0.05$. In figure \[fig:real\_qz\] (right panel), we plotted the results, $\overline{q}^{\text{XCDM}} (z)$ (red line) and the CI’s (red shaded areas), showing that XCDM model is also compatible within $99.73\%$ confidence level with the reconstructed curve in the entire redshift interval. It is worth noting that our conservative analyses provide only lower bounds for $z_{\tiny{T}} \simeq 0.19$ (JLA) and $\simeq 0.22$ \[JLA+BAO+$H(z)$\] within $99.73\%$. On the other hand, model-dependent works can estimate better constrained intervals for $z_{\tiny{T}}$ [@Capozziello2014; @Santos2015], as evinced by the XCDM results. However, as we discussed in sections \[sec:introduction\] and \[sec:review\], these estimates are accurate as long as the form-bias is small. Besides the results presented so far, we can also infer other kinematic quantities from the reconstructed function $q(z)$ (see sections \[sec:introduction\] and \[sec:def\_q\]), such as $H(z)$ and the jerk function $j(z) \equiv - \dddot{a}/(aH^3)$ (i.e., the third order term of the scale factor expansion). The functions $H(z)$ and $j(z)$ are estimated by integrating and differentiating, respectively, $q(z)$ with respect to z. Figure \[fig:real\_Hz\_jz\] shows the results obtained using only JLA[^18] (left panels) and JLA+BAO+$H(z)$ (right panels). The upper panels display the means $\overline{H}(z)$ (blue lines) and the $68.23\%$, $95.45\%$ and $99.73\%$ CI’s (blue shaded areas). As the $H(z)$ estimate is related with $q(z)$ by a numerical integration, it is consequently better constrained and the enhancement due to the combined data is even more apparent, as shown figure \[fig:real\_Hz\_jz\] (upper right panel), where the 21 $H(z)$ measurements are also plotted. As before, we compare these results with the respective outcomes assuming, respectively, $\Lambda$CDM and XCDM models (red lines and shaded areas). The decrease in the CI’s of the reconstructed $H(z)$ highlights its concordance with those cosmological models within $99.73\%$ level. The Planck best-fit $\overline{H}^{\text{Planck}}(z)$ (black lines) also lies inside the CI’s over the entire redshift interval for JLA+BAO+H(z), and it is outside the $99.73\% $ CI only in $0.0 \leq z \lesssim 0.03$ when we use JLA. Contrarily to the above scenario, $j(z)$ is obtained by numerical differentiation implying in high degenerated results, as displayed in the lower panels of figure \[fig:real\_Hz\_jz\]. The means $\overline{j}(z)$ and the CI’s are represented by the blue lines and blue shaded areas, respectively. The estimated jerk functions, obtained assuming $\Lambda$CDM (left panel) and XCDM models (right panel), correspond to the red lines and red shaded areas. Besides the redshift intervals $[0.14, 0.39]$ (left panel) and $[0.13, 0.45]$ (right panel), which provides $-6 \lesssim j(z) \lesssim 6$ with $\geq 99.73\%$ significance, the degenerated results do not provide relevant information about this kinematic function. ![$H(z)$ (upper panels) and $j(z)$ (lower panels) mean estimates (blue solid lines) and their $68.23\%$, $95.45\%$ and $99.73\%$ confidence intervals (blue shaded areas) obtained integrating and deriving, respectively, the reconstructed $q(z)$ using SNe Ia (left panels) and SNe Ia + BAO + $H(z)$ data (right panels). Similarly, the red line and contours are the $H(z)$ and $j(z)$ means and their CI’s obtained integrating/deriving the respective $q(z)$ where the $\Lambda$CDM (left panels) and XCDM (right panels) models were assumed. The black lines correspond to the Planck+BAO+JLA+$H_0$ best-fit assuming $\Lambda$CDM model. The black dots and error bars represent the 21 $H(z)$ measurements used in this work.[]{data-label="fig:real_Hz_jz"}](Hz_plot_real_data_s8_planckbf.pdf "fig:") ![$H(z)$ (upper panels) and $j(z)$ (lower panels) mean estimates (blue solid lines) and their $68.23\%$, $95.45\%$ and $99.73\%$ confidence intervals (blue shaded areas) obtained integrating and deriving, respectively, the reconstructed $q(z)$ using SNe Ia (left panels) and SNe Ia + BAO + $H(z)$ data (right panels). Similarly, the red line and contours are the $H(z)$ and $j(z)$ means and their CI’s obtained integrating/deriving the respective $q(z)$ where the $\Lambda$CDM (left panels) and XCDM (right panels) models were assumed. The black lines correspond to the Planck+BAO+JLA+$H_0$ best-fit assuming $\Lambda$CDM model. The black dots and error bars represent the 21 $H(z)$ measurements used in this work.[]{data-label="fig:real_Hz_jz"}](Hz_plot_real_data_s12_planck.pdf "fig:") ![$H(z)$ (upper panels) and $j(z)$ (lower panels) mean estimates (blue solid lines) and their $68.23\%$, $95.45\%$ and $99.73\%$ confidence intervals (blue shaded areas) obtained integrating and deriving, respectively, the reconstructed $q(z)$ using SNe Ia (left panels) and SNe Ia + BAO + $H(z)$ data (right panels). Similarly, the red line and contours are the $H(z)$ and $j(z)$ means and their CI’s obtained integrating/deriving the respective $q(z)$ where the $\Lambda$CDM (left panels) and XCDM (right panels) models were assumed. The black lines correspond to the Planck+BAO+JLA+$H_0$ best-fit assuming $\Lambda$CDM model. The black dots and error bars represent the 21 $H(z)$ measurements used in this work.[]{data-label="fig:real_Hz_jz"}](jz_plot_real_data_s8.pdf "fig:") ![$H(z)$ (upper panels) and $j(z)$ (lower panels) mean estimates (blue solid lines) and their $68.23\%$, $95.45\%$ and $99.73\%$ confidence intervals (blue shaded areas) obtained integrating and deriving, respectively, the reconstructed $q(z)$ using SNe Ia (left panels) and SNe Ia + BAO + $H(z)$ data (right panels). Similarly, the red line and contours are the $H(z)$ and $j(z)$ means and their CI’s obtained integrating/deriving the respective $q(z)$ where the $\Lambda$CDM (left panels) and XCDM (right panels) models were assumed. The black lines correspond to the Planck+BAO+JLA+$H_0$ best-fit assuming $\Lambda$CDM model. The black dots and error bars represent the 21 $H(z)$ measurements used in this work.[]{data-label="fig:real_Hz_jz"}](jz_plot_real_data_s12.pdf "fig:") At last, in this work we also obtain a model-independent estimate of the sound horizon, since $r_d$ is fitted throughout the reconstruction procedure. Thus, we obtain $r_d = 101.15 \pm 1.8 \, h^{-1}\text{Mpc}$ given the combined analysis in the 18-dimensional parametric space using JLA+BAO+$H(z)$ data. Our result is in accordance with that presented in reference [@Heavens2014], where they obtained $r_d = 100.7 \pm 2.0 \, h^{-1}\text{Mpc}$ (SNe+BAO+$H(z)$ without Hubble prior) and $r_d = 101.9 \pm 1.9 \, h^{-1}\text{Mpc}$ (idem with Hubble prior) assuming a linear spline for $h^{-1}(z) = 100 \, \text{km}\,\text{s}^{-1}\text{Mpc}^{-1}/ H(z)$, 6 knots equally-spaced and curved universe. It is worth noting that their reconstruction was directly applied to the data and no test regarding the assumed curve was performed. Conclusions {#sec:conclusions} =========== In this work we presented a general model-independent approach to reconstruct any one-variable function. In particular, we applied it to estimate the deceleration function $q(z)$ using the JLA SN Ia, BAO and $H(z)$ data sets. We performed a conservative analysis in the sense that we reconstructed $q(z)$ considering minimal assumptions (FLRW metric and flat universe) and also fitting simultaneously $H_0$, $r_d$ and the SN Ia astrophysical parameters ($\alpha$, $\beta$, $M_1$, $M_2$). The MC results presented in sections \[sec:mc\_sneia\] and \[sec:mc\_alldata\] show that it is crucial to estimate both the form- and the estimator-biases in order to validate any proposed reconstruction method, independently if it is parametric or non-parametric. They also reveal that the MSE’s are independent of the underlying fiducial model and are dominated by the variance of the current data sets, even for cases where the reconstructed functions are far to be a good representation of the true curves, i.e., highly biased. Nevertheless, a blindly minimization of the MSE could lead to a bias dominated reconstruction. Such reconstructions must be carefully analyzed and the bias added by hand or estimated in some way. Even so, the information about the shape of the curve is mostly lost. Thus, requiring the bias to be at most $10\%$ of the total MSE, we evaluated the bias-variance trade-off obtaining that, currently, the best penalty factor is given by ${\sigma_{\text{rel}}}= 30\%$. Even though the MSE is approximately independent of the fiducial model, analyzing MSE of only one model can lead to misleading conclusions. If we had only used the $\Lambda$CDM fiducial model, the smaller ${\sigma_{\text{rel}}}= 5\%$ would prove to be the best choice, i.e., smaller MSE and insignificant bias. In other words, if a method is good in reconstructing a $\Lambda$CDM shaped curve, this does not mean that the same method will be able to reconstruct other curves close to it. In addition, capping the bias at $10\%$ is a straightforward way to obtain the total uncertainty of the estimates, since in this case we can use directly the variance analysis without requiring additional bias estimation. Our main reconstructions are summarized in figure \[fig:real\_qz\], given that there are no assumptions on the gravitational dynamics nor on the matter content, and we also include the relevant astrophysical parameters into the study. The standard cosmological model agrees with the reconstructed results within $99.73\%$ CI in the entire redshift intervals. This is even further evident observing the estimate for the Hubble function in figure \[fig:real\_Hz\_jz\]. In this sense this work discards large deviations from the standard model with the caveat that we are assuming the FLRW metric and we are not testing this hypothesis. Notwithstanding, with more data available, we advocate that to measure departures from the standard model, one should repeat the process described in this work, evaluating the capability of the model to differentiate between a whole class of fiducial models. SDPV thanks CAPES for the grant 2649-13-6. MPL thanks CNPq (PCI/MCTI/INPE program and grant 202131/2014-9) for financial support. This research was performed using the Mesu-UV supercomputer of the Pierre & Marie Curie University – France (UPMC) and the computer cluster of the State University of the Rio Grande do Norte (UERN) – Brazil. We also thank Pierre Astier and Marc Betoule for kindly providing the complete covariance matrix necessary for the SNe Ia analysis, and Nicolás Busca for useful comments. Simulated data {#app:sampling} ============== In this appendix we describe the methodology to generate the SN Ia, BAO and $H(z)$ mock catalogs which we used to perform the MC study in section \[sec:meth\_valid\]. SN Ia sampling -------------- The SN Ia distance modulus $\mu$ is written in terms of its light-curve parameters $(m_B, X, \mathcal{C})$ as $$\label{eq:dist_mod} \mu = m_B - (M_{h_i} - \alpha X + \beta \mathcal{C}),$$ where $m_B$ is the observed peak magnitude in rest-frame B, $X$ and $\mathcal{C}$ are the observed stretching and the color at maximum brightness, respectively, and $\alpha$, $\beta$ and $M_{h_i} = \{M_1, M_2\}$ are nuisance parameters [@Betoule2014]. Thus, in order to generate SNe Ia samples, we must provide fiducial values of $(m_{B}, X, \mathcal{C})$, for each SN Ia, and their respective probability distributions. The fiducial values are usually determined choosing a specific set of the observable model parameters. In this case, from eq.  and the theoretical distance modulus, $\mu^{\text{th}} = 5\log_{10}(\mathcal{D}_L(z^{\text{hel}}, z^{{\text{cmb}}}) H_0/ c) + 25$, we can write the fiducial magnitude as function of $X_i$ and $\mathcal{C}_i$, $$\label{eq:mb_sneia} m_{B_i}^{\text{fid}}= 5\log_{10}\left[\mathcal{D}_L\left(z^{\text{hel}}_i, z^{{\text{cmb}}}_i, \hat{q}^{\text{fid}}\right)\right] - \alpha^{\text{fid}}X_i + \beta^{\text{fid}}\mathcal{C}_i + M_{h_i}^{\text{fid}}- 5\log_{10}(c/H_0^{\text{fid}}) + 25,$$ where $i$ denotes the SN Ia index and the SN Ia redshift values $\{z^{\text{hel}}_i, z^{{\text{cmb}}}_i\}$ are always kept fixed to their observable estimates. As described in section \[sec:mc\_sneia\], the fiducial parameters are $$\{\alpha^{\text{fid}}, \beta^{\text{fid}}, M_1^{\text{fid}}, M_2^{\text{fid}}, H_0^{\text{fid}}\} = \{0.141, 3.101, -19.05, -19.12, 73.0\},$$ and they correspond (besides $H_0^{\text{fid}}$) to the best-fitting values assuming $\Lambda$CDM model obtained in reference [@Betoule2014]. We consider the three different $q^{\text{fid}}(z)$ represented in figure \[fig:fiducial\_models\]. At this point, we still have to define the fiducial values for $X_i$ and $\mathcal{C}_i$. However, there is no astrophysical models for $X$ and $\mathcal{C}$. Therefore, we define the SNe Ia fiducial model $(m_{B_i}^{\text{fid}}, X_i^{\text{fid}}, \mathcal{C}_i^{\text{fid}})$ as being the best-fitting values obtained by maximizing, with respect to all $X_i$ and $\mathcal{C}_i$, the multivariate Gaussian distribution [@Betoule2014] $$\label{eq:sn_all_dist} G(\vec{\zeta}, \mathsf{C}_{cmp}) = \frac{1}{\sqrt{(2\pi)^{3N} \vert\mathsf{C}_{cmp}\vert}} e^{-\frac{1}{2}\vec{\zeta}^T \mathsf{C}_{cmp}^{-1} \vec{\zeta}},$$ where $\vert ... \vert$ denotes the determinant, $\mathsf{C}_{cmp}^{-1}$ is the complete inverse covariance matrix (with 2220 rows and columns), $N$ is the number of SNe Ia of the JLA sample (740) and $$\vec{\zeta} = \left(m_{B_1}, ..., m_{B_N}, X_{N+1}, ..., X_{2N}, \mathcal{C}_{2N+1}, ..., \mathcal{C}_{3N}\right),$$ with $\{m_{B_i}\}$ given by eq. .[^19] Finally, writing the SNe Ia variables as $$m_{B_i} = m_{B_i}^{\text{fid}}+ \delta m_{B_i}, \quad X_i = X^{\text{fid}}_i + \delta X_i \quad \text{and} \quad \mathcal{C}_i = \mathcal{C}^{\text{fid}}_i + \delta\mathcal{C}_i,$$ we create a SN Ia sample $\{m_{B_i}, X_i, \mathcal{C}_i\}$ by randomly generating 2220 values from the normal distribution with variance $\mathsf{C}_{cmp}$ and zero mean corresponding to $$\overrightarrow{\delta\zeta} = \left(\delta m_{B_1}, ..., \delta m_{B_N}, \delta X_{N+1}, ..., \delta X_{2N}, \delta\mathcal{C}_{2N+1}, ..., \delta\mathcal{C}_{3N}\right).$$ -------------------------- ------- -------------- -------------- ------- --------- --------- --------- --- -- Reference z $D_V(z)/r_d$ \[0.5ex\] Beutler et al. 0.106 $0.336$ $0.015^{-2}$ 0 0 0 0 0 Padmanabhan et al. 0.35 8.88 0 34.60 0 0 0 0 0.44 11.550 0 0 4.8116 -2.4651 1.0375 0 0.60 14.945 0 0 -2.4651 3.7697 -1.5865 0 0.73 16.932 0 0 1.0375 -1.5865 3.6498 0 Ross et al. 0.15 4.466 - - - - - - \[1ex\] -------------------------- ------- -------------- -------------- ------- --------- --------- --------- --- -- : BAO data[]{data-label="tab:bao"} In Ref. [@Beutler2011] the authors provide $r_d / D_V(z)$. Note that it is the reciprocal of the provided in the other references. As this data is not correlated with the others, we can include it directly in $L_{BAO}$ \[eq. \]. Refs. [@Kazin2014] and [@Ross2014] provide $D_V(z)(r_d^{{\text{fid}}}/r_d)$, where $r_d^{{\text{fid}}} = 148.6$ and $148.69 \, h^{-1}\text{Mpc}$, respectively. We consider these values to build $L_{\text{BAO}}$. $H(z)$ and BAO sampling ----------------------- Differently from the SN Ia sampling, the BAO and $H(z)$ mock catalogs can be directly generated, since we have theoretical models for the respective observables, i.e., $D_V(z)/r_d$ \[or $r_d/D_V(z)$\], $H(z)$ and $H(z)r_d /(1+z)$, as displayed in eqs. , and . Each $H^\text{obs}$ catalog correspond to $\{H_i^\text{obs}\}$, where $i = 1,..., 20$, and the point $\{H_r^{\text{obs}}\}$. The 21 redshift values are not altered in the sampling procedure and they are equal to the observed data $z$ of table \[tab:Hz\]. We obtain $H_i^\text{obs}$ writing it as $$H_i^\text{obs} = H^{\text{fid}}(z_i) + \delta H_i,$$ and, then, we randomly generate the quantity $\delta H_i$ from a Gaussian distribution with mean 0 and standard deviation $\sigma_i$ (last column of table \[tab:Hz\]), since $H^\text{obs}_i$ follows a Gaussian distribution. Analogously, we generate the point $H^\text{obs}_r$ using a zero mean Gaussian with standard deviation $\sigma_{21}$ and add the result to $H(z_{21})r_d /(1+z_{21})$. The value of $H^{\text{fid}}(z_i)$ correspond to the fiducial value, which is defined by $q^{\text{fid}}(z)$, $H_0^{\text{fid}}= 73.0 \, \text{km} \, \text{s}^{-1} \text{Mpc}^{-1}$ and $r_d^{\text{fid}}= 103.5 \, h^{-1}\text{Mpc}$. Applying the same methodology, we generate the first two points of the BAO set $\{D_{V_j}/r_d\}$, where $D_V(z_j)/r_d = D_V^{\text{fid}}(z_j)/r_d^{\text{fid}}+ \delta{}b_j$, the fiducial values are also defined by $q^{\text{fid}}(z)$, $H_0^{\text{fid}}$ and $r_d^{\text{fid}}$, and their respective inverse variances are presented in the first two rows of table \[tab:bao\]. The three correlated BAO points, related to Kazin et al. [@Kazin2014] data, are resampled randomly generating the 3 values $\delta{}b_j$ from a multivariate Gaussian distribution with means 0 and covariance matrix provided in reference [@Kazin2014], see table \[tab:bao\]. The last BAO point ($j = 6$) follows a different probability distribution, which is defined by the likelihood function $L_{Ross}$ provided by Ross et al. [@Ross2014] and centered in the observed value $(D_{V_j}/r_d)^{\text{obs}} = 4.466$. Thus, to produce a mock catalog from a fiducial model we perform the inverse transform sampling. First, we shift this function, $L_{Ross}^s$ such that the new center is $D_V^{\text{fid}}(z_j)/r_d^{\text{fid}}$. Then, we randomly generate a number $u$ from a uniform distribution $(0, 1)$ and, finally, we invert the equation $$u = \int_{\alpha_i}^\alpha {\mathrm{d}}\alpha^\prime L_{Ross}^s (\alpha^\prime)$$ obtaining $\alpha(u)$. The lower bound $\alpha_i$ corresponds to the lower bound described by the $L_{Ross}$ function. ------------------------------------- ------ -------- ---------- ----------- ------ -------- ---------- Reference z $H(z)$ $\sigma$ Reference z $H(z)$ $\sigma$ \[0.5ex\] Riess et al. [@Riess2011] 0.0 $73.8$ 2.4 0.1 69 12 0.18 75 4 0.17 83 8 0.20 75 5 0.27 77 14 0.35 83 14 0.4 95 17 0.59 104 13 0.48 97 60 0.68 92 8 0.88 90 40 0.78 105 12 0.9 117 23 0.88 125 17 1.3 168 17 1.04 154 20 1.43 177 18 1.53 140 14 Busca et al. [@Busca2013] - - - 1.75 202 40 ------------------------------------- ------ -------- ---------- ----------- ------ -------- ---------- : $H(z)$ data[]{data-label="tab:Hz"} The observable of this entry is related to $H(z_{21})r_d/(1+z_{21})$, instead of simply $H(z)$ as the other data points. 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[^1]: There are also works which explore the properties of the DE equation of state (i.e., assuming the general relativity) using kinematic quantities, e.g., [@Huterer2003; @Daly2008]. [^2]: For a discussion about the issues in deriving a kinematic function from a reconstructed one of another quantity, see [@Sahni2006]. [^3]: The second derivative of the distance, given by a linear spline, is zero everywhere. [^4]: For a detailed discussion of this problem in the context of the cluster number counts see [@Penna-Lima2014]. [^5]: In practice, once the value of the function at each knot is defined, the coefficients of each cubic polynomial are obtained from the solution of a linear system including not only the knots on its neighborhood but all knots. [^6]: One can also easily interpret, using the Bayesian point of view, the penalty factor as a prior on the fitted function. [^7]: The $\sigma_{abs}$ factor guarantees that, even if some $\bar{\hat{q}}_i \simeq 0$ the denominator in the penalty factor does not go to zero. [^8]: Note that here we are using the MC method to perform integrals in the data $\vec{D}$ given a set of parameters $\vec{\theta}^{\text{fid}}$. Therefore, we are sampling new data from the given likelihood. This procedure is similar but has different applications than the MC used to study the parameter space $\vec{\theta}$ given a data set $\vec{D}$. [^9]: The likelihoods used in this work were all implemented on top of the data description objects of the Numerical Cosmology library [@DiasPintoVitenti2014]. These objects automatically provide the resample feature for any likelihood using them. [^10]: There are methods to infer the bias without the knowledge of the true underlying model, e.g., bootstrap [@Efron1994]. In this work we do not explore these approaches, since their application is not straightforward for correlated data. [^11]: Following the discussion presented in the end of section \[sec:spline\], we also performed the analyses using $n +1 = 6$ and $n+1 = 10$ knots. The results showed a small improvement from 6 to 8 knots, and a negligible variation from 8 to 10 knots. [^12]: The densities $\Omega_c$ and $\Omega_b$ are the energy densities divided by the critical density today. [^13]: In the reference, the authors use a different parametrization $(M_B, \Delta_M)$ such that $M_1 = M_B$ and $M_2 = M_B + \Delta_M$. [^14]: In a more careful analysis, one could calculate the bias for several fiducial models and their respective upper limits. Then, this value could be added to the curve obtained from the real data. [^15]: This is natural since, in this case, we have more knots and we would need a smaller ${\sigma_{\text{rel}}}$ value to constrain into a linear curve. [^16]: The algorithm is describe in [@Goodman2010] and was implemented in C in the NumCosmo library [@DiasPintoVitenti2014]. Another unrelated implementation in Python is described in [@Foreman-Mackey2013]. [^17]: The MPSRF is a diagnose tool used to check the convergence of a Markov Chain, which was originally proposed in [@Brooks1998]. A value $<1.2$ indicates the convergence. [^18]: In this case, we recover $H(z)$ \[see eq. \] considering $H_0 = 70.0 \, \text{km}\,\text{s}^{-1}\text{Mpc}^{-1}$, which is the reference value used in [@Betoule2014]. [^19]: The covariance matrix $\mathsf{C}_{cmp}$ also depends on the intrinsic standard deviation of each SN Ia subsample, i.e., SDSS, SNLS3, low-z and HST (Hubble Space Telescope), $\{\sigma^1_{int}, \sigma^2_{int}, \sigma^3_{int}, \sigma^4_{int}\}$, respectively. In particular, we used original values calibrated in [@Betoule2014].

--- author: - | Robert Fleischer\ CERN, Switzerland\ E-mail: - | \ TU Muenchen, Germany\ E-mail: title: '$b \to d$ Penguins: CP Violation, General Lower Bounds on the Branching Ratios and Standard Model Tests' --- Introduction ============ Flavour-changing neutral-current (FCNC) processes, possible in the Standard Model (SM) only through loop diagrams, are an extremely important probe for new physics (NP). The good agreement between experiment and theory in processes induced by $b\to s$ FCNCs has already put important constraints on physics beyond the SM. Due to the excellent work of the $B$-factories, we are now entering the era where $b\to d$ penguin-induced processes – typically suppressed by a factor of 20 with respect to the corresponding $b\to s$ penguin transitions – can be used to test the SM more rigorously than it was possible before. The flavour structure of the SM, more specifically the order of magnitude of the individual elements of the Cabibbo–Kobayashi–Maskawa (CKM) matrix, allows us to derive certain relationships between different observables in $b\to d$-induced decays, and between $b\to s$- and $b\to d$-related observables. These relationships allow us to test the SM in those cases where the corresponding observables have already been measured and to make predictions where observations are still missing. $B_d^0\to K^0\bar K^0$: CP Violation and the Branching Ratio ============================================================ In the SM, we can write the amplitude for the decay $B_d^0\to K^0\bar K^0$ as $$\label{ampl-BdKK} A(B_d^0\to K^0\bar K^0)=\lambda^{(d)}_u {\cal P}_u^{K\!K} + \lambda^{(d)}_c {\cal P}_c^{K\!K} + \lambda^{(d)}_t {\cal P}_t^{K\!K},$$ where the $\lambda^{(d)}_q \equiv V_{qd}V_{qb}^\ast$ are CKM factors, and the ${\cal P}_q^{K\!K}$ denote the strong amplitudes of penguin topologies with internal $q$-quark exchanges, which receive tiny contributions from colour-suppressed electroweak (EW) penguins and are fully dominated by QCD penguin processes. Eliminating $\lambda^{(d)}_t$ with the help of the unitarity relation $\lambda^{(d)}_t=-\lambda^{(d)}_u-\lambda^{(d)}_c$ of the CKM matrix, we can write the amplitude as $$\label{ampl-BdKK-lamt} A(B^0_d\to K^0\bar K^0)=\lambda^3A{\cal P}_{tc}^{K\!K} \left[1-\rho_{K\!K} e^{i\theta_{K\!K}}e^{i\gamma}\right],$$ where ${\cal P}_{tc}^{K\!K}\equiv {\cal P}_t^{K\!K}-{\cal P}_c^{K\!K}$, and $\rho_{K\!K} e^{i\theta_{K\!K}}$ is a function of the ${\cal P}_q^{K\!K}$ that we treat as an unknown hadronic parameter. The direct and mixing-induced CP asymmetries ${\cal A}_{\rm CP}^{\rm dir}(B_d\to K^0\bar K^0)$ and ${\cal A}_{\rm CP}^{\rm mix}(B_d\to K^0\bar K^0)$ are functions of [*only*]{} $\rho_{K\!K}$, $\theta_{K\!K}$, the angle $\gamma$ of the unitarity triangle, and (in the latter case) the $B^0_d$–$\bar B^0_d$ mixing phase $\phi_d$; the same is true for the normalized branching ratio $\langle B \rangle$, where phase-space and CKM factors as well as $|{\cal P}_{tc}^{K\!K}|^2$ have been factored off. For fixed values of $\gamma$ and $\phi_d$, $\rho_{K\!K}$ and $\theta_{K\!K}$ then span a surface in the ${\cal A}_{\rm CP}^{\rm dir}$–${\cal A}_{\rm CP}^{\rm mix}$–$\langle B \rangle$ observable space, shown in Fig. 1 for $\phi_d = 47^\circ$ and $\gamma=65^\circ$. (The fringe is defined by $\rho_{K\!K}=1$, the numbers give the value for $\theta_{K\!K}$.) In the SM, any measurement of the three observables has to lie on this surface, which is theoretically clean. Sufficiently accurate measurements of the branching ratio will give strong constraints on possible values for the asymmetries. The form of the surface implies a theoretical [*lower*]{} bound for $\langle B \rangle$ that can be converted into a lower bound for $\mbox{BR}(B_d\to K^0\bar K^0)$ using input from $b\to s$ penguin decays (see [@FR1] for details). With the help of this lower bound, the recent measurement of $B_d\to K^0\bar K^0$ [@BK0K0exp] was correctly predicted in [@FR1]. Using the latest experimental input and the central values of the factorizable $SU(3)$-breaking parameters, we update the bound in (3) of [@FR1] to ${\rm BR}(B^0_d \to \bar K^0 K^0)> 1.43\,^{+0.17}_{-0.25}$, nicely consistent with the old result and the recent measurements (see Table \[BRtable\]). We observe that the measured ${\rm BR}(B^0_d \to \bar K^0 K^0)$ is right at the lower theoretical bound (bottom of the surface in Fig. 1). This implies a value of $\rho_{K\!K}$ significantly different from 0, with a small phase $\theta_{K\!K}$; $\rho_{K\!K}$ can be related to a hadronic $B\to \pi K$ parameter through $\rho_{\rm c}=\epsilon \rho_{K\!K}$, where $\epsilon\equiv\lambda^2/(1-\lambda^2)=0.053$. This quantity is usually neglected. However, a value of $\rho_{\rm c}\sim 0.05$, as suggested by ${\rm BR}(B^0_d \to \bar K^0 K^0)$, would be rather welcome in the analysis of the $B\to \pi K$ system [@UPDATE]. General Lower Bounds on the Branching Ratios of $b\to d$ Penguin Processes ========================================================================== The mechanism that provided the lower bound on ${\rm BR}(B^0_d \to \bar K^0 K^0)$ is actually more general. We will now first use it to derive lower bounds on $b\to d \gamma$ processes, and then discuss the general $b\to d$ penguin case. The amplitude for the decay $\bar B \to \rho\gamma$ can be written as $$\label{Ampl-Brhogam} A(\bar B \to \rho\gamma)=c_\rho \lambda^3 A {\cal P}_{tc}^{\rho\gamma} \left[1-\rho_{\rho\gamma}e^{i\theta_{\rho\gamma}}e^{-i\gamma}\right],$$ where $c_\rho=1/\sqrt{2}$ and 1 for $\rho=\rho^0$ and $\rho^\pm$, respectively, and $A=|V_{cb}|/\lambda^2$. Moreover, ${\cal P}_{tc}^{\rho\gamma}\equiv{\cal P}_t^{\rho\gamma}-{\cal P}_c^{\rho\gamma}$, where ${\cal P}_t^{\rho\gamma}$ and ${\cal P}_c^{\rho\gamma}$ are matrix elements of operators from the standard weak effective Hamiltonian (see [@FR2] for details). $\rho_{\rho\gamma}e^{i\theta_{\rho\gamma}}$ is again a hadronic parameter that we will treat as essentially unknown. Let us now use the information offered by the $b\to s$ counterpart of our $b\to d$ transition, which is well measured and takes an amplitude of the following form: $$\label{Ampl-BKastgam} A(\bar B \!\to\! K^\ast \!\gamma)\!=-\! \frac{\lambda^3 \! A {\cal P}_{tc}^{K^\ast\!\gamma}}{\sqrt{\epsilon}} \! \left[1\!+\!\epsilon\rho_{K\!^\ast\!\gamma}e^{i\theta_{K\!^\ast\!\gamma}} e^{-i\!\gamma}\right]\!,$$ where $\epsilon$ was introduced above. The ratio of the corresponding BRs is then given by $$\label{rare-ratio} \frac{\mbox{BR}(\bar B \to \rho \gamma)}{\mbox{BR}(\bar B \to K^\ast \gamma)}=\epsilon \left[\frac{\Phi_{\rho\gamma}}{\Phi_{K\!^\ast\gamma}}\right] \left|\frac{{\cal P}_{tc}^{\rho\gamma}}{{\cal P}_{tc}^{K\!^\ast\gamma}} \right|^2 H^{\rho\gamma}_{K\!^\ast\gamma},$$ where $\Phi_{\rho\gamma}$ and $\Phi_{K\!^\ast\gamma}$ denote phase-space factors, and $$H^{\rho\gamma}_{K\!^\ast\gamma}\equiv \frac{1-2\rho_{\rho\gamma}\cos\theta_{\rho\gamma}\cos\gamma+ \rho_{\rho\gamma}^2}{1+2\epsilon\rho_{K\!^\ast\gamma} \cos\theta_{K\!^\ast\gamma} \cos\gamma+\epsilon^2\rho_{K\!^\ast\gamma}^2}.$$ Although $\rho_{K\!^\ast\gamma}e^{i\theta_{K\!^\ast\gamma}}$ is here strongly suppressed by $\epsilon$, we can straightforwardly include the corresponding corrections by using the flavour-symmetry relation $\rho_{\rho\gamma}e^{i\theta_{\rho\gamma}}= \rho_{K\!^\ast\gamma}e^{i\theta_{K\!^\ast\gamma}}\equiv \rho e^{i\theta}$. Treating then $(\rho,\theta)$ as completely free parameters, we can derive the following lower bound: $$\label{H-bound} H^{\rho\gamma}_{K\!^\ast\gamma}\geq \left[1-2\epsilon \cos^2\gamma+{\cal O}(\epsilon^2)\right]\sin^2\gamma,$$ which can be converted into a lower bound for $\bar B \to \rho \gamma$ through (\[rare-ratio\]) and the measured value of the $\bar B \to K^\ast \gamma$ branching ratio. Taking into account phase-space effects and factorizable $SU(3)$-breaking corrections, we obtain the lower bounds given in Table 1. For comparision, we also show the bounds that result from neglecting the $SU(3)$-breaking corrections (“naïve bound”). The bounds are consistent with the experimental results for $B^0_d \to \rho^0 \gamma$ and $ B^+ \to \rho^+ \gamma$, although of course the well-known isospin-breaking puzzle of the Belle result remains. In a similar way we can also derive theoretical lower limits for other $b\to d$ penguin decays. We list bounds for $B^\pm\to K^{(\ast)\pm} K^{(\ast)}$ – together with the respective $b\to s$ decay that was used for the bound – in Table 1 (experimental data are taken from [@HFAG]); some more channels, including also $B^\pm \to \pi/\rho^\pm \ell^+ \ell^-$ modes, are discussed in [@FR2]. For the currently most interesting decays, the theoretical predictions and measurements by BaBar and Belle are also plotted in Fig. 2. It will be interesting to confront our bounds with future data. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- $b\to s$ process Exp. rate $b\to d$ process Naïve bound Bound Belle BaBar ------------------------------- -------------- ---------------------------------- ---------------------------- --------------------------- --------------------------- --------------------------- $B^+ \to K^0 \pi^+$ $24.1\pm1.3$ $ B^0_d \to \bar K^0 K^0$ $ 0.88\,^{+0.11}_{-0.15}$ $ 1.43\,^{+0.17}_{-0.25}$ $ 0.8\pm 0.32$ $ 1.19\,^{+0.42}_{-0.37}$ $B^+ \to K^0 \pi^+$ $24.1\pm1.3$ $ B^+ \to \bar K^0 K^+$ $ $ 1.69\,^{+0.19}_{-0.23}$ $ 1.0\pm 0.41$ $ 1.50\pm 0.51$ 1.03\,^{+0.11}_{-0.14}$ $B^+ \to K^{\ast 0} \pi^+$ $9.7\pm 1.2$ $B^+ \to \bar K^{\ast 0} K^+$ $0.46\,^{+0.06}_{-0.07}$ $0.76\,^{+0.10}_{-0.12}$ $B^+ \to K^{\ast 0} \rho^+$ $10.6\pm1.9$ $B^+ \to \bar K^{\ast 0} K^{\ast $0.46\,^{+0.09}_{-0.10}$ $0.73\,^{+0.15}_{-0.16}$ +}$ $B^0_d \to K^{\ast 0} \gamma$ $40.1\pm2.0$ $ B^0_d \to \rho^0 \gamma$ $ 0.86\,^{+0.10}_{-0.12}$ $ 0.51\,^{+0.13}_{-0.11}$ $ $0.0\pm0.22$ 1.17\,^{+0.36}_{-0.32}$ $B^+ \to K^{\ast +} \gamma$ $40.3\pm2.6$ $ B^+ \to \rho^+ \gamma$ $ 1.73\,^{+0.22}_{-0.26}$ $ 1.03\,^{+0.27}_{-0.23}$ $ $ 0.9\,^{+0.61}_{-0.51}$ 0.55\,^{+0.45}_{-0.39}$ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- : Post-LP2005: [ Belle]{} and [ BaBar]{} [ data]{}, [theory lower bounds.]{} \[BRtable\] [99]{} R. Fleischer and S. Recksiegel, Eur. Phys. J. C [**38**]{} (2004) 251 \[arXiv:hep-ph/0408016\]. R. Fleischer and S. Recksiegel, Phys. Rev. D [**71**]{} (2005) 051501 \[arXiv:hep-ph/0409137\]. B. Aubert [*et al.*]{} \[BABAR Collaboration\], arXiv:hep-ex/0408080;\ K. Abe [*et al.*]{} \[BELLE Collaboration\], arXiv:hep-ex/0506080. A. J. Buras, R. Fleischer, S. Recksiegel and F. Schwab, Acta Phys. Polon. B [**36**]{} (2005) 2015 \[arXiv:hep-ph/0410407\]. http://www.slac.stanford.edu/xorg/hfag/.

‘=11 \#1 =by60 = \#1[[bsphack@filesw [ gtempa[auxout[ ]{}]{}]{}gtempa @nobreak esphack]{} eqnlabel[\#1]{}]{} eqnlabel vacuum \#1 \#1[@underline\#1 $\@@underline{\hbox{#1}}$]{} ‘@=12 October 2001 PAR–LPTHE 01/?\ [**Universality of coupled Potts models**]{}\ [**Vladimir S. Dotsenko (1), Jesper Lykke Jacobsen (2),\ Xuan Son Nguyen (1), and Raoul Santachiara (1)**]{}\ [**(1)**]{} [*LPTHE*]{}[^1], [*Universit[é]{} Pierre et Marie Curie, Paris VI\ Universit[é]{} Denis Diderot, Paris VII\ Boîte 126, Tour 16, 1$^{\it er}$ [é]{}tage\ 4 place Jussieu, F-75252 Paris Cedex 05, France.*]{}\ [**(2)**]{} [*Laboratoire de Physique Théorique et Modèles Statistiques,\ Université Paris-Sud, Bâtiment 100, F-91405 Orsay, France.*]{} .15in **ABSTRACT** > [We study systems of $M$ Potts models coupled by their local energy density. Each model is taken to have a distinct number of states, and the permutational symmetry $S_M$ present in the case of identical coupled models is thus broken initially. The duality transformations within the space of $2^M-1$ multi-energy couplings are shown to have a particularly simple form. The selfdual manifold has dimension $D_M = 2^{M-1}-1$. Specialising to the case $M=3$, we identify a unique non-trivial critical point in the three-dimensional selfdual space. We compare its critical exponents as computed from the perturbative renormalisation group with numerical transfer matrix results. Our main objective is to provide evidence that at the critical point of three different coupled models the symmetry $S_3$ is restored.]{} Introduction {#sec:intro} ============ In the study of coupled models, and also of disordered models in their replica formulation, the permutation group symmetry $S_M$ is supposed to play an essential role [@ludwig; @djlp; @dns]. Namely, the interaction part of the action for a set of $M$ identical coupled models A\_[int]{} \^2x g \_[a b]{} \_[a]{}(x)\_[b]{}(x) is explicitly invariant with respect to any permutation of the models. Here $a$ and $b$ are replica indices, and $\{\varepsilon_a(x)\}$ designates the set of local energy operators. In the lattice definition of such coupled models the interaction part of the Hamiltonian takes a similar form, with only $\int {\rm d}^2x$ replaced by a summation and $\{\varepsilon_a(x)\}$ by an appropriate lattice expression[^2]. When one introduces asymmetric couplings, by generalising the common coupling constant $g$ to a matrix $g_{ab}$, A\_[int]{} \^2x \_[a b]{} g\_[ab]{} \_[a]{}(x) \_[b]{}(x) \[A\_int\] a perturbative renormalisation group (RG) analysis reveals that the $S_M$ symmetry is restored at the fixed point. A detailed study of this scenario, within the $\epsilon$-type perturbative RG for coupled Potts models, was carried out in [@ls]. Supposing all components of $g_{ab}$ to stay of order $\epsilon$, their initial values being all positive[^3], it was shown in [@ls] that the only non-trivial fixed point, having one attractive direction, all other directions being repulsive, is that with $g_{ab} \equiv g$.[^4] This type of restoration of the symmetry $S_M$ could be called “soft universality”. In this paper we are going to argue for a “strong universality” in the criticality of coupled models. We shall mainly be interested in coupling $M=3$ different models[^5], via (\[A\_int\]), with $\{\varepsilon_a\}$ belonging to Potts models with different number of states $\{q_1,q_2,q_3\}$. This breaks the permutational symmetry in a “strong” sense. Still, the RG calculations show the existence of a single fixed point with all $\{g_{ab}\}$ being positive, like it is the case for identical models. To our knowledge, Simon [@simon] was the first to apply the RG analysis to a set of different coupled Potts models. The most general model studied by this author was that of $M_1$ Potts models with $q_1$ states and $M_2$ Potts models with $q_2$ states ($q_1 \neq q_2$), all of them being coupled. After determining the fixed point structure, he computed the dimensions of the spin operators, as well as the RG equations for the energy operators to two-loop order. The effect of disorder on these coupled systems was also analysed. Here we generalise the RG calculations of [@simon] to the case of three different coupled Potts models $(q_1 \neq q_2 \neq q_3)$. We shall compute the dimensions of energy operators, with a special focus on the symmetry of the theory, at the non-trivial fixed point that generalises the one found in [@djlp] for three identical models. Within the space of couplings $\{g_{ab}\}$, the new fixed point is stable in one direction and unstable in the others, the topology of the RG flows being similar to those of [@djlp]. But there is also one apparent difference: The permutational symmetry has disappeared, the coupled models being different. The purpose of this paper will be to provide evidence, by using various methods, that at the fixed point of three different coupled models the apparently lost symmetry $S_3$ is restored, implying a “strong universality”. This symmetry restoration cannot be observed on the level of the initial action (\[A\_int\]), as discussed above, nor is it visible in the perturbative RG treatment, or in the Hamiltonian of the explicit lattice realisation. This is because $\{\varepsilon_{a}\}$ are the energy operators of different models, with different scaling dimensions in particular. The way we shall check for the restoration of the symmetry is by looking at the spectrum of scaling dimensions at the new fixed point, within the sector of energy operators. Like in the case of identical models [@ludwig; @djlp; @dns], the RG analysis implies that the three energy operators of the decoupled models $\{\varepsilon_1(x),\varepsilon_2(x),\varepsilon_3(x)\}$ will rearrange as three particular linear combinations so as to form the new primary operators at the fixed point of the coupled models. In the case of identical models the corresponding linear combinations are easy to guess on symmetry grounds. The irreducible representations (irreps) of the group $S_3$ in the basis $\{\varepsilon_1(x),\varepsilon_2(x),\varepsilon_3(x)\}$ consist of a (symmetric) singlet \_[S]{}(x)=\_[1]{}(x)+\_[2]{}(x)+ \_[3]{}(x) \[symm\] and an (antisymmetric) doublet \[antisym\] that act as the new primary operators at the fixed point [@ludwig; @djlp; @dns]. The fact that the operators $\varepsilon_{\rm A_1}$ and $\varepsilon_{\rm A_2}$ belong to the same two-dimensional irrep means that their dimensions must coincide: $\Delta(\varepsilon_{\rm A_1})=\Delta(\varepsilon_{\rm A_2})$. These are however in general different from the dimension $\Delta(\varepsilon_{\rm S})$ of the one-dimensional irrep. When coupling different models, the corresponding linear combinations of $\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}$ will have more complicated coefficients which have to be calculated by the RG technique. One will find something of the form: \[lin-comb\] Since the initial dimensions $\Delta(\varepsilon_{1}),\Delta(\varepsilon_{2}), \Delta(\varepsilon_{3})$ of the decoupled models differ, one may expect that the critical dimensions (RG eigenvalues) of the newly formed primary operators $\varepsilon_1^*,\varepsilon_2^*,\varepsilon_3^*$ might all be different. Our argument is that, in the case of coupling different models, it is not the linear combinations (\[lin-comb\]) which have to be examined to analyse the symmetry, but rather the spectrum of their critical dimensions. Permuting $\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}$ in the combinations (\[lin-comb\]) does not make much sense because they are different. To permute $\varepsilon_{1}^{\ast}, \varepsilon_{2}^{\ast}, \varepsilon_{3}^{\ast},$ one first has to know their properties, their scaling dimensions, to decide if it makes sense or not. The conclusion will be that one has to study the spectrum of dimensions at the new fixed point. This provides a representation independent information, independent of the way one has defined the theory initially, as by its action (Hamiltonian) in Eq. (\[A\_int\]). If the symmetry $S_{3}$ is restored then the dimensions (\^\_[1]{}),  (\^\_[2]{}), (\^\_[3]{}) should form a singlet and a doublet, as is the case when one couples initially identical models. The rest of the paper is organised as follows. In section \[sec:RG\] we analyse the case $M=3$ by perturbative RG calculations. As will be explained towards the end of that section, in this case the RG results alone are not sufficient to decide whether the $S_3$ symmetry is restored or not. We therefore propose to study the coupled models through a particular lattice realisation, following [@djlp]. To that end, in section \[sec:dual\], we extend the duality transformations derived in [@Jacobsen] for symmetrically coupled models to the asymmetric case. The resulting relations are valid for any number $M$ of coupled models, and for the most general asymmetric multi-energy couplings. In particular we determine the selfdual solutions, thus simplifying dramatically the subsequent numerical simulations. After recalling briefly, in section \[sec:alg\], the most efficient transfer matrix algorithm constructed in [@djlp], we show how it may be generalised to the case of asymmetrically coupled models. We then turn to the numerical analysis in section \[sec:num\]. We locate the non-trivial critical point on the selfdual manifold for various systems of $M=3$ coupled models, and we provide accurate values of the central charge and the energetic scaling dimensions. We shall find strong evidence for a singlet/doublet spectrum at the new fixed point. This we interpret as a signal of the restoration of the $S_{3}$ symmetry. Finally, section \[sec:disc\] is devoted to remarks and conclusions. Renormalization group analysis {#sec:RG} ============================== In the continuum limit, the three coupled Potts models can be represented by the action A &=& \^3\_[a=1]{} A\^[(a)]{}\_[0]{}+A\_[int]{}\ A\_[int]{} &=& \^2 x \^3\_[a b]{} g\^[0]{}\_[ab]{} \_[a]{}(x) \_[b]{}(x) \[A\_int2\] Here $\{A_0^{(a)}\}$ represents three decoupled models, or more precisely the three decoupled conformal field theories for these models at their respective critical points. The interaction term $A_{\rm int}$ makes them coupled. In general, the initial couplings $\{g^{0}_{ab}\}$ are taken to be different. Since $\{\varepsilon_{a}\}$ have different dimensions when the models are different, even if we start from identical couplings $g^{0}_{ab}=g_{0}$, they will become different in the course of renormalisation. We shall parametrise the dimensions of the energy operators $\{\varepsilon_{a}\}$ as in the papers [@djlp; @ls; @simon; @dpp1]: (\^[(a)]{}\_)\_[phys]{} = 1 - \_a where the physical dimension $(\Delta^{(a)}_{\varepsilon})_{\rm phys}$ corresponds to twice the conformal dimension, as usual. The quantity $\epsilon_{a}$, which measures the deviation from the Ising model, appears also in the Coulomb gas parameter (\^[(a)]{}\_[+]{})\^[2]{}=-\_a For the Ising model one has $\epsilon=0$, whence $\alpha_+^2=\frac43$ and $(\Delta_\varepsilon)_{\rm phys}=1$. The parameter $\alpha_+^2$ is useful in analytic calculations because it is simpler to use than the central charge of the corresponding conformal theory. The details of the RG calculations are the same as in Refs. [@simon; @dpp1]. In particular, all the necessary integrals have been calculated in these papers. The generalisation to the case of three different models is straightforward, and we shall therefore only give the final results. It turns out to be convenient to redefine the expansion parameters $\{\epsilon_{a}\}$ and the coupling constants $\{g_{ab}\}$ as follows: \_[a]{} = \_[a]{} g\_[ab]{} = \_[ab]{} \[conventions\] From now on we shall adopt this convention, omitting the tildes throughout for simplicity of notation. The $\beta$-functions for the interaction (\[A\_int2\]) are found in the form (to two-loop order): \[beta\] Here $\xi$ is the RG parameter, $g_{ab}=g_{ba}$ by symmetry of the action (\[A\_int2\]), and we have defined $\epsilon_{ab}\equiv\epsilon_{a}+\epsilon_{b}$. The renormalisation of the energy operators, to second order, is found to be given by the equations =-(1-\_[a]{})\_[a]{}- \_[ba]{}g\_[ab]{}\_[b]{}-(\_[da]{}(g\_[ad]{})\^[2]{}) \_[a]{} In matrix form this reads &=& -\^3\_[b=1]{} \_[ab]{} \_[b]{}\ \_[ab]{} &=& (1-\_[a]{})\_[ab]{}-\_[ab]{}({g\_[cd]{}})\ \_[ab]{} &=& ( [ccc]{} -(g\^[2]{}\_[12]{}+g\^[2]{}\_[13]{})& -g\_[12]{} & -g\_[13]{}\ -g\_[12]{} & -(g\^[2]{}\_[21]{}+g\^[2]{}\_[23]{}) & -g\_[23]{}\ -g\_[13]{} & -g\_[23]{} & -(g\^[2]{}\_[31]{}+g\^[2]{}\_[32]{}) ) To define the new primary operators and their scaling dimensions at the new fixed point we shall have to diagonalise the matrix $\Delta_{ab}$. It is however more convenient to regroup the terms $-\epsilon_{a}\delta_{ab}-\gamma_{ab}$ so that \_[ab]{} &=& \_[ab]{}-\_[ab]{} \[Lambda\]\ \_[ab]{} &=& ( [ccc]{} \_[1]{} - (g\^[2]{}\_[12]{} + g\^[2]{}\_[13]{}) & -g\_[12]{} & -g\_[13]{}\ -g\_[12]{} & \_[2]{} - (g\^[2]{}\_[21]{} + g\^[2]{}\_[23]{}) & -g\_[23]{}\ -g\_[13]{} & -g\_[23]{} & \_[3]{}-(g\^[2]{}\_[31]{} + g\^[2]{}\_[32]{}) ) and we need only diagonalise the matrix $\Lambda_{ab}$. The non-trivial zeros of the $\beta$-functions (\[beta\]) are found as \[zeros\] They correspond to the non-trivial fixed point of the coupled models which we are interested in. It is readily checked that the fixed point (\[zeros\]) is stable in one direction and unstable in the two others, as is the case when one couples identical models. Furthermore, the topology of the RG flows is similar. Substituting the fixed point values of the couplings in the matrix $\Lambda_{ab}$ and diagonalising it, one obtains, after some algebra, the following expressions for the eigenvalues: \_[1]{} &=& -+ \[lambda1\]\ \_[2,3]{} &=& -\ && { 6a\^[2]{}b\^[2]{}c\^[2]{}-2abc(a\^[2]{}b+ab\^[2]{}+ a\^[2]{}c+ac\^[2]{}+b\^[2]{}c+bc\^[2]{}+a\^[3]{}+b\^[3]{}+c\^[3]{}) .\ &+& . a\^[2]{}b\^[2]{}(a+b)\^[2]{}+a\^[2]{}c\^[2]{}(a+c)\^[2]{}+b\^[2]{}c\^[2]{}(b+c)\^[2]{} }\^[1/2]{} \[lambda23\] where we have simplified the notation by means of the abbreviations $a=\epsilon_{12}$, $b=\epsilon_{13}$ and $c=\epsilon_{23}$. According to the definition (\[Lambda\]) of $\Lambda_{ab}$, the dimensions of the new primary operators $\{\varepsilon^{\ast}_a\}$, cf. Eq. (\[lin-comb\]), are related to the above eigenvalues through (\^\_a) = 1- \_[a]{} \[dims\] for $a=1,2,3$. We recall that all these calculations have been done to second order in $\epsilon$. It should be remarked that in the special case of two of the parameters being equal, $b=c$ and $a \neq b$, the expressions for the dimensions simplify considerably. One finds: (\^\_[1]{}) &=& 1+ -\ (\^\_[2]{},\^\_[3]{}) &=& 1-+ At first order in $\epsilon$, the RG results for the dimensions (\^\_[1]{}) &=& 1+ \[order1a\]\ (\^\_[2]{})=(\^\_[3]{}) &=& 1- \[order1b\] form a singlet and a doublet, in accordance with the scenario for the restoration of the $S_{3}$ symmetry discussed in the Introduction. The degeneracy of the doublet is however lifted at order $\epsilon^2$, by the last term in (\[lambda23\]). This is true even when two of the models are identical, and only for $a=b=c$ does one recover the degeneracy $\Delta(\varepsilon^{\ast}_{2})=\Delta(\varepsilon^{\ast}_{3})$ [@djlp]. In section \[sec:num\], where the results of our numerical work are presented, we shall show that this last result of the perturbative RG is wrong[^6]. We provide evidence that the splitting between the dimensions of $\varepsilon^{\ast}_{2}$ and $\varepsilon^{\ast}_{3}$ is actually zero, and that the symmetry restoration scenario therefore holds true. We could suggest the following argument for the failure of the perturbative RG. The $\epsilon$-expansion calculations are valid for perturbed conformal theories because they respect the conformal symmetry, just as dimensional regularisation is valid in the context of pertubative calculations in a gauge theory because the method respects the gauge symmetry (otherwise the results would be dependent on the regularisation technique). In the present problem, the $\epsilon$-regularisation should be correct as far as the conformal symmetry alone is concerned. But in case of extra symmetries, such as $S_{3}$, the method might well give wrong results. The RG formulae (\[lambda1\])–(\[dims\]) for the scaling dimensions, and for the central charge which will be given below, are still quite useful. Apart from the degeneracy issue just discussed, they compare well with the numerical results of section \[sec:num\] when $\{\epsilon_a\}$ are small enough. As far as the dimensions $\Delta(\varepsilon^{\ast}_{2}), \Delta(\varepsilon ^{\ast}_{3})$, defined by Eqs. (\[lambda23\])–(\[dims\]), are concerned, it is their mean value which compares well with numerical results. It is worth noticing that the splitting, namely the third term in (\[lambda23\]), is numerically smaller than the principal $\epsilon^{2}$ term, the second term in (\[lambda23\]). This is because of the compensation of negative and positive terms in the third term of (\[lambda23\]).[^7] The central charge can be obtained in a simple way using Zamolodchikov’s $c$-theorem [@ctheorem]. This theorem provides us with a function of the couplings $c(g_{12},g_{13},g_{23})$ which decreases along the renormalisation flow and takes a value $c(g_{12}^{*},g_{13}^{*},g_{23}^{*})$ at the fixed point of the flow which equals the central charge of the associated conformal field theory. With the conventions (\[conventions\]) taken into account, the $c$-function is uniquely determined by &=& - \_[ab]{}(g\_[12]{},g\_[13]{},g\_[23]{}) \[betadif\] \[ccc1\]\ c(0,0,0) &=& c\_0 c\_1 + c\_2 + c\_3 \[ccc2\] where $c_0$ is the total central charge of three decoupled models. From Eq. (\[beta\]) and Eqs. (\[ccc1\])–(\[ccc2\]) the $c$-function turns out to be given by: c(g\_[12]{},g\_[13]{},g\_[23]{}) = c\_[0]{} - (a g\_[12]{}\^[2]{}+b g\_[13]{}\^[2]{}+c g\_[23]{}\^[2]{}) + g\_[12]{}g\_[13]{}g\_[23]{} +(g\_[12]{}\^[2]{}g\_[13]{}\^[2]{}+ g\_[12]{}\^[2]{} g\_[23]{}\^[2]{}+g\_[13]{}\^[2]{}g\_[23]{}\^[2]{}) \[c\_func\] At the fixed point we insert Eq. (\[zeros\]) into Eq. (\[c\_func\]); up to the order $\epsilon^{4}$ the correction $\Delta c$ of the central charge is: c = -abc +(abc\^[2]{}+bca\^[2]{}+cab\^[2]{}) \[c\_RG\] Duality transformations {#sec:dual} ======================= The possibility of endowing the Potts model with a duality transformation was one of the main motivations for introducing it [@Potts53]. The study of duality for several Potts models coupled by their local energy density was initiated by Domany and Riedel, who worked out the case of two models with $q_1$ and $q_2$ states [@Domany]. Technically, these authors used a particular version of the method of lattice Fourier transforms [@Savit], due to Wu and Wang [@Wu76]. Dotsenko [*et al.*]{} generalised the computations to three symmetrically coupled $q$-state models [@djlp]. However, it became clear that the complexity of the Fourier method grew rapidly as the number of models to be coupled was increased. Recently, it was pointed out by Jacobsen that trading the Potts spin variables for a formulation in terms of random clusters [@Kasteleyn], or loops [@Baxter82], the duality relations simplified dramatically. This observation made it possible to work out the case of $M$ coupled $q$-state Potts models, with the most general coupling by local energy densities consistent with an $S_M$ symmetry [@Jacobsen]. We now show how to treat the even more general case, where each model does not necessarily have the same number of states, and the coupling by local energy densities is the most general one. General case ------------ Consider a set ${\cal L}^M$ of $M$ identical planar lattices ${\cal L}$, which we imagine to be stacked on top of one another. On each lattice site $i \in {\cal L}$, and for each layer $\mu = 1,2,\ldots,M$, we define a Potts spin $\sigma^{(\mu)}_i$ taking the values $\sigma^{(\mu)}_i = 1,2,3,\ldots,q_\mu$. The layers interact by means of the reduced Hamiltonian = \_[ij ]{} [H]{}\_[ij]{}, where $\langle ij \rangle$ denotes the set of lattice edges, and the nearest-neighbour interaction is defined as \_[ij]{} = - \_ K\_ \_ ( \^[()]{}\_i,\^[()]{}\_j ). \[Hamil\] By definition, the Kronecker delta function $\delta(x,y) = 1$ if $x=y$, and zero otherwise. We have here defined ${\cal E}$ as the product set $\prod_{\mu=1}^M \{ \emptyset,{\cal L}^{(\mu)} \}$, so any one of the $2^M$ subsets in ${\cal E}$ can be interpreted as a certain subset of layer indices. One can also think of $\ell \subset {\cal E}$ as specifying the [*state*]{} of a given edge $\langle ij \rangle$, meaning that in the interaction term (\[Hamil\]), $\ell$ determines which of the layers contribute to the product $\prod_{\mu \in \ell}$ of the corresponding delta functions. The layers specified by some $\ell \subset {\cal E}$ then interact by means of the product of their local energy densities, through a coupling constant $K_{\ell}$. For later convenience, we shall represent a subset $\ell \subset {\cal E}$ as a list of $M$ open ($\circ$) or filled ($\bullet$) circles, the $\mu$th circle indicating respectively the absence (or presense) of the factor $\delta \big( \sigma^{(\mu)}_i,\sigma^{(\mu)}_j \big)$. Furthermore, we sometimes use a single open (resp. filled) circle as an abbreviation of a list of $M$ open (resp. filled) circles. When $M=1$, the model defined by Eq. (\[Hamil\]) reduces to the conventional Potts model, whilst for $M=2$ it is identical to the Ashkin-Teller like model considered in Ref. [@Domany]. For $M=3$, it is the asymmetric version of the model discussed in Ref. [@djlp], which had $q \equiv q_1=q_2=q_3$, and whose couplings possessed an $S_3$ symmetry upon permutation of the layers. Using the above symbolic notation, this symmetry can be expressed by the identities K\_1 && K\_ = K\_ = K\_\ K\_2 && K\_ = K\_ = K\_\ K\_3 && K\_ . By means of a generalised Kasteleyn-Fortuin transformation [@Kasteleyn] the local Boltzmann weights can be recast as (-[H]{}\_[ij]{}) = \_ , \[Boltzmann\] where we have simply used that $\exp(K \delta) = 1+[\exp(K)-1]\delta$, if $\delta$ can only take the values 0 and 1. Since furthermore $\delta^2 = \delta$, expanding the product over $\ell$ will lead to the equivalent form (-[H]{}\_[ij]{}) = b\_+ \_ b\_ \_ ( \^[()]{}\_i,\^[()]{}\_j ), \[Boltzmann1\] defining the coefficients $b_\ell$. The normalisation of Eq. (\[Boltzmann\]) is expressed by the fact that $b_\circ = 1$. To relate the $b_\ell$ to the physical coupling constants $K_\ell$, we evaluate Eqs. (\[Boltzmann\]) and (\[Boltzmann1\]) in the situation where $\delta(\sigma^{(\mu)}_i,\sigma^{(\mu)}_j)=1$ for $\mu \in \ell'$, and zero otherwise. This resulting equations (\_[’]{} K\_) = \_[’]{} b\_can readily be solved by applying the principle of inclusion-exclusion [@Birkhoff], yielding b\_= \_[’ ]{} (-1)\^[|| - |’|]{} ( \_[’]{} K\_). \[b-from-K\] The partition function in the spin representation Z = \_ \_[ij ]{} (-[H]{}\_[ij]{}) can now be transformed into the random cluster representation as follows. First, insert Eq. (\[Boltzmann1\]) on the right-hand side of the above equation, and imagine expanding the product over the lattice edges $\langle ij \rangle$. To each term in the resulting sum we associate an edge colouring ${\cal G}$ of ${\cal L}^M$, where an edge $\langle ij \rangle$ in layer $\mu$ is considered to be coloured (occupied) if the term contains the factor $\delta(\sigma^{(\mu)}_i,\sigma^{(\mu)}_j)$, and uncoloured (empty) if it does not. In other words, an edge colouring ${\cal G}$ is determined by specifying an edge state $\ell_{ij} \subset {\cal E}$ for every edge $\langle ij \rangle \in {\cal L}$. The summation over the spin variables $\{ \sigma \}$ is now trivially performed, yielding a factor of $q_\mu$ for each connected component (cluster) in layer $\mu$ of the colouring graph. Keeping track of the prefactors multiplying the $\delta$-functions, using Eq. (\[Boltzmann1\]), we conclude that Z = \_[G]{} \_ b\_\^[B\_]{} \_ q\_\^[C\_]{}, \[Z-cluster\] where $C_\mu$ is the number of clusters in layer $\mu$, and $B_\ell$ is the number of edges $\langle ij \rangle \in {\cal L}^M$ having the state $\ell$. It is worth noticing that the random cluster description of the model has the advantage that the $q_\mu$ only enter as parameters. By analytic continuation one can thus give meaning to a non-integer number of states. The price to be paid is that the $C_\mu$ are, a priori, non-local quantities. In terms of the edge variables $b_\ell$ the duality transformation of the partition function is easily worked out. For simplicity we shall assume that the couplings constants $K_\ell$ are identical between all nearest-neighbour pairs of spins. The generalisation to an arbitrary inhomogeneous distribution of couplings is trivial; it suffices to let $K_\ell$ depend on $\langle ij \rangle$ in Eq. (\[Boltzmann\]). By definition, we take the colouring state $\ell \subset {\cal E}$ of the edge $\langle ij \rangle \in {\cal L}$ to be dual to the complementary colouring $\ell^*$ of its intersecting dual edge $\widetilde{\langle ij \rangle} \in \tilde{\cal L}$. In the symbolic notation introduced above, the complementarity operation $*$ simply means replacing every $\bullet$ by a $\circ$, and vice versa. Also, note that we refer to dual quantities by a tilde throughout. To establish the duality transformations, we begin by postulating that the configuration ${\cal G}_{\bullet}$ with all lattice edges coloured, be dual to the configuration ${\cal G}_{\bullet}^* \equiv {\cal G}_{\circ}$ with no coloured (dual) edge. This requirement fixes the constant entering the duality transformation. Indeed, from Eq. (\[Z-cluster\]), we find that ${\cal G}_{\bullet}$ has weight $b_{\bullet}^E \prod_{\mu=1}^M q_\mu$, where $E$ is the total number of lattice edges, and ${\cal G}_{\circ}$ is weighted by $\tilde{b}_{\circ}^E \prod_{\mu=1}^M q_\mu^F$, where $F$ is the number of faces, including the exterior one. We thus seek for a duality transformation of the form = ( \_ / b\_ )\^E \_[=1]{}\^M q\_\^[F-1]{}, where for any configuration ${\cal G}$ the edge weights must transform so as to keep the same [*relative*]{} weight between ${\cal G}$ and ${\cal G}_{\bullet}$ as between ${\cal G}^*$ and ${\cal G}_{\circ}$. An arbitrary colouring configuration ${\cal G}$ entering Eq. (\[Z-cluster\]) can be generated by applying a finite number of changes to ${\cal G}_{\bullet}$, in which an edge of weight $b_\bullet$ is changed into an edge of weight $b_\ell$ for some $\ell \subset {\cal E}$. By such a change, in general, a pivotal bond is removed from the colouring graph in each of some subset $\ell' \subset \ell^*$ of layers, thus creating $|\ell'|$ new clusters in the corresponding layers. The weight relative to that of ${\cal G}_{\bullet}$ will therefore change by $(b_\ell / b_\bullet) \prod_{\mu \in \ell'} q_\mu$. On the other hand, in the dual configuration $\tilde{\cal G}$ a cluster will be lost in each of the layers $(\ell')^* \cap \ell^*$, since each of the $|\ell'|$ new clusters mentioned above will be accompanied by the formation of a [*loop*]{} in $\tilde{\cal G}$. The weight change relative to ${\cal G}_{\circ}$ therefore amounts to $(\tilde{b}_{\ell^*}/\tilde{b}_0) \prod_{\mu \in (\ell')^* \cap \ell^*} q_\mu^{-1}$. Comparing these two changes we see that the factor $\prod_{\mu \in \ell'} q_\mu$ cancels nicely, and the duality transformation takes the simple form \_= (b\_[\^\*]{}/b\_) \_ q\_, \[dual\] the relation with $\ell = \circ$ being trivial. Note in particular that Eq. (\[dual\]) with $\ell = \bullet$ implies that $b_\bullet \tilde{b}_\bullet = \prod_{\mu=1}^M q_\mu$, since $b_\circ=1$ by definition. Then, dualising Eq. (\[dual\]) once again yields = \_ q\_ \_[\^\*]{} q\_= b\_, so that the duality transformation is indeed involutive, as required. The duality relations (\[dual\]) can be recast in an even simpler form by trading the random clusters for the loops surrounding them (and their duals) on the lattice ${\cal L}_{\rm m}$ medial to ${\cal L}$ [@Baxter82]. Using the Euler relation, we find that Eq. (\[Z-cluster\]) must be replaced by Z = ( \_[=1]{}\^M q\_)\^[N/2]{} \_[G]{} \_ x\_\^[B\_]{} \_ q\_\^[L\_/2]{}, \[Z-loop\] where, by a slight abuse of notation, we use the same notation ${\cal G}$ for the loop and the cluster configurations, since they are in bijective correspondence. $L_\mu$ are now the number of closed loops in layer $\mu$, and $N$ is the total number of vertices in ${\cal L}$. Note that the bond weights $b_\ell$ have now been replaced by x\_= b\_\_ q\_\^[-1/2]{}. It is easily verified that the duality relations (\[dual\]) now simply read \_= x\_[\^\*]{}. \[dual-loop\] This is our main result. In the case of a lattice ${\cal L} = \tilde{\cal L}$ which is unchanged by the duality, such as the infinite square lattice, we can now search for selfdual solutions. These are obtained by imposing $\tilde{b}_\ell = b_\ell$, and read explicitly x\_\^[(s.d.)]{} = x\_[\^\*]{}\^[(s.d.)]{}. \[selfdual-loop\] Since $x_\circ = 1$ by normalisation, the selfdual manifold has dimension $D_M = 2^{M-1}-1$. Two special points always belong to the selfdual manifold. The first one is x\_= 1,     . \[special1\] It is straightforward to verify that in terms of the original couplings this means $K_{\circ \cdots \circ \bullet \circ \cdots \circ} = 1 + q_\mu^{1/2}$ (with the unique $\bullet$ at position $\mu$) for $\mu=1,2,\ldots,M$, all other $K_\ell$ being zero. In other words, this is just $M$ non-interacting selfdual (critical) Potts models. The other special point is x\_= x\_= 1,     . \[special2\] In terms of the original couplings this means $K_\bullet = 1 + \prod_{\mu=1}^M q_\mu^{1/2}$, all other $K_\ell$ being zero. In this case, the $M$ models effectively couple so as to form a [*single*]{} critical Potts model with $\prod_{\mu=1}^M q_\mu$ states. Two models ---------- Let us briefly show how to recover the result for $M=2$ [@Domany] from the compact formulation of Eq. (\[dual-loop\]). Introducing the shorthand notation $\delta_\mu = \delta \big( \sigma_i^{(\mu)},\sigma_j^{(\mu)} \big)$ for $\mu=1,2$, the Hamiltonian (\[Hamil\]) reads in this case -[H]{}\_[ij]{} = K\_ \_1 + K\_ \_2 + K\_ \_1 \_2. From Eq. (\[b-from-K\]) we have x\_ b\_ &=& [e]{}\^[K\_]{} - 1\ x\_ b\_ &=& [e]{}\^[K\_]{} - 1\ x\_ b\_ &=& [e]{}\^[K\_+K\_+K\_]{} - ( [e]{}\^[K\_]{} + [e]{}\^[K\_]{} ) + 1, and using the selfduality criteria $x_{\bullet \circ} = x_{\circ \bullet}$ and $x_{\bullet \bullet} = x_{\circ \circ} \equiv 1$ we readily find the solutions \^[K\_]{} &=& 1 + ( [e]{}\^[K\_]{}-1 )\ [e]{}\^[K\_]{} &=& . Three models ------------ For the case $M=3$, the Hamiltonian (\[Hamil\]) reads -[H]{}\_[ij]{} &=& K\_ \_1 + K\_ \_2 + K\_ \_3 +\ & & K\_ \_2 \_3 + K\_ \_1 \_3 + K\_ \_1 \_2 + K\_ \_1 \_2 \_3. From Eq. (\[b-from-K\]) we have b\_ &=& [e]{}\^[K\_]{} - 1\ b\_ &=& [e]{}\^[K\_ + K\_ + K\_]{} - ( [e]{}\^[K\_]{} + [e]{}\^[K\_]{} ) + 1\ b\_ &=& [e]{}\^[K\_+K\_+ K\_ + K\_ + K\_ + K\_ + K\_]{} -\ & &( [e]{}\^[K\_ + K\_ + K\_]{} + [e]{}\^[K\_ + K\_ + K\_]{} + [e]{}\^[K\_ + K\_ + K\_]{} ) +\ & & ( [e]{}\^[K\_]{} + [e]{}\^[K\_]{} + [e]{}\^[K\_]{} ) - 1, where the first and second line each represent three equations that can be obtained by cyclically rotating the layer indices. Imposing the selfduality criteria $\sqrt{q_1} \, b_{\circ \bullet \bullet}= \sqrt{q_2 q_3} \, b_{\bullet \circ \circ}$ and $b_{\bullet \bullet \bullet} = \sqrt{q_1 q_2 q_3}$ we obtain after a little algebra \^[K\_]{} &=&\ [e]{}\^[K\_]{} &=& \^[K\_+K\_+K\_]{}\ A &=& ( [e]{}\^[K\_]{} - 1 ) q\_2 q\_3 + ( [e]{}\^[K\_]{} - 1 ) q\_1 q\_3 + ( [e]{}\^[K\_]{} - 1 ) q\_1 q\_2 +\ & & ( [e]{}\^[K\_]{}+[e]{}\^[K\_]{}+ [e]{}\^[K\_]{}-2 ) + q\_1 q\_2 q\_3\ B &=&\ & &\ & & . These expressions generalise those given in Ref. [@djlp]. It should be clear that for higher $M$, manipulating such expressions by direct use of the Fourier method [@Wu76] becomes extremely cumbersome. Transfer matrix algorithm {#sec:alg} ========================= In Ref. [@djlp] it was shown that the transfer matrix $T(L)$ for coupled Potts models on semi-infinite strips of width $L$ may be written in a variety of ways. For a given $L$, the smaller the dimension of $T(L)$ the more efficient will be the computations, since both time and memory consumption increase roughly linearly with ${\rm dim} \; T(L)$. The best choice turns out to be to write the transfer matrix for the loop model on the medial lattice ${\cal L}_{\rm m}$ [@Baxter82], which was referred to as algorithm ${\tt alg4}$ in [@djlp]. This also gives us the advantage of having simple duality relations, cf. Eq. (\[dual-loop\]), and to be able to treat the numbers of states $\{q_1,q_2,q_3\}$ as continuous parameters. Let us recall from Eqs. (\[Z-loop\]) and (\[selfdual-loop\]) that the partition function for $M=3$ coupled models on the selfdual manifold can be written, up to a trivial multiplicative constant, as Z = \_[G]{} x\_1\^[B\_+B\_]{} x\_2\^[B\_+B\_]{} x\_3\^[B\_+B\_]{} q\_1\^[L\_1/2]{} q\_2\^[L\_2/2]{} q\_3\^[L\_3/2]{} \[Z3\] where we have put $x_1=x_{\bullet\circ\circ}$, $x_2=x_{\circ\bullet\circ}$ and $x_3=x_{\circ\circ\bullet}$ for brevity. For simplicity, we shall take the lattice ${\cal L}$ to be the square one, so that ${\cal L}_{\rm m}$ is once again a square lattice. In the loop picture, the symbols $\bullet$ and $\circ$ refer to the two different ways of splitting the vertex, with a definition that alternates between the even and the odd sublattice; see Fig. \[fig:medial\]. But thanks to the selfduality, Eq. (\[Z3\]) is invariant with respect to a global colour conjugation $\bullet \leftrightarrow \circ$, and so one might as well forget about the distinction between the sublattices[^8]. The number of occurrences of a given vertex splitting (and hence the $B$’s) can be counted locally, and are thus easily realised as local Boltzmann factors in the transfer matrix. The $L_\mu$ in (\[Z3\]) count the number of closed loops in each layer $\mu=1,2,3$. Despite of the non-local nature of the loops, these quantities do not obviate the construction of the transfer matrix. Rather, they can be counted locally by writing the transfer matrix in the basis of Catalan-like connectivities, as described in [@djlp]. Formally, if ${\cal C}_k$ designates the space of pairwise Catalan connectivities among a set of $L=2k$ points, $T(L)$ acts on the product space ${\cal S}_k = {\cal C}_k \bigotimes {\cal C}_k \bigotimes {\cal C}_k$. One has ${\rm dim}\; {\cal C}_k = \frac{(2k)!}{k!(k+1)!}$. Weighing the loops with a Boltzmann factor $\sqrt{q_\mu}$ that depends on the layer $\mu$ is a trivial modification of [@djlp]. An important difference, however, is that the layers are now distinguishable. Accordingly, one has simply ${\rm dim} \; {\cal S}_k = ({\rm dim}\; {\cal C}_k)^3$. As in Ref. [@djlp] we transfer along one of the main directions of the square lattice ${\cal L}_{\rm m}$, with periodic boundary conditions in the transverse direction. To ensure that ${\cal C}_k$ be well defined, we must take the strip width to be even, $L=2k$. We shall also need to consider the situation where all vertex weights $x$ tend to infinity in fixed ratios. Defining $x_2' = \frac{x_2}{x_1}$ and $x_3' = \frac{x_3}{x_1}$ as the relevant ratios, (\[Z3\]) can be rewritten, once again up to an irrelevant multiplicative constant, as Z’ = \_[G’]{} x\_2’\^[B\_+B\_]{} x\_3’\^[B\_+B\_]{} q\_1\^[L\_1/2]{} q\_2\^[L\_2/2]{} q\_3\^[L\_3/2]{} \[Z3’\] where the symbol ${\cal G'}$ means all colouring figurations in which the local colourings $\circ\circ\circ$ and $\bullet\bullet\bullet$ do not occur. In Ref. [@djlp], the special case $(x_2',x_3')=(1,1)$ with $q_1=q_2=q_3$ was identified as the non-trivial critical fixed point for three identical coupled models. Numerical results {#sec:num} ================= Using sparse matrix factorisation techniques, we have been able to numerically compute the first few eigenvalues of the transfer matrices $T(L)$ for $Z$ (\[Z3\]) and $T'(L)$ for $Z'$ (\[Z3’\]) for even strip widths up to $L_{\rm max}=12$. The largest matrices had dimension ${\rm dim} \; T(L_{\rm max}) = (132)^3$, but the sparse matrix factorisation necessitates the use of intermediate states with $L+2$ dangling loop segments, and so the largest sparse matrices involved were of dimension $(429)^3$. Phase diagram and central charge -------------------------------- As a first test of our algorithm, we checked that it gave the correct eigenvalues at the special points $(x_1,x_2,x_3)=(1,1,1)$ and $(0,0,0)$, cf. Eqs. (\[special1\]) and (\[special2\]). We then extracted the (effective) central charge from the leading eigenvalue $\lambda_0$ in the standard way [@bcn; @affleck]: f\_0(L) = f\_0() - + \[cc\] with $f_0(L) = - \frac{1}{L} \log \lambda_0$. Using the $c$-theorem [@ctheorem], we could readily establish the topology of the RG flows: - In the space $(x_1,x_2,x_3)$, there is a one-parameter curve along which the partial derivatives of the effective central charge with respect to the two perpendicular directions vanish identically. Moreover, the corresponding second derivatives are strictly negative, so that this curve acts as a “mountain ridge” for $c_{\rm eff}$. - The curve passes through the points $(0,0,0)$ and $(1,1,1)$, and then goes to infinity with some fixed ratios $(x_2',x_3') \equiv (x_2'^*,x_3'^*)$ that depend on $(q_1,q_2,q_3)$. - For various values $(q_1,q_2,q_3) \in [2,4]^3$, with at most one $q_a = 2$, we observe that $c_{\rm eff}$ is a monotonically decreasing function when going along the curve from the point $(1,1,1)$ towards either $(0,0,0)$ or $x_1(1,x_2'^*,x_3'^*)$, $x_1 \to \infty$. In the former case, the decrease is rapid and gets more pronounced with increasing system size $L$, signalling the first-order nature ($c_{\rm eff}=0$) of the phase transition in the $(q_1 q_2 q_3)$-state Potts model. In the latter case, the decrease is very slight, distinguishing the point $(x_2'^*,x_3'^*)$ as a candidate for the non-trivial fixed point of three coupled Potts models in the lattice realisation. Based on this evidence, we switched to the matrix $T'(L)$ in order to accurately locate the point $(x_2'^*,x_3'^*)$ and study its critical properties. Invoking again the $c$-theorem [@ctheorem], this was done by scanning the space $(x_2',x_3')$ for a maximum in $c_{\rm eff}$ for different system sizes. As usually [@djlp], we obtained the best precision by including a non-universal $1/L^4$ term in (\[cc\]), and so our fits are based on three consecutive strip widths $L$. The obtained maximum can be interpreted as a finite-size pseudo-critical point, which tends to $(x_2'^*,x_3'^*)$ as $L \to \infty$. We have concentrated the main part of our computation time on two important special cases: A tricritical Ising model and a three-state Potts model coupled with either an Ising model or with a tricritical three-state Potts model. In the language of minimal models, we refer to these two situations as ${\cal M}_{345} \equiv {\cal M}_3 \times {\cal M}_4 \times {\cal M}_5$ and ${\cal M}_{456} \equiv {\cal M}_4 \times {\cal M}_5 \times {\cal M}_6$ respectively. $L$ $x_2'^*$ $x_3'^*$ $(c_{\rm eff})_{\rm max}$ $\Delta c$ --------- ----------- ----------- --------------------------- ------------ 2,4,6 0.8523(1) 0.7133(1) 1.93540138 -0.00058 4,6,8 0.8476(1) 0.7136(1) 1.98040147 -0.00352 6,8,10 0.8481(1) 0.7076(1) 1.99047341 -0.00416 8,10,12 0.8483(1) 0.7061(1) 1.99348414 -0.00443 : \[tab:c345\]Maxima of the effective central charge for the system ${\cal M}_{345}$, and the deviation from the value at the decoupling point. $L$ $x_2'^*$ $x_3'^*$ $(c_{\rm eff})_{\rm max}$ $\Delta c$ --------- ----------- ----------- --------------------------- ------------ 2,4,6 0.9230(1) 0.8679(1) 2.27445933 -0.0115 4,6,8 0.9218(1) 0.8661(1) 2.32198052 -0.0185 6,8,10 0.9213(1) 0.8645(1) 2.33129236 -0.0190 8,10,12 0.9213(5) 0.8645(5) 2.333272 -0.0225 : \[tab:c456\]Maxima of the effective central charge for the system ${\cal M}_{456}$, and the deviation from the value at the decoupling point. The corresponding finite-size estimates for $(x_2'^*,x_3'^*)$ and for $(c_{\rm eff})_{\rm max}$ are given in Tables \[tab:c345\] and \[tab:c456\]. Rather than directly extrapolating $(c_{\rm eff})_{\rm max}$ to the $L\to\infty$ limit, we consider instead for any $L$ its deviation $\Delta c(L)$ with respect to the corresponding value found at the point where the three Potts models decouple. Since this deviation is numerically small, it is reasonable to expect that the finite-size corrections to $c$ at $(x_2'^*,x_3'^*)$ are similar to those at the decoupling point. Tables \[tab:c345\] and \[tab:c456\] confirm that this is indeed the case: the estimates for $\Delta c$ only depend very weakly on $L$. As $L\to\infty$, we obtain finally the following extrapolated values for $\Delta c$ and for the critical couplings: [lll]{} : & (x\_2’\^\*,x\_3’\^\*) = (0.8485(1),0.7053(5)) & c = -0.0050(3)\ : & (x\_2’\^\*,x\_3’\^\*) = (0.9210(5),0.862(2)) & c = -0.025(3)\ The values of $\Delta c$ are seen to compare quite favourably with the two-loop result of the perturbative RG (\[c\_RG\]), whose numerical values read respectively c([M]{}\_[345]{})\_[RG]{} = -0.0041 c([M]{}\_[456]{})\_[RG]{} = -0.0175 Actually, assuming that the RG series is alternating, the fifth-order term not present in (\[c\_RG\]) would supposedly lead to even better agreement with the numerical results. Higher exponents in the even sector ----------------------------------- Examining the higher eigenvalues of the transfer matrices, one has also access to the scaling dimensions by using the formula [@Cardy83] f\_i(L) - f\_0(L) = + \[scal\_dims\] We here work in the even sector of the transfer matrix, and besides the identity operator (the free energy) we expect to find the various energy-like primaries. In particular we are interested in the primary operators $\varepsilon_{1}^{*}$, $\varepsilon_{2}^{*}$, $\varepsilon_{3}^{*}$ discussed in section \[sec:RG\]; cf. Eq. (\[lin-comb\]). According to Eqs. (\[lambda1\])–(\[dims\]) we have $0<\Delta(\varepsilon_{2}^{*}),\Delta(\varepsilon_{3}^{*})< 1$ and $1<\Delta(\varepsilon_{1}^{*})<2$ for small enough $\{\epsilon_a\}$, and so we expect the first three gaps of the transfer matrix to be associated with these operators. As in Ref. [@djlp], the scaling dimensions of the operators have been extracted by adding to Eq. (\[scal\_dims\]) a non-universal $1/L^{4}$ term, and so the fits are based on two consecutive strip widths $L$. We consider first the case of coupling three identical models. At the non-trivial fixed point $(x_{2}',x_{3}')= (1,1)$ the second and third eigenvalues of the transfer matrix are degenerate. These eigenvalues are associated with the operators $\varepsilon_{\rm A_{1}}$, $\varepsilon_{\rm A_{2}}$ of (\[antisym\]) which generate the two-dimensional irrep of $S_3$, as discussed in the Introduction. It is important to notice that in the transfer matrix algorithm of [@djlp] the three layers were treated symmetrically and only the operators which are symmetric under permutation of the layer indices were accessible. In that case, the degenerate eigenvalues associated with the energy doublet were not present in the spectrum of the transfer matrix. The fact that we find them here is a strong confirmation of the antisymmetric nature of the corresponding operators. In Table \[tab3\] we report numerical values of the dimension $\Delta(\varepsilon_{\rm A}) \equiv \Delta(\varepsilon_{\rm A_{1}})=\Delta(\varepsilon_{\rm A_{2}})$ in the case of three coupled Potts models with $q=3$. The final result $\Delta(\varepsilon_{\rm A}) = 0.63(3)$ is in reasonable agreement with the RG value and also agrees well with the Monte Carlo result $\Delta(\varepsilon_{\rm A}) = 0.63 \pm 0.04$ given in [@djlp]. $L$ $\Delta(\varepsilon_{\rm A})$ --------------- ------------------------------- 4,6 0.6768 6,8 0.6648 8,10 0.6525 $L\to \infty$ 0.63(3) RG result 0.72 : \[tab3\]Dimension of the antisymmetric energy operator at $(x_{2}',x_{3}')= (1,1)$ for three coupled three-state Potts models. The fourth eigenvalue of the transfer matrix was already found in [@djlp] and it corresponds to the symmetric energy operator $\varepsilon_{\rm S}$ of Eq. (\[symm\]). It is a non-trivial check of our transfer matrices that the corresponding eigenvalue is identical to that of [@djlp]. We now turn our attention to the case of three different coupled models. In the space $(x_{2}',x_{3}')$ of the couplings we have found that for any given system size $L$ there is a unique point $(x_{2}', x_{3}')_{\rm deg}$, at which the second and third eigenvalues are degenerate. We have determined the location of this point for the models ${\cal M}_{345}$ and ${\cal M}_{456}$ with $L=4,6,8,10$ and it is shown on Figs. \[fig2\]–\[fig3\]. In contradistinction to the case of three coupled identical models, the points $(x_{2}'^*,x_{3}'^*)$ and $(x_{2}', x_{3}')_{\rm deg}$ do not coincide for finite systems. It is however evident from Figs. \[fig2\]–\[fig3\] that these two points become closer as the system size increases. This suggests that in the thermodynamic limit they actually coincide. If this scenario holds true, we have at the critical point that $\Delta(\varepsilon_{3}^{*})=\Delta(\varepsilon_{2}^{*})$. Since the finite-size effects in the sequence of points $(x_{2}', x_{3}')_{\rm deg}$ are much stronger than is the case for $(x_{2}'^*,x_{3}'^*)$, it is reasonable to measure numerical values of the dimensions at the best available estimate for the latter point. We display the corresponding results for $\Delta(\varepsilon_{1}^{*})$, $\Delta(\varepsilon_{2}^{*})$, $\Delta(\varepsilon_{3}^{*})$ in Tables \[tab4\]–\[tab5\], once again for the models ${\cal M}_{345}$ and ${\cal M}_{456}$ respectively. The numerical results are in quite good agreement with the two-loop RG calculation of section \[sec:RG\]. Notice that in the case of $\Delta(\varepsilon_{2}^{*})$, $\Delta(\varepsilon_{3}^{*})$ we compare with the RG values with the splitting ignored. $L$ $\Delta(\varepsilon_{2}^{*})$ $\Delta(\varepsilon_{3}^{*})$ $\Delta(\varepsilon_{1}^{*})$ ----------- ------------------------------- ------------------------------- ------------------------------- 4,6 0.75734 0.7677 1.2991 6,8 0.759 0.763 1.2803 8,10 0.7564 0.75647 1.2761 RG result 0.746 0.771 1.287 : \[tab4\]Dimensions of the energy operators at $(x_{2}'^*,x_{3}'^*)$ for the ${\cal M}_{345}$ model. $L$ $\Delta(\varepsilon_{2}^{*})$ $\Delta(\varepsilon_{3}^{*})$ $\Delta(\varepsilon_{1}^{*})$ ----------- ------------------------------- ------------------------------- ------------------------------- 4,6 0.6849 0.6857 1.2958 6,8 0.6731 0.6756 1.2765 8,10 0.6621 0.6637 1.2745 RG result 0.701 0.728 1.43 : \[tab5\]Dimensions of the energy operators at $(x_{2}'^*,x_{3}'^*)$ for the ${\cal M}_{456}$ model. In Tables \[tab4\]–\[tab5\] we have given the results for $\Delta(\varepsilon_{2}^{*})$ and $\Delta(\varepsilon_{3}^{*})$ separately, although we do in fact believe that the two dimensions coincide. This point of view is corroborated by the fact that the difference between the dimensions of $\varepsilon_{2}^{*}$ and $\varepsilon_{3}^{*}$ decreases rapidly with the lattice size. For $L=8,10$ the numerical results give $\Delta_{\rm splitting}\equiv |\Delta(\varepsilon_{3}^{*})-\Delta(\varepsilon_{2}^{*})|<0.0003$ for the ${\cal M}_{345}$ model and $\Delta_{\rm splitting}<0.002$ for the ${\cal M}_{456}$ model. In order to be sure that the very small values for $\Delta_{\rm splitting}$ obtained at finite size are not somehow accidental, we have determined an upper bound for $\Delta_{\rm splitting}$ by directly extrapolating the difference between the second and third eigenvalues of the transfer matrix at the point $(x_{2}'^*,x_{3}'^*)$ . The result is shown below together with the extrapolated values for the dimensions of the three energy operators: [llll]{} : &          (\_[1]{}\^[\*]{}) = 1.269 0.002 &\ & (\_[2]{}\^[\*]{}),(\_[3]{}\^[\*]{}) = 0.750 0.005 &  & \_[splitting]{} &lt; 0.003\ : &          (\_[1]{}\^[\*]{}) = 1.272 0.002 &\ & (\_[2]{}\^[\*]{}),(\_[3]{}\^[\*]{}) = 0.645 0.003 &  & \_[splitting]{} &lt; 0.002\ The difference $\Delta_{\rm splitting}$ is well below the one predicted by the RG calculation, which reads respectively $0.025$ and $0.027$ for the models ${\cal M}_{345}$ and ${\cal M}_{456}$. The numerical work thus provides clear evidence that at the non-trivial fixed point the splitting of the dimensions of $\varepsilon_{3}^{*}$ and $\varepsilon_{2}^{*}$ is actually zero. Discussion {#sec:disc} ========== In this paper we have shown that coupling $M=3$ different Potts models (with $q_1,q_2,q_3>2$ and not too large) one obtains a unique non-trivial critical point. The critical properties of this point, in particular its central charge and the values of various energetic scaling dimensions, have been determined quite accurately by a perturbative RG treatment and found to be consistent with large-scale numerical simulations. An exception is however the RG prediction that at two-loop order the degeneracy between the two antisymmetric energy operators should be lifted: this prediction has here been discarded on the basis of numerical evidence. An extension of our investigation to the case of spin-like operators will be published elsewhere [@djns]. This work forms part of a larger project [@djlp], in which we examine the possible universality classes of coupled Potts models, and eventually their relation to the random-bond Potts model. In particular, substantial evidence has been accumulated that in the random-bond case replica symmetry is not broken, and one can thus hope to make analytical progress by studying the [*unitary*]{} models that result from coupling a certain number of minimal models. We believe that the symmetry properties of the coupled models play an essential role. It is in the light of this belief that the present work appears to be interesting: even in the absense of an explicit $S_M$ symmetry in the initial action, this symmetry appears to be restored at the non-trivial critical point. As far as a putative CFT classification of $S_M$ symmetrical critical points goes, we therefore see that the number of models to be classified is potentially very large. Indeed, assuming the conclusions of the $M=3$ case to carry over to a general number $M$ of coupled models, one may expect a distinct $S_M$ symmetric universality class to arise from coupling any different set of $M$ minimal models. If this is true, it would call for a substantial number of new CFTs endowed with extended symmetries. Further research along these lines is currently in progress. [10]{} A. W. W. Ludwig, Nucl. Phys. B [**330**]{}, 639 (1990). Vl. S. Dotsenko, J. L. Jacobsen, M.-A. Lewis and M. Picco, Nucl. Phys. B [**546**]{}, 505 (1999). Vl. S. Dotsenko, X. S. Nguyen and R. Santachiara, hep-th/0104197, to appear in Nucl. Phys. B. M.-A. Lewis and P. Simon, Phys. Lett. B [**435**]{}, 159 (1998). P. Simon, Nucl. Phys. B [**515**]{}, 624 (1998). J. L. Jacobsen, Phys. Rev. E. [**62**]{}, R1 (2000). Vl. S. Dotsenko, M. Picco and P. Pujol, Nucl. Phys. B [**455**]{}, 701 (1995), A. B. Zamolodchikov, Pis’ma Zh. Eksp. Teor. Fiz. [**43**]{}, 565 (1986) \[English translation in JETP Lett. [**43**]{} (1986) 730\]; Sov. J. Nucl. Phys. [**46**]{} 1090 (1987). R. B. Potts, Proc. Cambr. Philos. Soc. [**48**]{}, 106 (1952). E. Domany and E. K. Riedel, Phys. Rev. B [**19**]{}, 5817 (1979). R. Savit, Rev. Mod. Phys. [**52**]{}, 453 (1980). F. Y. Wu and Y. K. Wang, J. Math. Phys. [**17**]{}, 439 (1976). P. W. Kasteleyn and C. M. Fortuin, J. Phys. Soc. Jpn. (suppl.) [**26**]{}, 11 (1969). R. J. Baxter, [*Exactly solved models in statistical mechanics*]{} (Academic Press, New York, 1982). G. D. Birkhoff, Ann. Math. [**14**]{}, 42 (1912). H. Bl[ö]{}te, J. L. Cardy and M. P. Nightingale, Phys. Rev. Lett. [**56**]{}, 742 (1986). I. Affleck, Phys. Rev. Lett. [**56**]{}, 746 (1986). J. L. Cardy, J. Phys. A [**16**]{}, L355 (1983). Vl. S. Dotsenko, J. L. Jacobsen, X. S. Nguyen and R. Santachiara, work in progress. [^1]: Unité Mixte de Recherche CNRS UMR 7589. [^2]: In the case of the Potts model with nearest-neighbour spins $\sigma^{(a)}_i$ and $\sigma^{(a)}_j$ in the replica $a$, one has $\varepsilon_a(x) \sim 1-\delta \big(\sigma^{(a)}_i,\sigma^{(b)}_j \big)$. [^3]: In [@ls] the interaction part of the action $A_{\rm int}$ (\[A\_int\]) was actually defined with an extra minus sign, so that initially all the components of $g_{ab}$ were taken to be negative. [^4]: Ref. [@ls] also identified other fixed points, which are related to the basic one (with all $g_{ab}$ equal and of the same sign) by changing the sign of some of its components. This is equivalent to switching the sign of a certain number of energy operators. This last operation is a symmetry of the individual Potts models, corresponding to their self-duality (note however such duality transformations on the individual models are not directly related to the global duality transformations on the entire coupled system to be discussed in section \[sec:dual\]). This argument implies that the critical properties of the various critical points classified in [@ls] should be equivalent. [^5]: The duality transformations of section \[sec:dual\] are however valid for general $M$. [^6]: We would still claim, of course, that our RG calculations are technically correct! [^7]: It is easy to check that the expression under the square-root sign of (\[lambda23\]) is always non-negative, so that $\lambda_{2,3}$ are well defined. [^8]: Using the language of Ref. [@Baxter82], coupled Potts models constitute a staggered vertex model, which however becomes homogeneous exactly on the selfdual manifold.

--- abstract: 'A new parameter-free method is proposed for treatment of single-particle resonances in the real-energy continuum shell model. This method yields quasi-bound states embedded in the continuum which provide a natural generalization of weakly bound single-particle states.' address: 'Grand Accélérateur National d’Ions Lourds (GANIL),CEA/DSM – CNRS/IN2P3, BP 55027, F-14076 Caen Cedex 05, France' author: - 'J.B. Faes and M. P[ł]{}oszajczak' title: 'New method for extracting quasi-bound states from the continuum' --- Introduction {#intro} ============ Many-body states of nuclear shell model (SM) are linear combinations of Slater determinants built of bound single-particle (s.p.) states. Thus, the SM describes bound many-body systems which are isolated from the external environment of scattering states and decay channels. A natural generalization of the SM for weakly bound or unbound many-body systems, the so-called Gamow shell model (GSM), has been formulated recently in Berggren ensemble consisting of bound s.p. states, s.p. resonant states (Gamow [@Gam28] or Siegert [@Sie39] states), and the complex-energy s.p. continuum states [@Mic02a; @Bet02]. A complete set of s.p. states can be defined in the Berggren ensemble [@Ber68; @Ber93]. The complete set of many-body states is then given by all Slater determinants spanned by s.p. states of the complete s.p. set in the Berggren ensemble [@Mic02a; @Bet02]. So defined theoretical framework gives the full description of interplay between scattering states, resonances and bound states in the many-body wave function, without imposing a limit on the number of particles in the scattering continuum. On the other hand, the asymptotic decay channels in GSM are not individually resolved. Hence, this approach cannot be applied for a description of nuclear reactions and remains the tool for nuclear structure studies. The unification of structure and reaction theories is possible in the continuum shell model (CSM) formalism [@Mah69; @Bar77; @Phi77], including the recently developed shell model embedded in the continuum (SMEC) [@Ben99; @Ben00; @Oko03; @Rot05]. In this formalism, the s.p. basis includes bound states and the real-energy continuum states. Feshbach’s projection technique [@Fes58], used in CSM and SMEC, allows to describe on the same footing the nuclear reactions, including the rearrangement term, and the nuclear structure of well-bound, weakly-bound or unbound nuclear states. Bound and scattering s.p. states define two orthogonal subspaces $q_0$ and $p_0$, respectively. The s.p. resonances do not belong to the Hilbert space and have to be regularized before including them in the SMEC framework. The regularization procedure consists of including s.p. resonances in a discrete part of the spectrum after removing the scattering tails which are included in the embedding scattering continuum. These regularized resonances are usually called the quasi-bound states embedded in the continuum (QBSEC). The new subspaces: $q$ including bound and QBSEC s.p. states, and $p$ including non-resonant scattering states and scattering tails of regularized resonances, are subsequently reorthogonalized. The many-body states in SMEC are given by all Slater determinants spanned by s.p. states in $q$ and $p$. Many-body states with all particles occupying bound and QBSEC s.p. states span the $\mathcal{Q} \equiv \mathcal{Q}_0$ subspace of the Hilbert space. The complement subspace $\mathcal{P}$ includes many-body states with one or more particles in the scattering states, hence: $\mathcal{P} \equiv \sum_{i=1}^{A}\mathcal{Q}_i$, where $\mathcal{Q}_{i}$ projects on the space with $i$ particles in the continuum. The number of particles in the scattering continuum provides a natural hierarchy of approximations in the CSM. Technical complications associated with inclusion of multiparticle continua and complex asymptotic channels are such that none of the early CSM or SMEC studies considered more than one particle in the continuum. Only recently, two-particle continuum has been included in the SMEC for the description of the two-proton radioactivity [@Rot05]. The general framework unifying the reaction theory and the structure theory for problems with any number of particles in the scattering continuum have been worked out as well [@Fae07]. One of key elements in the practical implementation of SMEC scheme is the definition of orthogonal subspaces $q$, $p$ and, consequently, $\mathcal{Q}_0, \mathcal{Q}_1, \mathcal{Q}_2, \dots$ many-body subspaces. This definition is associated with the extraction of regularized resonances from the s.p. scattering continuum. In all previous SMEC applications (for an extensive list of applications see Refs. [@Ben99; @Ben00; @Oko03; @Rot05]) as well as in CSM studies of Dresden group [@Bar77; @Rot91], the Wang and Shakin method was employed [@Wan70]. In this method, the construction of a QBSEC depends on few parameters: the (real) energy of a scattering wave function which is then regularized and removed from the continuum, and the parameters of the cutting function (Heaviside or Fermi functions) chopping off the tail of this wave function. The so defined QBSECs are auxiliary, artificial objects which do not correspond to any solution of the Schrödinger equation. Unfortunately, the CSM/SMEC results concerning, e.g., the partial decay widths, depend on the parameters of the cutting function. This ambiguity cannot be totally removed even if the condition on the s.p. width is applied in choosing the cutting radius [@Bar77]. In practical applications, the radius of the cutting function is selected close to the top of the Coulomb barrier [@Oko03; @Rot05] what yields a sensible prescription for narrow s.p. resonances. Problems with the choice of cutting function and the extraction of QBSECs appear for broad resonances. Moreover, the QBSECs obtained using Wang and Shakin method [@Wan70] do not have the correct asymptotic behavior. In this paper, we present the new method of regularizing s.p. resonances which is unambiguous, parameter-free and yields states with the correct bound-state asymptotics. These bound states embedded in the non-resonant continuum are called the [*anamneses*]{} of resonances in the space of square-integrable functions 9${\cal L}^2$-functions0. In Sect. \[radicons\], we provide a short discussion of different radial solutions of the Schrödinger equation which are used in the construction of a complete s.p. basis in SMEC. Qn unambiguous determination of the resonance anamneses is presented in Sect. \[new\_qbsec\], and Sect. \[non\_res\_cont\] is devoted to the discussion of consequences of the extraction of resonance anamneses on the non-resonant, real-energy continuum states. Together, bound s.p. states, anamneses of s.p. resonances and a regularized real-energy s.p. scattering states provide the complete s.p. basis in the Hilbert space. This basis can be used to obtain the complete many-body basis in CSM/SMEC studies. Applications of the new method are discussed in Sect. \[applications\]. We shall present examples of the anamneses for resonances of different widths and angular momenta $\ell$, both for neutrons and protons. General features of the resonance anamneses, such as the energy dependence of both the root-mean-square (RMS) radius or the matching point between inner and outer solutions of the Schrödinger solution, will be analyzed as well. Finally, main conclusions will be given in Sect. \[conclusion\]. Radial Schrödinger equation: the general considerations {#radicons} ======================================================= Let us consider a spherical potential $V(r)$ describing qn interaction between a nucleon and a target nucleus. In the center of mass coordinates, the one-body radial wave function $u(r)$ of the relative motion is the solution of the Schrödinger equation: $$\begin{aligned} \label{local_schr} \left[ \frac{d^{2}}{dr^{2}}+k^{2}-\frac{\ell(\ell+1)}{r^{2}}-\frac{2\mu}{\hbar^{2}}V(r)\right]u_{k,\ell}(r) = 0 ~ \ , \end{aligned}$$ where $\mu$ is the reduced mass and $\ell$ is the relative angular momentum. In the following, we shall omit the angular momentum index $\ell$ to simplify notations. We are interested in three kinds of solutions of Eq. (\[local\_schr\]): - The scattering solutions, which form a continuum for real and positive momenta $k$. These solutions are regular at $r=0$ and asymptotically take a form: $$\begin{aligned} \label{asym_scatt} u_{k}(r) &\sim& kr\Big{(}C^{-}h^{-}(kr)+C^{+}h^{+}(kr)\Big{)} ~ \ , \end{aligned}$$ where $h^{\pm}$ are irregular Coulomb wave functions or Hankel functions for protons and neutrons, respectively. $C^{\pm}$ in (\[asym\_scatt\]) are some constants. - The bound state solutions, which compose a discrete set for imaginary and positive values $k=k_{n}$. These solutions have an asymptotic of outgoing waves: $$\begin{aligned} \label{asym_bound} u_{n}(r) \sim Ak_{n}rh^{+}(k_{n}r) ~ \ .\end{aligned}$$ - The resonance solutions, which constitute a discrete set for $k=k_{n}^{res}$ such that ${\cal R}e(k_{n}^{res})>0$, ${\cal I}m(k_{n}^{res}<0)$ and ${\cal R}e(k_{n}^{res})>-{\cal I}m(k_{n}^{res})$. These solutions have also an outgoing wave asymptotic. The asymptotic form of irregular Coulomb wave functions (Eqs. (\[asym\_scatt\]), (\[asym\_bound\])) is: $$\begin{aligned} \label{coul_conv} h^{\pm}(kr)\sim \frac{i^{\mp(\ell+1)}}{kr}e^{\pm i(kr-\eta\log(2kr))} ~ \ , \end{aligned}$$ where $\eta=\mu e^2Z_1Z_2/{\hbar}^2k$ is the Sommerfeld parameter. The numerical integration for these three kinds of solution is carried out first by integrating the regular solution in the inner region $[0,R]$, where $R$ is some matching radius. On this interval, the solution of Eq. (\[local\_schr\]) is written as: $$\begin{aligned} \label{reg_sol} u(r)=Cu^{reg}_{k}(r)~ \ , \end{aligned}$$ where the regular solution $u^{reg}_{k}$ verifies: $$\begin{aligned} \label{reg_at_zero} \lim_{r\to 0}r^{-\ell-1}u^{reg}_{k}(r)=1 ~ \ , \end{aligned}$$ and $C$ is a constant. Then, we integrate the Jost functions $H_{k}^{\pm}$ which are the solutions of Eq. (\[local\_schr\]) with the asymptotic form when $r\to \infty$: $$\begin{aligned} \label{asym_jost} H_{k}^{\pm}(r)\sim krh^{\pm}(kr) ~ \ . \end{aligned}$$ In the outer region $[R,\infty[$, the scattering solutions are: $$\begin{aligned} \label{scatt_jost} u_{k}(r)=C^{-}H_{k}^{-}(r)+C^{+}H_{k}^{+}(r)~ \ .\end{aligned}$$ The continuity of the logarithmic derivative, which ensures the matching between the two solutions (Eqs. (\[reg\_sol\]) and (\[scatt\_jost\])), gives two conditions: $$\begin{aligned} \label{set_equa} Cu^{reg}_{k}(R) &=& C^{-}H_{k}^{-}(R) + C^{+}H_{k}^{+}(R) \nonumber \\ \\ C\frac{du^{reg}_{k}(r)}{dr}\Big{|}_{r=R} &=& C^{-}\frac{dH_{k}^{-}(r)}{dr}\Big{|}_{r=R} + C^{+}\frac{dH_{k}^{+}(r)}{dr}\Big{|}_{r=R} \nonumber\end{aligned}$$ for three unknown constants $C^{\pm}$ and $C$. The third equation is provided by the normalization condition for scattering states: $$\begin{aligned} \label{C+C-} C^{+}C^{-}=\frac{1}{2\pi} ~ \ . \end{aligned}$$ On the interval $[R,\infty[$, bound and resonance wave functions are written as: $$\begin{aligned} \label{bound_jost} u(r)=AH^{+}_{k}(r) ~ \ . \end{aligned}$$ The matching between the internal (\[reg\_sol\]) and external (\[bound\_jost\]) solutions is ensured for those discrete values of $k_{n}$ which nullify Wronskian of the regular solution with the Jost solution $H^{+}_{k}$: $$\begin{aligned} \label{wron_equa} W(u^{reg}_{k},H_{k}^{+})\Big{|}_{k=k_{n}}(r)\equiv0 ~ \ . \end{aligned}$$ For resonance states, the values of $k_n$ correspond to the poles of the scattering matrix ($S$-matrix), whose matrix elements are given by the ratio of outgoing and incoming waves in the asymptotic expression (\[asym\_scatt\]). Properties of the scattering continuum are very sensitive to the position of these poles in the complex $k$-plane. For a narrow resonance, the scattering states with $k$-values on the real axis just above the corresponding pole of the $S$-matrix are localized in the inner region of a potential and resemble bound states with, however, an oscillatory asymptotics instead of an exponential asymptotics expected for standard bound state wave functions. This generic feature of near-pole scattering wave functions helps to define the QBSEC function by cutting off the oscillatory tail with either the Heaviside function [@Wan70] or the Fermi [@Oko03] function. Determination of the anamneses of s.p. resonances {#new_qbsec} ================================================= The significance of a QBSEC function in the CSM/SMEC theoretical framework is due to the similarity between a near-pole scattering wave function in the inner region ($r<R$) and a bound state wave function. The behavior of QBSEC in the outer region ($r>R$) does not correspond to any solution of the Schrödinger equation and, in this sense, can be considered unphysical. In the following, we shall show that one can define an alternative to QBSEC functions, the anamneses of s.p. resonances which belong to the space of ${\cal L}^2$-functions. These resonance anamneses are bound s.p. states in the scattering continuum. They provide a natural continuation of weakly bound s.p. states for positive energies and extract all resonance features from the scattering continuum. Given a pole of the $S$-matrix at $k^{res}$ in the complex $k$-plane, this similarity is particularly striking for scattering states with $k$ close to: $$\begin{aligned} \label{value_kappa} \kappa=\sqrt{{\cal R}e\Big{(}(k^{res})^{2}\Big{)}} ~ \ . \end{aligned}$$ It is then quite natural to select the scattering state with a real eigenenergy $e^{res}=\hbar^{2}\kappa^{2}/2\mu$ for a construction of the resonance anamnesis. In the inner region $[0,R]$, where $R$ is yet arbitrary matching radius for inner and outer solutions, the resonance anamnesis should be proportional to the regular solution of Eq. (\[local\_schr\]) with $k=\kappa$: $$\begin{aligned} \label{qbsec_reg} v_{\kappa}(r)=Cu^{reg}_{\kappa}(r) ~ \ , \end{aligned}$$ where $C$ is some constant. In the external region ($[R,\infty[$), we require that the anamnesis of the resonance has the bound state asymptotic corresponding to $k=i\kappa$. Using the Jost solution $H^{+}_{i\kappa}$ of Eq. (\[local\_schr\]) with $k=i\kappa$, we write the wave function of resonance anamnesis in the interval $[R,\infty[$ as: $$\begin{aligned} \label{qbsec_jost} v_{\kappa}(r)=AH^{+}_{i\kappa}(r) ~ \ , \end{aligned}$$ where $A$ is some constant. Since the wave number is fixed, both in the inner $[0,R]$ and outer $[R,\infty[$ regions, therefore the continuity of the resonance-anamnesis wave function and its first derivative at $r=R$ is equivalent to the condition: $$\begin{aligned} \label{qbsec_cont} W(u^{reg}_{\kappa},H^{+}_{i\kappa})(r)\Big{|}_{r=R}=0~ \ . \end{aligned}$$ Since the wave number is fixed, both in the inner $[0,R]$ and outer $[R,\infty[$ regions, Eq. (\[qbsec\_cont\]) becomes a condition fixing the value of the matching radius $r_{m}\equiv R$. Hence, the wave function of the resonance anamnesis on $[0,\infty[$ can be as: $$\begin{aligned} \label{qbsec_wf} v_{\kappa}(r)=C\left[u^{reg}_{\kappa}(r) \Big{(}1-\Theta(r-r_{m})\Big{)}+\frac{u^{reg}_{\kappa}(r_{m})}{H^{+}_{i\kappa}(r_{m})}H^{+}_{i\kappa}(r)\Theta(r-r_{m})\right]_{r_m=R} ~ \ . \nonumber \\\end{aligned}$$ $\Theta$ in this equation denotes the Heaviside function and the constant $C$ is fixed by the normalization condition for $v_{\kappa}(r)$. The anamnesis of a s.p. resonance should be orthogonal to other bound and scattering states of a given $\tau_z,\ell,j$. If a bound state exists with the same angular momentum $\ell,j$, then the wave function of the resonance anamnesis $v_{\kappa}(r)$ (cf Eq. \[qbsec\_wf\]) has to be orthonormalized with respect to this state. Construction of an appropriate non-resonant scattering continuum will be discussed in Sect. \[non\_res\_cont\]. The above procedure yields for the resonance anamnesis a ${\cal C}(1)$-function with an appropriate bound state asymptotic. It is important to notice that the radius $r_{m}(\equiv R)$ in (\[qbsec\_wf\]) is determined by solving Eq. (\[qbsec\_cont\]), and does not correspond to any arbitrary cutting radius as in the procedure of Wang and Shakin [@Wan70]. We will see in Sect. \[applications\] that despite the fact that $r_m$ can in principle take any value in the interval $[0,\infty[$, the above method provides the resonance anamneses which maintain localized aspects of the corresponding resonance wave functions. Construction of the non-resonant continuum {#non_res_cont} ========================================== The radial wave functions corresponding to resonance anamneses are not eigenfunctions of the original Hamiltonian ${\hat h}$ which is used to define s.p. bound and scattering states. It is however possible to redefine this Hamiltonian in such a way that bound states of ${\hat h}$, resonance anamneses, and the states of a non-resonant scattering continuum are generated by one and the same Hamilton operator. Let us denote by $\{|u_{n}\rangle\}$ the state vectors corresponding to bound states, and by $\{|v_{m}\rangle\}$ the state vectors corresponding to the resonance anamneses defined in the previous section. The energy of each resonance anamnesis corresponds to the resonance energy: $e^{res}_{m}=\hbar^{2}\kappa_{m}^{2}/2\mu$, where $\kappa_{m}$ is given by (\[value\_kappa\]), and an index $m$ enumerates different poles. Bound states and resonance anamneses form together a discrete subset $\{|\tilde{u}_{n}\rangle\}$ of the complete set of basis states in Hilbert space [@New82; @Mic04]. Let us now define a new Hamilton operator ${\hat {\tilde{h}}}$: $$\label{new_ham} \hat{\tilde{h}}=\sum_{n}|\tilde{u}_{n}\rangle\tilde{e}_{n}\langle\tilde{u}_{n}|+\hat{p}{\hat h}\hat{p}~ \ ,$$ where $\hat{p}$ is a projection operator on the non-resonant s.p. continuum: $$\label{projector} \hat{p}=1-\sum_{n}|\tilde{u}_{n}\rangle\langle\tilde{u}_{n}| ~ \ ,$$ and $$\label{new_energies} \tilde{e}_{n} = \left\{ \begin{array}{ll} e_{n} & \rm{for~a~bound~state} \\[+0.3cm] e^{res}_{n} & \rm{for~an~anamnesis~of~the~resonance~state} \end{array} \right.$$ All vectors in the set $\{|\tilde{u}_{n}\rangle\}$ are the eigenstates of $\hat{\tilde{h}}$: $$\label{new_ham_equa} (\tilde{e}_{n}-\hat{\tilde{h}})|\tilde{u}_{n}\rangle=0 ~ \ .$$ The scattering continuum is renormalized by the second term on the r.h.s. of Eq. (\[new\_ham\]). Below, we shall discuss the construction of scattering states for the modified Hamiltonian $\hat{\tilde{h}}$ (cf Eq. (\[new\_ham\])). Such states, that we denote $|\tilde{u}\rangle$, have to be solutions of the equation: $$\label{a1} (\tilde{e}-\hat{p}\hat{h}\hat{p})|{\tilde u}\rangle=0 ~ \ .$$ Since $|\tilde{u}\rangle$ is orthogonal to all bound states including resonance anamneses, we can write: $$\label{a2} \hat{p}({\tilde e}-\hat{h})|\tilde{u}\rangle=0 ~ \ .$$ The general solution of Eq. (\[a1\]) can be written as: $$\label{a3} |\tilde{u}\rangle=|\tilde{u}^{h}\rangle+\sum_{n}\alpha_{n}|\tilde{U}_{n}\rangle ~ \ ,$$ where $|\tilde{u}^{h}\rangle$ is a solution of the homogeneous equation: $$\label{a4} (\tilde{e}-\hat{h})|\tilde{u}^{h}\rangle=0 ~ \ ,$$ $\{|\tilde{U}_{n}\rangle\}$ are solutions of the particular equations: $$\label{a5} (\tilde{e}-\hat{h})|\tilde{U}_{n}\rangle=|\tilde{u}_{n}\rangle ~ \ ,$$ and $\{\alpha_{n}\}$ are constants. The orthogonality of $|\tilde{u}\rangle$ with respect to all bound states and anamneses of resonances: $$\label{a6} \langle\tilde{u}_{m}|\tilde{u}\rangle=0 ~ \ ,$$ can be expressed as: $$\label{a7} \vec{A}+B\vec{\alpha}=\vec{0} ~ \ ,$$ where components of the vector $\vec{A}$ are given by: $$\label{a8} A_{m}=\langle\tilde{u}_{m}|\tilde{u}^{h}\rangle ~ \ ,$$ and those of the matrix $B$ by: $$\label{a9} B_{mn}=\langle\tilde{u}_{m}|\tilde{U}_{n}\rangle ~ \ .$$ Inverting the linear system of equations (\[a7\]), we obtain finally: $$\label{a10} |\tilde{u}\rangle=|\tilde{u}^{h}\rangle-\sum_{n,m}|\tilde{U}_{n}\rangle B^{-1}_{nm}A_{m} ~ \ .$$ The homogeneous solution has been discussed in Sect. \[radicons\]. Its asymptotic behavior is given in Eq. (\[asym\_scatt\]). The states $\{|\tilde{U}_{n}\rangle\}$ are solutions of inhomogeneous equations with outgoing wave asymptotics: $$\label{a11} \tilde{U}_{n}(r)\sim C_{n}krh^{+}(kr) ~ \ ,$$ where $\{C_{n}\}$ are constants. Inserting (\[asym\_scatt\]) and (\[a11\]) into the radial representation of Eq. (\[a10\]), one derives the asymptotic behavior of the general solution: $$\label{a12} \tilde{u}(r)=C^{-}i^{-(l+1)}\Big{[}(-1)^{l+1}e^{-i(kr-\eta\log(2kr))}+\tilde{S}e^{i(kr-\eta\log(2kr))}\Big{]} ~ \ ,$$ where the scattering matrix is given by: $$\label{a13} \tilde{S}=\frac{1}{C^{-}}\Big{(}C^{+}-\sum_{n,m}C_{n}B^{-1}_{nm}A_{m}\Big{)}=e^{2i\tilde{\delta}} ~ \ ,$$ and $\tilde{\delta}$ is the non-resonant phase shift. A possible test to see whether this procedure correctly removes the s.p. resonances from the s.p. scattering continuum is to compare the real-energy continuum phase shift for both Hamiltonians: $\hat{h}$ and $\hat{\tilde{h}}$. One demands that resonances are suppressed from the scattering continuum of $\hat{\tilde{h}}$. On the other hand, the background phase shift calculated for $\hat{h}$ and $\hat{\tilde{h}}$ should be essentially the same. Discussion of the results {#applications} ========================= In this section, we shall discuss examples of resonance anamneses constructed by the method described in Sects. \[radicons\] and \[new\_qbsec\]. We shall also analyze properties of an associated non-resonant scattering continuum described in Sect. \[non\_res\_cont\]. The starting point is the construction of an initial s.p. basis containing bound and scattering states. This basis is generated by the spherical Woods-Saxon (WS) potential consisting of both central and spin-orbit parts, and a Coulomb potential: $$\begin{aligned} \label{potential} V(r) &=& -V_0 f(r) - 4 V_{\rm so}~ (l \cdot s) \frac{1}{r} \frac{df(r)}{dr} + V_{\rm c}(r) ~ \ , \label{potential_1} \end{aligned}$$ where $$\begin{aligned} f(r) &=& \left[ 1 + \exp \left( \frac{r-R_0}{d} \right) \right]^{-1} ~ \ . \label{potential_2}\end{aligned}$$ In all examples, the WS potential has the radius $R_0$=3.5 fm, the diffuseness $d=$0.5 fm, and the spin-orbit strength $V_{\rm so}$=3.5 MeV. The mass of a target nucleus is 16 amu. The Coulomb potential $V_{\rm c}$ is assumed to be generated by a uniformly charged sphere of radius $R_0$ and charge number $Z$=8. The depth of the central part is varied to simulate different situations. The resonance anamneses are extracted from the scattering continuum of the WS potential (\[potential\]). Bound states, anamneses of resonances and remaining continuum states are subsequently reorthogonalized. Radial wave functions of resonance anamneses and associated phase shifts ------------------------------------------------------------------------ As an illustration of the method, we shall consider the $0d_{5/2}$ proton resonance in the WS potential (\[potential\_1\]) with $V_{0}$=40 MeV. The $0d_{5/2}$ resonance pole is situated at ${\cal R}e(k^{res})=0.265\,\rm{fm}^{-1}$, ${\cal I}m(k^{res})=-0.0014\,\rm{fm}^{-1}$, what corresponds to the energy $e^{res}$=1.55 MeV and the width $\Gamma^{res}$=32.692 keV. ![The radial wave function of a $d_{5/2}$ proton scattering state at an energy corresponding to the real part of a $0d_{5/2}$ resonance state energy. For more details, see the description in the text. []{data-label="scatt_0d5_proton"}](./FIGURES/d5_res.eps) The radial wave function of a scattering state with real energy $e^{res}$ is presented in Fig. \[scatt\_0d5\_proton\]. One can see the localization of the wave function in the inner region (up to $\sim$10 fm). Solving the matching condition (\[qbsec\_cont\]) yields the matching radius: $r_{m}$($\equiv R$)=18 fm. ![The resonance anamnesis corresponding to the $d_{5/2}$ proton scattering state shown in Fig. \[scatt\_0d5\_proton\]. []{data-label="qbsec_0d5_proton"}](./FIGURES/d5_qbsec.eps) The resonance anamnesis corresponding to the $0d_{5/2}$ proton resonance (see Fig. \[qbsec\_0d5\_proton\]) is extracted from the $d_{5/2}$ scattering wave function function at the energy $e^{res}$ corresponding to the real part of the $0d_{5/2}$ resonance energy (cf Fig. \[scatt\_0d5\_proton\]). One may notice a hump at larger distances ($10~{\rm fm}\leq r \leq20~{\rm fm}$) which is a characteristic feature of many resonance anamneses. ![Resonant (full line) and non-resonant (dashed line) $d_{5/2}$ phase shift as a function of the energy $e=\hbar^{2}k^{2}/2\mu$ in the center of mass. []{data-label="phase_0d5_proton"}](./FIGURES/d5_phase.eps) $d_{5/2}$ phase shifts of the resonant continuum generated by the Hamiltonian $\hat{h}_{\rm WS}$ with the WS potential (\[potential\_1\]), and of the non-resonant continuum generated by the modified Hamiltonian $\tilde{\hat h}_{\rm WS}$ (cf Sect. \[non\_res\_cont\]), are plotted in Fig. \[phase\_0d5\_proton\]. One can see that the resonant part of phase shift is fully removed by the $d_{5/2}$-resonance anamnesis and the phase shift calculated for the scattering continuum of $\tilde{\hat h}_{\rm WS}$ exhibits only a smooth energy dependence. ![Radial wave functions of resonance anamneses corresponding to $p_{1/2}$ neutron scattering states at three different energies of $1p_{1/2}$ neutron resonance. For more details, see the description in the text.[]{data-label="c3f4"}](./FIGURES/L_is_1_neutron.eps) Another examples of the radial wave functions of resonance anamneses and the respective phase shifts are shown in Figs. \[c3f4\]-\[c3f9\]. Fig. \[c3f4\] shows the resonance-anamnesis wave functions corresponding to $1p_{1/2}$ neutron resonances at three different energies. The depth of the WS potential is varied to yield the resonance energy $e^{res}=0.036$ MeV (the bottom part), 0.112 MeV (the middle part), and 0.47 MeV (the upper part). The widths of these resonances vary from $\sim10$ keV to $\sim560$ keV. The resonance anamneses are extracted from $p_{1/2}$ neutron scattering wave functions at energies $e^{res}$ corresponding to the real part of the $1p_{1/2}$ neutron resonance energy. In all these three cases, well pronounced hump at large distances can be seen. ![Resonant (full line) and non-resonant (dashed line) $p_{1/2}$ neutron phase shift for three resonances shown in Fig. \[c3f4\]. []{data-label="c3f5"}](./FIGURES/L_is_1_neutron_phase.eps) Phase shifts in resonant and non-resonant scattering continua corresponding to those three $1p_{1/2}$ neutron resonances shown are plotted in Fig. \[c3f5\]. The resonant continuum is generated by the $\hat{h}_{\rm WS}$ Hamiltonian, whereas the non-resonant continuum is given by the modified Hamiltonian $\tilde{\hat h}_{\rm WS}$. One can see that the regularization procedure fully subtracts the resonant part of the phase shift, both for broad and narrow resonances, and the phase shift corresponding to the continuum of $\tilde{\hat h}_{\rm WS}$ exhibits only a smooth energy dependence. ![Radial wave functions of resonance anamneses corresponding to $s_{1/2}$ proton scattering states at three different energies of $1s_{1/2}$ proton resonance. For more details, see the description in the text.[]{data-label="c3f8"}](./FIGURES/L_is_0_proton.eps) ![Resonant (full line) and non-resonant (dashed line) $s_{1/2}$ proton phase shift for three resonances shown in Fig. \[c3f8\]. []{data-label="c3f9"}](./FIGURES/L_is_0_proton_phase.eps) Radial wave functions of resonance anamneses and corresponding phase shifts for $1s_{1/2}$ proton resonance at three different energies are shown in Figs. \[c3f8\] and \[c3f9\]. The depth of the WS potential is varied to yield the resonance energy at $e^{res}=0.53$ MeV (the bottom part), 1.11 MeV (the middle part), and 1.29 MeV (the upper part). The resonance width in these cases vary from $\sim20$ keV to $\sim600$ keV. Also in this case, the extraction of the resonance anamnesis removes all resonant features from the scattering continuum of $\hat{h}_{\rm WS}$. In all cases, the anamnesis of a resonance constructed by the method proposed in this work provides an excellent description of localized aspects of the resonant wave function in a broad range of resonance widths. The so defined resonance anamnesis becomes an image of the resonance wave function in the subspace of ${\cal L}^2$-functions. Its removal from the subspace of scattering wave functions leaves only the non-resonant continuum with smoothly varying phase shifts. This feature of resonance-anamnesis wave functions give them a special significance in a decomposition of the scattering continuum. RMS radius and matching radius of the resonance-anamnesis wave functions ------------------------------------------------------------------------ In the following, we shall investigate the energy dependence of the resonance-anamnesis radial wave functions by calculating the RMS radius and the matching radius. The dependence of the mean-square radius of a bound state on the distance $e$ to the continuum threshold has been studied by Riisager et al. [@Rii92] for spherical nuclei and Misu et al. [@Mis97] for deformed nuclei. (For the discussion of three-body halo asymptotics, see Ref. [@Fed93].) The mean-square radius of a spherical neutron orbit varies as $(-e)^{-1}$ for $\ell=0$ and as $(-e)^{-1/2}$ for $\ell=1$, but remains finite for higher angular momenta due to the centrifugal barrier that confines the radial wave function. For proton bound states, the mean-square radius is finite for all $\ell$ at the continuum threshold because of the Coulomb barrier. The resonance-anamnesis wave functions constructed by the method proposed in this work have an asymptotic behavior of a bound state wave function with momentum $\sqrt{{\cal R}e\Big{(}(k^{res})^{2}\Big{)}}$. Consequently, it is likely that certain near-threshold features of resonance-anamnesis wave functions are similar to features of weakly bound s.p. states. In other words, one expects not only that the above method provides states which provide a faithful image of localized aspects of resonances, but also that it provides states which are a natural continuation of weakly bound s.p. states into the continuum. ![RMS radius of $1p_{1/2}$ neutron orbit as a function of the energy. The branch for positive energies is obtained for $1p_{1/2}$ resonance anamnesis constructed by the method described in Sect. \[new\_qbsec\]. For more details, see the description in the text.[]{data-label="1p1n_msr"}](./FIGURES/1p1n_msr_fig.eps) ![RMS radius of the $1s_{1/2}$ neutron orbit as a function of the energy. The curve for positive energies is obtained for the $1s_{1/2}$ resonance anamnesis constructed by the method described in Sect. \[new\_qbsec\]. For more details, see the description in the text.[]{data-label="1s1p_msr"}](./FIGURES/1s1p_msr_fig.eps) Figs. \[1p1n\_msr\] and \[1s1p\_msr\] present the RMS radius of a weakly bound s.p. state and a corresponding resonance anamnesis as a function of the energy for $1p_{1/2}$ neutron and $1s_{1/2}$ proton s.p. orbits, respectively. The curve for negative energies corresponds to the RMS radius for bound s.p. states: neutron $1p_{1/2}$ (Fig. \[1p1n\_msr\]) and proton $1s_{1/2}$ (Fig. \[1s1p\_msr\]) states. For positive energies, the curve shows the RMS radius of the corresponding resonance-anamnesis wave function extracted from the scattering wave function at the energy which is equal to the real part of the resonance energy, respectively. The corresponding bound and real-energy scattering states are found by varying the depth of the WS potential (\[potential\_1\]). For neutron $1p_{1/2}$ orbits (cf Fig. \[1p1n\_msr\]), the depth of the central part $V_0$ varies from 69.012 MeV to 61.44 MeV, whereas for proton $1s_{1/2}$ orbits, $V_0$ changes from 50.233 MeV to 39.06 MeV. The gap seen in Figs. \[1p1n\_msr\] and \[1s1p\_msr\] close to $e=0$ comes essentially from numerical limitations in the construction of reliable wave functions for a weakly-bound state and its resonance anamnesis at positive energies. One may notice a slight asymmetry in the energy dependence of RMS radius on both sides of $e=0$. This is essentially an effect of the centrifugal and/or Coulomb barrier for states with $e>0$. In all previous CSM or SMEC studies [@Ben99; @Ben00; @Oko03; @Rot05]), the QBSEC radial wave function was taken to be proportional to a regular solution of Eq. (\[local\_schr\]) up to a certain radius $r_{cut}$, and for $r>r_{cut}$ this solution was cut by a cutting function. The value of $r_{cut}$ in such an approach is a free parameter. For most of CSM/SMEC applications, the value of $r_{cut}$ was close to the position of the top of the centrifugal and/or Coulomb barrier. ![Matching radius $r_{m}$ (full line) in function of the energy $e$ of the resonant $1p_{1/2}$ neutron state. The depth $V_{0}$ of the potential is varied from $66.1$ MeV to $61.44$ MeV. For comparison, the external turing point (dashed line) is presented too. []{data-label="match_1p1n"}](./FIGURES//1p1n_re.eps) As discussed in Sect. \[new\_qbsec\], the resonance-anamnesis wave functions which replace QBSEC functions, are proportional to the regular solution of Eq. (\[local\_schr\]) up to the matching radius $r_{m}\equiv R$. For $r>r_{m}$, the resonance anamnesis takes an asymptotic form of a bound state with a well defined wave number. The matching radius is not a free parameter of the theory but its value is uniquely determined by solving Eq. (\[qbsec\_cont\]) for $r$. At $r=R$, inner and outer solutions join smoothly to form a ${\cal C}(1)$ radial wave function of the resonance anamnesis. The matching radius $r_m$, whose position depends on features of the studied resonance, is an interesting supplementary information about the resonance-anamnesis wave function. $r_m$ depends uniquely on resonance characteristics, such as the energy (width), angular momentum and $\tau_z$. The energy (width) dependence of $r_m$ is generic, depending only on the angular momentum and $\tau_z$. This dependence is illustrated in Fig. \[match\_1p1n\] on an example of neutron $1p_{1/2}$ wave function of the resonance anamnesis. We can see that $r_m$ grows with decreasing $e$. The dashed line in Fig. \[match\_1p1n\] shows also the variation of an external turning point $r_{etp}$ of the centrifugal barrier with the resonance energy. Except for a shift, both functions $r_m(e)$ and $r_{etp}(e)$ are rather similar, in particular for large $e$. In all studied cases, the matching radius lies outside of the external turning point of the potential barrier. Conclusions {#conclusion} =========== Construction of the many-body basis in Hilbert space is a key problem in the SM description of open quantum systems, such as the weakly bound nuclei, radioactive decays, or low-energy nuclear reactions. This construction is based on the determination of a complete s.p. basis which consists of discrete states (subspace $q$) and a scattering continuum (subspace $p$). A consistent unified formulation of nuclear structure and reactions is possible within the CSM/SMEC if the s.p. resonances are removed from the scattering continuum $p$ to be put in $q$ [@Bar77]. This regularization procedure for s.p. resonances is associated with an extraction of localized part of s.p. resonances from $p$ which leaves only non-resonant scattering states. In this paper, we have developed an unambiguous, parameter-free method of extracting the localized component of the s.p. resonance which allows to determine a new subspace $q^{'}$ of discrete s.p. states, consisting of bound states and resonance anamneses, and a new subspace $p^{'}$ of non-resonant scattering states. The resulting s.p. basis is complete and can be used for the construction of complete many-body basis in the Hilbert space. This new s.p. basis could also provide an interesting alternative for solving Hartree-Fock-Bogolyubov equations with finite-range density-dependent interactions for weakly-bound systems. The resonance anamneses provide a faithful image of localized features of resonances in the space of ${\cal L}^2$-functions, removing all resonant aspects from calculated phase shifts. These anamneses of the resonance states provide also a smooth continuation of weakly-bound s.p. states into the low-energy continuum. Appealing features of those states give a special significance to the s.p. basis which includes them in its discrete part. Future applications of this particular basis may open new and yet unexplored horizons for a direct calculation of multi-particle resonant states in nuclear systems. It may also be useful in extracting a resonance contribution from many-body observables in weakly-bound or unbound systems. \ One of us (M.P.) wish to thank Witek Nazarewicz for useful comments. 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--- author: - 'Steven V. Fuerst and Kinwah Wu' date: 'Received: ' title: 'Radiation Transfer of Emission Lines in Curved Space-Time' --- Introduction ============ The strong X-rays observed in active galactic nuclei (AGN) and some X-ray binaries are believed to be powered by accretion of material into black holes. The curved space-time around the black hole influences not only the accretion hydrodynamics but also the transport of radiation from the accretion flow. Emission lines from thin Keplerian disks around non-relativistic stellar objects generally have two symmetric peaks (Smak 1969), corresponding to the approaching and receding line-of-sight velocities due to disk rotation. Because of various relativistic effects, lines from accretion disks around black holes do not always have symmetrical double-peak profiles. The accretion flow near a black hole is often close to the speed of light, and emission is relativistically boosted. The blue peak of the line therefore becomes stronger and sharper. Moreover, the strong gravity near the black-hole event horizon causes time dilation, which shifts the line to lower energies. Emission lines from accretion disks around black holes appear to be broad, with a very extended red wing and a narrow, sharp blue peak (see e.g. the review by Fabian et al. (2000) and references therein). Furthermore, gravitational lensing can produce multiple images and self-occultation, further modifying the emission line profile. Various methods have been used to calculate the profiles of emission lines from accretion disks around black holes. The methods can be roughly divided into three categories. We now discuss each of them briefly. The first method uses a transfer function to map the image of the accretion disk onto a sky plane (Cunningham 1975, 1976). The accretion disk is assumed to reside in the equatorial plane. It is Keplerian and geometrically thin, but optically thick. The space-time metric around the black hole is first specified, and the energy shift of the emission (photons) from each point on the disk surface is then calculated. A parametric emissivity law for the disk emission is usually used — typically, a simple power-law which decreases radially outward. The specific intensity at each point in the sky plane is determined from the energy shift and the corresponding specific intensity at the disk surface, using the Lorentz-invariant property. The transfer-function formulation (Cunningham 1975, 1976) has been applied to line calculations in settings ranging from thin accretion rings (e.g. Gerbal & Pelat 1981) and accretion disks around Schwarzschild (e.g. Laor 1991) and rotating (Kerr) black holes (e.g. Bromley, Chen & Miller 1997). The second method makes use of the impact parameter of photon orbits around Schwarzschild black holes (e.g. Fabian et al. 1989; Stella 1990; Kojima 1991). The transfer function in this method is described in terms of elliptical functions, which are derived semi-analytically. The Jacobian of the transformation from the accretion disk to sky plane is, however, determined numerically via infinitesimal variations of the impact parameter (Bao 1992). The method can be generalized to the case of rotating black holes by using additional constants of motion (Viergutz 1993; Bao, Hadrava & Ostgaard 1994; Fanton et al. 1997; Cadez, Fanton & Calvani 1998). The third method simply considers direct integration of the geodesics to determine the photon trajectories and energy shifts (Dabrowski et al. 1997; Pariev & Bromley 1998; Reynolds et al. 1999). These calculations have shown how the dynamics of the accretion flow around the black hole and the curved space-time shape the line profiles. Various other aspects of the radiation processes, e.g. reverberation and reflection (Reynolds et al. 1999) and disk warping (Cadez et al. 2003) were also investigated using the methods described above. The results obtained from these calculations have provided us with a basic framework for interpreting X-ray spectroscopic observations, in particular, the peculiar broad Fe K$\alpha$ lines in the spectra of AGN, e.g. MCG-6-30-15 (Tanaka et al. 1995). While existing studies have put emphasis on the energy shift of the emission, transport effects such as extinction have been neglected. Resonant absorption (scattering) by ambient material can greatly modify the disk emission line profile. This effect was already demonstrated in a study by Ruszkowski & Fabian (2000), in which a simple rotating disk-corona provides the resonant scattering. Here, we present ray-tracing calculations of spectra from relativistic flows in curved space-time. We include line-of-sight extinction and emission explicitly in the formulation. The radiative-transfer equation is derived from the Lorentz-invariant form of the conservation law. It reduces to the standard classical radiative-transfer equation in the non-relativistic limit. The formulation can incorporate dynamical and geometric models for the line-of-sight absorbing and emitting material. As an illustration, we calculate the from thin accretion disks and thick accretion tori around rotating black holes. The emitted spectra include a power law continuum together with a line. This emission is resonantly scattered by the line-of-sight-material. We include the contribution from higher-order images and allow for self-occultation. We organize the paper as follows. In §2 we show the derivation of the transfer equation. In §3 we construct the equation of motion for free particles in a Kerr space-time and for force-constrained particles for some simple parametric models. In §4 we construct a thin disk and a thick torus model. In §5 we generalize this by adding in absorption due to a distribution of absorbing clouds. In §6 we present the results from the models where either emission geometry (tori), or absorption (clouds) are important. Radiative-Transfer Equation =========================== Throughout this paper, we adopt the usual convention $c=G=h=1$ for the speed of light, gravitational constant and Planck constant. The interval in space-time is specified by $$\label{metric} d\tau^2 = g_{\alpha \beta} dx^{\alpha} dx^{\beta} $$ where $g_{\alpha \beta}$ is the metric. Consider a bundle of particles which fill a phase-space volume element $$\label{bundle} {d\cal{V}} \equiv dx\,dy\,dz\,dp^x\,dp^y\,dp^z\ ,$$ where $dx\,dy\,dz (\equiv dV)$ is the three-volume and $dp^x\ dp^y\ dp^z$ is the momentum range, at a given time $t$. Liouville’s Theorem reads $$\frac{d{\cal V}}{d\lambda} = 0$$ (see Misner, Thorne & Wheeler 1973), with $\lambda$ here being the affine parameter for the central ray in the bundle. The volume element $d{\cal V}$ is thus Lorentz invariant. The distribution function for the particles in the bundle, $F(x^i, p^i)$ is given by $$F(x^i, p^i) = {dN \over d{\cal V}}\ ,$$ where $dN$ is the number of particles in the three-volume. Since $dN/d{\cal V}$ is Lorentz invariant, $F(x^i, p^i)$ is Lorentz invariant. From equation (\[bundle\]), we have $$\label{rawfluxdefn} F={dN\over p^2 dV\,dp\,d\Omega}\ ,$$ where $p^2\,dp\,d\Omega=dp^x\,dp^y\,dp^z$. For massless particles, $v = c = 1$ and $\vert p\vert=E$. The number of photons in the given spatial volume is therefore the number of photons flowing through an area $dA$ in a time $dt$. It follows that $$\label{flux_inv} F={dN\over E^2dA\,dt\,dE\,d\Omega}\ .$$ Recall that the specific intensity of the photons is $$\label{inten_inv} I_\nu={E dN\over dA\,dt\,dE\,d\Omega}\ ,$$ By inspection of equations (\[flux\_inv\]) and (\[inten\_inv\]), we obtain $$F=\frac{I_\nu}{E^3}=\frac{I_\nu}{\nu^3}\ ,$$ where $\nu ~(= E)$ is the frequency of the photon. We will use this Lorentz invariant intensity, ${\cal I}\equiv F$, in the radiative transfer formulation. In a linear medium, extinction is proportional to the intensity, and the emission is independent of the intensity of the incoming radiation. The radiative transfer equation is therefore $$\label{classradtrans} \frac{d{\cal I}}{d s}=-\chi{\cal I} + \eta\left(\frac{\nu_0}{\nu}\right)^3 \ ,$$ where $\chi$ is the absorption coefficient, $\eta$ is the emission coefficient and $ds$ is the length element the ray traverses. The equation in this form is defined in the observer’s frame, and the absorption and emission coefficients are related to their counterparts in the rest frame with respect to the medium via $$\begin{aligned} \label{chframe} \chi&=&\left(\frac{\nu_0}{\nu}\right)\chi_0 \ , \\ \eta&=&\left(\frac{\nu}{\nu_0}\right)^2\eta_0 \ , \end{aligned}$$ where the subscript “0” denotes quantities in the local rest frame. The relative energy / frequency shift in a moving medium with respect to an observer at infinity is given by $$\label{freqshift} \frac{E_0}{E}= \frac{\nu_0}{\nu}= \frac{p^\alpha u_\alpha\vert_\lambda}{p^\alpha u_\alpha\vert_\infty}\ ,$$ where $u^\alpha$ is the four-velocity of the medium as measured by an observer, and $$\begin{aligned} \label{chframe2} \frac{ds}{d\lambda}&=&-p^\alpha u_\alpha\vert_\infty\ . \end{aligned}$$ The radiative transfer equation (equation \[\[classradtrans\]\]) in the co-moving frame is therefore $$\label{radtrans} \frac{d{\cal I}}{d\lambda} =-p^\alpha u_\alpha\vert_\lambda \left[-\chi_0(x^\beta,\nu){\cal I}+\eta_0(x^\beta, \nu)\right]$$ (see Baschek et al. 1997). The results in the co-moving frame can be used to determine the intensity and frequency in the other reference frames. The ray is specified by choosing $x^\alpha(\lambda_0)$ and $p^\alpha(\lambda_0)$. From the geodesic equation, we have $d p^\alpha/d \lambda+\Gamma^\alpha_{\beta\gamma}p^\beta p^\gamma=0$, where we have scaled $\lambda$ by $m$ for massive, and by $1$ for massless particles. The derivative of $\cal I$ is therefore $$\begin{aligned} \label{fullideriv} \frac{d{\cal I}}{d\lambda} &=& \frac{\partial{\cal I}}{\partial x^\alpha}\frac{d x^\alpha}{d \lambda} + \frac{\partial{\cal I}}{\partial p^\alpha} \frac{d p^\alpha}{d \lambda} \nonumber\ , \\ &=&p^\alpha\frac{\partial{\cal I}}{\partial x^\alpha} -\Gamma^\alpha_{\beta\gamma}p^\beta p^\gamma \frac{\partial{\cal I}}{\partial p^\alpha} \ .\end{aligned}$$ This, combined with equation (\[radtrans\]), yields $$\begin{aligned} \label{noray} & & -p^\alpha u_\alpha\vert_\lambda \left[-\chi_0(x^\beta,\nu){\cal I}+\eta_0(x^\beta, \nu)\right] \nonumber\\ & &\hspace{3cm}=p^\alpha\frac{\partial{\cal I}}{\partial x^\alpha} -\Gamma^\alpha_{\beta\gamma}p^\beta p^\gamma \frac{\partial{\cal I}}{\partial p^\alpha}\ , \end{aligned}$$ which is the same as that derived by Lindquist (1966) from the Boltzmann Equation. The metric and the initial conditions define the rays (the photon trajectories in 3D space), and the solution can be obtained by direct integration along the ray. For simplicity, we assume the refractive index $n=1$ throughout the medium. The solution to equation (\[radtrans\]) is then $$\begin{aligned} \label{anaray} {\cal I}(\lambda) &=& {\cal I}(\lambda_0) \exp\left(\int_{\lambda_0}^{\lambda} \chi_0(\lambda',\nu_0) u_\alpha p^\alpha d\lambda'\right)\\ & &\hspace{-1cm} -\int_{\lambda_0}^{\lambda} \exp\biggl(\int_{\lambda'}^{\lambda}\chi_0(\lambda'', \nu_0) u_\alpha p^\alpha d\lambda''\biggr) \eta_0(\lambda', \nu_0) u_\alpha p^\alpha d\lambda'\ . \nonumber \end{aligned}$$ In the non-relativistic limit, $u_\alpha p^\alpha = 1$, and the equation recovers the conventional form (see Rybicki & Lightman 1979). Particle Trajectories ===================== Free particles -------------- To determines the photon trajectories we need to specify the metric of the space-time. We consider the Boyer-Lindquist coordinates: $$\begin{aligned} d\tau^2 &=& \biggl( 1- {{2Mr} \over \Sigma}\biggr)dt^2 + {{4aMr \sin^2\theta} \over \Sigma}dtd\phi - {\Sigma \over \Delta}dr^2 \nonumber \\ & & \hspace{-0.25cm} - \Sigma d\theta^2 - \biggl( r^2+a^2 + {{2a^2Mr \sin^2\theta} \over \Sigma} \biggr) \sin^2\theta d\phi^2 \ , \end{aligned}$$ where $M$ is the black hole mass, $\Sigma = r^2+a^2\cos^2\theta$ and $\Delta = r^2 -2Mr +a^2$. The dimensionless parameter $a/M$ specifies the spin of the black hole, with $a/M = 0$ corresponding to a Schwarzschild (non-rotating) black hole and $a/M = 1$ to a maximally rotating Kerr black hole. The motion of a free particle is described by the Lagrangian: $$\begin{aligned} {\cal L} & = & \frac{1}{2} \biggl[ -\left(1-\frac{2Mr}{\Sigma}\right)\dot{t}^2 - \frac{4aMr\sin^2\theta}{\Sigma}\dot{t} \dot{\phi} + \frac{\Sigma}{\Delta}\dot{r}^2 \nonumber \\ && + \Sigma \dot{\theta}^2 + \left( r^2+a^2 + \frac{2a^2M r\sin^2\theta}{\Sigma} \right) \sin^2\theta \dot{\phi}^2 \biggr] \ \label{lagkerreqn}\end{aligned}$$ (here $\dot x^{\alpha} = d x^{\alpha}/ d\lambda$). The Lagrangian does not depend explicitly on the $t$ and $\phi$ coordinates. The momenta in the four coordinates are therefore $$\begin{aligned} p_{\rm t} = \partial{\cal L}/\partial \dot t &=& -E\ ,\\ \label{pr} p_{\rm r} = \partial{\cal L}/\partial \dot r &=& \frac{\Sigma}{\Delta} \dot r\ , \\ \label{ptheta} p_{\rm \theta} = \partial{\cal L}/\partial \dot \theta &=& \Sigma \dot \theta\ , \\ p_{\rm \phi} = \partial{\cal L}/\partial \dot \phi &=& L\ .\end{aligned}$$ with $E$ being the energy of the particle at infinity and $L$ the angular momentum in the $\phi$ direction. The corresponding equations of motion are $$\begin{aligned} \label{tdot} \dot t & = & E + \frac{2 r(r^2+a^2)E-2a L}{\Sigma\Delta}\ ,\\ \dot r^2 & = & {\Delta \over \Sigma} \big(H + E \dot t - L \dot \phi- \Sigma \dot \theta^2 \big)\ , \\ \dot \theta^2 & = & {1 \over \Sigma^2} \big( Q + (E^2 + H)a^2 \cos^2\theta - L^2 \cot^2 \theta \big)\ , \\ \label{phidot} \dot \phi & = & {{2a rE + (\Sigma - 2 r)L/\sin^2\theta} \over {\Sigma\Delta} } \ , \end{aligned}$$ where $Q$ is Carter’s constant (Carter 1968), and $H$ is the Hamiltonian, which equals 0 for photons and massless particles and equals $-1$ for particles with a non-zero mass. (See Reynolds et al. (1999) for more details.) For simplicity, we have set the black-hole mass equal to unity ($M =1$) in the equations above. This is equivalent to normalizing the length to the gravitational radius of the black hole (i.e., set $R_{\rm g} \equiv GM/c^2 = 1$), and we will adopt this normalization hereafter. There are square terms of $\dot r$ and $\dot \theta$ in two equations of motion. They could cause problems when determining the turning points at which $\dot r$ and $\dot \theta$ change sign in the numerical calculations. To overcome this, we consider the second derivatives of $r$ and $\theta$ instead. From the Euler-Lagrange equation, we obtain $$\begin{aligned} \ddot r & = & \frac{\Delta}{\Sigma} \bigg\{\frac{\Sigma-2r^2}{\Sigma^2}\dot t^2+\frac{(r-1) \Sigma-r\Delta}{\Delta^2}\dot r^2+ r\dot \theta^2 \nonumber \\ & & \hspace{1cm} +\sin^2\theta \left(r+\frac{\Sigma-2r^2}{\Sigma^2} a^2\sin^2\theta\right)\dot \phi^2 \nonumber \\ & & \hspace{1cm} -2a\sin^2\theta\frac{\Sigma-2r^2}{\Sigma^2}\dot t \dot \phi +\frac{2a^2\sin\theta\cos\theta}{\Delta}\dot r \dot \theta\bigg\}\ , \\ \ddot \theta & = & {1 \over \Sigma} \bigg\{ \sin\theta\cos\theta \bigg[ \frac{2a^2r}{\Sigma}\dot t^2 -\frac{4ar(r^2+a^2)}{\Sigma}\dot t \dot \phi-\frac{a^2}{\Delta}\dot r^2\nonumber \\ & & \hspace*{1cm}+a^2\dot \theta^2 +\frac{\Delta +2r(r^2+a^2)^2}{\Sigma^2}\dot \phi^2 \bigg] -2r\dot r\dot \theta \bigg\}\ .\end{aligned}$$ In terms of the momenta and the Hamiltonian, the equations above can be expressed as $$\begin{aligned} \label{rdot} \dot p_{\rm r} &=& \frac{1}{\Sigma \Delta} \left[(r-1)\left((r^2+a^2)H-\kappa\right) +r\Delta H\right.\nonumber\\ & &\hspace{0.5cm}\left.+2r(r^2+a^2)E^2-2aEL\right]-\frac{2{p_r}^2(r-1)}{\Sigma}\ ,\\ \label{thetadot} \dot p_{\rm \theta} &=& \frac{\sin\theta\cos\theta}{\Sigma} \left[\frac{L^2}{\sin^4\theta}-a^2(E^2+H)\right]\ ,\end{aligned}$$ where $\kappa = Q+L^2+a^2(E^2+H)$. Equations (\[pr\]), (\[ptheta\]), (\[tdot\]), (\[phidot\]), (\[rdot\]) and (\[thetadot\]) are the equations of motion. Motion in the presence of external forces ----------------------------------------- The equations of motion obtained in the previous section are applicable to free particles only. In a general situation external (non-gravitational) forces may be present and we need to specify the external force explicitly in deriving the equations of motion. However, in the setting of accretion disks around black holes we can often treat the effect of the external force implicitly which we will discuss in more detail in the following subsections. ### Rotational and Pressure Supported Model Here we consider a simple model such that $$\dot{t} > \dot{\phi} \gg \dot{r} \gg \dot{\theta}\ .$$ As $\dot{r}$ and $\dot{\theta}$ are small in comparison with other quantities, they can be neglected as a first approximation. The equation of motion reads $$\frac{d^2 x^\nu}{d\lambda^2} +\Gamma^{\nu}_{\alpha \beta}u^{\alpha}u^{\beta}=a^{\nu}\ ,$$ where $a^{\nu}$ is the four-acceleration due to an external force per unit mass. For axisymmetry (which is a sensible assumption for accretion onto rotating black holes), $d/d\phi=0$ and $a^{\phi}=0$. The identity $u^\alpha a_\alpha = 0$ together with $\dot r = 0$ and $\dot \theta=0$ imply that $a^{t}=0$. We may also set $a^{r}=0$ for simplicity. Because we have an extra equation from the identity $u^{\alpha}u_{\alpha}=-1$, $a^{\theta}$ can be determined self-consistently under the approximation $\dot{\theta}=0$. This scenario thus corresponds to flows supported by rotation in the $\hat{r}$ direction and by pressure in the $\hat{\theta}$ direction. Inserting the affine connection coefficients for the Kerr metric into the equation of motion yields quantities identical to zero on the left hand side of the equations for the $\hat{t}$ and $\hat{\phi}$ directions. This leaves only the non-trivial momentum equation in the radial direction: $$\begin{aligned} 0&=&-\left(\frac{\Sigma-2r^2}{\Sigma^2}\right)\dot{t}^2 +2\left(\frac{\Sigma-2r^2}{\Sigma^2}\right) a\sin^2\theta\dot{t}\dot{\phi}\nonumber\\ &&\hspace{0.5cm}- \left(r+a^2\sin^2\theta\left(\frac{\Sigma-2r^2}{\Sigma^2}\right)\right) \sin^2\theta\dot{\phi}^2\ ,\end{aligned}$$ which further simplifies to $$\frac{r\Sigma^2\sin^2\theta}{2r^2-\Sigma}\dot{\phi}^2 =\left(\dot{t}-a\sin^2\theta\dot{\phi}\right)^2 \ .$$ Here we choose the positive solution $$\label{flowtphi} \dot{t}=\left(\frac{\sqrt{r}\Sigma\sin\theta} {\sqrt{2r^2-\Sigma}}+a\sin^2\theta\right)\dot{\phi} \ ,$$ which corresponds to the same rotation as the black hole. This solution thus allows the flow to match the rotation of a prograde accretion disk. From the metric we have $$\begin{aligned} \label{mertic1} 1&=&\left(1-\frac{2r}{\Sigma}\right)\dot{t}^2 +\frac{4ar\sin^2\theta}{\Sigma}\dot{t}\dot{\phi}\nonumber\\ &&\hspace{0.5cm}-\left(r^2+a^2+\frac{2a^2r\sin^2\theta}{\Sigma}\right) \sin^2\theta\dot{\phi}^2 \ .\end{aligned}$$ Combining equations (\[flowtphi\]) and (\[mertic1\]) yields $$\Sigma\sin^2\theta\left(\frac{r(\Sigma-2r)}{2r^2-\Sigma} +\frac{2\sqrt{r}a\sin\theta} {\sqrt{2r^2-\Sigma}}-1\right)\dot{\phi}^2=1 \ .$$ It follows that the components of the four-velocity are $$\begin{aligned} \label{mediummotion} \dot{t}&=&\frac{1}{\zeta} \left(\Sigma\sqrt{r}+a\sin\theta \sqrt{2r^2-\Sigma}\right)\ , \nonumber\\ \dot{r}&=&0\ , \nonumber\\ \dot{\theta}&=&0\ ,\nonumber\\ \dot{\phi}&=&\frac{\sqrt{2r^2-\Sigma}}{\zeta\sin\theta} \ ,\end{aligned}$$ where $$\label{zeta} \zeta=\sqrt{\Sigma\left(\Sigma(r+1) -4r^2+2a\sin\theta\sqrt{r(2r^2-\Sigma)}\right)} \ .$$ The marginally stable orbit for particles is defined by the surface where $$\label{margin_stable} \frac{\partial E}{\partial r} = 0 \ .$$ From equations (\[tdot\]), (\[phidot\]) and (\[mediummotion\]), we have $$E = \frac{1}{\zeta}\left((\Sigma-2r) \sqrt{r}+a\sin\theta\sqrt{2r^2-\Sigma}\right) \ .$$ After differentiation, we remove the non-zero factors in the expression and obtain the condition $$\label{marg_stable} \Delta\Sigma^2-4r(2r^2-\Sigma) \left(\sqrt{2r^2-\Sigma}-a\sin\theta\sqrt{r}\right)^2=0\ .$$ Setting $a=0$ gives $r=6$, which is often regarded as the limit for the inner boundary of an accretion disk around a Schwarzschild black hole. This value is the same as that derived by Bardeen, Press & Teukolsky (1972) using $\partial^2 p_{\rm r}^2 / \partial r^2 = 0$. Before we proceed further, we must note that the expressions for the velocity components in equations (\[mediummotion\]) hold only for regions “sufficiently” far from the black-hole event horizon. The approximation that we adopt in the model breaks down when the square root in the denominator approaches zero. This occurs at the light circularisation radius $r_{\rm cir}$, which is given by $\zeta = 0$, or equivalently $$\label{light_circ} \Sigma(r+1)-4r^2+2a\sin\theta\sqrt{r(2r^2-\Sigma)}\ \bigg|_{r = r_{\rm cir}} = 0 \ .$$ Moreover, the assumption of $\dot{\theta}=0$ is also invalid for radii smaller than the radius of the marginally stable orbit — the flow is neither rotational nor pressure supported and it follows a geodesic into the event horizon. ### Isobaric Surfaces In a stationary accretion flow, the acceleration must be balanced by some forces, e.g. the gradient of gas or radiation pressure. As the local acceleration $a^\alpha$ can be calculated from the rotation law $\omega(r,\theta)$ we can derive a set of isobaric surfaces when a rotation law is given. For a barotropic equation of state of the accreting matter the isobaric surfaces coincide with the isopicnic (constant-density) surfaces. The accelerations in the ${\hat r}$ and ${\hat \theta}$ directions are $$\begin{aligned} -\frac{\Sigma}{\Delta}a^r&=&\frac{\Sigma-2r^2}{\Sigma^2} \left(\dot{t}-a\sin\theta\dot{\phi}\right)^2+r \sin^2\theta \dot{\phi}^2\ , \\ -\Sigma a^\theta&=&\sin\theta\cos\theta \left[\frac{2r}{\Sigma^2}\left(a\dot{t}-(r^2+a^2)\dot{\phi}\right)^2 +\Delta\dot{\phi}^2\right]\ .\end{aligned}$$ The surface of constant acceleration is given by $$\label{tsurface} a_\alpha \frac{dx_{\rm surf}^\alpha}{d\lambda} = 0$$ (here and hereafter $dx_{\rm surf}^{\alpha}/{d\lambda} \equiv dx^{\alpha}/{d\lambda}|_{x_{\rm surf}}$). The stationary condition implies $d t/{d\lambda}= 0$, and axisymmetry implies $d \phi/{d\lambda}=0$. Without losing generality, we can choose $t = \phi = 0$ on the surface. Thus, equation (\[tsurface\]) becomes $$\begin{aligned} \label{tsurface2} 0 &=& \frac{\Sigma a^r}{\Delta} \frac{d r_{\rm surf}}{d \lambda} + \Sigma a^\theta \frac{d \theta_{\rm surf}}{d \lambda}\ ,\nonumber\\ &=& \beta_1 \frac{d r_{\rm surf}}{d \lambda} + \beta_2 \frac{d \theta_{\rm surf}}{d \lambda}\ ,\end{aligned}$$ where $$\begin{aligned} \beta_1&=&\frac{\Sigma-2r^2}{\Sigma^2} \left(\frac{1}{\omega}-a\sin\theta\right)^2+r\sin^2\theta \ , \nonumber\\ \beta_2&=&\sin\theta\cos\theta \left[\Delta+\frac{2r}{\Sigma^2} \left(\frac{a}{\omega}-(r^2+a^2)\right)^2\right]\ , \end{aligned}$$ and $d r_{\rm surf}/d \lambda$ and $d \theta_{\rm surf}/d \lambda$ determine the intersection of the isobaric surfaces and the $(r,\theta)$ plane. By rescaling equation (\[tsurface2\]) with a factor of $\sqrt{\Delta/\Sigma}$ and making use of the invariance $$- \left(\frac{d\tau}{d\lambda}\right)^2 = \frac{\Sigma}{\Delta}{\dot r}^2 + \Sigma {\dot \theta}^2 \ ,$$ we obtain $$\begin{aligned} \frac{d r_{\rm surf}}{d \lambda'} &=& \frac{\beta_1}{\sqrt{\beta_2^2+\Delta\beta_1^2}}\ , \nonumber \\ \frac{d \theta_{\rm surf}}{d \lambda'} &=& \frac{-\beta_2}{\sqrt{\beta_2^2+\Delta\beta_1^2}}\ . \label{isobaric} \end{aligned}$$ These two differential equations can be solved numerically and yield the isobaric surface as a parametric function of $\lambda'$. Model Accretion Disks and Tori ============================== We now demonstrate using the equations of motion above to construct the emitter models. The first is a geometrically thin accretion disk, in which the emitting particles are in Keplerian motion. The second is a torus, a 3-dimensional object with non-negligible thickness. Accretion Disk -------------- When space-time curvature is important, the Keplerian angular velocity of a test particle around a gravitating object is no longer $\omega_k= r^{-3/2}$, the expression in flat space-time. Instead, the Keplerian angular velocity in a plane containing the gravitating object can be obtained by setting $\theta=\pi/2$ in equations (\[mediummotion\]) and (\[zeta\]). Hence, the components of the four-velocity of the particles in the disk are $$\begin{aligned} \label{diskvel} \dot{t}&=&\frac{r^2+a\sqrt{r}}{r\sqrt{r^2-3r+2a\sqrt{r}}}\ , \nonumber \\ \dot{r}&=&0 \ ,\nonumber \\ \dot{\theta}&=&0 \ ,\nonumber \\ \dot{\phi}&=&\frac{1}{\sqrt{r}\sqrt{r^2-3r+2a\sqrt{r}}}\ ,\end{aligned}$$ and the rotational velocity of a Keplerian accretion disk around a black hole is $$\omega_k=\frac{1}{r^{3/2}+a},$$ (Bardeen, Press & Teukolsky 1972). The relative energy shift of the emission between the disk particle and an observer at a large distance is determined by equation (\[freqshift\]), with $u^\alpha$ as given in equation (\[diskvel\]). Keplerian disk images can be found in many existing works (e.g.  Bromley, Miller & Pariev 1998), and we do not show disk images here. The general characteristics are that a disk image is asymmetric, with the separatrix for the energy shift of the emission no longer bisecting the disk image into two equal sectors, one for red shift and another for blue shift. The whole disk appears to be reddened, especially at the inner rim. Accretion Torus --------------- To determine the geometry and structure of an accretion torus self-consistently is beyond the scope of this paper. Here, we consider a simple parametric model, with an angular velocity profile given by $$\omega=\frac{1}{(r\sin\theta)^{3/2}+a} \left(\frac{r_{\rm k}}{r\sin\theta}\right)^n \ . \label{rk-equation}$$ The quantity $r_{\rm k}$ is the radius (on the equatorial plane) at which the material moves with a Keplerian velocity. The parameter $n$ adjusts the force term, such as a pressure gradient, to keep the disk particles in their orbits, and it determines the thickness of the torus. In this study we just take $n=0.21$ without losing generality. If the torus is supported by radiation pressure, its inner edge is determined by the intersection of the isobaric surface with either one of two surfaces. These two surfaces provide the constraints, inside which the pressure-supported solution does not hold. The first is a surface defined by the orbits of marginal stability. For $\omega(r,\theta)$, it is given by $$\begin{aligned} 0 &=& 2a\sin^4\theta\left[\frac{r^2}{\Sigma} -\left(r^2+a^2+\frac{a^2r\sin^2\theta}{\Sigma} \right) \frac{\Sigma-2r^2}{\Sigma^2} \right]\omega^3 \nonumber\\ && +\sin^2\theta \bigg[\left(\frac{6r(r^2+a^2)}{\Sigma}+3\Delta-\Sigma\right) \frac{\Sigma-2r^2}{\Sigma^2} \nonumber \\ && \hspace*{0.5cm} +r\left(1-\frac{2r}{\Sigma}\right)\bigg]\omega^2 -\frac{6ar\sin^2\theta}{\Sigma} \left(\frac{\Sigma-2r^2}{\Sigma^2}\right)\omega \nonumber\\ && +\Delta\sin^2\theta ~\omega\frac{\partial\omega}{\partial r} -\left(1-\frac{2r}{\Sigma}\right)\frac{\Sigma-2r^2}{\Sigma^2} \ . \end{aligned}$$ The second is the limiting surface where the linear velocity approaches the speed of light. It is given by $$\begin{aligned} 0 &=& \Sigma-2r+4ar\omega\sin^2\theta \nonumber\\ & &\hspace{1cm} -\left((r^2+a^2)\Sigma+2a^2r\sin^2\theta \right)\omega^2\sin^2\theta \ . \end{aligned}$$ Usually the former is larger than the latter. The outermost of these two surfaces determines the inner boundary and hence the critical surface of the torus. Figure \[various\_surface\] shows the critical density surfaces of two tori. The first torus is around a Schwarzschild black hole and the second torus is around a maximally rotating black hole. The tori are constructed such that their specific angular momentum has a profile similar to those of the simulated accretion disks in Fig. 3. of Hawley & Balbus (2002). In our model the boundary surface of the torus is determined by a single parameter, $n$, which specifies the index of the angular-velocity power law. Its value is selected such that the angular-velocity profile matches the profiles obtained by the numerical simulations — here we consider that of Hawley & Balbus (2002). Model tori can be constructed using various different methods. An example is that in a study of dynamical stability of tori around a Schwarzschild black hole carried out by Kojima (1986), the model parametrizes the angular momentum instead of the angular velocity. We note that the aspect ratios of the torus surfaces obtained by Kojima (1986) and those shown in Fig. \[various\_surface\]. are similar. Extinction ========== The generic setting of the system under our investigation is that emitters with various strengths are distributed in space in a curved space-time, and the radiation is attenuated, and may be re-emitted, when propagating. The emitters and the line-of-sight material are in relativistic motion with respect to the observer and also with respect to each other. An example is that shown in Fig. \[cloud\_model\]., in which the emitters are the surface elements of an accretion disk and the absorbers are some clouds in the vicinity of the disk. The photon trajectories and the motion of the emitters and absorbers are affected by the space-time distorted by the central black hole. To construct the model we need to determine - the rays that connect the emitters, absorbers and observer, - the four-velocities of the emitters and absorbers /scatterers, - the spatial distributions of the emitters and the absorbers/scatterers, and - the effective cross section of the absorbers/scatterers. In the previous section, we have shown how to obtain (i) and (ii); in this section we incorporate (iii) and (iv) into the radiative-transfer calculations. An illustrative model --------------------- We consider a model with the geometry shown in Fig. \[cloud\_model\]. The photons are emitted from the elements on the top and bottom surfaces of a geometrically thin disk in a Keplerian rotation around a Kerr black hole. The radiation is resonantly scattered (absorbed) by plasma clouds and is attenuated in its propagation. The size of the clouds is small in comparison with the length scale of the system. They are not confined to be in the equatorial plane and are in orbital motion, supported by some implicit forces (which may be radiation, kinematic or magnetic pressure gradients). These clouds have a large (thermal) distribution of velocities, in addition to their collective bulk velocity. We assume that the radiation scattered into the energies of the lines is insignificant and ignore the photons that are scattered into the line-of-sight. Under this approximation, scattering simply removes the line photons and causes extinction similar to true absorption. Thus, for simplicity, hereafter we do not distinguish between scattering and absorption, [^1] and the two terms are interchangeable, unless otherwise stated explicitly. The rays originating from the accretion-disk surface that can reach the observer are determined by the 4-momenta of the photons, which are calculated using equations (\[tdot\]), (\[phidot\]) and (\[rdot\]). The 4-velocities of the emitting surface elements on the accretion disk are given by equations (\[diskvel\]). These determine the relative energy shifts of the photons between the emitters and the observer. What we need next is to determine the relative energy shifts between the emitters and the absorbers. Then, we need a model mechanism by which the absorption takes place, and to derive the resonant absorption condition for the absorption coefficient. Now we construct a model for the spatial distribution and the velocities of the absorbers. Consider a parametric model in which the bulk 4-velocities of the clouds are given by equations (\[mediummotion\]) and (\[zeta\]). In this model the bulk velocities of the clouds in the equatorial plane matches the 4-velocities of the accretion disk. The clouds themselves are cold, and the thermal velocity of the gas particles inside are much smaller than their bulk motion and root-mean-square velocity dispersion. However, the clouds have a large velocity dispersion, given by the local virial temperature, which is comparable to the energy of the emission lines of interest. Therefore, the clouds can be considered as relativistic particles in the calculation. Using the bulk-motion velocities obtained by (\[mediummotion\]) and (\[zeta\]) together with the virial theorem, we can derive this temperature and determine the velocity distribution of the absorbing clouds. The clouds fill most of space, with a radially dependent number density. However, close to the black hole, the assumptions above break down, and the axial force cannot support the clouds out of the equatorial plane. When this happens, they will flow along geodesics directly into the hole. In the numerical calculation, we determine the asymptotic boundaries at which the left hand sides of equations (\[light\_circ\]) and (\[marg\_stable\]) vanish. This is done by evaluating these expressions and testing to see if they are negative at each point along the photon rays. Inside that surface, the number density of the clouds will be much less than that outside, which we approximate by setting it to zero in this zone. The absorption coefficient -------------------------- We assume that the absorption is due to “cold” cloudlets with high virial velocities. The absorption coefficient of individual cloudlets is $$\begin{aligned} \chi_{\rm i} & \propto & \sigma ~ \delta\left(\frac{u^\alpha k_\alpha+{E_{\rm line}}}{{E_\gamma}}\right) \end{aligned}$$ (with $\sigma$ as the effective absorption cross section of the cloudlet, and $k_\alpha$ the photon four-momentum). The absorption rest frequency is ${E_{\rm line}}$, and ${E_\gamma}$ is the energy of the photon in the bulk rest frame. The total effective absorption coefficient $\chi_0$ is the sum of the contribution of these cloudlets, i.e., $$\begin{aligned} \chi_0 & = & \sum_{\rm i} ~\chi_{\rm i} \ . \end{aligned}$$ Converting the sum into an integral in momentum space yields the absorption per unit length in the rest frame as $$\begin{aligned} \label{chieqn} \chi_0&=&\frac{-2\pi\lambda\sigma}{{E_{\rm line}}^2}\times \nonumber\\ &&\!\!\!\int\!\!\!\int\!\! p^2dpd\mu \exp(-E/\Theta) u^\alpha k_\alpha \delta\left(\frac{u^\alpha k_\alpha + {E_{\rm line}}}{{E_\gamma}}\right),\end{aligned}$$ where we have defined $\mu = \cos\theta$, $\lambda$ is a normalization constant, $\Theta$ is the temperature in relativistic units (with $k_B=1$), and $E$ and $p$ are the energy and momentum of a gas particle in the bulk rest frame. This form assumes isotropic thermal motion in the rest frame. There are three terms that we need to determine before we can evaluate the integral, and they are the normalisation parameter $\lambda$, the temperature $\Theta$ and the photon energy in the rest frame of the absorbing particle $u^\alpha k_\alpha$. When these variables are determined, we can then parametrise $\sigma$ after integration is carried out. ### The normalisation parameter $\lambda$ We now derive $\lambda$ starting from $$N=4\pi\lambda\int^\infty_0 p^2dp \exp\left(-E/\Theta\right)\ ,$$ where $N$ is the number density of absorbing clouds. Integrating yields $$\lambda=\frac{N\frac{m}{\Theta}}{4\pi m^3K_2(\frac{m}{\Theta})}\ ,$$ where $K_\nu(x)$ is a modified Bessel function, and $m$ is the average cloud mass. This is called the Jüttner distribution and corresponds to Maxwell’s distribution except in the case of a relativistically high temperature. ### The temperature $\Theta$ The total energy in the distribution of clouds is given by $$E_{\rm tot}=4\pi\lambda\int^\infty_0 p^2dp E \exp\left(-E/\Theta\right)\ .$$ Integrating this yields the energy per unit mass as $$\label{Etot2} \frac{E_{\rm tot}}{Nm}=\frac{K_3(\frac{m}{\Theta})} {K_2(\frac{m}{\Theta})}-\frac{\Theta}{m}\ .$$ Using conservation of energy and angular momentum, we calculate the thermal energy of the virialised relativistic gas of absorbing clouds. At infinity the medium has $$\begin{aligned} E_{\rm init}&=&Nm \ , \\ L_{\rm init}&=&L_{\rm fin} \ .\end{aligned}$$ Close to the black hole it has $$\begin{aligned} E_{\rm fin} &=&\frac{Nm}{\zeta}\left[(\Sigma-2r)\sqrt{r} +a\sin\theta\sqrt{2r^2-\Sigma}\right]\ , \\ L_{\rm fin}&=&\frac{Nm}{\zeta}\left[2ar\sqrt{r}\sin^2\theta -(r^2+a^2)\sin\theta\sqrt{2r^2-\Sigma}\right]\end{aligned}$$ The energy released by the gas falling from infinity and slamming into a wall moving with a velocity given by equation (\[mediummotion\]) is $$-E_{\rm tot}=u^\alpha p_\alpha =-E_{\rm fin}\dot{t}_{\rm init}-L_{\rm fin}\dot{\phi}_{\rm init}\ .$$ After simplification, this becomes $$\begin{aligned} \label{Etot1} -\frac{E_{\rm tot}}{Nm}\!&=&\!\frac{1}{\zeta^2} \left[(2r^2-\Sigma)(r^2+a^2) -2ar\sin\theta\sqrt{r}\sqrt{2r^2-\Sigma}\right. \nonumber\\ &&\hspace{0.5cm}\left.-\left(a\sin\theta\sqrt{2r^2-\Sigma} +\Sigma\sqrt{r}\right)\zeta\right]\ .\end{aligned}$$ Thus the temperature of the media can be derived using equations (\[Etot2\]) and (\[Etot1\]). Unfortunately, this yields an implicit relation of $m/\Theta$ that contains transcendental functions. The modified Bessel functions can be expanded in the limit where $\Theta \ll m$ which corresponds to an “almost relativistic” gas. Since the potential energy released in accretion is of the order of a few percent of the rest mass of the infalling material, this approximation should hold in AGN. Expanding to second order in $\Theta / m$, cancelling the exponential factors, and then solving the resulting quadratic yields $$\frac{\Theta}{m} = \frac{2}{5}\left(-1+\sqrt{1+\frac{10}{3} \left(\frac{E_{\rm tot}}{Nm}-1\right)}\right)\ .$$ Thus we have an explicit description of how the kinematic temperature varies with position. ### The photon energy $u^\alpha k_\alpha$ In the rest frame, the motion of the thermalised medium is isotropic. Thus we can simplify the problem by aligning an axis along the photon propagation vector and working in a local Lorentz frame so that $$k_\alpha = {E_\gamma}(-1, 1, 0, 0)\ ,$$ and $$p^\alpha = m u^\alpha =(E, p \mu, p_{\rm y}, p_{\rm z})\ .$$ Thus, $$u^\alpha k_\alpha = \frac{{E_\gamma}}{m}(p\mu - E)\ .$$ ### Evaluation of the $\delta-$function Using the relation that $$\frac{d}{d\mu}\ \left[\frac{u^\alpha k_\alpha + {E_{\rm line}}}{{E_\gamma}}\right] = -\frac{p}{m} \ ,$$ we obtain $$\begin{aligned} \int d\mu \ u^\alpha k_\alpha \delta\left(\frac{u^\alpha k_\alpha + {E_{\rm line}}}{{E_\gamma}}\right) & & \nonumber \\ & & \hspace{-2cm} = {\frac{u^\alpha k_\alpha}{\big\vert \frac{d}{d\mu}\left[\frac{u^\alpha k_\alpha + {E_{\rm line}}}{{E_\gamma}}\right]\big\vert}}\bigg\vert_{u^\alpha k_\alpha =- {E_{\rm line}}} \nonumber \\ & & \hspace{-2cm} = -{E_{\rm line}}\frac{m}{p}\bigg\vert_{u^\alpha k_\alpha =- {E_{\rm line}}} \ . \end{aligned}$$ We change the variable $p dp$ to $EdE$. The integral in equation (\[chieqn\]) can now be simplified to $$\chi_0=\frac{2\pi\lambda \sigma m}{{E_{\rm line}}}\int^\infty_{\frac{m}{2}\left(\frac{{E_\gamma}}{{E_{\rm line}}}+\frac{{E_{\rm line}}}{{E_\gamma}}\right)} EdE \exp(-E/\Theta)$$ Integrating this gives the absorption coefficient in the rest frame of the gas $$\begin{aligned} \label{abschi0} \chi_0&=&\frac{N\sigma}{2 K_2(\frac{m}{\Theta})}\left[\frac{1}{2}\left(\frac{{E_\gamma}}{{E_{\rm line}}}+\frac{{E_{\rm line}}}{{E_\gamma}}\right)+\frac{\Theta}{m}\right]\times\nonumber\\ &&\hspace{1.5cm}\exp\left[-\left(\frac{m}{2\Theta}\right)\left(\frac{{E_\gamma}}{{E_{\rm line}}}+\frac{{E_{\rm line}}}{{E_\gamma}}\right)\right]\end{aligned}$$ This equation can be recast in terms of $\chi$ by using equations (\[chframe\]) and (\[freqshift\]). ### The effective absorption $N\sigma$ The absorption coefficient depends upon $N$, the number density of the clouds, and on $\sigma$, the absorption cross-section per cloud. These are in general a function of position. Since the pressure gradient and the inflow velocity in the $\hat{r}$ and $\hat{\theta}$ directions are ignored in our approximation of the flow, the density profile cannot be determined in a fully self-consistent manner via the mass continuity equation. To overcome this, we assume a simple two-parameter profile $$N\sigma=\sigma_0 r^{-\beta}\ .$$ Where $\sigma_0$ can be considered as a proportionality constant fixing the density and opacity scales. Without losing generality, we adopt a value $\beta = 3/2$. The results can then easily be generalized to other values of $\beta$. For the case of a swarm of absorbing clouds around a black hole, the optical depth is approximately one. This corresponds to the case where $\sigma_0$ is of order $0.5$ or greater, with the integration proceeding radially to the event horizon. The effective optical depth depends greatly on the paths the photons take. Spectral Calculations ===================== No absorption ------------- ### Accretion disks We calculate the observed energy of the flux from a point on the planar disk using equation (\[freqshift\]). We ignore absorption. The line emissivity has a power law profile, which decreases radially from disk centre. The intensity is proportional to the third power of the relative shift; the flux has an extra factor of $\nu_0/\nu$ due to time dilation, i.e. it is proportional to the fourth power of the relative frequency shift. (Note that $F$ in §2 is the distribution function, not the flux of the emission.) Our calculation reproduces the line profiles of direct images of accretion disks as those shown in Fabian et al (1989), Kojima (1991), Fanton et al (1997), Bromley et al (1997), and Reynolds et al (1999). Figure \[fantonfig\] shows two example line spectra calculated using the method described above. The spectra contain only emission from the direct image. We also show line obtained using the method by Fanton et al (1997) for comparison. The results in the two ca lculations are in excellent argeement. We also carry out spectral line calculations which include contribution from higher-order disk images. (Here and hereafter we assume that the emissivity powerlaw has an index of $-$2, except where otherwise stated explicitly. ) Our calculations show that the contribution of the higher-order images are significant only at high inclination angles (see Fig. \[multi\_orders\]). The emission from high-order images is mostly at frequencies close to the rest frequency of the line, because the region where highly red and blue-shifted emission originates is obscured. ### Rotational Torus We now investigate the emission from an accretion torus. We consider a model in which the inner radius of the torus is determined by the marginally stable orbits of the particles. These marginally stable orbits, which depend on $\omega$ and $d \omega/dr$, form a surface in a three-dimensional space. The marginally stable orbit for particles in Keplerian motion in the equatorial plane is 6 $R_{\rm g}$ around a Schwarzschild black hole and is 1.23 $R_{\rm g}$ around a Kerr black hole with $a = 0.998$. We use a surface-finding algorithm to determine the boundary of the torus (see Appendix \[surface\_finding\]). In Fig. \[torus\_image2\]. we show three-dimensional images of the model torus around a Kerr black hole. We include the first four image orders. The inclusion of high order images is mandatory due to the ‘mixing’ caused by the extension out of the equatorial plane (Viergutz 1993). The torus is viewing inclination angles of $45^\circ$ and $85^\circ$ (top and bottom panels respectively). The left-right asymmetry is caused by inertial-frame dragging. The multiple images are consequences of gravitational lensing. At small inclination angles, only the surface above the equatorial plane of the torus is seen in the direct image. At very large inclination angles, the surface below the equatorial plane is severely lensed and also becomes visible. The false-colour map laid on the torus surface show the energy shift of the emitted photons (determined by equation \[\[freqshift\]\]), as viewed by a distant observer. The separatrix, which corresponds to zero energy shift, divides the torus surface into regions of blue energy shift and regions of red energy shift. At large inclination angles the inner surface of the near side of the torus is not visible, and the inner surface of the far side is obscured by the near side of the torus. Thus, emission with the largest energy shifts is hidden. This is very different to the situation for a planar accretion disk – regardless of the viewing inclination and the visual distortion, the emission from the innermost part of the disk is always visible. Figure \[torus\_line1\]. shows the resulting line profiles obtained by integrating the emission over the images shown in Fig. \[torus\_image2\]. The line profile of a torus viewed at $45^\circ$ is similar to that of the planar accretion disk. It has a sharp blue peak and smaller red peak. It also has an extended red wing. This is due to the fact that the projections of a torus and a disk on the sky plane are very similar at low inclination angles. However, our calculations show that geometric effects are very important for large viewing inclination angles. When some part of the emission region is self-obscured. the resulting profile, as observed from infinity, to be completely different from that of the flat disk (see Fig. \[torus\_line2\].). For a torus with large viewing inclination angles, the inner surface of the torus tends to be obscured. This corresponds to the region where the most redshifted flux of the line is emitted (due to large transverse red shift and gravitational red shift). This makes the red wing less prominent. The outer surface of the torus is visible from all inclinations. As a result the line profile tends to be singly peaked, with the maximum at the unshifted line frequency due to the emission from the outer surface dominating. By altering the geometry of the emitter a wide variety of emission profiles can be obtained. Resonant Scattering ------------------- We use a disk model to illustrate the resonant scattering effects. We assume the inner edge of the accretion disk is given by the marginally stable orbit. We use an outer disk radius of $20 R_{\rm g}$ in all the disk simulations. This was chosen to accentuate the relativistic effects. The emission line profile is assumed to be a delta function. We collate the light from the first four image orders of the accretion disk. The line profiles are obtained from $750\times750$ pixel images. The intensity scale on the graphs is in arbitrary units. (It is just the log of the sum of pixel intensity over the image, as a function of frequency.) The x-axis of the graph is in units of the line rest energy, with $E/E_0 = 1$ corresponding to the unshifted line. We bin the injection spectrum and the absorption coefficient linearly with energy. There are 1000 bins from $E=0$ to $E=2E_0$. We investigated two space-time models. One with $a=0$ corresponding to a Schwarzschild black hole, and one with $a=0.998$, corresponding to a maximally spinning Kerr black hole. We have plotted spectra containing the continuum and the continuum plus line as we have varied the opacity of the absorbing clouds. We have also shown how the spectra change with inclination due to geometric effects. We have included a nearly edge-on model ($i=85^\circ$), and a model with a moderate inclination of $45^\circ$. See Fig. \[contnoline\]. for the results of absorbing the power law continuum, and Fig. \[contline\]. for the results of absorbing both the line and continuum. Discussion ========== We have modelled line profiles from accretion disks to demonstrate the use of a general formulation for transfer of radiation through relativistic media in arbitrary space-times. In this paper, we used the transfer of emission from AGN as an illustration. In this model, we parametrized the disk/torus to describe the emitters and the space-density distribution of the absorbers. We took into account relativistic effects on the bulk dynamics and the microscopic kinematic properties of the absorbing medium. Resonant absorption/scattering of line emission from accreting black holes in the general relativistic framework had been investigated previously by Ruszkowski & Fabian (2000). In their study, a thin Keplerian disk was assumed and the absorbing medium is a spherical corona of constant density centred on the black hole. The corona is rotating, with local rates obtained by linear interpolation from the rotation rate of a planar Keplerian accretion disk and the rotation rate at the polar region caused by frame dragging due to the Kerr black hole. The Sobolov approximation was used in the resonant absorption calculations, and a Monte Carlo method determined the re-emission/scattering. Our calculation is different to that of Ruszkowski & Fabian (2000) in the following ways. Firstly, the emitters are not confined to the equatorial plane, i.e. they can be thin accretion disks or thick tori. Secondly, the absorbing medium is a collection of (cold) clouds with relativistic motions. The number density distribution of the clouds is parametrized by a powerlaw decreasing radially. The local bulk (rotational) velocity of the clouds is determined by general relativistic dynamics, and the velocity dispersion is calculated from the Virial theorem. Thirdly, we do not assume the Sobolov approximation. The resonant condition for the absorption coefficient is derived directly from the kinematics of the absorbing cloud particles. Fourthly, we ignore the contribution from re-emission to the line flux. However, we include emission from higher order disk images, in addition to the direct image. One of the main differences between the two studies is the treatment of resonant absorption. In the Sobolov approximation, the absorption takes place locally (see Rybicki and Lightman 1979). The line profile is practically a delta function; otherwise, the assumption of quasi-local absorption breaks down. Moreover, it requires that the absorbing medium is a radial flow. The emission lines are because of relativistic effects, and the motion of the medium is rotationally dominated. The locality of the absorption (required by the Sobolov approximation) therefore breaks down. To overcome these difficulties, we abandon the Sobolov approximation but, instead, employ full ray tracing. In modelling the bulk flow, we derived the equation of motion of the absorbers using the rotational-support approximation for the accretion disk. In addition, we assume that the flow is supported out of the equatorial plane. There is negligible radial force in the equations of motion, and the radial velocity can be neglected. To include outflowing as a wind, or inflowing, requires additional components in the equation of motion. This complicates the formulation, in particular, when matching the flow boundary condition at the surfaces of the accretion disk/torus. A self-consistent boundary condition requires a dissipation mechanism in the boundary regions of the disk/torus and absorbing clouds. The inclusion of such dissipation is beyond the scope of this paper and this issue will be addressed in future works. As the rays propagate from the accretion disk to the observer, they experience position-dependent absorption. The absorption depends upon the velocity profile of the material as well as its density and the line profile function of the absorption coefficient. Since the absorbers are moving in relativistic speeds, the bulk velocity is an important factor. Lorentz contraction increases the absorption coefficient accordingly, and Doppler shift alters the frequency of the emission as seen by the absorbers. The potential energy liberated by material infalling into a black hole is of order a few percent of the material’s rest mass, and the energy corresponding to ‘thermal’ kinematic velocity dispersion approaches this rest mass energy. Thus, we replace the conventional Maxwellian distribution by the Jüttner distribution for relativistic particles in deriving the resonant line absorption coefficient. This distribution does not give a Gaussian absorption profile even in the bulk rest frame of the absorbers. The line profiles of the direct images show a dip around the line rest frequency (energy) $(E/E_0 =1)$. Absorption can change the line profile significantly. The flux around the rest frequencies is, however, augmented by the flux from shifted lines from the higher order images, especially at high disk inclination. These two effects compete, and when the line-of-sight optical depth is high, the contribution of the high-order images is masked. Conclusion ========== We present a numerical ray-tracing method for radiative-transfer calculations in curved space time and apply the method to calculate line emission from accretion disks and tori around black holes. Our calculations have shown that lines from relativistic accretion tori have profiles very different to lines from relativistic thin planar accretion disks for the same system parameters, such as the spin of the black hole and the viewing inclination. The self-obscuration of the inner region of the accretion torus leads to weaker red wing in the emission lines when compared with the lines emitted from a thin planar accretion disk. At high inclination angles the strong blue peak is also absent in the line emission from the tori. We also investigate the effects of resonant absorption/scattering by the line-of-sight material in relativistic motion with respect to the emitters in the disk/torus, and the observer. Our method does not invoke the Sobolov approximation, and the resonant absorption/scattering condition is derived directly. We have shown that absorption effects are important in shaping the profiles of emission lines. The interpretation of observations of relativistic lines from AGN is non-trivial when absorption is present. We thank Mat Page for discussion, and Alex Blustin for comments on the manuscript. SVF acknowledges the support by a UK government Overseas Research Students Award and a UCL Graduate School Scholarship. Bao, G. 1992, A&A, 257, 594 Bao, G., Hadrava, P., & Ostgaard, E. 1994, ApJ, 435, 55 Bardeen, J. M., Press, W. H., & Teukolsky, S. 1972, ApJ, 178, 347 Baschek, B., Efimov, G. V., Waldenfels, W. von, & , R. 1997, A&A, 317, 630 Bromley, B. C., Chen, K., & Miller, W. A. 1997, ApJ, 475, 57 Cadez, A., Fanton, C., & Calvani, M. 1998, New Astronomy, 3, 647 Cadez, A., Brajnik, M., Comboc, A., Calvani, M., & Fanton, C. 2003, A&A, 403, 29 Carter, B. 1968, Phys. Rev., 174, 1559 Cunningham, C. T. 1975, ApJ, 202, 788 Cunningham, C. T. 1976, ApJ, 208, 534 Dabrowski, Y., Fabian, A. C., Iwasawa, K., Lasenby, A. N., & Reynolds, C. S. 1997, MNRAS, 288, L11 Fabian, A. C., Rees, M. 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C., 1999, ApJ, 514, 164 Ruszkowski, M., & Fabian, A. C. 2000, MNRAS, 315, 223 Rybicki, G. B., & Lightman, A. P. 1979, Radiative Processes in Astrophysics, Wiley, New York Smak, J. 1969, Acta Astronomica, 19, 155 Stella, L. 1990, Nature, 344, 747 Tanaka, Y., et al. 1995, Nature, 375, 659 Viergutz, S. U. 1993, A&A, 272, 355 Zane, S., Turolla, R., Nobili, L., & Erna, M. 1996, ApJ, 466, 871 Frequency Bins ============== Using an astrophysical model together with the metric, we can obtain an equation describing the flow of the medium through which the ray is propagating; $u^\alpha(x^\alpha(\lambda))$. The properties and distribution of the medium can be used to derive the absorption and emission terms $\chi_0(x^\alpha(\lambda))$ and $\eta_0(x^\alpha(\lambda))$. These can then be used to calculate what is observed from infinity via equation (\[anaray\]). In practice this simple method is altered with a ray-tracing numerical algorithm. The radiative transfer equation is not time-symmetric, so if one follows rays back from the observer to the emitter the above formulation cannot be used. The emission frequency and intensity is not known until the emitter is reached along the path. This problem can be solved using one of two approaches. The first is to collate the path from emitter to observer, and then to integrate along that path forward in time using equation (\[anaray\]). The second approach is to collate the optical depth for $\cal I$ for a table of frequencies as one travels back in time. This optical depth can then be used to calculate the observed intensity, $I$, for each binned frequency. The results for each emission element along the path are simply summed. Since, in general, the resulting frequencies are binned anyway to get a spectrum, we choose the second method. It requires less storage, and is a simple extension of the grey-absorber case where $\chi_0$ and $\eta_0$ are no longer functions of $\nu$. In our treatment, we sum over the observed frequencies rather than those in the rest frame, avoiding the need to transform the distribution of the frequencies along the path due to the gravitational redshift factor. The act of going from the emission frame to the observer’s frame also affects the intensity $I$, so we use the invariant $\cal I$ instead, only converting to specific intensity just before outputting the results. Binning the optical depth with frequency along a path also allows us to model the radiative transport of the continuum as well as the lines, provided that one is in the limit where scattering is unimportant. (Stimulated emission can be modeled by using a negative absorption coefficient.) Depending on the relative intensities of the continuum and the line, absorption may cause the emission feature to be converted into an absorption feature. We assume that the continuum is a power law, which we parametrize by a slope and intensity at the line rest frequency. We have assumed that the continuum is emitted with an intensity that scales with the line emissivity. This means that the equivalent width of the emission line is constant across the disk. We set this to be $0.05$ of the rest line frequency. (This value is roughly what is seen in AGN.) To simplify things further, we will fix the power law index of the continuum to be $\gamma=0.5$, where $\gamma$ is defined by $$I=C E^{-\gamma},$$ in which $I$ is the continuum intensity as a function of energy, $E$, and $C$ is derived from the given equivalent width. This rather hard spectrum was chosen to emphasise the line. The treatment assumes that the continuum is created in a relatively thin planar structure above the accretion disk. In effect, we treat the emission from the disk corona as part of the injection spectrum, together with the emission line, which we propagate through the absorbing material suspended much higher above. Ray Tracing Algorithm {#rtalgorithm} ===================== A direct ray-tracing method is used instead of the conventional transfer-function method, as it is easier to incorporate the numerical radiative-transport calculations. The ray-tracing algorithm is as follows: 1. Integrate the equations governing the geodesics, and those describing the optical depth for each frequency, from the observer to the emitting surface; 2. At each crossing of the equatorial plane / torus surface, collate the position and the direction of the photon; 3. Construct the image, and determine the observed frequency/energy shift; 4. Integrate the emission over the images of each order to produce the line profiles. The foot points of the null geodesics (photon trajectories) on the disk surface are calculated by a root-finding algorithm (see e.g. Press et al. 1992, p.343). The first four intersections of the null geodesics and the disk plane (corresponding to the direct and first three higher-order images), and the four-vectors of the photons emitted from there are recorded. The incorporation of an emissivity law is therefore straightforward, as it is defined in terms of the spatial coordinates on the disk plane. Since the trajectory of the photon is also saved at each crossing point, it is also possible to include the effects of limb darkening, and to model a semi-transparent disk. Since the disk is imaged upon a sky plane, all gravitational lensing effects on the intensity of the light are implicitly included in the calculation of the image itself. If a region of the disk is magnified then it will cover more area in the image, and thus will appear brighter than a non-magnified region. Inclination effects are also included implicitly. An inclined disk will cover less pixels than a face-on disk, where the number of pixels is roughly proportional to $\cos i$, where $i$ is the inclination angle. (Light bending causes this Euclidean formula to be only an approximation.) Gravitational lensing does not alter the observed surface brightness of a point. This implicit inclusion of changing areas of photon flux-tubes linking the observer to the emitter vastly simplifies the calculation of the observed flux. All that is required is to integrate over each pixel on the image, taking into account the redshift of the emission regions corresponding to the pixels. If one were integrating over the surface of the disk, instead of over the image, then the Jacobian of the transformation from the disk to the image plane coordinates would be required. This is numerically difficult to obtain, and would require a separate transformation for each image order. Surface Finding Algorithm {#surface_finding} ========================= We consider the following algorithm to determine the torus surface. We integrate equation (\[isobaric\]) and tabulate the resulting points ($r,\theta$) along the path of the integration. Then we interpolate ($r,\theta$) and construct the torus surface, where the emission originates. We use spline interpolation between the surface points. When ray tracing the photon paths, we determine the intersection of the trajectory and the torus surface. As the photon trajectory calculations may take large spatial steps, there is a possibity that the torus is not ’detected’. To prevent this from happening, we consider the following procedure. We take note of the region where the photon is located during the trajectory calculation: either inside or outside the torus and either above or below the equatoral plane. Whenever the photon leaves one of the four regions, and enters another region, we use a boundary-searching algorithm to find the exact location where the transit occurs. If the trajectory hits the equatorial plane outside the torus, the integration will continue. If the trajectory hits the torus, then integration is terminated. By taking smaller steps required by the boundary-finding algorithm, we can prevent the integrator from missing the torus entirely. This algorithm can be used for more complicated surfaces. It works well, because the integrator will only miss intersections when the trajectory is close to tagential to a surface. This happens close to the equatorial plane in the torus models. Adding in a fake boundary there, and thus decreasing the step size, helps in preventing missed intersections. [^1]:

--- abstract: 'We study the $\beta$ analogue of the nonintersecting Poisson random walks. We derive a stochastic differential equation of the Stieltjes transform of the empirical measure process, which can be viewed as a dynamical version of the Nekrasov’s equation in [@MR3668648 Section 4]. We find that the empirical measure process converges weakly in the space of c[á]{}dl[á]{}g measure-valued processes to a deterministic process, characterized by the quantized free convolution, as introduced in [@MR3361772]. For suitable initial data, we prove that the rescaled empirical measure process converges weakly in the space of distributions acting on analytic test functions to a Gaussian process. The means and the covariances are universal, and coincide with those of $\beta$-Dyson Brownian motions with the initial data constructed by the Markov-Krein correspondence. Our proof relies on integrable features of the generators of the $\beta$-nonintersecting Poisson random walks, the method of characteristics, and a coupling technique for Poisson random walks.' address: | Harvard University\ E-mail: jiaoyang@math.harvard.edu author: - Jiaoyang Huang bibliography: - 'References.bib' title: '$\beta$-Nonintersecting Poisson Random Walks: Law of Large Numbers and Central Limit Theorems' ---   Introduction ============ $\beta$-nonintersecting Poisson random walks -------------------------------------------- Let $\tilde{{\bm{x}}}(t)=(\tilde x_1(t),\tilde x_2(t),\cdots, \tilde x_n(t))$ be the continuous-time *Poisson random walk* on ${\mathbb{Z}}_{{\geqslant}0}^n$, i.e. particles independently jump to the neighboring right site with rate $n$. The generator of ${{\bm{x}}}(t)$ is given by $$\begin{aligned} \tilde {{{\mathcal}L}}^n f(\tilde {{\bm{x}}})=\sum_{i=1}^nn\left(f(\tilde{{\bm{x}}}+{\bm{e}}_i)-f(\tilde {{\bm{x}}})\right),\end{aligned}$$ where $\{{\bm{e}}_i\}_{1{\leqslant}i{\leqslant}n}$ is the standard vector basis of ${{\mathbb R}}^n$. $\tilde {{\bm{x}}}(t)$ conditioned never to collide with each other is the *nonintersecting Poisson random walk*, denoted by $\tilde {{\bm{x}}}(t)=(x_1(t),x_2(t),\cdots,x_n(t))$. The nonintersecting condition has probability zero, and therefore, needs to be defined through a limit procedure which is performed in [@MR1887625]. The nonintersecting Poisson random walk is a continuous time Markov process on $$\begin{aligned} {\mathbb{W}}^n_1=\{({\lambda}_1+(n-1),{\lambda}_2+(n-2),\cdots,{\lambda}_n): ({\lambda}_1,{\lambda}_2,\cdots,{\lambda}_n)\in {\mathbb{Z}}_{{\geqslant}0}^n, {\lambda}_1{\geqslant}{\lambda}_2{\geqslant}\cdots{\geqslant}{\lambda}_n{\geqslant}0\},\end{aligned}$$ with generator $$\begin{aligned} {{{\mathcal}L}}_1^n f({{\bm{x}}})=n\sum_{i=1}^n\frac{V({{\bm{x}}}+{\bm{e}}_i)}{V({{\bm{x}}})}\left(f({{\bm{x}}}+{\bm{e}}_i)-f({{\bm{x}}})\right)=n\sum_{i=1}^{n}\left(\prod_{j:j\neq i}\frac{x_i-x_j+1}{x_i-x_j}\right)\left(f({{\bm{x}}}+{\bm{e}}_i)-f({{\bm{x}}})\right),\end{aligned}$$ where $V({{\bm{x}}})=\prod_{1{\leqslant}i<j{\leqslant}n}(x_i-x_j)$ is the Vandermond determinant in variables $x_1,x_2,\cdots,x_n$. If instead of the Poisson random walk, we start from $n$ independent Brownian motions with mean $0$ and variance $t/n$, then the same conditioning leads to the celebrated *Dyson Brownian motion* with $\beta=2$, which describes the stochastic evolution of eigenvalues of a Hermitian matrix under independent Brownian motion of its entries. For general $\beta>0$, the *$\beta$-Dyson Brownian motion* ${{\bm{y}}}(t)=(y_1(t), y_2(t),\cdots, y_n(t))$ is a diffusion process solving $$\begin{aligned} \label{e:DBM1} {{\rm d}}y_i(t)=\sqrt{\frac{2}{\beta n}}{{\rm d}}{{\mathcal B}}_i(t)+\frac{1}{n}\sum_{j\neq i}\frac{1}{y_i(t)-y_j(t)}{{\rm d}}t,\quad i=1,2,\cdots, n,\end{aligned}$$ where $\{({{\mathcal B}}_1(t), {{\mathcal B}}_2(t),\cdots, {{\mathcal B}}_n(t))\}_{t{\geqslant}0}$ are independent standard Brownian motions, and $\{{{\bm{y}}}(t)\}_{t>0}$ lives on the Weyl chamber ${\mathbb{W}}^n=\{({\lambda}_1,{\lambda}_2,\cdots,{\lambda}_n): {\lambda}_1>{\lambda}_2>\cdots>{\lambda}_n\}$. The nonintersecting Poisson random walk can be viewed as a discrete version of the Dyson Brownian motion with $\beta=2$. For general $\beta>0$, we fix $\theta=\beta/2$ and define the *$\beta$-nonintersecting Poisson random walk*, denoted by ${{\bm{x}}}(t)=(x_1(t),x_2(t),\cdots,x_n(t))$, as a continuous time Markov process on $$\begin{aligned} \label{e:defWtheta} {\mathbb{W}}^n_\theta=\{({\lambda}_1+(n-1)\theta,{\lambda}_2+(n-2)\theta,\cdots,{\lambda}_n): ({\lambda}_1,{\lambda}_2,\cdots,{\lambda}_n)\in {\mathbb{Z}}_{{\geqslant}0}^n, {\lambda}_1{\geqslant}{\lambda}_2{\geqslant}\cdots{\geqslant}{\lambda}_n{\geqslant}0\},\end{aligned}$$ with generator $$\begin{aligned} \label{e:generator} {{{\mathcal}L}}^n_\theta f({{\bm{x}}})=\theta n\sum_{i=1}^n\frac{V({{\bm{x}}}+{\bm{e}}_i)}{V({{\bm{x}}})}\left(f({{\bm{x}}}+{\bm{e}}_i)-f({{\bm{x}}})\right)=\theta n\sum_{i=1}^{n}\left(\prod_{j:j\neq i}\frac{x_i-x_j+\theta}{x_i-x_j}\right)\left(f({{\bm{x}}}+{\bm{e}}_i)-f({{\bm{x}}})\right).\end{aligned}$$ In the beautiful article [@MR3418747], Gorin and Shkolnikov constructed certain multilevel discrete Markov chains whose top level dynamics coincide with the $\beta$-nonintersecting Poisson random walks. However, we use slightly different notations, and speed up time by $n$. In [@MR3418747], the $\beta$-nonintersecting Poisson random walks are constructed as stochastic dynamics on Young diagrams. We recall that a Young diagram $\bm\lambda$, is a non-increasing sequence of integers $$\begin{aligned} \bm\lambda=({\lambda}_1,{\lambda}_2,{\lambda}_3, \cdots), \quad \lambda_1{\geqslant}\lambda_2{\geqslant}{\lambda}_3{\geqslant}\cdots{\geqslant}0.\end{aligned}$$ We denote $\ell_{{\bm{\lambda}}}$ the number of non-empty rows in ${\bm{\lambda}}$, i.e. ${\lambda}_{\ell_{{\bm{\lambda}}}}>0, {\lambda}_{\ell_{{\bm{\lambda}}}+1}={\lambda}_{\ell_{{\bm{\lambda}}}+2}=\cdots =0$, and $|{\bm{\lambda}}|=\sum_{i=1}^{\ell_{{\bm{\lambda}}}}{\lambda}_i$ the number of boxes in ${\bm{\lambda}}$. Let ${\mathbb{Y}}^n$ denote the set of all Young diagrams with at most $n$ rows, i.e. $\ell_{{\bm{\lambda}}}{\leqslant}n$. A box $\Box\in {\bm{\lambda}}$ is a pair of integers, $$\begin{aligned} \Box=(i,j)\in {\bm\lambda}, \text{ if and only if } 1{\leqslant}i{\leqslant}\ell_\lambda, 1{\leqslant}j{\leqslant}\lambda_i.\end{aligned}$$ We denote ${\bm{\lambda}}'$ the transposed diagram of $\bm\lambda$, defined by $$\begin{aligned} \lambda_j'=|\{i: 1{\leqslant}j{\leqslant}\lambda_i\}|, \quad 1{\leqslant}j{\leqslant}{\lambda}_1.\end{aligned}$$ For a box $\Box=(i,j)\in {\bm{\lambda}}$, its arm $a_\Box$, leg $l_\Box$, co-arm $a_\Box'$ and co-leg $l_\Box'$ are $$\begin{aligned} a_\Box=\lambda_i-j,\quad l_\Box=\lambda_j'-i,\quad a_\Box'=j-1,\quad l_\Box'=i-1.\end{aligned}$$ Given a $\beta$-nonintersecting Poisson random walk ${{\bm{x}}}(t)$, we can view it as a growth process on ${\mathbb{Y}}^n$, by defining ${\bm{\lambda}}(t)$ by $$\begin{aligned} \label{e:defla} {\lambda}_i(t)=x_i(t)-(n-i)\theta, \quad 1{\leqslant}i{\leqslant}n.\end{aligned}$$ Since ${{\bm{x}}}(t)\in {\mathbb{W}}_\theta^n$, we have ${\lambda}_1(t){\geqslant}{\lambda}_2(t){\geqslant}\cdots{\geqslant}{\lambda}_n(t){\geqslant}0$, and thus ${\bm{\lambda}}(t)$ is a continuous time Markov process on ${\mathbb{Y}}^n$. Its jump rate is given in [@MR3418747 Proposition 2.25] rescaled by $n$, which, after simplification, is the same as . There is a simple formula for the transition probability of ${\bm{\lambda}}(t)$ with zero initial data [@MR3418747 Proposition 2.9, 2.28]. However, there are no simple formulas for the transition probabilities of ${\bm{\lambda}}(t)$ with general initial data. \[t:density\] Suppose the initial data of ${\bm{\lambda}}(t)$ is the empty Young diagram. Then for any fixed $t>0$, the law of ${\bm{\lambda}}(t)$ is given by $$\begin{aligned} \label{e:density} {\mathbb{P}}_t({\lambda}_1,{\lambda}_2,\cdots, {\lambda}_n)= e^{-\theta t n^2}(\theta tn)^{|{\bm{\lambda}}|}\prod_{\Box\in {\bm{\lambda}}}\frac{\theta n+a_\Box'-\theta l_\Box'}{(a_\Box+\theta l_\Box+\theta)(a_\Box+\theta l_\Box +1)}.\end{aligned}$$ It is proven in [[@MR3418747 Theorem 3.2]]{} that the Markov process $\bm{\lambda}(t)$ converges in the diffusive scaling limit to the $\beta$-Dyson Brownian motion. Fix $\theta=\beta/2{\geqslant}1/2$ and let $\varepsilon>0$ be a small parameter. Let ${{\bm{x}}}(t)$ be the $\beta$-nonintersecting Poisson random walk starting at ${{\bm{x}}}(0)\in {\mathbb{W}}^n_\theta$. We define $\bm{\lambda}(t)$ as in and the rescaled stochastic process $\bm{\lambda}^\varepsilon(t)=({\lambda}_1^{{\varepsilon}}(t),{\lambda}_2^{{\varepsilon}}(t),\cdots, {\lambda}_n^{{\varepsilon}}(t))$ be defined through, $$\begin{aligned} {\lambda}_i^{{\varepsilon}}(t){\mathrel{\mathop:}=}\varepsilon ^{1/2}\left(\frac{{\lambda}_i(t/\varepsilon)}{\theta n}-\frac{t}{\varepsilon }\right),\quad i=1,2,\cdots, n. $$ Suppose that as ${{\varepsilon}}\rightarrow 0$, the initial data $\bm{\lambda}^{{\varepsilon}}(0)$ converges to a point ${{\bm{y}}}(0)\in {\mathbb{W}}^n$. Then the process $\bm{\lambda}^{{\varepsilon}}(t)$ converges in the limit ${{\varepsilon}}\rightarrow 0$ weakly in the Skorokhod topology towards the $\beta$-Dyson Brownian motion ${{\bm{y}}}(t)=(y_1(t), y_2(t),\cdots, y_n(t))$ as in . Notations --------- Throughout this paper, we use the following notations: We denote ${{\mathbb R}}$ the set of real numbers, ${{\mathbb C}}$ the set of complex numbers, ${{\mathbb C}}_+$ the set of complex numbers with positive imaginary parts, ${{\mathbb C}}_-$ the set of complex numbers with negative imaginary parts, ${\mathbb{Z}}$ the set of integers, and ${\mathbb{Z}}_{{\geqslant}0}$ the set of non-negative integers. We denote $M_1({{\mathbb R}})$ the space of probability measures on ${{\mathbb R}}$ equipped with the weak topology. A metric compatible with the weak topology is the *L[é]{}vy metric* defined by $$\begin{aligned} {\rm{dist}}(\mu,\nu){\mathrel{\mathop:}=}\inf_{\varepsilon}\{\mu(-\infty, x-\varepsilon)-\varepsilon{\leqslant}\nu(-\infty, x){\leqslant}\mu(-\infty, x+\varepsilon)+\varepsilon \text{ for all $x$}\}.\end{aligned}$$ Let $({\mathbb{M}},\operatorname{dist}(\cdot,\cdot))$ be a metric space, either ${{\mathbb C}}^m$ or ${{\mathbb R}}^m$ with the Euclidean metric or $M_1({{\mathbb R}})$ with the L[é]{}vy metric. The set of c[á]{}dl[á]{}g functions, i.e. functions which are right continuous with left limits, from $[0,T]$ to ${\mathbb{M}}$ is denoted by $D([0,T],{\mathbb{M}})$ and is called the *Skorokhod space*. Let $\Lambda$ denote the set of all strictly increasing, continuous bijections from $[0,T]$ to $[0,T]$. The *Skorokhod metric* on $D([0,T],{\mathbb{M}})$ is defined by $$\begin{aligned} \operatorname{dist}(f,g)=\inf_{{\lambda}\in \Lambda}\max\left\{\sup_{0{\leqslant}t{\leqslant}T}|{\lambda}(t)-t|, \sup_{0{\leqslant}t{\leqslant}T}\operatorname{dist}(f(t), g({\lambda}(t)))\right\}.\end{aligned}$$ Let $Z^n$, $Z$ be random variables taking value in the Skorokhod space $D([0,T],{\mathbb{M}})$. We say $Z_n$ converges *almost surely* towards $Z$, if $Z_n\rightarrow Z$ in $D([0,T], {\mathbb{M}})$ for the Skorokhod metric almost surely. We say $Z_n$ *weakly converges* towards $Z$, denoted by $Z^n\Rightarrow Z$, if for all bounded Skorokhod continuous functions $f$, $$\begin{aligned} \lim_{n\rightarrow \infty}{\mathbb{E}}[f(Z^n)]\rightarrow {\mathbb{E}}[f(Z)].\end{aligned}$$ We refer to [@MR1431297 Chapter 1] and [@MR1876437 Chapter 3] for nice presentations on weak convergence of stochastic processes in the Skorokhod space. A *random field* is a collection of random variables indexed by elements in a topological space. If the collection of random variables are jointly Gaussian, we call it a *Gaussian random field*. Let $(g^n(z))_{z\in \Omega}$, $(g(z))_{z\in \Omega}$ be ${{\mathbb C}}$-valued random fields indexed by an open subset $\Omega\subset {{\mathbb C}}\setminus{{\mathbb R}}$. We say $(g^n(z))_{z\in \Omega}$ weakly converges towards $(g(z))_{z\in \Omega}$ in the sense of finite dimensional distributions, if for any $z_1,z_2,\cdots, z_m\in \Omega$ the random vector $(g^n(z_j))_{1{\leqslant}j{\leqslant}m}$ weakly converges to $(g(z_j))_{1{\leqslant}j{\leqslant}m}$. Let $\{(g_t^n(z))_{z\in \Omega}\}_{0{\leqslant}t{\leqslant}T}$, $\{(g_t(z))_{z\in \Omega}\}_{0{\leqslant}t{\leqslant}T}$ be random field valued random processes. We say $\{(g_t^n(z))_{z\in \Omega}\}_{0{\leqslant}t{\leqslant}T}$ weakly converges towards $\{(g_t(z))_{z\in \Omega}\}_{0{\leqslant}t{\leqslant}T}$ in the sense of finite dimensional processes, if for any $z_1,z_2,\cdots, z_m\in \Omega$ the random process $\{(g_t^n(z_j))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ weakly converges to $\{(g_t(z_j))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ in the Skorokhod space $D([0,T], {{\mathbb C}}^m)$. Main results ------------ In this paper, we study the asymptotic behavior of the $\beta$-nonintersecting Poisson random walks, as the number of particles $n$ goes to infinity. We consider $\beta$-nonintersecting Poisson random walks ${{\bm{x}}}(t)=(x_1(t), x_2(t),\cdots, x_n(t))$, with initial data ${{\bm{x}}}(0)=(x_1(0), x_2(0),\cdots, x_n(0))$. We define the empirical measure process $$\begin{aligned} \label{e:defemp} \mu^n_t=\frac{1}{n}\sum_{i=1}^n \delta_{x_i(t)/\theta n},\end{aligned}$$ which can be viewed as a random element in $D([0,T], M_1({{\mathbb R}}))$, the space of right-continuous with left limits processes from $[0,T]$ into the space $M_1({{\mathbb R}})$ of probability measures on ${{\mathbb R}}$. The law of large numbers theorem states that the empirical measure process $\{\mu_t^n\}_{0{\leqslant}t{\leqslant}T}$ converges in $D([0,T], M_1({{\mathbb R}}))$. We need to assume that the initial empirical measure $\mu_0^n$ converges in the L[é]{}vy metric as $n$ goes to infinity in $M_1({{\mathbb R}})$. We denote the Stieltjes transform of the empirical measure at time $t$ as $$\begin{aligned} \label{e:defmt} m^n_t(z)=\frac{1}{n}\sum_{i=1}^{n}\frac{1}{x_i(t)/\theta n-z}=\int \frac{{{\rm d}}\mu^n_t(x)}{x-z},\end{aligned}$$ where $z\in {{\mathbb C}}\setminus {{\mathbb R}}$. \[t:LLN\] Fix $\theta>0$. We assume that there exists a measure $\mu_0\in M_1({{\mathbb R}})$, such that the initial empirical measure $\mu_0^n$ converges in the L[é]{}vy metric as $n$ goes to infinity towards $\mu_0$ almost surely (in probability). Then, for any fixed time $T>0$, $\{\mu^n_t\}_{0{\leqslant}t{\leqslant}T}$ converges as $n$ goes to infinity in $D([0,T], M_1({{\mathbb R}}))$ almost surely (in probability). Its limit is the unique measure-valued process $\{\mu_t\}_{0{\leqslant}t{\leqslant}T}$, so that the density satisfies $0{\leqslant}{{\rm d}}\mu_t(x)/{{\rm d}}x{\leqslant}1$, and its Stieltjes transform $$\begin{aligned} \label{e:defmt} m_t(z)=\int \frac{{{\rm d}}\mu_t(x)}{x-z},\end{aligned}$$ satisfies the equation $$\begin{aligned} \label{e:limitMST} m_t(z)=m_0(z)-\int_0^t e^{- m_s(z)}{\partial}_z m_s(z){{\rm d}}s,\end{aligned}$$ for $z\in {{\mathbb C}}\setminus {{\mathbb R}}$. Assumption in Theorem \[t:LLN\] is equivalent to that the Stieltjes transform of the initial empirical measure $$\begin{aligned} \lim_{n\rightarrow \infty} m^n_0(z)=\int \frac{{{\rm d}}\mu_0(x)}{x-z}{=\mathrel{\mathop:}}m_0(z),\end{aligned}$$ almost surely (in probability), for any $z\in {{\mathbb C}}\setminus {{\mathbb R}}$. The Stieltjes transform of $ \mu_t$ is characterized by , $$\begin{aligned} \label{e:dif1} {\partial}_t m_t(z)=-e^{-m_t(z)}{\partial}_z m_t(z)={\partial}_z (e^{- m_t(z)}).\end{aligned}$$ This is a complex Burgers type equation, and can be solved by the method of characteristics. We define the characteristic lines, $$\begin{aligned} \label{e:dif2} {\partial}_t z_t(z)=e^{- m_t( z_t(z))}, \quad z_0(z)=z.\end{aligned}$$ If the context is clear, we omit the parameter $z$, i.e. we simply write $z_t$ instead of $z_t(z)$. Plugging into , and applying the chain rule we obtain $ {\partial}_t m_t(z_t)=0. $ It implies that $ m_t(z)$ is a constant along the characteristic lines, i.e. $m_t(z_t(z))=m_0(z_0(z))=m_0(z)$. And the solution of the differential equation is given by $$\begin{aligned} \label{e:deft} z_t(z)=z+te^{-m_0(z)}, \quad 0{\leqslant}t< -\frac{{\mathop{\mathrm{Im}}}[z]}{{\mathop{\mathrm{Im}}}[e^{-m_0(z)}]}{=\mathrel{\mathop:}}{{\frak t}}(z).\end{aligned}$$ We conclude that the Stieltjes transform $m_t(z)$ is given by $$\begin{aligned} \label{e:tmtrelation} m_t(z+te^{-m_0(z)})=m_0(z).\end{aligned}$$ Later we will prove that for any time $t{\geqslant}0$, there exists an open set $\Omega_t\subset{{\mathbb C}}\setminus {{\mathbb R}}$ defined in , such that $z_t(z)=z+te^{-m_0(z)}$ is conformal from $\Omega_t$ to ${{\mathbb C}}\setminus {{\mathbb R}}$, and is a homeomorphism from the closure of $\Omega_t\cap {{\mathbb C}}_+$ to ${{\mathbb C}}_+\cup {{\mathbb R}}$, and from the closure of $\Omega_t\cap {{\mathbb C}}_-$ to ${{\mathbb C}}_-\cup {{\mathbb R}}$. The central limit theorem states that the rescaled empirical measure process $\{n(\mu_t^n-\mu_t)\}_{0{\leqslant}t{\leqslant}T}$ weakly converges in the space of distributions acting on analytic test functions to a Gaussian process. We need to assume that the rescaled initial empirical measure $n(\mu_0^n-\mu_0)$ weakly converges to a measure. We define the rescaled fluctuation process $$\begin{aligned} \label{e:defgt} g_t^n(z)=n(m_t^n(z)- m_t(z))=n\int \frac{{{\rm d}}(\mu_t^n(x)- \mu_t(x))}{x-z},\end{aligned}$$ which characterizes the behaviors of the rescaled empirical measure process $\{n(\mu_t^n-\mu_t)\}_{0{\leqslant}t{\leqslant}T}$. \[a:initiallaw\] We assume there exists a constant ${{\frak a}}$, such that that for any $z\in {{\mathbb C}}\setminus{{\mathbb R}}$, $$\begin{aligned} {\mathbb{E}}\left[|g_0^n(z)|^2\right]{\leqslant}{{\frak a}}({\mathop{\mathrm{Im}}}[z])^{-2},\end{aligned}$$ and the random field $(g_0^n(z))_{z\in {{\mathbb C}}\setminus{{\mathbb R}}}$ weakly converges to a deterministic field $(g_0(z))_{z\in {{\mathbb C}}\setminus {{\mathbb R}}}$, in the sense of finite dimensional distributions. Assumption \[a:initiallaw\] implies that the initial empirical measure $\mu_0^n$ converges in the L[é]{}vy metric as $n$ goes to infinity towards $\mu_0$ in probability. \[t:CLT\] Fix $\theta>0$. We assume Assumption \[a:initiallaw\]. Then for any fixed time $T>0$, the process $\{(g^n_t(z_t(z)))_{z\in\Omega_T}\}_{0{\leqslant}t{\leqslant}T}$ converges weakly towards a Gaussian process $\{(g_t(z_t(z)))_{z\in \Omega_T}\}_{0{\leqslant}t{\leqslant}T}$, in the sense of finite dimensional processes, with initial data $(g_0(z))_{z\in {{\mathbb C}}\setminus{{\mathbb R}}}$ given in Assumption \[a:initiallaw\], means $$\begin{aligned} \label{e:mean0} {\mathbb{E}}[g_t(z_t(z))]&=\mu(t, z){\mathrel{\mathop:}=}\frac{g_0(z)}{1-t{\partial}_zm_0(z)e^{-m_0(z)}} +\left(\frac{1}{2}-\frac{1}{2\theta}\right)\frac{t(({\partial}_z m_0(z))^2-{\partial}_z^2 m_0(z))e^{-m_0(z)}}{(1-t{\partial}_z m_0(z)e^{-m_0(z)})^2},\end{aligned}$$ and covariances $$\begin{aligned} \begin{split}\label{e:variance0} {{\rm{cov}}}[g_s(z_s(z)), g_t(z_t(z'))]&=\sigma(s, z, t, z'){\mathrel{\mathop:}=}\frac{1}{\theta}\frac{1}{(1-s{\partial}_zm_0(z)e^{-m_0(z)})(1-t{\partial}_zm_0(z')e^{-m_0(z')})}\\ &\times\left(\frac{1}{(z-z')^2}-\frac{(1-(s\wedge t){\partial}_zm_0(z)e^{-m_0(z)})(1-(s\wedge t){\partial}_zm_0(z')e^{-m_0(z')})}{(z-z'+(s\wedge t)(e^{-m_0(z)}-e^{-m_0(z')}))^2}\right)\\ {{\rm{cov}}}[g_s(z_s(z)), \overline{g_t(z_t(z'))}]&=\sigma(s, z,t,\bar z'), \end{split}\end{aligned}$$ where $$\begin{aligned} \sigma(s, z,t,z){\mathrel{\mathop:}=}\lim_{z'\rightarrow z}\sigma(s,z,t,z') &=\frac{(s\wedge t)e^{-m_0(z)}(2({\partial}_z m_0(z))^3-6{\partial}_z m_0(z){\partial}_z^2 m_0(z)+2{\partial}_z^3m_0(z))}{12\theta(1-(s\wedge t){\partial}_z m_0(z)e^{-m_0(z)})^3(1-(s\vee t){\partial}_z m_0(z)e^{-m_0(z)})}\\ &+\frac{(s\wedge t)^2e^{-2m_0(z)}(({\partial}_z m_0(z))^4+3({\partial}_z^2 m_0(z))^2-2{\partial}_z m_0(z){\partial}_z^3m_0(z))}{12\theta(1-(s\wedge t){\partial}_z m_0(z)e^{-m_0(z)})^3(1-(s\vee t){\partial}_z m_0(z)e^{-m_0(z)})}.\end{aligned}$$ We can rewrite the means and covariances in terms of the characteristic lines $z_t(z)$: $$\begin{aligned} \begin{split}\label{e:meanandvar} \mu(t,z)&=\frac{g_0(z)}{{\partial}_z z_t(z)}+\left(\frac{1}{2}-\frac{1}{2\theta}\right)\frac{{\partial}_z^2 z_t(z)}{({\partial}_z z_t(z))^2},\\ \sigma(s, z, t,z')&=\frac{1}{\theta}\frac{1}{{\partial}_zz_s(z){\partial}_zz_t(z')}\left(\frac{1}{(z-z')^2}-\frac{{\partial}_z z_{s\wedge t}(z){\partial}_z z_{s\wedge t}(z')}{(z_{s\wedge t}(z)-z_{s\wedge t}(z'))^2}\right). \end{split}\end{aligned}$$ We will prove in Section \[s:compareDBM\], the means and the covariances are universal, and coincide with those of $\beta$-Dyson Brownian motions with initial data constructed by the Markov-Krein correspondence. To study the fluctuation of the rescaled empirical measure process $\{n(\mu_t^n-\mu_t)\}_{0{\leqslant}t{\leqslant}T}$ with analytic functions as test functions, we need to assume that the extreme particles are bounded. \[a:ibound\] We assume there exists a large number ${{\frak b}}$, such that $$\begin{aligned} \label{e:ibound} {{\frak b}}n{\geqslant}x_1(0){\geqslant}x_2(0){\geqslant}\cdots {\geqslant}x_n(0).\end{aligned}$$ \[t:CLT2\] Fix $\theta>0$. We assume Assumptions \[a:initiallaw\] and \[a:ibound\]. Then for any fixed time $T>0$ and real analytic functions $f_1, f_2, \cdots, f_m$ on ${{\mathbb R}}$, the random process $$\begin{aligned} \left\{\left(n\int f_j(x){{\rm d}}(\mu_{t}^{n}(x)-\mu_{t}(x))\right)_{1{\leqslant}j{\leqslant}m}\right\}_{0{\leqslant}t{\leqslant}T},\end{aligned}$$ converges as $n$ goes to infinity in $D([0,T], {{\mathbb R}}^m)$ weakly towards a Gaussian process $\{({{{\mathcal}F}}_j(t))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$, with means and covariances $$\begin{aligned} {\mathbb{E}}[{{{\mathcal}F}}_j(t)]&=\frac{1}{2\pi{\mathrm{i}}}\oint_{{{\mathcal}C}}\mu(t, z_t^{-1}(w))f_j(w){{\rm d}}w,\\ {{\rm{cov}}}[{{{\mathcal}F}}_j(s), {{{\mathcal}F}}_k(t)]&=-\frac{1}{4\pi^2}\oint_{{{\mathcal}C}}\oint_{{{\mathcal}C}}\sigma(s, z_s^{-1}(w),t, z_t^{-1}(w'))f_j(w)f_k(w'){{\rm d}}w{{\rm d}}w',\end{aligned}$$ where the contours are sufficiently large depending on ${{\frak b}}$. We prove in Proposition \[p:extremePbound\] that with exponentially high probability all particles $x_i(t)$ are inside an interval $[0, {{\frak c}}n]$. The contours in Theorem \[t:CLT2\] encloses a neighborhood of $[0,{{\frak c}}/\theta]$. Further, it is enough to assume in Theorem \[t:CLT2\] that $f_j$ are analytic only in a neighborhood of $[0,{{\frak c}}/\theta]$. Related results --------------- For the $\beta$-Dyson Brownian motion , the asymptotic behavior of the empirical measure process was studied in [@MR1217451; @MR1176727]. They found that the empirical measure process $$\begin{aligned} \label{e:empirical} \tilde \mu_t^n=\frac{1}{n}\sum_{i=1}^n \delta_{y_i(t)},\quad 1{\leqslant}i{\leqslant}n,\end{aligned}$$ converges weakly in the space of continuous measure-valued processes to a deterministic process $\tilde \mu_t$, characterized by the free convolution with semi-circle distributions. It was proven in [@MR1819483], that the rescaled empirical measure process $n(\tilde \mu_t^n-\tilde \mu_t)$ converges weakly in the space of distributions acting on a class of $C^6$ test functions to a Gaussian process, provided that the initial distributions $n(\tilde\mu_0^n-\tilde\mu_0)$ converge. The explicit formulas of the means and the covariances of the limit Gaussian process was derived in [@MR2418256]. More generally, the $\beta$-Dyson Brownian motion ${{\bm{y}}}(t)=(y_1(t), y_2(t),\cdots, y_n(t))$ with potential $V$ is given by $$\begin{aligned} {{\rm d}}y_i(t)=\sqrt{\frac{2}{\beta n}}{{\rm d}}{{\mathcal B}}_i(t)+\frac{1}{n}\sum_{j\neq i}\frac{1}{y_i(t)-y_j(t)}{{\rm d}}t -\frac{1}{2}V'(y_i(t)){{\rm d}}t,\quad i=1,2,\cdots, n,\end{aligned}$$ where $\{({{\mathcal B}}_1(t), {{\mathcal B}}_2(t), \cdots, {{\mathcal B}}_n(t))\}_{t{\geqslant}0}$ are independent standard Brownian motions. It was proven in [@GDBM1; @GDBM2], that under mild conditions on $V$, the empirical measure process converges to a $V$-dependent measure-valued process, which can be realized as the gradient flow of the Voiculescu free entropy on the Wasserstein space over ${{\mathbb R}}$. The central limit theorem of the rescaled empirical measure process was proven in [@Un] for $\beta>1$ and sufficiently regular convex potential $V$. The Wigner-Dyson-Mehta conjecture stated that the eigenvalue correlation functions of a general class of random matrices converge to the corresponding ones of Gaussian matrices. The Dyson Brownian motion plays a central role in the three-step approach to the universality conjecture in a series of works [@Landon2016; @fix; @MR2919197; @MR3429490; @MR3372074; @MR2810797; @kevin3], developed by Erd[ő]{}s, Yau and their collaborators. Parallel results were established in certain cases in [@MR2669449; @MR2784665], with a four moment comparison theorem. The transition probability of the $\beta$-nonintersecting Poisson random walks with the fully-packed initial data $x_i(0)=(n-i)\theta$, $1{\leqslant}i{\leqslant}n$, is a discrete $\beta$ ensemble with Charlier weight. The discrete $\beta$ ensembles with general weights were introduced in [@MR3668648], which is a probability distribution $$\begin{aligned} \label{e:disc} {\mathbb{P}}_n(\ell_1,\ell_2,\cdots, \ell_n)=\frac{1}{Z_n}\prod_{1{\leqslant}i<j{\leqslant}n}\frac{\Gamma(\ell_i-\ell_j+1)\Gamma(\ell_i-\ell_j+\theta)}{\Gamma(\ell_i-\ell_j)\Gamma(\ell_i-\ell_j+1-\theta)}\prod_{i=1}^nw(\ell_i; N),\end{aligned}$$ on ordered $n$-tuples $\ell_1>\ell_2>\cdots \ell_n$ such that $\ell_i={\lambda}_i+(n-i)\theta$ and ${\lambda}_1{\geqslant}{\lambda}_2{\geqslant}\cdots{\geqslant}{\lambda}_N$ are integers. The discrete $\beta$ ensembles are discretizations for the $\beta$ ensembles of random matrix theory, which are probability distributions on $n$ tuples of reals $y_1>y_2>\cdots>y_n$, $$\begin{aligned} \label{e:betaensemble} {\mathbb{P}}_n(y_1,y_2,\cdots, y_n)=\frac{1}{Z_n}\prod_{1{\leqslant}i<j{\leqslant}n}|y_i-y_j|^\beta\prod_{i=1}^ne^{-nV(y_i)},\end{aligned}$$ where the potential $V$ is a continuous function. Under mild assumptions on the potential $V$, the $\beta$ ensembles exhibit a law of large number, i.e., the empirical measure $$\begin{aligned} \mu^n=\frac{1}{n}\sum_{i=1}^n \delta_{y_i},\end{aligned}$$ converges to a non-random equilibrium measure $\mu$. For $\beta=1,2,4$ and $V(y)=y^2$, this statement dates back to the original work of Wigner [@MR0077805; @MR0083848]. We refer to [@MR2760897 Chapter 2.6] for the study of the $\beta$ ensembles with general $V$. In the breakthrough paper [@MR1487983], Johansson introduced the loop (or Dyson-Schwinger) equations to the mathematical community, and proved that the rescaled empirical measure satisfies a central limit theorem, i.e., for sufficiently smooth functions $f(y)$ the random variable $$\begin{aligned} n\int f(x) ({{\rm d}}\mu^n(x)-{{\rm d}}\mu(x)).\end{aligned}$$ converges to a Gaussian random variable. We refer to [@MR3010191; @borot-guionnet2; @KrSh] for further development. The law of large numbers and the central limit theorems of the discrete $\beta$ ensemble were proven in [@MR3668648], using a discrete version of the loop equations [@Nekrasov]. In the special case when $\beta=2$, the central limit theorem for the global fluctuations of the nonintersecting Poisson random walk were obtained by various methods. For the fully-packed initial data, the central limit theorem was established in [@MR3148098] by the technique of determinantal point processes, in [@MR3552537; @MR3263029] by computations in the universal enveloping algebra of $U(N)$ and in [@Dui; @MR3556288] by employing finite term recurrence relations of orthogonal polynomials. For general initial data, the law of large numbers and the central limit theorems were proven in [@MR3361772; @BuGo], where the Schur generating functions were introduced to study random discrete models. Our results give a new proof of these results based on the dynamical approach. Organization of the paper ------------------------- In Section \[s:qfc\], we recall the quantized free convolution as introduced in [@MR3361772]. We show that the limit measure-valued process $\mu_t$ is characterized by the quantized free convolution. In Section \[s:compareDBM\], we compare the central limit theorems of the $\beta$-nonintersecting Poisson random walks with those of the $\beta$-Dyson Brownian motions. It turns out that the means and covariances of the limit fluctuation process coincide under Markov-Krein correspondence. In Section \[s:sde\], we collect some properties of the generator ${{{\mathcal}L}}_\theta^n$ of the $\beta$-nonintersecting Poisson random walks, and derive a stochastic differential equation of the Stieltjes transform of the empirical measure process, which relies on the integrable features of the generator. The stochastic differential equation can be viewed as a dynamical version of the Nekrasov’s equation in [@MR3668648 Section 4], which is crucial for the proof of central limit theorems of the discrete $\beta$ ensembles. In Section \[s:LLN\] and \[s:CLT\], we prove the law of large numbers and central limit theorem of the $\beta$-nonintersecting Poisson random walks. We directly analyze the stochastic differential equation satisfied by the Stieltjes transform using the method of characteristics as in [@HL], where the method of characteristics was used to derive the rigidity of the Dyson Brownian motion. Since the $\beta$-nonintersecting Poisson random walks are jump processes, the analysis is more sophisticated than that of the Dyson Brownian motion. In Section \[s:extremep\] we derive an estimate of the locations of extreme particles, by a coupling technique, and prove the central limit theorem with analytic test functions. Finally we remark that by analyzing the stochastic differential equation of the Stieltjes transform of the empirical measure process as in [@HL], one can prove the optimal rigidity estimates and a mesoscopic central limit theorem for the $\beta$-nonintersecting Poisson random walks. **Acknowledgement.** The author heartily thanks Vadim Gorin for constructive comments on the draft of this paper. Law of large numbers for the empirical measure process ====================================================== Quantized free convolution {#s:qfc} -------------------------- In this section we study the limit measure-valued process $ \mu_t$. To describe it, we need the concept of quantized free convolution as introduced in [@MR3361772]. The quantized free convolution is a quantized version of the free convolution originally defined by Voiculescu [@MR799593; @MR839105] in the setting of operator algebras. Given a probability measure $\mu$, we denote its Stieltjes transform by $m_\mu(z)=\int {{\rm d}}\mu(x)/(x-z)$, for any $z\in {{\mathbb C}}\setminus{{\mathbb R}}$. The $R$-transform is defined as $$\begin{aligned} \label{e:defRf} R_\mu(z){\mathrel{\mathop:}=}m_\mu^{-1}(-z)-\frac{1}{z},\end{aligned}$$ where $m_\mu^{-1}(z)$ is the functional inverse of $m_\mu(z)$, i.e. $m_\mu(m_{\mu}^{-1}(z))=m_{\mu}^{-1}(m_\mu(z))=z$. The free convolution is a unique operation on probability measures $(\mu, \nu)\mapsto \mu\boxplus\nu$, which agrees with the addition of the $R$-transforms: $$\begin{aligned} R_\mu(z)+R_{\nu}(z)=R_{\mu\boxplus\nu}(z)\end{aligned}$$ It was proven in [@MR1094052] that the asymptotic distribution of eigenvalues of sums of independent random matrices is given by the free convolution. The quantized free convolution is an operation on probability measures which have bounded by $1$ density with respect to the Lebesgue measure. One gets the quantized free convolution by replacing the $R$-transform in with the quantized $R$-transform $$\begin{aligned} \label{e:defR} R_\mu^{\text{quant}}(z){\mathrel{\mathop:}=}m_\mu^{-1}(-z)-\frac{1}{1-e^{-z}}.\end{aligned}$$ The quantized free convolution is a unique operation on probability measures $(\mu, \nu)\mapsto \mu\otimes\nu$, which agrees with the addition of the quantized $R$-transforms: $$\begin{aligned} R_\mu^{\text{quant}}(z)+R_{\nu}^{\text{quant}}(z)=R_{\mu\otimes\nu}^{\text{quant}}(z)\end{aligned}$$ It was proven in [@MR3361772 Theorem 1.1] that the quantized free convolution characterizes the tensor product of two irreducible representation of unitary group. The Markov-Krein correspondence [@MR1618739; @MR0167806] gives an exact relationship between the free convolution and the quantized free convolution. \[t:MKcor\] For every probability measure $\mu$ on ${{\mathbb R}}$ which has bounded by $1$ density with respect to the Lebesgue measure, there exists a probability measure $Q(\mu)$ such that $$\begin{aligned} \label{e:defQ} m_{Q(\mu)}(z)=1-e^{-m_{\mu}(z)},\end{aligned}$$ where $m_\mu(z)$ and $m_{Q(\mu)}(z)$ are Stieltjes transforms of $\mu$ and $Q(\mu)$ respectively. We denote the operator $$\begin{aligned} \tilde Q(\mu)=r\circ Q \circ r(\mu),\end{aligned}$$ where $r$ is the reflection of a measure with respect to the origin. The operator $\tilde Q$ intertwines the free convolution and the quantized free convolution, i.e. for any two probability measures $\mu_1, \mu_2$ as above, we have $$\begin{aligned} \tilde Q(\mu_1\otimes\mu_2)=\tilde Q(\mu_1)\boxplus \tilde Q(\mu_2).\end{aligned}$$ Theorem \[t:MKcor\] essentially reduces the quantized free convolution to the free convolution. Properties of the quantized free convolution, e.g., existence and uniqueness, follow from their counterparts of the free convolution. We sketch the construction of the operator $Q$ in Remark \[r:proofMKcor\] in Section \[s:sde\]. The limit measure-valued process $ \mu_t$ can be described by the quantized free convolution. We denote $ R_t^{\text{quant}}(z)$ the quantized $R$-transform of the measure $ \mu_t$. From , we have $$\begin{aligned} \left(m_t\right)^{-1}(z)=\left(m_0\right)^{-1}(z)+te^{-z},\end{aligned}$$ and $$\begin{aligned} R_t^{\text{quant}}(z)= R_0^{\text{quant}}(z)+te^{z}.\end{aligned}$$ There exists a family of measures $ \nu_t$ such that the quantized $R$-transform of $ \nu_t$ is given by $te^{z}$. The Stieltjes transform $m_{\nu_t}(z)$ of $\nu_t$ is given by $$\begin{aligned} z e^{2m_{\nu_t}(z)}+(1-t-z)e^{m_{\nu_t}(z)}+t=0.\end{aligned}$$ We can solve for $m_{\nu_t}(z)$, and the density of $ \nu_t$ is given by for $t{\leqslant}1$, $$\begin{aligned} \label{e:defnu1} {{\rm d}}\nu_t(x)/{{\rm d}}x= \left\{ \begin{array}{cc} \frac{1}{\pi} \text{arccot} \left(\frac{x+t-1}{\sqrt{4xt-(x+t-1)^2}}\right), &(1-\sqrt{t})^2{\leqslant}x{\leqslant}(\sqrt{t}+1)^2, \\ 1,& x< (1-\sqrt{t})^2,\\ 0, & x> (\sqrt{t}+1)^2, \end{array} \right.\end{aligned}$$ for $t>1$, $$\begin{aligned} \label{e:defnu2} {{\rm d}}\nu_t(x)/{{\rm d}}x= \left\{ \begin{array}{cc} \frac{1}{\pi} \text{arccot} \left(\frac{x+t-1}{\sqrt{4xt-(x+t-1)^2}}\right), &(1-\sqrt{t})^2{\leqslant}x{\leqslant}(\sqrt{t}+1)^2, \\ 0, & x< (\sqrt{t}-1)^2 \text{ or } x> (\sqrt{t}+1)^2. \end{array} \right.\end{aligned}$$ We can conclude from the discussion above, The limit measure $ \mu_t$ is the quantized free convolution of the initial measure $ \mu_0$ with the measure $ \nu_t$ as defined in and : $$\begin{aligned} \mu_t= \mu_0\otimes \nu_t.\end{aligned}$$ In the rest of this section, we collect some properties of the Stieltjes transform $m_t$, the characteristic lines $z_t$ and the logarithmic potential $h_t$ of the measure $\mu_t$, $$\begin{aligned} \label{e:defht} h_t(z)=\int\log(x-z){{\rm d}}\mu_t(x), \quad z\in {{\mathbb C}}\setminus {{\mathbb R}}.\end{aligned}$$ We remark that ${\partial}_z h_t(z)=-m_t(z)$. \[p:ztp\] For any time $t{\geqslant}0$, we define an open set $\Omega_t\subset{{\mathbb C}}\setminus {{\mathbb R}}$ $$\begin{aligned} \label{e:defOmega} \Omega_t{\mathrel{\mathop:}=}\left\{z\in {{\mathbb C}}\setminus {{\mathbb R}}: \int \frac{{{\rm d}}Q(\mu_0)(x)}{|x-z|^2}<\frac{1}{t}\right\},\end{aligned}$$ where the operator $Q$ is defined in Theorem \[t:MKcor\]. Then, $z_t(z)=z+te^{-m_0(z)}$ is conformal from $\Omega_t$ to ${{\mathbb C}}\setminus {{\mathbb R}}$, and is a homeomorphism from the closure of $\Omega_t\cap {{\mathbb C}}_+$ to ${{\mathbb C}}_+\cup {{\mathbb R}}$, and from the closure of $\Omega_t\cap {{\mathbb C}}_-$ to ${{\mathbb C}}_-\cup {{\mathbb R}}$. Moreover for any $z\in \Omega_t$, $|{\mathop{\mathrm{Im}}}[z_s]|$ is monotonically decreasing for $0{\leqslant}s{\leqslant}t$, i.e., $|{\mathop{\mathrm{Im}}}[z_s]|{\geqslant}|{\mathop{\mathrm{Im}}}[z_t]|$. Thanks to Theorem \[t:MKcor\], we have $$\begin{aligned} z_t(z)=z+te^{-m_0(z)}=z+t-tm_{Q(\mu_0)}(z),\end{aligned}$$ and the proposition follows from [@MR1488333 Lemma 4]. \[p:estimatemt\] Fix $T>0$. For any $0{\leqslant}t{\leqslant}T$ and $z\in \Omega_T$ as defined in , we have $$\begin{aligned} \begin{split}\label{e:mtestimate} &({\partial}_z m_t)(z_t(z))=\frac{{\partial}_zm_0(z)}{1-t{\partial}_z m_0(z)e^{-m_0(z)}},\quad ({\partial}^2_z m_t)(z_t(z))=\frac{{\partial}_z^2 m_0(z)-t({\partial}_zm_0(z))^3e^{-m_0(z)}}{(1-t{\partial}_z m_0(z)e^{-m_0(z)})^3},\\ &({\partial}_t m_t)(z_t(z))=-\frac{{\partial}_z m_0(z)e^{-m_0(z)}}{1-t{\partial}_z m_0(z)e^{-m_0(z)}},\quad ({\partial}_t h_t)(z_t(z))=-e^{-m_0(z)}. \end{split}\end{aligned}$$ The first three relations follow directly by taking derivative of . For the last relation, we have $$\begin{aligned} {\partial}_z(({\partial}_t h_t)(z_t(z))) =-({\partial}_t m_t)(z_t){\partial}_z z_t(z)={\partial}_z(-e^{-m_0(z)}).\end{aligned}$$ The last relation follows by noticing that $\lim_{z\rightarrow\infty}({\partial}_t h_t)(z_t(z))=0$. Comparing with $\beta$-Dyson Brownian motion {#s:compareDBM} -------------------------------------------- In this section, we compare the central limit theorems of the $\beta$-nonintersecting random walks with those of the $\beta$-Dyson Brownian motion. For general $\beta>0$, we recall the $\beta$-Dyson Brownian motion ${{\bm{y}}}(t)=(y_1(t), y_2(t),\cdots, y_n(t))$ is a diffusion process solving $$\begin{aligned} \label{e:DBM3} {{\rm d}}y_i(t)=\sqrt{\frac{2}{\beta n}}{{\rm d}}{{\mathcal B}}_i(t)+\frac{1}{n}\sum_{j\neq i}\frac{1}{y_i(t)-y_j(t)}{{\rm d}}t +{{\rm d}}t,\quad i=1,2,\cdots, n,\end{aligned}$$ where $\{({{\mathcal B}}_1(t), {{\mathcal B}}_2(t),\cdots, {{\mathcal B}}_n(t))\}_{t{\geqslant}0}$ are independent standard Brownian motions, and $\{{{\bm{y}}}(t)\}_{t>0}$ lives on the Weyl chamber ${\mathbb{W}}^n=\{({\lambda}_1,{\lambda}_2,\cdots,{\lambda}_n): {\lambda}_1>{\lambda}_2>\cdots>{\lambda}_n\}$. The expression is slightly different from . We add a constant drift term in , so that it matches with the dynamics of the $\beta$-nonintersecting Poisson random walks. We denote the empirical measure process of , $$\begin{aligned} \tilde \mu_t^n {\mathrel{\mathop:}=}\frac{1}{n}\sum_{i=1}^n \delta_{y_i(t)}.\end{aligned}$$ It follows from [@MR1217451; @MR1176727], if the initial empirical measure $\tilde \mu_0^n$ converges in the L[é]{}vy metric as $n$ goes to infinity towards a probability measure $\tilde \mu_0$ almost surely (in probability), then, for any fixed time $T>0$, $\{\tilde \mu^n_t\}_{0{\leqslant}t{\leqslant}T}$ converges as $n$ goes to infinity in $D([0,T], M_1({{\mathbb R}}))$ almost surely (in probability). The Stieltjes transform of the limit measure-valued process $\{\tilde\mu_t\}_{0{\leqslant}t{\leqslant}T}$ $$\begin{aligned} \tilde m_t(z)=\int \frac{{{\rm d}}\tilde \mu_t(x)}{x-z},\quad z\in {{\mathbb C}}\setminus {{\mathbb R}},\end{aligned}$$ is characterized by $$\begin{aligned} \begin{split}\label{e:stDBM} & \tilde m_t(\tilde z_t(z))=\tilde m_0(z),\\ & \tilde z_t(z)=z+t-t\tilde m_0(z), \end{split}\end{aligned}$$ where $\tilde z_t(z)$ is well-defined on the domain $$\begin{aligned} \tilde \Omega_t{\mathrel{\mathop:}=}\left\{z\in {{\mathbb C}}\setminus {{\mathbb R}}: \int \frac{{{\rm d}}\tilde \mu_0(x)}{|x-z|^2}<\frac{1}{t}\right\}.\end{aligned}$$ We recall the limit empirical measure process $\mu_t$ of the $\beta$-nonintersecting Poisson random walks from Theorem \[t:LLN\], its Stieltjes transform $m_t(z)$ in , and the key relations and . We also recall the Markov-Krein correspondence operator $Q$ from Theorem \[t:MKcor\]. If we take the $\tilde \mu_0=Q(\mu_0)$, by the defining relation of $Q$, we have $$\begin{aligned} \tilde m_0(z)=1-e^{-m_0(z)}.\end{aligned}$$ Therefore, the characteristic lines for $m_t(z)$ and $\tilde m_t(z)$ are the same: $$\begin{aligned} \label{e:cline} \tilde z_t(z)=z+t-t\tilde m_0(z)=z+te^{-m_0(z)}=z_t(z).\end{aligned}$$ The Stieltjes transforms $m_t(z)$ and $\tilde m_t(z)$ satisfy $$\begin{aligned} \tilde m_t(\tilde z_t(z)) =\tilde m_0(z) =1-e^{-m_0(z)} =1-e^{-m_t(z_t(z))} =1-e^{-m_t(\tilde z_t(z))}.\end{aligned}$$ Since $\tilde z_t(z)$ is a surjection onto ${{\mathbb C}}\setminus {{\mathbb R}}$, we get $\tilde \mu_t=Q(\mu_t)$. We denote the rescaled fluctuation process $$\begin{aligned} \tilde g_t^n(z)=n(\tilde m_t^n(z)- \tilde m_t(z)).\end{aligned}$$ It follows from [@MR1819483; @MR2418256], if there exists a constant ${{\frak a}}$, such that for any $z\in {{\mathbb C}}\setminus{{\mathbb R}}$, $$\begin{aligned} {\mathbb{E}}\left[|\tilde g_0^n(z)|^2\right]{\leqslant}{{\frak a}}({\mathop{\mathrm{Im}}}[z])^{-2},\end{aligned}$$ and the random field $(\tilde g_0^n(z))_{z\in {{\mathbb C}}\setminus{{\mathbb R}}}$ weakly converges to a deterministic field $(\tilde g_0(z))_{z\in {{\mathbb C}}\setminus {{\mathbb R}}}$, in the sense of finite dimensional distributions, then, for any fixed time $T>0$, the process $\{(\tilde g^n_t(\tilde z_t(z)))_{z\in\tilde \Omega_T}\}_{0{\leqslant}t{\leqslant}T}$ converges weakly towards a Gaussian process $\{(\tilde g_t(\tilde z_t(z)))_{z\in \tilde \Omega_T}\}_{0{\leqslant}t{\leqslant}T}$, in the sense of finite dimensional processes, with initial data $(\tilde g_0(z))_{z\in {{\mathbb C}}\setminus{{\mathbb R}}}$, means and covariances $$\begin{aligned} \begin{split}\label{e:meanandvar2} {\mathbb{E}}[\tilde g_t(\tilde z_t(z))]&=\tilde \mu(t, z)=\frac{\tilde g_0(z)}{ {\partial}_z\tilde z_t(z)}+\left(\frac{1}{2}-\frac{1}{2\theta}\right)\frac{{\partial}_z^2\tilde z_t(z)}{({\partial}_z \tilde z_t(z))^2},\\ {{\rm{cov}}}[\tilde g_s(\tilde z_s(z)), \tilde g_t(\tilde z_t(z'))]&=\tilde \sigma(s, z, t, z')=\frac{1}{\theta}\frac{1}{{\partial}_z\tilde z_s(z){\partial}_z\tilde z_t(z')}\left(\frac{1}{(z-z')^2}-\frac{{\partial}_z \tilde z_{s\wedge t}(z){\partial}_z \tilde z_{s\wedge t}(z')}{(\tilde z_{s\wedge t}(z)-\tilde z_{s\wedge t}(z'))^2}\right),\\ {{\rm{cov}}}[\tilde g_s(\tilde z_s(z)), \overline{\tilde g_t(\tilde z_t(z'))}]&=\tilde \sigma(s, z,t,\bar z'). \end{split}\end{aligned}$$ The statements in [@MR1819483; @MR2418256] are for $\beta$-Dyson Brownian motions with quadratic potential, which differ from by a rescaling of time and space. follows from [@MR2418256 Theorem 2.3] by a change of variable. By comparing with , if we replace the characteristic lines $\tilde z_t(z)$ in the expressions of the means and covariances of the random field $\{(\tilde g_t(\tilde z_t(z)))_{z\in\tilde \Omega_T}\}_{0{\leqslant}t{\leqslant}T}$ by $z_t(z)$, we get the means and variances of the random field $\{(g_t( z_t(z)))_{z\in \Omega_T}\}_{0{\leqslant}t{\leqslant}T}$. If we take the $\tilde \mu_0=Q(\mu_0)$ and $\tilde g_0(z)=g_0(z)$, then from the discussion above, we have $\tilde \mu_t=Q(\mu_t)$ and $\tilde z_t(z)=z_t(z)$ from . Thus, $$\begin{aligned} \{(\tilde g_t(\tilde z_t(z)))_{z\in\tilde \Omega_T}\}_{0{\leqslant}t{\leqslant}T}\stackrel{d}{=}\{(g_t( z_t(z)))_{z\in \Omega_T}\}_{0{\leqslant}t{\leqslant}T},\end{aligned}$$ in the sense of finite dimensional processes. From the discussion above, we have that the following diagram commutes \[e:commute\] (\_0, g\_0(z)) & & & & & &&(\_t, g\_t(z\_t(z)))\ (\_0, g\_0(z)) & & & & & &&(\_t, g\_t(z\_t(z))), where $Q$ is the Markov-Krein correspondence from Theorem \[t:MKcor\], and $I$ is the identity map. Stochastic differential equation for the Stieltjes transform {#s:sde} ------------------------------------------------------------ In this section we derive a stochastic differential equation for the Stieltjes transform of the empirical measure process $\mu_t^n$. \[p:msde\] The Stieltjes transform of the empirical measure process satisfies the following stochastic differential equation $$\begin{aligned} \begin{split}\label{e:msde} m^n_t(z_t)=m^n_0(z) +\int_0^t {\partial}_z m^n_s(z_s) e^{-m_s(z_s)}{{\rm d}}s&+\theta n\int_{0}^t\left(\prod_{j=1}^n\left(1+\frac{1}{n}\frac{1}{z_s-x_j(s)/\theta n}\right)\right.\\ &-\left.\prod_{j=1}^n\left(1+\frac{1}{n}\frac{1}{(z_s-1/\theta n)-x_j(s)/\theta n}\right)\right){{\rm d}}s+M^n_t(z), \end{split}\end{aligned}$$ where $z_t$ is defined in with $z_0=z$ and $M^n_t(z)$ is a Martingale starting at $0$, with quadratic variations, $$\begin{aligned} \begin{split}\label{e:qv} [ M^n(z), M^n(z)]_t=\sum_{0{\leqslant}s{\leqslant}t}(m^n_s(z_s)-m^n_{s-}(z_s))^2, \\ [ M^n(z), \overline{M^n(z)}]_t=\sum_{0{\leqslant}s{\leqslant}t}|m^n_s(z_s)-m^n_{s-}(z_s)|^2. \end{split}\end{aligned}$$ We remark that the integrand in also appears in the Nekrasov’s equation in [@MR3668648 Section 4], which is crucial for the proof of the central limit theorems of the discrete $\beta$ ensembles. We can view as a dynamical version of the Nekrasov’s equation. The $\beta$-nonintersecting random walk ${{\bm{x}}}(t)$ is a continuous time Markov jump process. We recall its generator ${{{\mathcal}L}}^n_\theta $ from . By It[ó]{}’s formula, $$\begin{aligned} \label{e:ito} M^n_t(z){\mathrel{\mathop:}=}m^n_t(z_t)-m^n_0(z_t)-\int_0^t {\partial}_z m^n_s(z_s){\partial}_t z_s{{\rm d}}s-\int_{0}^t {{{\mathcal}L}}^n_\theta m^n_s(z_s){{\rm d}}s,\end{aligned}$$ is a martingale with quadratic variations given by . To estimate the integrand ${{{\mathcal}L}}_\theta^n m_s^n(z_s)$ in , we need some algebraic facts about the generator ${{{\mathcal}L}}_\theta^n$. \[l:average\] For any $\theta\in {{\mathbb R}}$, we have $$\begin{aligned} \label{e:average} \sum_{i=1}^{n} \prod_{j:j\neq i}\frac{x_i-x_j+\theta}{x_i-x_j}=n.\end{aligned}$$ We can rewrite the left hand side of in terms of the Vandermond determinant in variables $x_1,x_2,\cdots, x_n$, $$\begin{aligned} \sum_{i=1}^{n} \prod_{j:j\neq i}\frac{x_i-x_j+\theta}{x_i-x_j} =\sum_{i=1}^n \frac{V({{\bm{x}}}+ \theta {\bm{e}}_i)}{V({{\bm{x}}})}.\end{aligned}$$ We notice that $\sum_{i=1}^n V({{\bm{x}}}+\theta {\bm{e}}_i)$ is a degree $n(n-1)/2$ polynomial in variables $x_1,x_2,\cdots,x_n$. More importantly, it is antisymmetric. Therefore, there exists a constant $C(\theta, n)$ depending on $\theta$ and $n$ such that $$\begin{aligned} \label{e:average2} \sum_{i=1}^n V({{\bm{x}}}+\theta {\bm{e}}_i)=C(\theta, n)V({{\bm{x}}}).\end{aligned}$$ We conclude from by comparing the coefficient of the term $x_1^{n-1}x_2^{n-2}\cdots x_{n-1}$. The following identity will be crucial for the derivation of the stochastic differential equation of the Stieltjes transforms of the empirical measure process $\mu_t$. \[c:average3\] For any $\theta\in {{\mathbb R}}$, we have $$\begin{aligned} \label{e:average3} \sum_{i=1}^{n} \left(\prod_{j:j\neq i}\frac{x_i-x_j+\theta}{x_i-x_j}\right)\frac{1}{n}\frac{1}{x_i/\theta n-z}=1-\prod_{j=1}^n\left(1+\frac{1}{n}\frac{1}{z-x_j/\theta n}\right).\end{aligned}$$ We use Lemma \[l:average\] for the vector $(x_1,x_2,\cdots, x_n, \theta n z)$, $$\begin{aligned} \begin{split} n+1&=\sum_{i=1}^{n} \left(\prod_{j:j\neq i}\frac{x_i-x_j+\theta}{x_i-x_j}\right)\frac{x_i-\theta n z+\theta}{x_i-\theta n z} +\prod_{j=1}^n\frac{\theta n z-x_j+\theta}{\theta n z-x_j}\\ &=\sum_{i=1}^{n} \left(\prod_{j:j\neq i}\frac{x_i-x_j+\theta}{x_i-x_j}\right)+\sum_{i=1}^{n} \left(\prod_{j:j\neq i}\frac{x_i-x_j+\theta}{x_i-x_j}\right)\frac{\theta}{x_i-\theta n z} +\prod_{j=1}^n\left(1+\frac{\theta}{\theta n z-x_j}\right)\\ &=n+\sum_{i=1}^{n} \left(\prod_{j:j\neq i}\frac{x_i-x_j+\theta}{x_i-x_j}\right)\frac{\theta}{x_i-\theta n z} +\prod_{j=1}^n\left(1+\frac{\theta}{\theta n z-x_j}\right). \end{split}\end{aligned}$$ The claim follows by rearranging. \[r:proofMKcor\] We can use Corollary \[c:average3\] to give a construction of the operator $Q$ in Theorem \[t:MKcor\]. Let $\mu$ be a measure as in Theorem \[t:MKcor\]. For any large integer $m>0$, we discretize $\mu$ on the scale $1/m$ and define $$\begin{aligned} \mu^m{\mathrel{\mathop:}=}\frac{1}{m}\sum_{i=1}^m \delta_{y_i^m},\quad \frac{i-1/2}{m}=\int_{y_i^m}^{\infty}{{\rm d}}\mu(x), \quad 1{\leqslant}i{\leqslant}m.\end{aligned}$$ As $m$ goes to infinity, $\mu^m$ weakly converges to $\mu$. Since the density of $\mu$ is bounded by $1$, we have $y_{i}^m-y_{i+1}^m{\geqslant}1/m$ for all $1{\leqslant}i{\leqslant}m-1$. The Perelomov-Popov measure is defined as $$\begin{aligned} Q^m(\mu^m){\mathrel{\mathop:}=}\frac{1}{m}\sum_{i=1}^m \prod_{j:j\neq i}\frac{y_i^m-y_{j}^m+1/m}{y_i^m-y_j^m}\delta_{y_i^m}.\end{aligned}$$ Since $y_{i}^m-y_{i+1}^m{\geqslant}1/m$, $Q^m(\mu^m)$ is a positive measure. Moreover, thanks to Lemma \[l:average\], $Q^m(\mu^m)$ is a probability measure. We denote the Stieltjes transform of $\mu$, $\mu^m$ and $Q^m(\mu^m)$ by $m_{\mu}(z)$, $m_{\mu^m}(z)$ and $m_{Q^m(\mu^m)}(z)$ respectively. Since $\mu^m$ weakly converges to $\mu$ as $m$ goes to infinity, $$\begin{aligned} \lim_{m\rightarrow \infty}m_{\mu^m}(z)=m_\mu(z),\quad z\in {{\mathbb C}}\setminus{{\mathbb R}}.\end{aligned}$$ For the Stieltjes transform $m_{Q^m(\mu^m)}(z)$, we use Corollary \[c:average3\], $$\begin{aligned} m_{Q^m(\mu^m)}(z) &=\frac{1}{m}\sum_{i=1}^m \prod_{j:j\neq i}\frac{y_i^m-y_{j}^m+1/m}{y_i^m-y_j^m}\frac{1}{y_i^m-z}\\ &=1-\prod_{j=1}^m\left(1-\frac{1}{m}\frac{1}{y_j^m-z}\right) =1-\exp\left\{-\frac{1}{m}\sum_{j=1}^m\frac{1}{y_j^m-z}+\operatorname{O}\left(\frac{1}{m}\right)\right\}\\ &=1-\exp\left\{-m_{\mu^m}(z)+\operatorname{O}\left(\frac{1}{m}\right)\right\}\rightarrow 1-e^{-m_\mu(z)},\end{aligned}$$ as $m$ goes to infinity. Since $$\begin{aligned} \lim_{y\rightarrow \infty}{\mathrm{i}}y\left(1-e^{-m_\mu({\mathrm{i}}y)}\right)=-1,\end{aligned}$$ by [@MR1962995 Theorem 1], there exists a probability measure $Q(\mu)$ with Stieltjes transform $1-e^{-m_\mu(z)}$, and $Q^m(\mu^m)$ weakly converges to $Q(\mu)$. For the integrand ${{{\mathcal}L}}^n_\theta m^n_s(z)$, we have $$\begin{aligned} \begin{split} {{{\mathcal}L}}^n_\theta m^n_s(z) &=\theta n\sum_{i=1}^{n}\left(\prod_{j:j\neq i}\frac{x_i(s)-x_j(s)+\theta}{x_i(s)-x_j(s)}\right)\left(\frac{1}{n}\frac{1}{(x_i(s)+1)/\theta n-z_s}-\frac{1}{n}\frac{1}{x_i(s)/\theta n-z_s}\right)\\ &=\theta n\left(\prod_{j=1}^n\left(1+\frac{1}{n}\frac{1}{z_s-x_j(s)/\theta n}\right) -\prod_{j=1}^n\left(1+\frac{1}{n}\frac{1}{(z_s-1/\theta n)-x_j(s)/\theta n}\right)\right). \end{split}\end{aligned}$$ where we used Corollary \[c:average3\]. Combining with , this finishes the proof Proposition \[p:msde\] Law of large numbers {#s:LLN} -------------------- In this section we analyze , and prove the Law of large numbers for the $\beta$-nonintersecting Poisson random walk. We define an auxiliary process $N^n_t$, which counts the number of jumps for the $\beta$-nonintersecting Poisson random walk ${{\bm{x}}}(t)$, $$\begin{aligned} \label{e:defNt} N^n_t=\sum_{i=1}^n \left(x_i(t)-x_i(0)\right).\end{aligned}$$ The Poisson process $N^n_t$ will be used later to control the martingale term $M^n_t(z)$ in . $N^n_t$ is a Poisson process, starting at $0$, with jump rate $\theta n^2$. According to the generator of the $\beta$-nonintersecting Poisson random walk, the process $N^n_t$ increases $1$ with rate $$\begin{aligned} \theta n\sum_{i=1}^n\sum_{j:j\neq i}\frac{x_i-x_j+\theta}{x_i-x_j}=\theta n^2,\end{aligned}$$ where we used Proposition \[e:average\]. In the following we simplify the stochastic differential equation of $m_t(z_t)$, i.e. the second integrand in . By Proposition \[p:ztp\], for any $z\in \Omega_T$, and $0{\leqslant}t{\leqslant}T$, we have $|{\mathop{\mathrm{Im}}}[z_t]|{\geqslant}|{\mathop{\mathrm{Im}}}[z_T]|>0$. Therefore, we have the trivial bound $1/|x_j(s)/\theta n-z_t|=\operatorname{O}(1)$, where the implicit constant depends on ${\mathop{\mathrm{Im}}}[z_T]$. For the first term in the second integrand in , by the Tylor expansion $$\begin{aligned} \begin{split}\label{e:t1} &\phantom{{}={}}\prod_{j=1}^n\left(1+\frac{1}{n}\frac{1}{z_s-x_j(s)/\theta n}\right) =\exp\left(\sum_{j=1}^n \ln\left(1+\frac{1}{n}\frac{1}{z_s-x_j(s)/\theta n}\right)\right)\\ &=\exp\left(\sum_{j=1}^n \frac{1}{n}\frac{1}{z_s-x_j(s)/\theta n}-\frac{1}{2n^2}\frac{1}{(z_s-x_j(s)/\theta n)^2}+\frac{1}{3n^3}\frac{1}{(z_s-x_j(s)/\theta n)^3}+\operatorname{O}\left(\frac{1}{n^4}\right)\right)\\ &=e^{-m^n_s(z_s)}\left(1-\frac{1}{2}\frac{{\partial}_z m^n_s(z_s)}{n}+\frac{1}{8}\frac{({\partial}_z m^n_s(z_s))^2}{n^2}-\frac{1}{6}\frac{{\partial}_z^2 m^n_s(z_s)}{n^2}+\operatorname{O}\left(\frac{1}{n^3}\right)\right). \end{split}\end{aligned}$$ Similarly, for the second term in the integrand, $$\begin{aligned} \begin{split}\label{e:t2} &\phantom{{}={}}\prod_{j=1}^n\left(1+\frac{1}{n}\frac{1}{(z_s-1/\theta n)-x_j(s)/\theta n}\right) =\exp\left(\sum_{j=1}^n \ln\left(1+\frac{1}{n}\frac{1}{(z_s-1/\theta n)-x_j(s)/\theta n}\right)\right)\\ &=e^{-m^n_s(z)}\left(1-\left(\frac{1}{2}-\frac{1}{\theta}\right)\frac{{\partial}_z m^n_s(z)}{n}+\frac{1}{2}\left(\frac{1}{\theta}-\frac{1}{2}\right)^2\frac{({\partial}_z m^n_s(z_s))^2}{n^2}-\right.\\ &\phantom{{}=e^{-m^n_s(z_s)}\left(1-\left(\frac{1}{2}-\frac{1}{\theta}\right)\frac{{\partial}_z m^n_s(z_s)}{n}\right.}\left.-\frac{1}{2}\left(\frac{1}{\theta^2}-\frac{1}{\theta}+\frac{1}{3}\right)\frac{{\partial}_z^2 m^n_s(z_s)}{n^2}+\operatorname{O}\left(\frac{1}{n^3}\right)\right). \end{split}\end{aligned}$$ The difference of and is $$\begin{aligned} \begin{split}\label{e:t3} &\phantom{{}={}}\prod_{j=1}^n\left(1+\frac{1}{n}\frac{1}{(z_s-1/\theta n)-x_j(s)/\theta n}\right)-\prod_{j=1}^n\left(1+\frac{1}{n}\frac{1}{(z_s-1/\theta n)-x_j(s)/\theta n}\right)\\ &=e^{-m^n_s(z_s)}\left(-\frac{{\partial}_z m^n_s(z_s)}{\theta n}+\left(\frac{1}{2}-\frac{1}{2\theta}\right)\frac{({\partial}_z m^n_s(z_s))^2-{\partial}_z^2 m^n_s(z_s)}{\theta n^2}+\operatorname{O}\left(\frac{1}{n^3}\right)\right). \end{split}\end{aligned}$$ We can use to simplify the stochastic differential equation , $$\begin{aligned} \begin{split}\label{e:newsde} m^n_t(z_t)=m^n_0(z) &+\int_0^t {\partial}_z m^n_s(z_s) \left(e^{-m_s(z_s)}-e^{-m^n_s(z_s)}\right){{\rm d}}s+\\ &+\left(\frac{1}{2}-\frac{1}{2\theta}\right)\int_{0}^t\frac{(({\partial}_z m^n_s(z_s))^2-{\partial}_z^2 m^n_s(z_s))e^{-m_s^n(z_s)}}{n}{{\rm d}}s+M^n_t(z)+\operatorname{O}\left(\frac{t}{n^2}\right), \end{split}\end{aligned}$$ where the implicit constant depends on $\theta$ and ${\mathop{\mathrm{Im}}}[z_T]$. In the following we estimate the martingale $M^n_t(z)$ using the Burkholder-Davis-Gundy inequality. For the quadratic variation of $M^n_t(z)$, we have $$\begin{aligned} \begin{split}\label{e:quadvar} &\phantom{{}={}}[ M^n(z), \overline{M^n(z)}]_t =\sum_{0< s{\leqslant}t} |m^n_s(z_s)-m^n_{s-}(z_s)|^2\\ &=\frac{1}{n^2}\sum_{0< s{\leqslant}t\atop \Delta x_i(s)>0}\left|\frac{1}{x_i(s)/\theta n-z_s}-\frac{1}{x_i(s-)/\theta n-z_s}\right|^2 = \operatorname{O}\left(\frac{N^n_t}{ n^4}\right), \end{split}\end{aligned}$$ where the implicit constant depends on $\theta$ and ${\mathop{\mathrm{Im}}}[z_T]$. It follows from the Burkholder-Davis-Gundy inequality, for any $p{\geqslant}1$, we have $$\begin{aligned} \begin{split}\label{e:BDG} {\mathbb{E}}\left[\left(\sup_{0{\leqslant}t{\leqslant}T}|M^n_t(z)|\right)^p\right]^{1/p} &{\leqslant}Cp {\mathbb{E}}\left[[ M^n(z), \overline{M^n(z) }]_T^{p/2}\right]^{1/p} \\ &=\operatorname{O}\left(\frac{p}{ n^2}{\mathbb{E}}\left[\left(N^n_T\right)^{p/2}\right]^{1/p} \right)=\operatorname{O}\left(\frac{T^{1/2}p^{3/2}}{n}\right). \end{split}\end{aligned}$$ By the Markov’s inequality, we have $$\begin{aligned} {\mathbb{P}}\left(\sup_{0{\leqslant}t{\leqslant}T}|M^n_t(z)|{\geqslant}\varepsilon\right){\leqslant}\left(\frac{CT^{1/2}p^{3/2}}{\varepsilon n}\right)^p.\end{aligned}$$ Therefore it follows by taking $p>1$, $\sup_{0{\leqslant}t{\leqslant}T}|M^n_t(z)|$ converges to zero almost surely as $n$ goes to infinity. For any $0{\leqslant}t{\leqslant}T$, we can rewrite as $$\begin{aligned} \begin{split} |m^n_t(z_t)-m^n_0(z)| &{\leqslant}|m^n_0(z)-m_0(z)|+C\left(\int_0^t |m_s(z_s)-m_0(z)|{{\rm d}}s+\frac{t}{n}\right)+\sup_{0{\leqslant}t{\leqslant}T}|M_t^n|\\ &=C\left(\int_0^t |m_s(z_s)-m_0(z)|{{\rm d}}s\right)+|m^n_0(z)-m_0(z)|+\operatorname{o}(1) \end{split}\end{aligned}$$ where the constant $C$ depends on $\theta$, $T$ and ${\mathop{\mathrm{Im}}}[z_T]$, and the term $\operatorname{o}(1)$ converges to zero almost surely and is uniform for $0{\leqslant}t{\leqslant}T$. Thus, it follows from Gronwell’s inequality, $$\begin{aligned} \sup_{0{\leqslant}t{\leqslant}T}|m_t(z_t)-m_0(z)|{\leqslant}C|m_0^n(z)-m_0(z)|+o(1),\end{aligned}$$ which converges to zero almost surely (in probability), if $|m_0^n(z)-m_0(z)|$ converges to zero almost surely (in probability). This finishes the proof of Theorem \[t:LLN\]. Central limit theorems for the empirical measure process ======================================================== In this section, we prove the central limit theorems for the rescaled empirical measure process $\{n(\mu^n_t-\mu_t)\}_{0{\leqslant}t {\leqslant}T}$ with analytic test functions. Central limit theorems {#s:CLT} ---------------------- Theorem \[t:CLT\] follows from the following proposition. \[p:CLT\] We assume Assumption \[a:initiallaw\]. Then for any values $z_1,z_2,\cdots, z_m\in {{\mathbb C}}\setminus{{\mathbb R}}$ and time $T<\min\{{{\frak t}}(z_1),{{\frak t}}(z_2),\cdots, {{\frak t}}(z_m)\}$ as defined in , the random processes $\{(g^n_t(z_t(z_j)))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ converge weakly in the Skorokhod space $D([0,T], {{\mathbb C}}^m)$ towards a Gaussian process $\{(\mathcal{G}_j(t))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$, which is the unique solution of the system of stochastic differential equations $$\begin{aligned} \begin{split}\label{e:limitprocess} {{{\mathcal}G}}_j(t)&={{{\mathcal}G}}_j(0)+ \int_0^t \frac{{\partial}_zm_0(z_j)e^{-m_0(z_j)}}{1-s{\partial}_zm_0(z_j)e^{-m_0(z_j)}} {{{\mathcal}G}}_j(s){{\rm d}}s\\ &+\left(\frac{1}{2}-\frac{1}{2\theta}\right)\int_0^t\frac{(({\partial}_z m_0(z_j))^2-{\partial}_z^2 m_0(z_j))e^{-m_0(z_j)}}{(1-s{\partial}_z m_0(z_j)e^{-m_0(z_j)})^3}{{\rm d}}s+{{\mathcal W}}_j(t),\quad 1{\leqslant}j{\leqslant}m, \end{split}\end{aligned}$$ with initial data $(g_0(z_j))_{1{\leqslant}j{\leqslant}m}$ given in Assumption \[a:initiallaw\], and $\{({{\mathcal W}}_j(t))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ is a centered Gaussian process independent of $({{{\mathcal}G}}_j(0))_{1{\leqslant}j{\leqslant}m}$, and $$\begin{aligned} \begin{split}\label{e:covW1} \langle {{\mathcal W}}_j, {{\mathcal W}}_k\rangle_t=-\frac{1}{\theta}\int_0^t\frac{({\partial}_s m_s)(z_s(z_j))+({\partial}_sm_s)(z_s(z_k))}{(z_s(z_j)-z_s(z_k))^2}-\frac{2(e^{-m_0(z_j)}-e^{-m_0(z_k)})}{(z_s(z_j)-z_s(z_k))^3}{{\rm d}}s,\\ \langle{{\mathcal W}}_j, \bar{{{\mathcal W}}}_k\rangle_t=-\frac{1}{\theta}\int_0^t\frac{({\partial}_s m_s)(z_s(z_j))+({\partial}_sm_s)(z_s(\bar{z}_k))}{(z_s(z_j)-z_s(\bar{z}_k))^2}-\frac{2(e^{-m_0(z_j)}-e^{-m_0(\bar{z}_k)})}{(z_s(z_j)-z_s(\bar{z}_k))^3}{{\rm d}}s, \end{split}\end{aligned}$$ where $$\begin{aligned} \begin{split}\label{e:covW2} \langle {{\mathcal W}}_j, {{\mathcal W}}_j\rangle_t &=\lim_{z_k\rightarrow z_j}-\frac{1}{\theta}\int_0^t\frac{({\partial}_s m_s)(z_s(z_j))+({\partial}_sm_s)(z_s(z_k))}{(z_s(z_j)-z_s(z_k))^2}-\frac{2(e^{-m_0(z_j)}-e^{-m_0(z_k)})}{(z_s(z_j)-z_s(z_k))^3}{{\rm d}}s\\ &=-\frac{1}{6\theta }\int_0^t ({\partial}_s{\partial}^2_z m_s)(z_s(z_j)){{\rm d}}s. \end{split}\end{aligned}$$ We can solve for $\{({{{\mathcal}G}}_j(t))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ explicitly. From , we have $$\begin{aligned} \begin{split}\label{e:rearrange} {{\rm d}}(1-t{\partial}_zm_0(z_j)e^{-m_0(z_j)}){{{\mathcal}G}}_j(t)&=\left(\frac{1}{2}-\frac{1}{2\theta}\right)\frac{(({\partial}_z m_0(z_j))^2-{\partial}_z^2 m_0(z_j))e^{-m_0(z_j)}}{(1-t{\partial}_z m_0(z_j)e^{-m_0(z_j)})^2}{{\rm d}}t\\ &+(1-t{\partial}_zm_0(z_j)e^{-m_0(z_j)}){{\rm d}}{{\mathcal W}}_j(t). \end{split}\end{aligned}$$ We integrate both sides of , $$\begin{aligned} {{{\mathcal}G}}_j(t)=\frac{{{{\mathcal}G}}_j(0)}{1-t{\partial}_zm_0(z_j)e^{-m_0(z_j)}} +\left(\frac{1}{2}-\frac{1}{2\theta}\right)\frac{t(({\partial}_z m_0(z_j))^2-{\partial}_z^2 m_0(z_j))e^{-m_0(z_j)}}{(1-t{\partial}_z m_0(z_j)e^{-m_0(z_j)})^2} + {{\mathcal B}}_j(t),\end{aligned}$$ where $$\begin{aligned} {{\mathcal B}}_j(t)=\frac{1}{(1-t{\partial}_zm_0(z_j)e^{-m_0(z_j)})}\int_0^t (1-s{\partial}_zm_0(z_j)e^{-m_0(z_j)}){{\rm d}}{{\mathcal W}}_j(s).\end{aligned}$$ By a straightforward (but tedious and lengthy) calculation, using , and , we get the covariances of $\{({{\mathcal B}}_j(t))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$, $$\begin{aligned} {{\rm{cov}}}[ {{\mathcal B}}_j(s), {{\mathcal B}}_k(t)] &=\frac{\int_0^{s\wedge t} (1-u{\partial}_zm_0(z_j)e^{-m_0(z_j)})(1-u{\partial}_zm_0(z_k)e^{-m_0(z_k)}){{\rm d}}\langle{{\mathcal W}}_j, {{\mathcal W}}_k\rangle_u}{(1-s{\partial}_zm_0(z_j)e^{-m_0(z_j)})(1-t{\partial}_zm_0(z_k)e^{-m_0(z_k)})}\\ &=\frac{1}{\theta}\frac{1}{ (1-s{\partial}_zm_0(z_j)e^{-m_0(z_j)})(1-t{\partial}_zm_0(z_k)e^{-m_0(z_k)})}\\ &\times\left(\frac{1}{(z_j-z_k)^2}-\frac{(1-(s\wedge t){\partial}_zm_0(z_j)e^{-m_0(z_j)})(1-(s\wedge t){\partial}_zm_0(z_k)e^{-m_0(z_k)})}{(z_j-z_k+(s\wedge t)(e^{-m_0(z_j)}-e^{-m_0(z_k)}))^2}\right)=\sigma(s, z_j,t,z_k)\\ {{\rm{cov}}}[ {{\mathcal B}}_j(s), \overline{{{\mathcal B}}_k(t)}]&=\sigma(s,z_j,t, \bar z_k),\end{aligned}$$ and $$\begin{aligned} &\phantom{{}={}}{{\rm{cov}}}[{{\mathcal B}}_j(s), {{\mathcal B}}_j(t)]=\frac{\int_0^{s\wedge t} (1-u{\partial}_zm_0(z_j)e^{-m_0(z_j)})^2{{\rm d}}\langle{{\mathcal W}}_j, {{\mathcal W}}_j\rangle_u}{(1-s{\partial}_zm_0(z_j)e^{-m_0(z_j)})(1-t{\partial}_zm_0(z_j)e^{-m_0(z_j)})}\\ &=\frac{(s\wedge t)e^{-m_0(z_j)}(2({\partial}_z m_0(z_j))^3-6{\partial}_z m_0(z_j){\partial}_z^2 m_0(z_j)+2{\partial}_z^3m_0(z_j))}{12\theta(1-(s\wedge t){\partial}_z m_0(z_j)e^{-m_0(z_j)})^3(1-(s\vee t){\partial}_z m_0(z_j)e^{-m_0(z_j)})}\\ &+\frac{(s\wedge t)^2e^{-2m_0(z_j)}(({\partial}_z m_0(z_j))^4+3({\partial}_z^2 m_0(z_j))^2-2{\partial}_z m_0(z_j){\partial}_z^3m_0(z_j))}{12\theta(1-(s\wedge t){\partial}_z m_0(z_j)e^{-m_0(z_j)})^3(1-(s\vee t){\partial}_z m_0(z_j)e^{-m_0(z_j)})}=\sigma(s,z_j,t, z_j)=\lim_{z_k\rightarrow z_j}\sigma(s, z_j,t, z_k). $$ This finishes the proof of Theorem \[t:CLT\]. We divide the proof of Proposition \[p:CLT\] into three steps. In Step one we prove the tightness of the processes $\{(g^n_t(z_t(z_j)))_{1{\leqslant}j{\leqslant}m})\}_{0{\leqslant}t{\leqslant}T}$ as $n$ goes to infinity. In Step two, we prove that the martingale term $\{(nM_t^n(z_j))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ converges weakly to a centered complex Gaussian process. In Step three, we prove that the subsequential limits of $\{(g^n_t(z_t(z_j)))_{1{\leqslant}j{\leqslant}m})\}_{0{\leqslant}t{\leqslant}T}$ solve the stochastic differential equation . Proposition \[p:CLT\] follows from this fact and the uniqueness of the solution to . [*Step one: tightness.*]{} We first prove the tightness of the martingale term. We assume the assumptions of Proposition \[p:CLT\]. Then as $n$ goes to infinity, the random processes $\{(nM^n_t(z_j))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$, and $\{(n^2[ M^n(z_j), M^n(z_k)]_t)_{1{\leqslant}j,k{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ are tight. We apply the sufficient condition for tightness of [@MR1943877 Chapter 6, Proposition 3.26]. We need to check the modulus conditions: for any $\varepsilon>0$ there exists a $\delta>0$ such that $$\begin{aligned} \label{e:modulus1} &{\mathbb{P}}\left(\sup_{1{\leqslant}j{\leqslant}m}\sup_{0{\leqslant}t{\leqslant}t'{\leqslant}T, t'-t{\leqslant}\delta}|n(M^n_{t'}(z_j)-M^n_t(z_j))|{\geqslant}\varepsilon\right){\leqslant}\varepsilon,\\ \label{e:modulus2}&{\mathbb{P}}\left(\sup_{1{\leqslant}j,k{\leqslant}m}\sup_{0{\leqslant}t{\leqslant}t'{\leqslant}T, t'-t{\leqslant}\delta}\left|n^2([ M^n(z_j), M^n(z_k)]_{t'}-[ M^n(z_j), M^n(z_k)]_t)\right|{\geqslant}\varepsilon\right){\leqslant}\varepsilon.\end{aligned}$$ For , since $\{M^n_{t'}(z_j)-M^n_t(z_j)\}_{t{\leqslant}t'{\leqslant}T\vee t+\delta}$ is a martingale, it follows from the Burkholder-Davis-Gundy inequality, for any $p{\geqslant}1$, we have $$\begin{aligned} \begin{split} &\phantom{{}={}}{\mathbb{E}}\left[\left(\sup_{t{\leqslant}t'{\leqslant}T\vee t+\delta}|n(M^n_{t'}(z_j)-M^n_t(z_j))|\right)^p\right]^{1/p}\\ &{\leqslant}Cpn {\mathbb{E}}\left[[ M^n(z_j)-M^n_t(z_j), \overline{M^n(z_j)-M_t^n(z_j) }]_{T\vee t+\delta}^{p/2}\right]^{1/p} \\ &=\operatorname{O}\left(\frac{p}{ n}{\mathbb{E}}\left[\left(N^n_{T\vee t+\delta}-N^n_t\right)^{p/2}\right]^{1/p} \right)=\operatorname{O}\left(\delta^{1/2}p^{3/2}\right). \end{split}\end{aligned}$$ where the implicit constant depends on $\theta$ and $\min_{1{\leqslant}j{\leqslant}m}|{\mathop{\mathrm{Im}}}[z_T(z_j)]|$. By the Markov’s inequality, we have $$\begin{aligned} {\mathbb{P}}\left(\sup_{t{\leqslant}t'{\leqslant}T\vee t+\delta}|n(M^n_{t'}(z_j)-M^n_t(z_j))|{\geqslant}\varepsilon\right){\leqslant}\left(\frac{C\delta^{1/2}p^{3/2}}{\varepsilon }\right)^p.\end{aligned}$$ Let $t_k=(k-1)\delta\vee T$ for $1{\leqslant}k{\leqslant}\lfloor1/\delta\rfloor$. By a union bound $$\begin{aligned} \begin{split} &\phantom{{}={}}{\mathbb{P}}\left(\sup_{1{\leqslant}j{\leqslant}m}\sup_{0{\leqslant}t{\leqslant}t'{\leqslant}T, t'-t{\leqslant}\delta}|n(M^n_{t'}(z_j)-M^n_t(z_j))|{\geqslant}\varepsilon\right)\\ &{\leqslant}\frac{m}{\delta} \sup_{1{\leqslant}j{\leqslant}m}\sup_{1{\leqslant}k{\leqslant}\lfloor 1/\delta \rfloor}{\mathbb{P}}\left(\sup_{t_k{\leqslant}t{\leqslant}T\vee(t_k+\delta)}|n(M^n_{t}(z_j)-M^n_{t_k}(z_j))|{\geqslant}\varepsilon/2\right)\\ &{\leqslant}\frac{m}{\delta}\left(\frac{2C\delta^{1/2}p^{3/2}}{\varepsilon}\right)^p{\leqslant}\varepsilon. \end{split}\end{aligned}$$ if we take $p>2$ and $\delta$ small enough. This finishes the proof of . The modulus of the process $n^2[M^n(z_j), M^n(z_k)]_{t}$ is dominated by the Poisson process $n^{-2}N_t^n$ in the following sense. For any $0{\leqslant}t{\leqslant}T$ and $t{\leqslant}t'{\leqslant}T\vee(t+\delta)$, $$\begin{aligned} \begin{split}\label{e:control} &\phantom{{}={}}\left|n^2([ M^n(z_j), M^n(z_k)]_{t'}-[ M^n(z_j), M^n(z_k)]_t)\right|\\ &{\leqslant}n^2\sum_{t< s{\leqslant}t'} |m^n_s(z_s(z_j))-m^n_{s-}(z_s(z_j))||m^n_s(z_s(z_k))-m^n_{s-}(z_s(z_k))|\\ &=\sum_{t< s{\leqslant}t'\atop \Delta x_i(s)>0}\left|\frac{1}{x_i(s)/\theta n-z_s(z_j)}-\frac{1}{x_i(s-)/\theta n-z_s(z_j)}\right|\left|\frac{1}{x_i(s)/\theta n-z_s(z_k)}-\frac{1}{x_i(s-)/\theta n-z_s(z_k)}\right|\\ &= \operatorname{O}\left(\frac{N^n_{t'}-N^n_t}{ n^2}\right), \end{split}\end{aligned}$$ where the implicit constant depends on $\theta$ and $\min_{1{\leqslant}j{\leqslant}m}|{\mathop{\mathrm{Im}}}[z_T(z_j)]|$. By the same argument as for , we have that $n^{-2}N_t^n$ satisfies the modulus condition: $$\begin{aligned} \label{e:modulus3}{\mathbb{P}}\left(\sup_{0{\leqslant}t{\leqslant}t'{\leqslant}T, t'-t{\leqslant}\delta}\left|\frac{N_{t'}^n-N_t^n}{n^2}\right|{\geqslant}\varepsilon\right){\leqslant}\varepsilon.\end{aligned}$$ The claim follows from combining and . We assume the assumptions of Proposition \[p:CLT\]. Then as $n$ goes to infinity, the random processes $\{(g^n_t(z_t(z_j)))_{1{\leqslant}j{\leqslant}m})\}_{0{\leqslant}t{\leqslant}T}$ are tight. We apply the sufficient condition for tightness of [@MR1943877 Chapter 6, Proposition 3.26], and check the modulus condition: for any $\varepsilon>0$ there exists a $\delta>0$ such that $$\begin{aligned} \label{e:modulus} \sup_{n{\geqslant}1}{\mathbb{P}}\left(\sup_{1{\leqslant}i{\leqslant}m}\sup_{0{\leqslant}t{\leqslant}t'{\leqslant}T, t'-t{\leqslant}\delta}|g^n_{t'}(z_{t'}(z_j))-g^n_t(z_t(z_j))|{\geqslant}\varepsilon\right){\leqslant}\varepsilon.\end{aligned}$$ Before we prove the modulus condition , we first prove that as $n$ goes to infinity, the random processes $\{(g_t^n(z_t(z_j)))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ are stochastically bounded, i.e. for any $\varepsilon>0$, there exists $M>0$ such that $$\begin{aligned} \label{e:stBound} \sup_{n{\geqslant}1}{\mathbb{P}}\left(\sup_{1{\leqslant}j{\leqslant}m}\sup_{0{\leqslant}t{\leqslant}T}|g^n_t(z_t(z_j))|{\geqslant}M\right){\leqslant}\varepsilon.\end{aligned}$$ By rearranging , for any $1{\leqslant}i {\leqslant}m$, we get $$\begin{aligned} \begin{split}\label{e:gsde1} g_t^n(z_t(z_j))&=g_0^n(z_j)+\int_0^t {\partial}_z m^n_s(z_s(z_j))e^{-m_0(z_j)} n\left(1-e^{-(m^n_s(z_s(z_j))-m_0(z_j))}\right){{\rm d}}s+\\ &+\left(\frac{1}{2}-\frac{1}{2\theta}\right)\int_{0}^t\left(({\partial}_z m^n_s(z_s(z_j)))^2-{\partial}_z^2 m^n_s(z_s(z_j))\right)e^{-m_s^n(z_s(z_j))}{{\rm d}}s+nM^n_t(z_j)+\operatorname{O}\left(\frac{t}{n}\right). \end{split}\end{aligned}$$ For the second and third terms on the righthand side of , we have $$\begin{aligned} \begin{split}\label{e:bound1} \left|{\partial}_z m^n_s(z_s(z_j))e^{-m_0(z_j)} n\left(1-e^{-(m^n_s(z_s(z_j))-m_0(z_j))}\right)\right|&{\leqslant}C|g_s^n(z_s(z_j))|,\\ \left|({\partial}_z m^n_s(z_s(z_j)))^2-{\partial}_z^2 m^n_s(z_s(z_j))e^{-m_s^n(z_s(z_j))}\right|&{\leqslant}C, \end{split}\end{aligned}$$ where the constants $C$ depends on $\min_{1{\leqslant}j{\leqslant}m}|{\mathop{\mathrm{Im}}}[z_T(z_j)]|$. From , we have $$\begin{aligned} \begin{split}\label{e:bound2} {\mathbb{E}}\left[\left(\sup_{0{\leqslant}t{\leqslant}T}|nM^n_t(z_j)|\right)^p\right]^{1/p} &{\leqslant}CT^{1/2}p^{3/2}, \end{split}\end{aligned}$$ where the constant $C$ depends on $\theta$ and $\min_{1{\leqslant}j{\leqslant}m}|{\mathop{\mathrm{Im}}}[z_T(z_j)]|$. Combining , with the Gronwall’s inequality, we get that the process $\{(g_t^n(z_t(z_j))_{1{\leqslant}j{\leqslant}m}\}_{1{\leqslant}t{\leqslant}T}$ is stochastically bounded . On the event $\{\sup_{1{\leqslant}j{\leqslant}m}\sup_{0{\leqslant}t{\leqslant}T}|g^n_t(z_t(z_j))|{\leqslant}M\}$, for any $0{\leqslant}t{\leqslant}T$ and $t{\leqslant}t'{\leqslant}T\vee t+\delta$, we have $$\begin{aligned} |g^n_{t'}(z_{t'}(z_j))-g^n_t(z_t(z_j))|{\leqslant}C(M+1)\delta+n(M_{t'}^n(z_j)-M_t^n(z_j))+\operatorname{O}\left(\frac{t}{n}\right).\end{aligned}$$ The claim follows from the tightness of $\{(nM_t^n(z_j))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$. *Step two: weak convergence of the martingale term.* We define a sequence of stopping times $\tau_0^n, \tau^n_1, \tau^n_2, \tau^n_3, \cdots$, where $\tau_0^n=0$ and for $l{\geqslant}1$, $\tau^n_l$ is the time of the $l$-th jump of the Poisson process $N_t^n$. The following estimate follows from the tail estimate of the exponential random variables. \[c:waittime\] Fix $\theta>0$ and time $T>0$. For any $\varepsilon>0$, there exists a $M>0$ such that $$\begin{aligned} \label{e:waittime} \sup_{n{\geqslant}1}{\mathbb{P}}\left(\sup_{0<\tau^n_j{\leqslant}t} |\tau^n_j-\tau^n_{j-1}|{\geqslant}\frac{M\ln n}{n^2}\right){\leqslant}\varepsilon\end{aligned}$$ Since the waiting time of $N_t^n$ is an exponential random variable of rate $\theta n^2$, for any $j{\geqslant}1$, we have $$\begin{aligned} \label{e:waittime2} {\mathbb{P}}\left(|\tau^n_j-\tau^n_{j-1}|{\geqslant}\frac{M\ln n}{n^2}\right)=\exp\left\{-\theta n^2\frac{M\ln n}{n^2}\right\}=n^{-\theta M}.\end{aligned}$$ The claim follows from and a union bound. \[c:Mconverge\] We assume the assumptions of Proposition \[p:CLT\]. Then as $n$ goes to infinity, the complex martingales $\{(nM_t^n(z_j))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ converge weakly in $D([0,T], {{\mathbb C}}^m)$ towards a centered complex Gaussian process $\{({{\mathcal W}}_j(t))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$, with quadratic variation given by and . We notice that $\overline{M_t^n(z_j)}=M_t^n(\bar{z}_j)$. Claim \[c:Mconverge\] follows from [@MR838085 Chapter 7, Theorem 1.4] and the weak convergence of the quadratic variations, $$\begin{aligned} \label{e:var}n^2[ M^n(z_j), M^n(z_j) ]_t &\Rightarrow -\frac{1}{6\theta }\int_0^t ({\partial}_s{\partial}^2_z m_s)(z_s(z_j)){{\rm d}}s, \\ \label{e:cov}n^2[ M^n(z_j), M^n(z_k) ]_t &\Rightarrow-\frac{1}{\theta}\int_0^t\frac{({\partial}_s m_s)(z_s(z_j))+({\partial}_sm_s)(z_s(z_k))}{(z_s(z_j)-z_s(z_k))^2}-\frac{2(e^{-m_0(z_j)}-e^{-m_0(z_k)})}{(z_s(z_j)-z_s(z_k))^3}{{\rm d}}s.\end{aligned}$$ Thanks to , we know that the processes $\{(n^2[M^n_t(z_j), M^n_t(z_k)]_t)_{1{\leqslant}j,k{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ are tight. For and , it remains to prove the weak convergence of any fixed time. By definition, the quadratic variation $n^2[M^n_t(z_j), M^n_t(z_k)]_t$ is given by $$\begin{aligned} \begin{split}\label{e:quadvar2} n^2[ M^n(z_j), M^n(z_k) ]_t &= \sum_{0< s{\leqslant}t\atop \Delta x_i(s)>0}\left(\frac{1}{x_i(s)/\theta n-z_s(z_j)}-\frac{1}{x_i(s-)/\theta n-z_s(z_j)}\right)\\ &\phantom{{}= \sum_{0< s{\leqslant}t\atop \Delta x_i(s)>0}}\left(\frac{1}{x_i(s)/\theta n-z_s(z_k)}-\frac{1}{x_i(s-)/\theta n-z_s(z_k)}\right)\\ &=\sum_{0< s{\leqslant}t\atop \Delta x_i(s)>0}\frac{1}{(\theta n)^2}\frac{1}{(x_i(s)/\theta n-z_s(z_j))^2(x_i(s)/\theta n-z_s(z_k))^2}+\operatorname{O}\left(\frac{N^n_t}{ n^3}\right), \end{split}\end{aligned}$$ where the implicit constant depends on $\theta$ and $\min_{1{\leqslant}j{\leqslant}m}|{\mathop{\mathrm{Im}}}[z_T(z_j)]|$. We can further rewrite as a sum of differences. For , we have $$\begin{aligned} \begin{split}\label{e:quadvarexp1} &\phantom{{}={}}n^2[ M^n(z_j), M^n(z_j) ]_t =\sum_{0< s{\leqslant}t\atop \Delta x_i(s)>0}\frac{1}{(\theta n)^2}\frac{1}{(x_i(s)/\theta n-z_s(z_j))^4}+\operatorname{O}\left(\frac{N^n_t}{ n^3}\right)\\ &=\sum_{0< s{\leqslant}t\atop \Delta x_i(s)>0}-\frac{1}{3\theta n}\left(\frac{1}{(x_i(s)/\theta n-z_s(z_j))^3}-\frac{1}{(x_i(s-)/\theta n-z_s(z_j))^3}\right)+\operatorname{O}\left(\frac{N^n_t}{ n^3}\right)\\ &=-\frac{1}{6\theta }\sum_{0< s{\leqslant}t\atop \Delta x_i(s)>0}\left({\partial}_z^2 m_{s}^n(z_s(z_j)) -{\partial}_z^2 m_{s-}^n(z_s(z_j))\right)+\operatorname{O}\left(\frac{N^n_t}{ n^3}\right) \end{split}\end{aligned}$$ We recall the stopping times defined above Claim \[c:waittime\], and rewrite as $$\begin{aligned} \begin{split}\label{e:quadvarexp2} &\phantom{{}={}}n^2[ M^n(z_j), M^n(z_j) ]_t =-\frac{1}{6\theta }\sum_{0<l{\leqslant}N_t^n}\left({\partial}_z^2 m_{\tau^n_l}^n(z_{\tau^n_l}(z_j)) -{\partial}_z^2 m_{\tau^n_{l-1}}^n(z_{\tau^n_l}(z_j))\right)+\operatorname{O}\left(\frac{N^n_t}{ n^3}\right)\\ &=-\frac{1}{6\theta }\sum_{0<l{\leqslant}N_t^n}\left({\partial}_z^2 m_{\tau^n_l}^n(z_{\tau^n_l}(z_j)) -{\partial}_z^2 m_{\tau^n_{l-1}}^n(z_{\tau^n_{l-1}}(z_j))-{\partial}^3_zm_{\tau^n_{l-1}}^n(z_{\tau^n_{l-1}}(z_j))(z_{\tau^n_l}(z_j)-z_{\tau^n_{l-1}}(z_j))\right)\\ &+\operatorname{O}\left(N_t^n\left(\frac{1}{ n^3}+\sup_{0<l{\leqslant}N_t^n} |z_{\tau^n_l}-z_{\tau^n_{l-1}}|^2\right)\right)\\ &=-\frac{1}{6\theta}\left({\partial}_z^2 m_{t}^n(z_{t}(z_j))-{\partial}_z^2 m_{0}^n(z_{0}(z_j))-\int_0^t{\partial}_z^3m_s^n(z_s(z_j)){{\rm d}}z_s(z_j)\right)\\ &+\operatorname{O}\left(N_t^n\left(\frac{1}{ n^3}+\sup_{0<l{\leqslant}N_t^n} |z_{\tau^n_l}-z_{\tau^n_{l-1}}|^2\right)+|z_t(z_j)-z_{\tau^n_{N_t^n}}(z_j)|\right)\\ &\Rightarrow -\frac{1}{6\theta }\int_0^t ({\partial}_s{\partial}^2_z m_s)(z_s(z_j)){{\rm d}}s, \end{split}\end{aligned}$$ where in the last line we used Claim and that $z_t$ is Lipschitz with respect to $t$. This finishes the proof of . For , we have $$\begin{aligned} &\phantom{{}={}}n^2[ M^n(z_j), M^n(z_k) ]_t =\sum_{0< s{\leqslant}t\atop \Delta x_i(s)>0}\frac{1}{(\theta n)^2}\frac{1}{(x_i(s)/\theta n-z_s(z_j))^2(x_i(s)/\theta n -z_s(z_k))^2}+\operatorname{O}\left(\frac{N^n_t}{ n^3}\right)\\ &=\frac{1}{(\theta n)^2}\sum_{0< s{\leqslant}t\atop \Delta x_i(s)>0}\left(\frac{1}{(z_s(z_j)-z_s(z_k))^2}\left(\frac{1}{(x_i(s)/\theta n-z_s(z_j))^2}+\frac{1}{(x_i(s)/\theta n-z_s(z_k))^2}\right)\right.\\ &\left.-\frac{2}{(z_s(z_j)-z_s(z_k))^3}\left(\frac{1}{(x_i(s)/\theta n-z_s(z_j))}-\frac{1}{(x_i(s)/\theta n-z_s(z_k))}\right)\right)+\operatorname{O}\left(\frac{N^n_t}{ n^3}\right)\\ &=-\frac{1}{\theta}\sum_{0<s{\leqslant}t\atop \Delta x_i(s)>0} \frac{(m^n_s(z_s(z_j))-m^n_{s-}(z_s(z_j)))+(m^n_s(z_s(z_k))-m^n_{s-}(z_s(z_k)))}{(z_s(z_j)-z_s(z_k))^2}\\ &- \frac{2}{\theta}\sum_{0<s{\leqslant}t\atop \Delta x_i(s)>0}\frac{(h^n_s(z_s(z_j))-h^n_{s-}(z_s(z_j)))-(h^n_s(z_s(z_k))-h^n_{s-}(z_s(z_k)))}{ (z_s(z_j)-z_s(z_k))^3}+\operatorname{O}\left(\frac{N^n_t}{ n^3}\right),\end{aligned}$$ where $h_t^n(z)$ is the logarithmic potential of the empirical measure $\mu_t^n$, $$\begin{aligned} h_t^n(z)=\int \ln(x-z){{\rm d}}\mu_t^n(x)=\frac{1}{n}\sum_{i=1}^n \ln (x_i(t)-z),\quad z\in {{\mathbb C}}\setminus{{\mathbb R}}.\end{aligned}$$ Thanks to Theorem , we have $$\begin{aligned} h_t^n(z)\Rightarrow h_t(z)=\int \ln (x-z){{\rm d}}\mu_t,\quad z\in {{\mathbb C}}\setminus{{\mathbb R}},\end{aligned}$$ where the logarithmic potential $h_t(z)$ is defined in . By the same argument as in , we get $$\begin{aligned} n^2\langle M^n(z_j), M^n(z_k) \rangle_t &\Rightarrow -\frac{1}{\theta}\int_0^t \frac{({\partial}_s m_s)(z_s(z_j))+({\partial}_sm_s)(z_s(z_k))}{(z_s(z_j)-z_s(z_k))^2}+\frac{2(({\partial}_s h_s)(z_s(z_j))-({\partial}_sh_s)(z_s(z_k)))}{(z_s(z_j)-z_s(z_k))^3}{{\rm d}}s\\ &=-\frac{1}{\theta}\int_0^t\frac{({\partial}_s m_s)(z_s(z_j))+({\partial}_sm_s)(z_s(z_k))}{(z_s(z_j)-z_s(z_k))^2}-\frac{2(e^{-m_0(z_j)}-e^{-m_0(z_k)})}{(z_s(z_j)-z_s(z_k))^3}{{\rm d}}s.\end{aligned}$$ This finishes the proof of . *Step three: subsequential limit.* In the first step, we have proven that as $n$ goes to infinity, the random processes $\{(g^n_t(z_t(z_j)))_{1{\leqslant}j{\leqslant}m})\}_{0{\leqslant}t{\leqslant}T}$ are tight. Without loss of generality, by passing to a subsequence, we assume that they weakly converge towards to a random process $\{({{{\mathcal}G}}_j(t))_{1{\leqslant}j{\leqslant}m})\}_{0{\leqslant}t{\leqslant}T}$. We check that the limit process satisfies the stochastic differential equation . The random process $\{(g^n_t(z_t(z_j)))_{1{\leqslant}j{\leqslant}m})\}_{0{\leqslant}t{\leqslant}T}$ satisfies the stochastic differential equation . For the first term on the righthand side of , by our assumption $(g_0^n(z_j))_{1{\leqslant}j{\leqslant}m}\Rightarrow (g_0(z_j))_{1{\leqslant}j{\leqslant}m}$. For the second term, by Theorem \[t:LLN\], we have $$\begin{aligned} &\phantom{{}={}}\int_0^t {\partial}_z m^n_s(z_s(z_j))e^{-m_0(z_j)} n\left(1-e^{-(m^n_s(z_s(z_j))-m_0(z_j))}\right){{\rm d}}s\\ &=\int_0^t {\partial}_z m^n_s(z_s(z_j))e^{-m_0(z_j)} g^n_t(z_s(z_j)){{\rm d}}s +\operatorname{O}\left(\int_0^tn|m_s^n(z_s(z_j))-m_0(z_j)|^2{{\rm d}}s\right)\\ &\Rightarrow \int_0^t {\partial}_z m_s(z_s(z_j))e^{-m_0(z_j)} {{{\mathcal}G}}_j(s){{\rm d}}s = \int_0^t \frac{{\partial}_zm_0(z_j)e^{-m_0(z_j)}}{1-s{\partial}_zm_0(z_j)e^{-m_0(z_j)}} {{{\mathcal}G}}_j(s){{\rm d}}s\end{aligned}$$ where the last term vanishes, because the processes $\{(g^n_t(z_t(z_j)))_{1{\leqslant}j{\leqslant}m})\}_{0{\leqslant}t{\leqslant}T}$ are stochastically bounded, i.e. . For the third term, by Theorem \[t:LLN\] and Proposition \[p:estimatemt\], we have $$\begin{aligned} \begin{split} \int_{0}^t\left(({\partial}_z m^n_s(z_s(z_j)))^2-{\partial}_z^2 m^n_s(z_s(z_j))\right)e^{-m_s^n(z_s(z_j))}{{\rm d}}s&\Rightarrow \int_{0}^t\left(({\partial}_z m_s(z_s(z_j)))^2-{\partial}_z^2 m_s(z_s(z_j))\right)e^{-m_0(z_j)}{{\rm d}}s\\ &=\int_0^t\frac{(({\partial}_z m_0(z_j))^2-{\partial}_z^2 m_0(z_j))e^{-m_0(z_j)}}{(1-s{\partial}_z m_0(z_j)e^{-m_0(z_j)})^3}{{\rm d}}s. \end{split}\end{aligned}$$ For the fourth term, in Step two we have proven that $\{(nM_t^n(z_j))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ converges weakly towards a centered complex Gaussian process $\{({{\mathcal W}}_j(t))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$, which is characterized by and . This finishes the proof of Theorem \[p:CLT\]. Extreme particles {#s:extremep} ----------------- In the following we first derive a large deviation estimate of the extreme particles of the $\beta$-nonintersecting random walks. Then Theorem \[t:CLT2\] follows from Theorem \[t:CLT\] by a contour integral. \[p:extremePbound\] Suppose the initial data ${{\bm{x}}}(0)$ satisfies Assumption \[a:ibound\]. For any time $t>0$, there exists a constant ${{\frak c}}$ depending on ${{\frak b}}$ and $t$, such that $$\begin{aligned} \label{e:extremePbound} {{\frak c}}n{\geqslant}x_1(t){\geqslant}x_2(t){\geqslant}\cdots {\geqslant}x_n(t), \end{aligned}$$ with probability at least $1-\exp(-cn)$. We notice that the $\beta$-nonintersecting Poisson random walks are shift invariant. Suppose that the $\beta$-nonintersecting Poisson random walk ${{\bm{y}}}(t)$ starts from $(a+(n-1)\theta, a+(n-2)\theta, \cdots, a)$, where $a= a(n)\in {\mathbb{Z}}_{{\geqslant}0}$. Then it follows from Theorem \[t:density\] that for any fixed $t>0$, the law of ${{\bm{y}}}(t)$ is given by $$\begin{aligned} \label{e:defPt} {\mathbb{P}}_{t}(y_1,y_2,\cdots, y_n)=\frac{1}{Z_n}\prod_{1{\leqslant}i <j{\leqslant}n}\frac{\Gamma(y_i-y_j+1)\Gamma(y_i-y_j+\theta)}{\Gamma(y_i-y_j)\Gamma(y_i-y_j+1-\theta)}\prod_{i=1}^n\frac{(\theta t n)^{y_i-a}}{\Gamma(y_i-a+1)},\end{aligned}$$ where the partition function $Z_n$ is given by $$\begin{aligned} Z_n=e^{\theta t n^2}(\theta t n)^{\theta(n-1)n/2}\prod_{i=1}^n\frac{\Gamma(i\theta)}{\Gamma(\theta)}.\end{aligned}$$ The measure ${\mathbb{P}}_t(y_1,y_2,\cdots, y_n)$ is a discrete $\beta$ ensemble studies in [@MR3668648]. The next proposition follows from [@MR3668648 Theorem 7.1]. \[p:0initialbound\] Take $a=\lceil {{\frak b}}n\rceil$ and $ t>0$. There exits a constant ${{\frak c}}$ depending ${{\frak b}}$ and $t$, such that the measure ${\mathbb{P}}_ t$ as in satisfies $$\begin{aligned} {\mathbb{P}}_{ t}\left(y_1{\leqslant}{{\frak c}}n\right){\geqslant}1-\exp(-cn). \end{aligned}$$ Let ${{\bm{x}}}(t)$ be a $\beta$-nonintersecting Poisson random walk with initial data ${{\bm{x}}}(0)\in {\mathbb{W}}_\theta^n$ satisfying , and ${{\bm{y}}}(t)$ another independent $\beta$-nonintersecting Poisson random walk with initial data ${{\bm{y}}}(0)=(\lceil {{\frak b}}n\rceil+(n-1)\theta, \lceil{{\frak b}}n\rceil+(n-2)\theta, \cdots, \lceil {{\frak b}}n \rceil)$. Let ${{\frak c}}$ be as in Proposition \[p:0initialbound\], we prove by constructing a coupling of ${{\bm{x}}}(t)$ and ${{\bm{y}}}(t)$, that $$\begin{aligned} \label{e:PBound} {\mathbb{P}}(x_1( t){\leqslant}{{\frak c}}n){\geqslant}{\mathbb{P}}(y_1( t){\leqslant}{{\frak c}}n).\end{aligned}$$ Then the claim follows from combining Proposition \[p:0initialbound\] and . We define the coupling $(\hat{{\bm{x}}}(t), \hat{{\bm{y}}}(t))$ as a Poisson random walk on ${\mathbb{W}}^n_\theta\times {\mathbb{W}}^n_\theta$, with initial data $(\hat{{\bm{x}}}(0), \hat{{\bm{y}}}(0))=({{\bm{x}}}(0), {{\bm{x}}}(0))$, and generator $$\begin{aligned} \begin{split} \hat{{{\mathcal}L}}^n_\theta f({{\bm{x}}}, {{\bm{y}}})&=\theta n\sum_{i=1}^n\left[\frac{V({{\bm{x}}}+\theta {\bm{e}}_i)}{V({{\bm{x}}})}-\frac{V({{\bm{y}}}+\theta {\bm{e}}_i)}{V({{\bm{y}}})}\right]_+\left(f({{\bm{x}}}+{\bm{e}}_i, {{\bm{y}}})-f({{\bm{x}}}, {{\bm{y}}})\right)\\ &+\theta n\sum_{i=1}^n\left[\frac{V({{\bm{y}}}+\theta {\bm{e}}_i)}{V({{\bm{y}}})}-\frac{V({{\bm{x}}}+\theta {\bm{e}}_i)}{V({{\bm{x}}})}\right]_+\left(f({{\bm{x}}}, {{\bm{y}}}+{\bm{e}}_i)-f({{\bm{x}}}, {{\bm{y}}})\right)\\ &+\theta n\sum_{i=1}^n\min\left\{\frac{V({{\bm{x}}}+\theta {\bm{e}}_i)}{V({{\bm{x}}})},\frac{V({{\bm{y}}}+\theta {\bm{e}}_i)}{V({{\bm{y}}})}\right\}\left(f({{\bm{x}}}+{\bm{e}}_i, {{\bm{y}}}+{\bm{e}}_i)-f({{\bm{x}}}, {{\bm{y}}})\right). \end{split}\end{aligned}$$ where $[x]_+=\max\{x,0\}$. The marginal distributions of $\hat{{\bm{x}}}(t)$ and $\hat{{\bm{y}}}(t)$ coincide with those of ${{\bm{x}}}(t)$ and ${{\bm{y}}}(t)$ respectively, $$\begin{aligned} \{\hat{{\bm{x}}}(s)\}_{0{\leqslant}s{\leqslant}t}\overset{d}{=}\{{{\bm{x}}}(s)\}_{0{\leqslant}s{\leqslant}t},\quad \{\hat{{\bm{y}}}(s)\}_{0{\leqslant}s{\leqslant}t}\overset{d}{=}\{{{\bm{y}}}(s)\}_{0{\leqslant}s{\leqslant}t}.\end{aligned}$$ For the initial data, we have $$\begin{aligned} \hat x_i(0){\leqslant}{{\frak b}}n{\leqslant}\hat y_i(0), \quad 1{\leqslant}i{\leqslant}n.\end{aligned}$$ In the following we prove that the coupling process $(\hat {{\bm{x}}}(t), \hat{{\bm{y}}}(t))$ satisfies $$\begin{aligned} \label{e:comparison1} {\mathbb{P}}\left(\text{for all $t{\geqslant}0$ and $1{\leqslant}i{\leqslant}n$, }\hat x_i(t){\leqslant}\hat y_i(t)\right)=1.\end{aligned}$$ We define a sequence of stopping times, $\tau^n_1, \tau^n_2, \tau^n_3, \cdots$, where $\tau^n_k$ is the time of the $k$-th jump of the coupling process $(\hat {{\bm{x}}}(t), \hat {{\bm{y}}}(t))$. We prove by induction that $$\begin{aligned} \label{e:comparison2} {\mathbb{P}}\left(\text{for all $0{\leqslant}t{\leqslant}\tau^n_k$ and $1{\leqslant}i{\leqslant}n$, }\hat x_i(t){\leqslant}\hat y_i(t)\right)=1.\end{aligned}$$ Then follows by noticing that $\lim_{k\rightarrow\infty}\tau^n_k=\infty$. We assume that holds for $k$, we prove it for $k+1$. If $\hat x_i(\tau^n_k)<\hat y_i(\tau^n_k)$, then with probability one, $\hat x_i(\tau^n_{k+1}){\leqslant}\hat y_i(\tau^n_{k+1})$. If $\hat x_i(\tau^n_k)=\hat y_i(\tau^n_k)$, by our assumptions, $\hat{{\bm{x}}}(\tau^n_k), \hat {{\bm{y}}}(\tau^n_k)\in {\mathbb{W}}^n_\theta$ and $\hat x_j(\tau^n_k){\leqslant}\hat y_j(\tau^n_k)$ for all $1{\leqslant}j{\leqslant}n$, we have $$\begin{aligned} 0{\leqslant}\frac{\hat x_i(\tau^n_k)-\hat x_j(\tau^n_k)+\theta}{\hat x_i(\tau^n_k)-\hat x_j(\tau^n_k)}{\leqslant}\frac{\hat y_i(\tau^n_k)-\hat y_j(\tau^n_k)+\theta}{\hat y_i(\tau^n_k)-\hat y_j(\tau^n_k)},\quad j\neq i.\end{aligned}$$ Thus the jump rate from $(\hat {{\bm{x}}}(\tau^n_k), \hat{{\bm{y}}}(\tau^n_k))$ to $(\hat {{\bm{x}}}(\tau^n_k)+{\bm{e}}_i, \hat{{\bm{y}}}(\tau^n_k))$, $$\begin{aligned} \begin{split} &\phantom{{}={}}\theta n\left[\frac{V(\hat {{\bm{x}}}(\tau^n_k)+\theta {\bm{e}}_i)}{V(\hat {{\bm{x}}}(\tau^n_k))}-\frac{V(\hat{{\bm{y}}}(\tau^n_k)+\theta {\bm{e}}_i)}{V(\hat {{\bm{y}}}(\tau^n_k))}\right]_+\\ &=\theta n\left[\prod_{j:j\neq i}\frac{\hat x_i(\tau^n_k)-\hat x_j(\tau^n_k)+\theta}{\hat x_i(\tau^n_k)-\hat x_j(\tau^n_k)}-\prod_{j:j\neq i}\frac{\hat y_i(\tau^n_k)-\hat y_j(\tau^n_k)+\theta}{\hat y_i(\tau^n_k)-\hat y_j(\tau^n_k)}\right]_+= 0. \end{split}\end{aligned}$$ vanishes. Therefore with probability one, $\hat x_i(\tau^n_{k+1}){\leqslant}\hat y_i(\tau^n_{k+1})$. This finishes the proof of and . It follows from , $$\begin{aligned} \label{e:PBound2} {\mathbb{P}}(\hat x_1( t ){\leqslant}{{\frak c}}n){\geqslant}{\mathbb{P}}(\hat y_1( t ){\leqslant}{{\frak c}}n).\end{aligned}$$ Since the marginal distributions of $\hat{{\bm{x}}}(t)$ and $\hat{{\bm{y}}}(t)$ coincide with those of ${{\bm{x}}}(t)$ and ${{\bm{y}}}(t)$ respectively, follows from combining Proposition \[p:0initialbound\] and . This finishes the proof of Propostion \[p:extremePbound\]. We take a contour ${{{\mathcal}C}}$ which encloses a neighborhood of $[0,{{\frak c}}/\theta]$. Then with exponentially high probability we have $$\begin{aligned} n\int f_j(x){{\rm d}}(\mu_t^n(x)-\mu_t(x))=\frac{1}{2\pi{\mathrm{i}}}\oint_{{{\mathcal}C}}g^n_t(w)f_j(w){{\rm d}}w, \quad 1{\leqslant}j{\leqslant}m.\end{aligned}$$ By Proposition \[e:defOmega\], $z_t(z)$ is a homeomorphism from the closure of $\Omega_t\cap {{\mathbb C}}_+$ to ${{\mathbb C}}_+\cup {{\mathbb R}}$, and from the closure of $\Omega_t\cap {{\mathbb C}}_-$ to ${{\mathbb C}}_-\cup {{\mathbb R}}$. By a change of variable, we have $$\begin{aligned} \frac{1}{2\pi{\mathrm{i}}}\oint_{{{\mathcal}C}}g^n_t(w)f_j(w){{\rm d}}w=\frac{1}{2\pi{\mathrm{i}}}\oint_{z_t^{-1}({{{\mathcal}C}})} g^n_t(z_t(z))f_j(z_t(z)){{\rm d}}z_t(z), \quad 1{\leqslant}j{\leqslant}m.\end{aligned}$$ By the continuous mapping theorem of weak convergence, it follows from Theorem \[t:CLT\] $$\begin{aligned} &\left\{\left(\frac{1}{2\pi{\mathrm{i}}}\oint_{z_t^{-1}({{{\mathcal}C}})} g^n_t(z_t(z))f_j(z_t(z)){{\rm d}}z_t(z)\right)_{1{\leqslant}j{\leqslant}m}\right\}_{0{\leqslant}t{\leqslant}T}\Rightarrow \{({{{\mathcal}F}}_j(t))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}\\ &{{{\mathcal}F}}_j(t){\mathrel{\mathop:}=}\frac{1}{2\pi{\mathrm{i}}}\oint_{z_t^{-1}({{{\mathcal}C}})} g_t(z_t(z))f_j(z_t(z)){{\rm d}}z_t(z),\quad 1{\leqslant}j{\leqslant}m,\end{aligned}$$ and the means and the covariances of the Gaussian process $\{({{{\mathcal}F}}_j(t))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ are given by $$\begin{aligned} {\mathbb{E}}[{{{\mathcal}F}}_j(t)]&=\frac{1}{2\pi{\mathrm{i}}}\oint_{z_t^{-1}({{{\mathcal}C}})}\mu(t, z) f_j(z_t(z)){{\rm d}}z_t(z)=\frac{1}{2\pi{\mathrm{i}}}\oint_{{{{\mathcal}C}}}\mu(t, z_t^{-1}(w))f_j(w){{\rm d}}w\\ {{\rm{cov}}}[{{{\mathcal}F}}_j(s), {{{\mathcal}F}}_{k}(t)]&=-\frac{1}{4\pi^2}\oint_{z_s^{-1}({{{\mathcal}C}})}\oint_{z_t^{-1}({{{\mathcal}C}})}\sigma(s, z,t,z') f_j(z_s(z))f_k(z_t(z')){{\rm d}}z_s(z){{\rm d}}z_t(z')\\ &=-\frac{1}{4\pi^2}\oint_{{{{\mathcal}C}}}\oint_{{{{\mathcal}C}}}\sigma(s, z_s^{-1}(w),t,z_t^{-1}(w')) f_j(w)f_k(w'){{\rm d}}w{{\rm d}}w'.\end{aligned}$$ where $\mu(t,z)$ and $\sigma(s, z,t,z')$ are as defined in and . This finishes the proof of Theorem \[t:CLT2\].

--- abstract: 'We construct a class of anomaly-free supersymmetric $U(1)^\prime$ models that are characterized by family non-universal $U(1)^\prime$ charges motivated from $E_6$ embeddings. The family non-universality arises from an interchange of the standard roles of the two $SU(5)$ ${\bf 5}^*$ representations within the ${\bf 27}$ of $E_6$ for the third generation. We analyze $U(1)^\prime$ and electroweak symmetry breaking and present the particle mass spectrum. The models, which include additional Higgs multiplets and exotic quarks at the TeV scale, result in specific patterns of flavor-changing neutral currents in the $b\to s$ transitions that can accommodate the presently observed deviations in this sector from the SM predictions.' --- EFI-09-25\ MADPH-09-1546\ 1.5cm [**Phenomenological Implications of Supersymmetric Family Non-universal $U(1)^\prime$ Models**]{} 2.0cm [Lisa L. Everett$^{a}$, Jing Jiang$^{a}$, Paul G. Langacker$^{b}$ and Tao Liu$^{c}$ ]{} 1.0cm Introduction ============ Extensions of the Standard Model of particle physics (SM) with an additional anomaly-free gauged $U(1)^\prime$ symmetry broken at the TeV scale are arguably some of the most well-motivated candidates for new physics (for a review, see [@Langacker:2008yv]). Such symmetries are theoretically motivated, as they represent the simplest augmentations of the SM gauge sector and are ubiquitous within string and/or grand unified theories. While the phenomenology of such $Z^\prime$ gauge bosons depends on the details of the couplings of the $Z^\prime$ to the SM fermions, current limits from direct and indirect searches indicate typical lower bounds of order $800-900$ GeV on the $Z^\prime$ mass and an upper bound of $\sim 10^{-3}$ on the $Z-Z^\prime$ mixing angle [@Erler:2009jh]. For a reasonable range of couplings, the presence of such TeV scale $Z^\prime$ bosons should be easily discernable at present and forthcoming colliders such as the Tevatron and the Large Hadron Collider (LHC). Within the context of supersymmetric theories, a plethora of $U(1)^\prime$ models have been proposed, including scenarios motivated by grand unified theories (GUTs) such as $SO(10)$ and $E_6$ and scenarios motivated from string compactifications of heterotic and/or Type II theories (see [@Langacker:2008yv] for a review). Recent models also include scenarios in which the $U(1)^\prime$ mediates supersymmetry breaking [@Langacker:2007ac], plays a role in the generation of neutrino masses [@durmus] and/or spontaneous $R$-parity violation [@pavel], or provides a portal to a hidden/secluded sector (for reviews, see [@Langacker:2009im; @Goodsell:2009xc]). Though the details of the $U(1)^\prime$ charge assignments are model-dependent, generically the cancellation of $U(1)^\prime$ anomalies requires an enlargement of the matter content to include SM exotics and SM singlets with nontrivial $U(1)^\prime$ charges. In these theories, the SM singlets also typically play an important role in triggering the low-scale breaking of the $U(1)^\prime$ gauge symmetry. In most models of this type, the $U(1)^\prime$ charges of the quarks and leptons are family universal. Though this feature is desirable for the first and second generations due to the strong constraints from flavor-changing neutral currents (FCNCs), there is still room for departures from family universality for the charges of the third generation. In fact, this often occurs in string constructions if the families result from different embeddings (see e.g., [@Cleaver:1998gc; @Blumenhagen:2005mu]). Indeed, though many of the results from the $B$ factories have indicated a strong degree of consistency with the Cabibbo-Kobayashi-Maskawa (CKM) predictions of the SM, there are hints of non-SM FCNC patterns within the $b\to s$ transitions for both $\Delta B=1$ and $\Delta B=2$ processes at the level of a few standard deviations [@Bona:2008jn]. Of the many options for new physics models that can explain this discrepancy, family non-universal $U(1)^\prime$ models are interesting in that they are theoretically well-motivated scenarios that lead to tree-level FCNC, as opposed to scenarios in which the new physics contributions are loop-suppressed [@Langacker:2000ju]. A recent model-independent analysis of $Z^\prime$-mediated FCNC in the $b\to s$ transitions showed that this general framework can accommodate the data [@Barger:2009eq; @Barger:2009qs]. (Related analyses include [@zprimefcnc; @zprimefcnc2]). However, it is optimal to consider the bounds on specific family non-universal $U(1)^\prime$ models in addition to the fully model-independent results. Our purpose in this paper is to construct and analyze supersymmetric anomaly-free family non-universal $U(1)'$ models (which we will denote as NUSSM models). Our strategy in building this class of NUSSM models is to exploit the well-known fact that in $E_6$ models, there are two options for embedding the down quarks and lepton doublets in the ${\bf 5^*}$ representation of $SU(5)$, which is related to the fact that the down-type Higgs and the lepton doublets have the same gauge quantum numbers. By choosing one embedding for the first and second generations and the alternative embedding for the third generation, we can obtain anomaly-free models in which the additional family non-universal $U(1)^\prime$ is given by a particular linear combination of the usual $U(1)_\psi$ and $U(1)_\chi$ of $E_6$-inspired models. This paper is structured as follows. We begin by outlining our basic procedure and presenting the resulting classes of anomaly-free family non-universal $U(1)^\prime$ models. In the following section, we analyze the gauge symmetry breaking and comment on general features of the mass spectrum. We next turn to an analysis of the implications of these models for FCNC in the $b\to s $ transitions, then provide our concluding remarks. $E_6$-Motivated Family Non-universal $U(1)^\prime$ Models (NUSSMs) ================================================================== \[table1\] In $U(1)^\prime$ models, the cancellation of gauge anomalies generally implies that additional fermions are present in the theory (see e.g., [@Langacker:2008yv]). To motivate the presence of these additional fermions and construct simple anomaly-free family non-universal models, our approach is to exploit the properties of $E_6$ embeddings of the SM fermions and Higgs fields in grand unified theories. Recall that in $E_6$ models, the SM particles are embedded in the fundamental ${\bf 27}$ representations. With respect to the two-step breaking scheme of $E_6$ to its $SO(10)$ and $SU(5)$ subgroups $$E_6\rightarrow SO(10)\times U(1)_\psi \rightarrow SU(5)\times U(1)_\chi \times U(1)_\psi,$$ the ${\bf 27}$ has the decomposition $${\bf 27}= {\bf 16}+{\bf 10}+{\bf 1}=({\bf 10}+{\bf 5^*}+{\bf 1})+({\bf 5}+{\bf 5^*})+{\bf 1},$$ with respect to the representations of $SO(10)$ and $SU(5)$, respectively. Hence, the ${\bf 27}$ has two ${\bf 5^*}$ multiplets; these representations are used to embed the down-type $SU(2)$-singlet quarks with the lepton doublets and exotic $SU(2)$-singlet quarks with down-type Higgs doublets. A standard choice for model-building is to have the down-type quarks and lepton doublets of all three SM families in the ${\bf 5^*}$ of the ${\bf 16}$, though models with the SM down-type quarks and lepton doublets in the other ${\bf 5^*}$ have also been considered in the literature [@alternative; @Athron:2009bs]. We will assign the down-type quark singlets and lepton doublets of the first and second generations to be in the ${\bf 5^*}$ of the ${\bf 16}$, and the associated particles of the third generation to be in the ${\bf 5^*}$ of the ${\bf 10}$, as shown in Table \[table1\]. The matter content of these theories thus includes the following fields: (i) the SM first and second families $\{ \Psi_{10}^i, \Psi_{5^*}^i, \Psi_1^i \}$, Higgs plus exotic fields $\{\sigma^i_5,\sigma^i_{5^*}\}$, and singlets $\sigma_0^i$ ($i=1,2$ is a family index), and (ii) the SM third family $\{ \Phi_{10}, \Sigma_{5^*}, \Sigma_0\}$, Higgs and exotics $\{ \Sigma_5,\Phi_{5^*} \}$, and singlet $\Phi_1$. The Higgs sector of the theory thus generically has multiple Higgs doublets and singlets beyond those of the MSSM.[^1] The additional family non-universal $U(1)^\prime$ in these NUSSM models is then a linear combination of the $U(1)_\chi$ and the $U(1)_\psi$ gauge groups: $$\begin{aligned} Q^{\prime} = \cos\theta \ Q_{\chi} + \sin\theta \ Q_{\psi} \label{202}\end{aligned}$$ (the assumption is that the orthogonal linear combination of $U(1)_\chi$ and $U(1)_\psi$ is either absent or broken at a high scale). The familiar $U(1)_\eta$ group, which has $\tan\theta=-\sqrt{5/3}$, is family universal and therefore is not useful for our purposes.[^2] Two viable options for the additional $U(1)^\prime$ group are: - ${\bf U(1)_I}$ ($\tan\theta=\sqrt{3/5}$). In this model (the [*inert*]{} model) the $U(1)'$ gauge boson couplings to the up-type quarks vanish [@Langacker:1984dc] . Hence, the production of the associated $Z^\prime$ boson is suppressed at hadron colliders. This is especially the case at the Tevatron, since in high-energy $p\bar{p}$ collisions the $Z'$ production via down quarks is suppressed by an order of magnitude relative to up quarks [@Leike:1998wr]. - ${\bf U(1)_S}$ ($\tan\theta=\sqrt{5/27}$). This symmetry is motivated by models with a secluded $U(1)'$ breaking sector and a large supersymmetry breaking $A$-term that have (1) an approximately flat potential that results in an appropriate $Z$–$Z'$ mass hierarchy [@Erler:2002pr]; (2) a strong first order electroweak phase transition and large spontaneous CP-violation, which can result in viable electroweak baryogenesis [@Kang:2004pp]. While we use the $E_6$ framework to motivate the matter content and $U(1)^\prime$ charges of these models, we do not work within a full grand unified theory. More precisely, we do not impose the $E_6$ Yukawa coupling relations. This allows for a TeV-scale $U(1)^\prime$ without the danger of rapid proton decay.[^3] The allowed superpotential terms of NUSSM models (assuming a conserved $R$-parity) are the couplings that are consistent with the SM and $U(1)^\prime$ gauge symmetries. An inspection of Table \[table1\] shows that in the language of the $SU(5)$ decomposition, the usual ${\bf 10\ 5^* 5^*}$ and ${\bf 10\ 10\ 5}$ terms that give rise to quark and lepton Yukawa couplings for all three families (including mixing terms between the third family and the other two families) are allowed by both $U(1)_\chi$ and $U(1)_\psi$. Similarly, these symmetries allow the generation of Yukawa interactions for the exotic quarks and for the Higgs doublets with the SM singlets ([*i.e.*]{}, mass terms for the exotic quarks and effective $\mu$ terms for the Higgs fields, which will be of importance for gauge symmetry breaking). For simplicity, we assume that only the neutral Higgs bosons from the third family ($H_{u,d}$ and $S$) and one of the first two families ($h_{u,d}$ and $s$) acquire vacuum expectation values (VEVs).[^4] In this limit, the Higgs bosons and Higgsinos in the other family have no mixing at leading order with the other particles. The mass eigenvalues of these particles are determined by the VEVs of ${h_{u,d}, s, H_{u,d}, S}$ as well as the Yukawa couplings and soft parameters which are not directly involved in the electroweak symmetry breaking. In this article, therefore, we will not discuss them in detail. The relevant superpotential terms are then given by ($I,J=1,2,3$ and $i, j = 1,2$ are family indices): $$\begin{aligned} W_Y&=& (f^{IJ}_{d1} h_d + f^{IJ}_{d2} H_d) Q_L^I d_R^J + (h^{IJ}_1 h_u+ h^{IJ}_2 H_u )Q_L^I u_R^J +\nonumber \\&& (f^{IJ}_{e1} h_d+ f^{IJ}_{e2} H_d) L^I e_R^J + (y^{IJ}_1 h_u+ y^{IJ}_2H_u) L^I \nu^J_R \label{205a} \\ W_{H} &=& \lambda_1 sh_dh_u + \lambda_2 sh_dH_u+\lambda_3 S H_dh_u +\lambda_4 S H_dH_u + \Delta W_H, \label{205b}\end{aligned}$$ in which the Yukawa couplings satisfy the relations $ f_{d1}^{i3}=f_{d1}^{33} \equiv 0$, $f_{d2}^{ij}=f_{d2}^{3i} \equiv 0$, $f_{e1}^{3i}=f_{e1}^{33} \equiv 0$, $f_{e2}^{ij}=f_{e2}^{i3} \equiv 0$, $y_1^{3i}=y_1^{i3} \equiv 0$, and $y_2^{3i}=y_2^{i3} \equiv 0$. In Eq. (\[205b\]), $\Delta W_H$ represents additional superpotential terms that are consistent with $U(1)_I$ or $U(1)_S$, but explicitly break the orthogonal linear combination of $U(1)_\chi$ and $U(1)_\psi$. These terms are needed to avoid the appearance of undesirable light axions in the low energy theory (see [@Langacker:2008dq] for a recent discussion). For the $U(1)_I$ model, $\Delta W_H$ is a bilinear term: $$\begin{aligned} \Delta W_H=\lambda_5 s S. \label{206a}\end{aligned}$$ Although the coupling $\lambda_5$ in $\Delta W_H$ is dimensionful, there is no associated $\mu$ problem in the traditional sense. This term is not necessary for $U(1)'$ or electroweak symmetry breaking, so its mass scale need not be connected with the electroweak scale. The Giudice-Masiero mechanism [@Giudice:1988yz] therefore can be implemented in both gravity- and gauge-mediated breaking frameworks to produce such a term, even though the $\lambda_5$ in the latter case is typically small. In the $U(1)_S$ model, $\Delta W_H$ consists of the trilinear term $$\begin{aligned} \Delta W_H=\lambda_5 ss S. \label{211}\end{aligned}$$ In what follows, we will focus on the $U(1)_I$ model as a concrete and minimal example, and defer the $U(1)_S$ model for future study. Particle Mass Spectrum and Gauge Symmetry Breaking {#spectrum} ================================================== The gauge group of the $U(1)_I$ NUSSM model is given by $SU(3)_c\times SU(2)_L\times U(1)_Y\times U(1)_I$, with gauge couplings $g_3$, $g_2$, $g_1$, and $g'$, respectively. The matter content, which was presented in Table \[table1\], includes three sets of Higgs fields (one pair of doublets and one singlet per family) and three sets of exotic down-type quarks in addition to the MSSM fields. As previously discussed, we assume that only two of the three Higgs doublet pairs ($H_{u,d}$ and $h_{u,d}$) acquire vacuum expectation values, and hence focus on the couplings of these Higgs fields only. Restricting to this set of terms, the superpotential for the model is given in Eq. (\[205a\]), Eq. (\[205b\]), and Eq. (\[206a\]). The tree-level Higgs potential (for the neutral components of the fields) is given by $V=V_F+V_D+V_H$, in which $$\begin{aligned} V_F &=& |\lambda_1sh_u^0+\lambda_2sH_u^0|^2 + |\lambda_1sh_d^0+\lambda_3SH_d^0|^2 +|\lambda_3Sh_u^0+\lambda_4SH_u^0|^2+ |\lambda_2sh_d^0+\lambda_4SH_d^0|^2\nonumber \\&& + |\lambda_1h_u^0h_d^0+\lambda_2H_u^0h_d^0+\lambda_5S|^2 + |\lambda_3h_u^0H_d^0+\lambda_4H_u^0H_d^0+\lambda_5s|^2~,~ \label{206} \end{aligned}$$ $$\begin{aligned} V_D &=& {{G^2}\over 8} \left(|h_u^0|^2 - |h_d^0|^2+|H_u^0|^2 - |H_d^0|^2\right)^2 +\frac{g'^2}{8}\left(- |h_d^0|^2 + |s|^2 + |H_d^0|^2 -|S|^2\right)^2 ~,~\, \label{207} \nonumber \\\end{aligned}$$ $$\begin{aligned} V_S &=& m_{h_d}^2 |h_d^0|^2 + m_{h_u}^2 |h_u^0|^2 + m_{H_d}^2 |H_d^0|^2 + m_{H_u}^2 |H_u^0|^2 + m_s^2 |s|^2 + m_S^2 |S|^2 \nonumber\\&& +(A_{\lambda_1} \lambda_1 s h_d^0 h_u^0 + A_{\lambda_2} \lambda_2 s h_d^0 H_u^0 + A_{\lambda_3} \lambda_3 S H_d^0 h_u^0 \nonumber \\&& + A_{\lambda_4} \lambda_4 S H_d^0 H_u^0 + B_{\lambda_5} \lambda_5 sS + {\rm H. C.} )~,~\ \label{208}\end{aligned}$$ where $G^2=g_1^{2} +g_2^2$. We also include the one-loop contribution to the potential: $$\Delta V = \frac{1}{64\pi^2} \mbox{STr} {\cal M}^4 (H_i) \left(\ln \frac{{\cal M}^2 (H_i)}{\Lambda_{\overline{\rm MS}}^2} - \frac{3}{2} \right), \label{211b}$$ in which $ {\cal M}^2 (H_i)$ denotes the field-dependent mass-squared matrices of the theory, and $\Lambda_{\overline{\rm MS}}$ is the $\overline{\rm MS}$ renormalization scale. We will only consider the dominant one-loop contributions that arise from the top quark sector: $$\begin{aligned} \Delta V &=& \frac{3}{32\pi^2} \left[ m_{\tilde{t}_1}^4 (H_i) \left( \ln \frac{m_{\tilde{t}_1}^2 (H_i)}{\Lambda_{\overline{\rm MS}}^2} - \frac{3}{2} \right) + m_{\tilde{t}_2}^4 (H_i) \left( \ln \frac{m_{\tilde{t}_2}^2 (H_i)}{\Lambda_{\overline{\rm MS}}^2} - \frac{3}{2} \right) \right. \nonumber \\ && \left. - 2 m_t^4 (H_i) \left( \ln \frac{m_t^2 (H_i)}{\Lambda_{\overline{\rm MS}}^2} - \frac{3}{2} \right) \right] . $$ The Higgs potential allows for a rich structure of CP-violating effects, including explicit CP violation (for complex couplings) and spontaneous CP violation. In this work, we will assume that all couplings are real and let the potential parameters satisfy some necessary constraints such that spontaneous CP violation can be avoided. In this case, the vacuum expectation values of the neutral Higgs components can be taken to be real: $$\langle h_d^0 \rangle =v_1,\;\; \langle h_u^0 \rangle =v_2, \;\; \langle H_d^0 \rangle =V_1,\;\; \langle H_u^0\rangle =V_2,\;\; \langle s \rangle =s_1,\;\; \langle S \rangle =s_2.$$ Before turning to a numerical analysis, we begin with a general discussion of the particle mass spectrum, starting with the gauge bosons. The $Z-Z^\prime$ mass-squared matrix is $$\begin{aligned} M_{Z-Z'} =\left(\matrix{M_{Z}^2 & M_{Z Z'}^2\cr M_{Z Z'}^2 & M_{Z'}^2\cr}\right),~ \,\end{aligned}$$ in which $$\begin{aligned} M_{Z}^2 &=& {G^2\over 2}(v_1^2 +v_2^2+V_1^2 + V_2^2)\equiv {G^2\over 2}v^2, \nonumber \\ M_{ Z'}^2 &=& 2 g'^2 (Q'^2_{h_d} v_1^2 + Q'^2_{h_u}v_2^2 + Q'^2_s s_1^2 + Q'^2_{H_d} V_1^2 + Q'^2_{H_u} V_2^2 + Q'^2_S s_2^2),~\nonumber \\ M_{Z Z'}^2 &=& g'G ( Q'_{h_d} v_1^2 - Q'_{h_u} v_2^2+Q'_{H_d}V_1^2 - Q'_{H_u} V_2^2), \end{aligned}$$ with $v^2=v_1^2 +v_2^2+V_1^2 + V_2^2=(174\, {\rm GeV})^2$. The mass-squared eigenvalues are $$\begin{aligned} M_{Z_1, Z_2}^2 = {1\over 2} \left(M_Z^2 + M_{Z'}^2 \mp \sqrt {(M_Z^2-M_{Z'}^2)^2 + 4 M_{Z Z'}^4 } \right),\end{aligned}$$ and the $Z-Z'$ mixing angle $\alpha_{Z-Z'}$ is $$\begin{aligned} \alpha_{Z-Z'} = {1\over 2} {\rm arctan} \left({{2 M_{ZZ'}^2} \over\displaystyle {M_{Z'}^2 - M_Z^2}}\right),\end{aligned}$$ which is bounded to be less than a few times $10^{-3}$ (see [@Erler:2009jh] for a recent discussion). This typically requires that the singlet vacuum expectation values $s_{1,2} \gg 1$ TeV, resulting in a TeV-scale $Z^\prime$ mass. The charged gauge boson mass is given as usual by $M_{W^{\pm}}= g_2 v/\sqrt{2}$. In the basis $\{ {\tilde B}^{\prime}, \tilde B, \tilde W_3^0, \tilde h_d^0, \tilde h_u^0, \tilde s, \tilde H_d^0, \tilde H_u^0, \tilde S\}$, the neutralino mass matrix is $$\begin{aligned} M_{\tilde \chi^{0}} =\left(\matrix{ M_{\tilde \chi^{0}}(3, 3) & M_{\tilde \chi^{0}}(3, 6) \cr M_{\tilde \chi^{0}}(3,6)^T& M_{\tilde \chi^{0}}(6, 6) \cr}\right),~ \,\end{aligned}$$ in which $$\begin{aligned} M_{\tilde \chi^{0}} (3, 3)= \left(\matrix{M_1^{\prime} &0&0\cr 0&M_1&0\cr 0&0&M_2\cr}\right) ,\ \nonumber \end{aligned}$$ $$\begin{aligned} M_{\tilde \chi^{0}} (3, 6)= \left(\matrix{\Gamma_{h_d}&\Gamma_{h_u}&\Gamma_{s}&\Gamma_{H_d}&\Gamma_{H_u}&\Gamma_{S}\cr -{1\over \sqrt 2} g_1 v_1 & {1\over \sqrt 2} g_1 v_2 &0& -{1\over \sqrt 2} g_1 V_1 & {1\over \sqrt 2} g_1 V_2 &0\cr {1\over \sqrt 2} g_2 v_1 & -{1\over \sqrt 2} g_2 v_2&0& {1\over \sqrt 2} g_2 V_1 & -{1\over \sqrt 2} g_2 V_2&0\cr}\right) ,\, $$ $$\begin{aligned} M_{\tilde \chi^{0}} (6, 6)= \left(\matrix{0& \lambda_1 s_1& \lambda_1v_2+ \lambda_2V_2 &0& \lambda_2 s_1 &0 \cr \lambda_1 s_1 &0& \lambda_1v_1 &\lambda_3s_2 &0&\lambda_3 V_1 \cr \lambda_1 v_2+ \lambda_2V_2 & \lambda_1 v_1 &0& & \lambda_2v_1 & \lambda_5 \cr 0&\lambda_3 s_2& 0&0& \lambda_4 s_2&\lambda_3 v_2+ \lambda_4 V_2 \cr \lambda_2 s_1 &0& \lambda_2v_1& \lambda_4 s_2 &0& \lambda_4 V_1 \cr 0&\lambda_3 V_1& \lambda_5 &\lambda_3 v_2+ \lambda_4 V_2& \lambda_4 V_1 &0 \cr}\right).\ \nonumber $$ In the above, $\Gamma_{\phi} \equiv \sqrt 2 g' Q_{\phi} \langle \phi^*\rangle$, and $M_1^{\prime}$, $M_1$, and $M_2$ are the gaugino mass parameters for $U(1)^{\prime}$, $U(1)_Y$, and $SU(2)_L$, respectively. The chargino mass matrix is $$\begin{aligned} M_{\tilde \chi^{\pm}} =\left(\matrix{M_2 & \frac{g_2}{\sqrt 2} v_2& \frac{g_2}{\sqrt 2} V_2 \cr \frac{g_2 }{\sqrt 2}v_1 & \lambda_1 s_1& \lambda_2 s_1\cr \frac{g_2 }{\sqrt 2}V_1& \lambda_3 s_2&\lambda_4 s_2\cr}\right). $$ Since $s_{1,2}\gg v_{1,2}, V_{1,2}$ because of the experimental bounds on $\alpha_{Z-Z'}$, the charginos and neutralinos are typically heavy unless the $\lambda$’s are small or the gaugino masses are light. In the latter situation, the lightest chargino and neutralino will be gaugino-like. The mass-squared matrices of the sfermions (denoted collectively as $\phi$) are $$\begin{aligned} M_{\phi}^2 =\left(\matrix{ (M_{\phi}^2)_{11}& (M_{\phi}^2)_{12}\cr (M_{\phi}^2)_{21}& (M_{\phi}^2)_{22} \cr}\right).\end{aligned}$$ With the definitions $$\begin{aligned} \Delta_{\phi} &\equiv& \frac{G^2}{2}(T_3^{\phi} - Q_{EM}^{\phi} \sin^2\theta_W)(v_1^2-v_2^2+V_1^2-V_2^2)~,~\, \\ \Delta'_{\phi} &\equiv& Q'_{\phi} g'^2 ( Q'_{h_d}v_1^2 + Q'_{h_u} v_2^2 + Q'_s s_1^2+ Q'_{H_d} V_1^2 + Q'_{H_u} V_2^2 + Q'_S s_2^2 )~,~\ \end{aligned}$$ the entries for example of the up-type squark mass-squared matrix are: $$\begin{aligned} (M_{\tilde u}^2)_{11}& =&m^2_{{\tilde Q}_L} + m^2_{u} + \Delta_{{\tilde u}_L} + \Delta'_{{\tilde u}_L} ~,~\, \nonumber \\ (M_{\tilde u}^2)_{12}& = & h_1(\lambda_1v_1s_1 + \lambda_3V_1 s_2 ) +h_2(\lambda_2 v_1 s_1 + \lambda_4 V_1 s_2 ) -(A_{h_1}h_1v_2 + A_{h_2} h_2 V_2) \nonumber \\ (M_{\tilde u}^2)_{21}& = & (M_{\tilde u}^2)_{12}\nonumber \\ (M_{\tilde u}^2)_{22} &=&m^2_{{\tilde u}_R} + m^2_{u} + \Delta_{{\tilde u}_R} + \Delta'_{{\tilde u}_R}.~\,\end{aligned}$$ Analogous expressions can be written for the down-type squarks, sleptons, and sneutrinos. The physical Higgs spectrum consists of 6 CP-even neutral Higgs bosons, 4 CP-odd neutral Higgs bosons, and 6 charged Higgs bosons (not including the second family). The tree-level charged Higgs boson mass-squared matrix is given in the Appendix. Next we turn to a numerical analysis of this sector of the model, taking into account the constraints on the $Z'$ gauge boson. We explore the viable regions of parameter space in which (i) $s_1,s_2 \gg v_{1,2}, V_{1,2} $, which is needed for a TeV scale $Z'$, and (ii) $V_2 > V_1 > v_{1,2}$, which is motivated by the observed hierarchies in the SM fermion mass spectrum. To obtain an acceptable minimum, typically we need the Higgs soft mass parameters to satisfy $m_s^2$ or $m_S^2 \ll m_{h_u}^2, m_{H_u}^2 < m_{h_d}^2, m_{H_d}^2$. We also set the $U(1)_I$ gauge coupling to $g'=\sqrt{\frac{5}{3}} g_1$ and enforce the following constraints on the Yukawa couplings:[^5] $$\begin{aligned} h_1^{33}=h_2^{33} , \qquad h_1^{3i} = h_1^{i3}= h_2^{3i} = h_2^{i3} = 0.\end{aligned}$$ which result in the condition $$\begin{aligned} h_1^{33}=h_2^{33} = \frac{165\, {\rm GeV}}{v_2+V_2},\end{aligned}$$ in which we have included the one-loop QCD corrections to the top quark mass. We consider one typical numerical example; the relevant input parameters and results are summarized in Tables \[table2\]–\[table4\]. The mass spectrum of the neutral Higgs bosons are calculated at one-loop level, and the mass spectra of the other particles are calculated at tree-level. As a check, we estimate the $Z'$ mass and the $Z-Z'$ mixing angle by using the Higgs VEVs given in Table \[table2\], as follows: $$\begin{aligned} M_{Z_2} \approx M_{Z'} \approx \sqrt{0.18(s_1^2 +s_2^2)} \sim 1.9\, {\rm TeV},\;\; \alpha_{Z-Z'} \approx \frac{M^2_{Z-Z'}}{M_{Z'}^2} \sim 0.0003, \end{aligned}$$ which is consistent with the detailed results. The lightest CP-even Higgs boson $H_1$ is a linear combination of the real parts of the four Higgs doublets, with a negligible singlet admixture; the orthogonal linear combinations of these four states are the $H_3$, $H_4$, and $H_6$ bosons. These heavier Higgs bosons fall into $SU(2)$ multiplets together with the three heaviest CP-odd states and the set of charged Higgs bosons, as follows: $(H_3, A_2, H_1^\pm)$, $(H_4, A_3, H_2^\pm)$, and $(H_6, A_4, H_3^\pm)$. The second lightest and second heaviest CP-even states are admixtures of the two singlet Higgs fields, as is the lightest CP-odd boson (which has a mass that controlled by the Higgs bilinear terms). The chargino and neutralino mass spectrum is highly model-dependent, as it is sensitive to the electroweak and hypercharge gaugino masses, which do not strongly impact the gauge symmetry breaking. Hence, the physics of the lightest superparticle (LSP) can vary greatly depending on the exact structure of the gaugino sector, though the gauge and Higgs sectors can remain almost the same in this case. In our numerical example, in which the gaugino masses are light and obey GUT relations, the LSP is a predominantly bino-like neutralino that can be an acceptable dark matter candidate in regions of the parameter space.[^6] Finally, we comment on the exotic colored particles $\{D^i, D^{ci}\}$ and $\{\Delta, \Delta^c\}$. The exotic scalars do not obtain VEVs. They and their superpartners influence the gauge symmetry breaking only at loop level. These exotic particles are chiral, so their tree-level masses can be produced only through Yukawa interactions. The superpotential terms that describe their interactions with the Higgs fields and the corresponding soft supersymmetry breaking terms are $$\begin{aligned} W_{E} &=& \tilde \lambda_1^{ij} s D^{ci} D^j + \tilde \lambda_2^i s D^{ci} \Delta+ \tilde \lambda_3^i S \Delta^c D^i + \tilde \lambda_4 S \Delta^c \Delta, \\ V_{E} &=& A_{\tilde \lambda_1^{ij}} \tilde \lambda_1^{ij} s \tilde{D}^{ci} \tilde{D}^j + A_{\tilde \lambda_2^i} \tilde \lambda_2^i s \tilde{D}^{ci} \tilde{\Delta}+ A_{\tilde \lambda_3^i} \tilde \lambda_3^i S \tilde{\Delta}^c \tilde{D}^i + A_{\tilde \lambda_4}\tilde \lambda_4 S \tilde{\Delta}^c \tilde{\Delta}.\end{aligned}$$ It is straightforward to determine the mass matrices for these states for given Yukawa couplings and $A$ parameter values. For ${\mathcal O} (\tilde \lambda_{1,2,3,4}) \sim 0.1$ (where their contributions to the effective neutral Higgs potential can be neglected) and not large $A_{\tilde \lambda}$ values, the exotic particles will typically obtain masses of the order of several hundred GeV. $\lambda_1$ $\lambda_2$ $\lambda_3$ $\lambda_4$ $\lambda_5$ $g'$ -------------------- -------------------- ------------------- -------------------- --------------------- --------------------- 0.10 0.30 0.10 $-0.45$ 449 0.60 $M_1'$ $M_1$ $M_2$ $M_3$ $m_{\tilde Q_3}^2$ $m_{\tilde T_R}^2$ 112 112 224 673 $1.21 \times 10^5$ $1.21 \times 10^5$ $m_{h_d}^2$ $m_{h_u}^2$ $m_s^2$ $m_{H_d}^2$ $m_{H_u}^2$ $m_S^2$ $9.06 \times 10^5$ $7.04 \times 10^5$ $1.21\times 10^6$ $9.06 \times 10^5$ $-1.01 \times 10^6$ $-1.21 \times 10^6$ $A_{\lambda_1}$ $A_{\lambda_2}$ $A_{\lambda_3}$ $A_{\lambda_4}$ $B_{\lambda_5}$ $A_{\tilde T}$ $-1350$ $ -1350$ $-449$ 1080 $-359$ 897 $v_1$ $v_2$ $s_1$ $V_1$ $V_2$ $s_2$ 54.8 83.7 2000 93.3 108 4010 : Parameter values and Higgs VEVs. The dimensional parameter values are given in GeV or GeV$^2$. The Higgs VEVs are given in GeV. \[table2\] $M_{Z_2}$ $\sin\theta_{Z-Z'}$ $m_{\tilde t_1}$ $m_{\tilde t_2}$ $m_{\tilde \chi_1^\pm}$ $m_{\tilde \chi_1^0}$ / ----------- ---------------------- ------------------ ------------------ ------------------------- ----------------------- --------------- 1900 $3.10\times 10^{-4}$ 275 609 219 114 / $m_{H_1}$ $m_{H_2}$ $m_{H_3}$ $m_{H_4}$ $m_{H_5}$ $m_{H_6}$ / 173 369 1100 1640 1970 2360 / $m_{A_1}$ $m_{A_2}$ $m_{A_3}$ $m_{A_4}$ $m_{H_1^\pm}$ $m_{H_2^\pm}$ $m_{H_3^\pm}$ 633 1080 1650 2340 1060 1630 2330 : The particle mass spectrum and $Z-Z'$ mixing angle of the NUSSM model (all masses are in GeV). \[table3\] $h_{dr}^0$ $h_{ur}^0$ $s_{r}$ $H_{dr}^0$ $H_{ur}^0$ $S_{r}$ ------- ------------ ------------ --------- ------------ ------------ --------- $H_1$ 0.31 0.48 -0.09 0.53 0.61 -0.07 $H_2$ 0.07 0.06 0.87 0.06 0.05 0.49 $H_3$ -0.28 0.86 -0.02 -0.34 -0.25 0.03 $H_4$ -0.88 -0.11 0.02 0.13 0.43 0.04 $H_5$ 0.03 -0.01 -0.49 0.04 -0.01 0.87 $H_6$ -0.20 0.06 0.01 0.76 -0.61 -0.03 $h_{di}^0$ $h_{ui}^0$ $s_{i}$ $H_{di}^0$ $H_{ui}^0$ $S_{i}$ $A_1$ -0.04 -0.02 0.89 -0.01 -0.01 0.45 $A_2$ 0.26 0.87 0.03 0.35 -0.24 0.02 $A_3$ 0.89 -0.10 0.04 -0.12 0.43 0.01 $A_4$ -0.20 -0.08 -0.01 0.76 0.62 0.02 $G_1$ 0.30 -0.33 0.20 0.59 -0.52 -0.39 $G_2$ -0.12 0.21 -0.40 -0.25 0.26 0.81 : The composition of the neutral Higgs mass eigenstates at the one-loop level. \[table4\] $h_d^-$ $h_u^{+*}$ $H_d^-$ $H_u^{+*}$ --------- --------- ------------ --------- ------------ $H_1^-$ 0.26 0.87 0.34 -0.23 $H_2^-$ 0.89 -0.10 -0.12 0.42 $H_3^-$ -0.20 -0.07 0.75 0.62 $G_1^-$ 0.31 -0.47 0.55 -0.62 : The composition of the charged Higgs mass eigenstates at tree level. \[table5\] $Z^\prime$-mediated FCNC Effects {#FCNC} ================================ In this section, we analyze the $Z'$-induced FCNC effects After a brief review of the formalism, we will show the results of our correlated analysis of the $\Delta B =1, 2$ processes via $b\to s$ transitions and discuss the resulting parameter space constraints. The processes of interest include $B_s - \bar B_s$ mixing and the time-dependent CP asymmetries of the penguin-dominated neutral $B_d \to (\phi, \eta', \pi, \rho, \omega, f_0)K_S$ decays. The FCNC effects in general NUSSM models include both $Z^\prime$-mediated FCNC processes and contributions to FCNC from the soft supersymmetry breaking parameters. In this work, we assume for simplicity that the soft terms do not result in large FCNC effects (this can be easily achieved; see e.g. [@Martin:1997ns]) and consider only the $Z^\prime$ contributions. We now briefly discuss the formalism for addressing such $Z^\prime$ effects in the NUSSM (for a model-independent discussion, see [@Langacker:2000ju; @Barger:2009eq; @Barger:2009qs]). For the SM fermions $\psi_{L,R}$ with $U(1)^\prime$ charges $\tilde \e_{\psi_{L,R}}$, the fermion mass matrices are diagonalized by the biunitary transformation $M_{\psi,{\rm diag}}=V_{\psi_R} M_{\psi} V_{\psi_L}^{\dg}$ (the CKM matrix is $ V_{\rm CKM} = V_{u_L} V_{d_L}^{\dg}$). The chiral $Z'$ couplings in the fermion mass eigenstate basis are $$\begin{aligned} B^{\psi_L}\equiv V_{\psi_L}\tilde \e^{\psi_L} V_{\psi_L}^{\dg}\;, \qquad B^{\psi_R} \equiv V_{\psi_R} \tilde \e^{\psi_R} V_{\psi_R}^{\dg}\;. \label{453}\end{aligned}$$ In our $U(1)_I$ model, the only SM fields with nontrivial $U(1)^\prime$ charges are the down-type quark singlets and the lepton doublets: $$\begin{aligned} \tilde \e^{d_R}=-\frac{1}{2}\begin{pmatrix} {1 &0&0 \cr 0 & 1 & 0 \cr 0 & 0& -1 } \end{pmatrix},\;\;\; \tilde \e^{L_L}&=&\frac{1}{2}\begin{pmatrix} {1 &0&0 \cr 0 & 1 & 0 \cr 0 & 0& -1 } \end{pmatrix}. \label{454}\end{aligned}$$ With the unitary matrices $V_{d_R,L_L}$ written as $$\begin{aligned} V_{d_R,L_L} &=&\begin{pmatrix} { W_{d_R,L_L} & X_{d_R,L_L} \cr Y_{d_R,L_L} & Z_{d_R,L_L} } \end{pmatrix}, \label{455}\end{aligned}$$ where $W_{d_R,L_L}$ is a $2\times 2$ submatrix, one obtains $$\begin{aligned} B^{d_R} &=&-\frac{1}{2}\begin{pmatrix} { W_{d_R}^\dagger W_{d_R} - Y_{d_R}^\dagger Y_{d_R} & W_{d_R}^\dagger X_{d_R} -Y_{d_R}^\dagger Z_{d_R} \cr X_{d_R}^\dagger W_{d_R} - Z_{d_R}^\dagger Y_{d_R} & X_{d_R}^\dagger X_{d_R} - Z_{d_R}^\dagger Z_{d_R} } \end{pmatrix},\nonumber \\ B^{L_L} &=&\frac{1}{2}\begin{pmatrix} { W_{L_L}^\dagger W_{L_L} - Y_{L_L}^\dagger Y_{L_L} & W_{L_L}^\dagger X_{L_L} -Y_{L_L}^\dagger Z_{L_L} \cr X_{L_L}^\dagger W_{L_L} - Z_{L_L}^\dagger Y_{L_L} & X_{L_L}^\dagger X_{L_L} - Z_{L_L}^\dagger Z_{L_L} } \end{pmatrix}. \label{456}\end{aligned}$$ To avoid the constraints on non-universality for the first two families from $K- \bar K$ mixing and $\mu-e$ conversion in muonic atoms, we assume small fermion mixing angles or small $X_{d_R,L_L}$, $Y_{d_R,L_L}$ elements. The $Z'$ couplings then take the form $$\begin{aligned} && B^{d_R}_{11}, B^{d_R}_{22} \approx -\frac{1}{2}, \ \ B^{d_R}_{33} \approx \frac{1}{2} , \ \ B^{d_R}_{13}, B^{d_R}_{23} \sim {\mathcal O} (X_{d_R},Y_{d_R}), \nonumber \\ && B^{L_L}_{11}, B^{L_L}_{22} \approx \frac{1}{2}, \ \ B^{L_L}_{33} \approx -\frac{1}{2} , \ \ B^{L_L}_{13}, B^{L_L}_{23} \sim {\mathcal O} (X_{L_L},Y_{L_L}).\label{457}\end{aligned}$$ Here $B^{d_R,L_L}_{13}$ and $B^{d_R,L_L}_{23}$ generically are complex. The $Z'$-induced corrections to the Wilson coefficients in the $U(1)_I$ model[^7] are given by (for the associated operators, see e.g. [@Barger:2009qs]): $$\begin{aligned} \Delta \tilde C^{B_s}_1& =& - (B_{bs}^R)^2, \;\; \Delta \tilde C_{3} = - \frac{4}{3 V_{tb} V_{ts}^*} B_{bs}^R B_{dd}^R, \;\; \Delta \tilde C_{9} = \frac{4}{3 V_{tb} V_{ts}^*} B_{bs}^R B_{d d}^R , \nonumber \\ \Delta \tilde C_{9V} &=& -\Delta \tilde C_{10A} = - \frac{2}{V_{tb} V_{ts}^*} B_{bs}^RB_{ll}^L. \label{458} \end{aligned}$$ To achieve sufficient precision, we need to have an accurate knowledge of the relevant Wilson coefficients at the $b$ quark mass scale $m_b=4.2$ GeV (for general discussions, see e.g. [@Buchalla:1995vs]). The parameter values used in our calculations are summarized in \[Parameters\].\ $\bullet$ [**$B_s-\bar B_s$ mixing.**]{} The new physics (NP) contributions to the off-diagonal mixing matrix element are parametrized as $$\begin{aligned} M_{12}^{B_s}=(M_{12}^{B_s})_{\rm SM} C_{B_s} e^{2 i \phi_{B_s}^{\rm NP}}, \label{402}\end{aligned}$$ where $C_{B_s}=1$ and $\phi_{B_s}^{\rm NP}=0$ in the SM limit. Although the data indicate that $C_{B_s}\simeq 1$, a recent analysis [@Bona:2008jn] suggests that $\phi_{B_s}^{\rm NP}$ deviates from zero at the $2-3 \sigma$ level (see Table \[table6\]); an earlier discussion was given in [@Lenz:2006hd]. The analysis of [@Bona:2008jn] includes all available results on $B_s$ mixing, including the tagged analyses of $B_s \to \psi \phi$ by CDF [@Aaltonen:2007he] and D$\emptyset$ [@:2008fj]. As discussed for example in [@Tarantino:2008pb], this discrepancy disfavors scenarios with minimal flavor violation (MFV), though no single measurement yet has a $3\sigma$ significance. In our $U(1)_I$ NUSSM model, $C_{B_s}$ and $\phi_{B_s}^{\rm NP}$ at the $m_b$ scale are given by $$\begin{aligned} C_{B_s} e^{2i\phi_{B_s}} &=& 1 - 3.59 \times 10^5 (\Delta C_1^{B_s} + \Delta \tilde C_1^{B_s}) + 2.04 \times 10^6 \Delta \tilde C_3^{B_s}. \label{403}\end{aligned}$$ The large coefficients of the correction terms in Eq. (\[403\]) are due to the fact that the NP is introduced at tree-level while the SM limit is a loop-level effect.\ Observable $1 \sigma$ C.L. $2 \sigma$ C.L. ------------------------------------- ----------------- ----------------- $\phi_{B_s}^{\rm NP} [^\circ]$ (S1) -20.3 $\pm$ 5.3 \[-30.5,-9.9\] $\phi_{B_s}^{\rm NP} [^\circ]$ (S2) -68.0 $\pm$ 4.8 \[-77.8,-58.2\] $C_{B_s}$ 1.00 $\pm$ 0.20 \[0.68,1.51\] : The fit results for the $B_s - \bar B_s$ mixing parameters [@Bona:2008jn]. The two $\phi_{B_s}^{\rm NP}$ solutions (“S1” and “S2”) result from measurement ambiguities; see [@Bona:2008jn] for details.[]{data-label="table6"} $\bullet$ [**$B_d \to (\psi,\pi, \phi, \eta', \rho, \omega, f^0)K_S$ decays.**]{} $f_{CP}$ $-\eta_{CP} {\mathcal S}_{f_{CP}}$ (1$\sigma$ C.L.) ${\mathcal C}_{f_{CP}}$(1$\sigma$ C.L.) ------------------- ----------------------------------------------------- ----------------------------------------- $\psi K_S$ $+0.672\pm0.024 $ $+0.005\pm0.019 $ $\phi K_S$ $+0.44^{+0.17}_{-0.18} $ $-0.23\pm0.15 $ $\eta^\prime K_S$ $+0.59\pm0.07 $ $-0.05\pm0.05$ $\pi K_S$ $+0.57\pm0.17$ $+0.01\pm0.10$ $\rho K_S$ $+0.63^{+0.17}_{-0.21}$ $-0.01\pm0.20 $ $\omega K_S$ $+0.45\pm0.24$ $-0.32\pm0.17 $ $f_0 K_S$ $+0.62^{+0.11}_{-0.13}$ $0.10\pm0.13$ : The world averages of the experimental results for the CP asymmetries in $B_d$ decays via $b\to\bar qq s$ transitions [@Barberio:2008fa].[]{data-label="table7"} The direct and the mixing-induced CP asymmetries in hadronic $B_d$ decays are parametrized as follows: $$\begin{aligned} {\mathcal C}_{f_{CP}} = \frac{1-|\lambda_{f_{CP}}|^2}{1+|\lambda_{f_{CP}}|^2} ~, \qquad {\mathcal S}_{f_{CP}} = \frac{2{\rm Im} \left[ \lambda_{f_{CP}} \right]}{1+|\lambda_{f_{CP}}|^2}. \label{404}\end{aligned}$$ in which $$\begin{aligned} \lambda_{f_{CP}} \equiv \eta_{f_{CP}} e^{-2i\phi_{B_d}} \frac{\bar{A}_{f_{CP}}}{A_{f_{CP}}}. \label{405}\end{aligned}$$ Here $\phi_{B_d}$ is the $B_d-\bar B_d$ mixing angle, $A_{f_{CP}}$ is the decay amplitude of $B_d \to f_{CP}$ ($\bar A_{f_{CP}}$ is its CP conjugate), and $\eta_{f_{CP}}=\pm1$ is the CP eigenvalue for the final state $f_{CP}$. In the SM, $\phi_{B_d} = \beta \equiv \arg\left[-(V_{cd}V_{cb}^*)/(V_{td}V_{tb}^*)\right]$, and a non-trivial weak phase enters $A_{f_{CP}} $ only at ${\mathcal O}(\lambda^2)$. This implies the following SM relation between the decays proceeding via $b\to s \bar qq(q=u,d,c,s)$ and the penguin-dominated modes such as $B_d \to (\pi,\phi, \eta', \pi, \rho, $ $\omega, f^0 ) K_S$: $$\begin{aligned} -\eta_{f_{CP}}{\mathcal S}_{f_{CP}} = \sin 2 \beta + {\mathcal O}(\lambda^2), \ \ \ \ {\mathcal C}_{f_{CP}} = 0 + {\mathcal O}(\lambda^2). \label{416}\end{aligned}$$ However, the experimental values of $\sin 2 \beta$ obtained from the penguin-dominated modes are below the SM prediction and the results from the charmed $B_d \to \psi K_S$ mode. The central values of the direct CP asymmetries of $B_d\to \phi K_S$ and $B_d \to \omega K_S$ are also small compared to the $B_d \to \psi K_S$ mode, as shown in Table \[table7\]. Since $B_d\to \psi K_S$ is a tree-level process in the SM, large values for $\Delta {\mathcal S}_{f_{CP}}=-\eta_{f_{CP}}{\mathcal S}_{f_{CP}}+ \eta_{\psi K_S}{\mathcal S}_{\psi K_S}$ and $\Delta {\mathcal C}_{f_{CP}}={\mathcal C}_{f_{CP}} - {\mathcal C}_{\psi K_S}$ may indicate the presence of NP in the $b\to s$ transitions. In NUSSM models, $Z'$-induced FCNC effects can provide dramatic changes to the results, since a new weak phase can enter $A_{f_{CP}}$ at tree level. Following [@Ali:1998eb], the $\lambda_{f_{CP}}$ parameters of $B_d \to (\psi, \phi, \eta', \pi, \rho, \omega, f^0)K_S$ at the $m_b$ scale are given by $$\begin{aligned} \lambda_{\psi K_S} &=&(-0.63+ 0.74 i) \label{406}\\ {\nonumber}&& [1 - (2.93 -2.61 i )(\Delta C_3 + \Delta \tilde C_3)^{*} - (2.94 -2.95 i)(\Delta C_5 + \Delta \tilde C_5)^{*}\nonumber \\ && + (0.18 - 0.01 i )(\Delta C_7 + \Delta \tilde C_7)^{*} - (0.06 - 0.04 i)(\Delta C_9 + \Delta \tilde C_9)^{*}]\nonumber \\ &&/[ 1 - (2.80+2.61 i )(\Delta C_3 + \Delta \tilde C_3) - ( 2.74+2.99 i)(\Delta C_5 + \Delta \tilde C_5)\nonumber \\ &&+ (0.17 + 0.01 i )(\Delta C_7 + \Delta \tilde C_7) - (0.04 + 0.05 i)(\Delta C_9 + \Delta \tilde C_9)]~, \nonumber \end{aligned}$$ $$\begin{aligned} \lambda_{\pi K_S} &=&(-0.70+ 0.70 i) \label{407}\\ {\nonumber}&& [1 - (1.09 +0.50 i )(\Delta C_3 + \Delta \tilde C_3)^{*} - (6.73 +2.79 i)(\Delta C_5 + \Delta \tilde C_5)^{*}\nonumber \\ && - (9.68 +3.21 i )(\Delta C_7 + \Delta \tilde C_7)^{*} + (13.86 +4.48 i)(\Delta C_9 + \Delta \tilde C_9)^{*}]\nonumber \\ &&/[ 1 + (1.08+0.48 i )(\Delta C_3 + \Delta \tilde C_3) - ( 6.66+2.70 i)(\Delta C_5 + \Delta \tilde C_5)\nonumber \\ &&- (9.58 + 3.09 i )(\Delta C_7 + \Delta \tilde C_7) +(13.71 + 4.31 i)(\Delta C_9 + \Delta \tilde C_9)]~, \nonumber \end{aligned}$$ $$\begin{aligned} \lambda_{\phi K_S} &=& (-0.70 + 0.70 i) \label{408}\\ {\nonumber}&& [1 - ( 28.62+11.37i )(\Delta C_3 + \Delta \tilde C_3)^{*} - (24.08 + 10.41 i)(\Delta C_5 + \Delta \tilde C_5)^{*}\nonumber \\ &&+ (14.57 + 5.88 i) (\Delta C_7 + \Delta \tilde C_7)^{*} + (15.08 + 5.92 i) (\Delta C_9 + \Delta \tilde C_9)^{*}]\nonumber \\ &&/[ 1 - (28.27+10.89 i )(\Delta C_3 + \Delta \tilde C_3) - ( 23.80+10.00 i)(\Delta C_5 + \Delta \tilde C_5)\nonumber \\ &&+ (14.39 + 5.64 i) (\Delta C_7 + \Delta \tilde C_7) + (14.90 + 5.67 i) (\Delta C_9 + \Delta \tilde C_9)]~, \nonumber \end{aligned}$$ $$\begin{aligned} \lambda_{\eta^\prime K_S} &=& (-0.70 + 0.69 i) \label{409}\\ {\nonumber}&& [1 - (10.88+3.29 i )(\Delta C_3 + \Delta \tilde C_3)^{*} +(8.26+2.06 i)(\Delta C_5 + \Delta \tilde C_5)^{*}\nonumber \\ &&+ (2.11 + 0.67 i)(\Delta C_7 + \Delta \tilde C_7)^{*} + (2.10 + 0.54 i)(\Delta C_9 + \Delta \tilde C_9)^{*}]\nonumber \\ &&/[ 1 - (10.73+3.21 i )(\Delta C_3 + \Delta \tilde C_3) + ( 8.14+2.00 i)(\Delta C_5 + \Delta \tilde C_5)\nonumber \\ &&+ (2.08 + 0.65 i)(\Delta C_7 + \Delta \tilde C_7) + (2.07 + 0.52 i)(\Delta C_9 + \Delta \tilde C_9)]~, \nonumber \end{aligned}$$ $$\begin{aligned} \lambda_{\rho K_S} &=& (-0.74 + 0.65 i) \label{411}\\ {\nonumber}&& [1 + (0.26 + 0.06 i )(\Delta C_3 + \Delta \tilde C_3)^{*} - (19.62+1.81 i)(\Delta C_5 + \Delta \tilde C_5)^{*}\nonumber \\ &&- (39.11 + 3.31 i )(\Delta C_7 + \Delta \tilde C_7)^{*} - (48.28 + 4.12 i)(\Delta C_9 + \Delta \tilde C_9)^{*}]\nonumber \\ &&/[ 1 + (0.25+0.07 i )(\Delta C_3 + \Delta \tilde C_3) - ( 19.28+2.79 i)(\Delta C_5 + \Delta \tilde C_5)\nonumber \\ &&- (38.46 + 5.28 i )(\Delta C_7 + \Delta \tilde C_7) - ( 47.48 + 6.55 i)(\Delta C_9 + \Delta \tilde C_9)]~, \nonumber \end{aligned}$$ $$\begin{aligned} \lambda_{\omega K_S} &=& (-0.71 + 0.70 i) \label{412}\\ {\nonumber}&& [1 + (90.48+13.54 i )(\Delta C_3 + \Delta \tilde C_3)^{*} + (85.24+12.50 i)(\Delta C_5 + \Delta \tilde C_5)^{*}\nonumber \\ &&+ (32.21+4.80i )(\Delta C_7 + \Delta \tilde C_7)^{*} + (19.07 + 2.79 i)(\Delta C_9 + \Delta \tilde C_9)^{*}]\nonumber \\ &&/[ 1+ (90.01+13.29 i )(\Delta C_3 + \Delta \tilde C_3) + ( 84.80+12.26 i)(\Delta C_5 + \Delta \tilde C_5)\nonumber \\ && + (32.04 + 4.71 i )(\Delta C_7 + \Delta \tilde C_7) + (18.97 + 2.74 i)(\Delta C_9 + \Delta \tilde C_9)], \nonumber \end{aligned}$$ $$\begin{aligned} \lambda_{f^0 K_S} &=& (-0.70 + 0.70 i) \label{413}\\ {\nonumber}&& [1 + (1.02+0.42i )(\Delta C_3 + \Delta \tilde C_3)^{*} - (1.67+0.97 i)(\Delta C_5 + \Delta \tilde C_5)^{*}\nonumber \\ &&+ (3.19 + 0.93 i )(\Delta C_7 + \Delta \tilde C_7)^{*} - (0.12 + 0.15 i)(\Delta C_9 + \Delta \tilde C_9)^{*}] \nonumber \\ && /[1 + (1.01+0.40 i )(\Delta C_3 + \Delta \tilde C_3) - ( 1.65+0.95 i)(\Delta C_5 + \Delta \tilde C_5)\nonumber \\ &&+ (3.16 + 0.90 i )(\Delta C_7 + \Delta \tilde C_7) - ( 0.12 + 0.15 i)(\Delta C_9 + \Delta \tilde C_9)]~. \nonumber \end{aligned}$$ These results are more general than those of  [@Barger:2009eq; @Barger:2009qs], as they include the $Z'$ contributions to both the QCD and electroweak penguins. At the leading order, the deviations for ${\mathcal C}_{f_{CP}}$ and ${\mathcal S}_{f_{CP}}$ from their SM predictions are a linear combination of these two classes of $Z'$ contributions. This discussion is independent of the details of the $U(1)'$ charges, so it can be applied to other family non-universal models as well; however, in the $U(1)_I$ model, the only non-trivial corrections are $\Delta \tilde C_3$ and $\Delta \tilde C_9$. ![Correlated constraints on $|B_{bs}^R|$ and $\phi_{bs}^R$. Random values for $C_{B_s}$ and $\phi_{B_s}^{\rm NP}$ from the experimentally allowed regions at different C.L. (see Table \[table1\]) are mapped to the $|B_{bs}^R|-\phi_{bs}^R$ plane using Eq. (\[403\]).[]{data-label="figure1"}](Figures/Bbs_RR.png){width="60.00000%"} ![ The NP contributions to ${\mathcal C}_{(\phi, \eta', \rho, \omega, f_0)K_S}$ and ${\mathcal S}_{(\phi, \eta', \pi, \rho, \omega, f_0)K_S}$, with $|B_{bs}^R|$, $\phi_{bs}^R$ constrained by $B_s-\bar B_s$ mixing. The colors specify the C.L. that their inverse image points represent in Fig. \[figure1\] (yellow for $1 \sigma$ and blue for $2 \sigma$). The boxes specify the allowed regions at 1$\sigma$ and $1.5\sigma$, and the dark points denote the SM limit. []{data-label="figure2"}](Figures/BdPhiKs_RR.png "fig:"){width="45.00000%"} ![ The NP contributions to ${\mathcal C}_{(\phi, \eta', \rho, \omega, f_0)K_S}$ and ${\mathcal S}_{(\phi, \eta', \pi, \rho, \omega, f_0)K_S}$, with $|B_{bs}^R|$, $\phi_{bs}^R$ constrained by $B_s-\bar B_s$ mixing. The colors specify the C.L. that their inverse image points represent in Fig. \[figure1\] (yellow for $1 \sigma$ and blue for $2 \sigma$). The boxes specify the allowed regions at 1$\sigma$ and $1.5\sigma$, and the dark points denote the SM limit. []{data-label="figure2"}](Figures/BdEtapKs_RR.png "fig:"){width="45.00000%"} ![ The NP contributions to ${\mathcal C}_{(\phi, \eta', \rho, \omega, f_0)K_S}$ and ${\mathcal S}_{(\phi, \eta', \pi, \rho, \omega, f_0)K_S}$, with $|B_{bs}^R|$, $\phi_{bs}^R$ constrained by $B_s-\bar B_s$ mixing. The colors specify the C.L. that their inverse image points represent in Fig. \[figure1\] (yellow for $1 \sigma$ and blue for $2 \sigma$). The boxes specify the allowed regions at 1$\sigma$ and $1.5\sigma$, and the dark points denote the SM limit. []{data-label="figure2"}](Figures/BdPiKs_RR.png "fig:"){width="45.00000%"} ![ The NP contributions to ${\mathcal C}_{(\phi, \eta', \rho, \omega, f_0)K_S}$ and ${\mathcal S}_{(\phi, \eta', \pi, \rho, \omega, f_0)K_S}$, with $|B_{bs}^R|$, $\phi_{bs}^R$ constrained by $B_s-\bar B_s$ mixing. The colors specify the C.L. that their inverse image points represent in Fig. \[figure1\] (yellow for $1 \sigma$ and blue for $2 \sigma$). The boxes specify the allowed regions at 1$\sigma$ and $1.5\sigma$, and the dark points denote the SM limit. []{data-label="figure2"}](Figures/BdOmgKs_RR.png "fig:"){width="45.00000%"} ![ The NP contributions to ${\mathcal C}_{(\phi, \eta', \rho, \omega, f_0)K_S}$ and ${\mathcal S}_{(\phi, \eta', \pi, \rho, \omega, f_0)K_S}$, with $|B_{bs}^R|$, $\phi_{bs}^R$ constrained by $B_s-\bar B_s$ mixing. The colors specify the C.L. that their inverse image points represent in Fig. \[figure1\] (yellow for $1 \sigma$ and blue for $2 \sigma$). The boxes specify the allowed regions at 1$\sigma$ and $1.5\sigma$, and the dark points denote the SM limit. []{data-label="figure2"}](Figures/BdRhoKs_RR.png "fig:"){width="45.00000%"} ![ The NP contributions to ${\mathcal C}_{(\phi, \eta', \rho, \omega, f_0)K_S}$ and ${\mathcal S}_{(\phi, \eta', \pi, \rho, \omega, f_0)K_S}$, with $|B_{bs}^R|$, $\phi_{bs}^R$ constrained by $B_s-\bar B_s$ mixing. The colors specify the C.L. that their inverse image points represent in Fig. \[figure1\] (yellow for $1 \sigma$ and blue for $2 \sigma$). The boxes specify the allowed regions at 1$\sigma$ and $1.5\sigma$, and the dark points denote the SM limit. []{data-label="figure2"}](Figures/BdFKs_RR.png "fig:"){width="45.00000%"} ![The $|B_{bs}^{L,R}|$, $\phi_{bs}^{L,R}$ and $B_{dd}^R$ distributions, with values constrained by $B_s-\bar B_s$ mixing at $x \sigma$ C.L. and selected by ${\mathcal C}_{(\phi, \eta', \pi, \rho, \omega, f_0)K_S}$ and ${\mathcal S}_{(\phi, \eta', \pi, \rho, \omega, f_0)K_S}$ at $y \sigma$ C.L.. Here $x=2.0$ and $y=1.7$ for the purple points, $x=1.0$ and $y=1.7$ for the blue points, $x=1.0$ and $y=1.4$ for the dark points. The red lines represent the vacuum considered in Section \[spectrum\], in which the values of $|B_{bs}^R|$ and $\phi_{bs}^R$ are not fixed.[]{data-label="figure4"}](Figures/GPD2_RR.png "fig:"){width="45.00000%"} ![The $|B_{bs}^{L,R}|$, $\phi_{bs}^{L,R}$ and $B_{dd}^R$ distributions, with values constrained by $B_s-\bar B_s$ mixing at $x \sigma$ C.L. and selected by ${\mathcal C}_{(\phi, \eta', \pi, \rho, \omega, f_0)K_S}$ and ${\mathcal S}_{(\phi, \eta', \pi, \rho, \omega, f_0)K_S}$ at $y \sigma$ C.L.. Here $x=2.0$ and $y=1.7$ for the purple points, $x=1.0$ and $y=1.7$ for the blue points, $x=1.0$ and $y=1.4$ for the dark points. The red lines represent the vacuum considered in Section \[spectrum\], in which the values of $|B_{bs}^R|$ and $\phi_{bs}^R$ are not fixed.[]{data-label="figure4"}](Figures/GPD3_RR.png "fig:"){width="45.00000%"} ![The $|B_{bs}^{L,R}|$, $\phi_{bs}^{L,R}$ and $B_{dd}^R$ distributions, with values constrained by $B_s-\bar B_s$ mixing at $x \sigma$ C.L. and selected by ${\mathcal C}_{(\phi, \eta', \pi, \rho, \omega, f_0)K_S}$ and ${\mathcal S}_{(\phi, \eta', \pi, \rho, \omega, f_0)K_S}$ at $y \sigma$ C.L.. Here $x=2.0$ and $y=1.7$ for the purple points, $x=1.0$ and $y=1.7$ for the blue points, $x=1.0$ and $y=1.4$ for the dark points. The red lines represent the vacuum considered in Section \[spectrum\], in which the values of $|B_{bs}^R|$ and $\phi_{bs}^R$ are not fixed.[]{data-label="figure4"}](Figures/GPD1_RR.png "fig:"){width="45.00000%"} We now turn to a numerical analysis of the FCNC constraints with the $U(1)_I$ model, for which the three free parameters are $|B_{bs}^R|$, $\phi_{bs}^R$ and $B_{dd}^R$. First, we consider $B_s-\bar{B}_s$ mixing, which involves two of these parameters, $|B_{bs}^R|$ and $\phi_{bs}^R$. The experimental constraints on these two parameters are illustrated in Fig. \[figure1\], where the various colors of the points specify the different confidence levels (C.L.) that the relevant $C_{B_s}$ and $\phi_{B_s}^{\rm NP}$ values represent. There are two separate shaded regions in this figure. The left one corresponds to the $\phi_{B_s}^{\rm NP}$ solution “S1” and the right one corresponds to “S2” (see Tab. \[table5\]). $\phi_{bs}^R$ varies within the ranges $-80^\circ \sim -20^\circ$ and $-90^\circ \sim -70^\circ$ in the two regions, respectively. This is similar to what happens to $\phi_{bs}^L$ in the LL limit in [@Barger:2009qs], since the $\Delta \tilde C_3^{B_s}$ contributions to $C_{B_s} e^{2i\phi_{B_s}}$ in Eq. (\[403\]) are absent in both cases. In addition, to explain the observed discrepancy in $B_s-\bar B_s$ mixing from the SM prediction, $|B_{bs}^R|$ is required to be $\sim 10^{-3}$. As discussed in [@Barger:2009eq; @Barger:2009qs], there are two reasons for this feature. First, $C_{B_s}$ does not deviate significantly from its SM prediction (the anomaly in $B_s -\bar B_s$ mixing is mainly caused by the phase $\phi_{B_s}^{\rm NP}$). Second, the corrections of a family non-universal $Z'$ arise at tree level, so only a small coupling is needed to explain this small deviation, according to Eq. (\[403\]). The smallness of $|B_{bs}^R|$ is consistent with our assumption of small fermion mixing angles, since $B_{bs}^R$ is proportional to them (see Eq. (\[457\])) as well as to $g' M_{Z_1}/(g_Z M_{Z_2})$. The constraints from the branching ratio ${\rm Br}(B_s \to \mu^+ \mu^-)$ and ${\rm Br}(B_d \to K^{(*)}\mu^+\mu^-)$ can be easily satisfied due to the smallness of $|B_{bs}^R|$ [@Barger:2009qs]. With the constrained values of $|B_{bs}^R|$ and $\phi_{bs}^R$ by $B_s-\bar B_s$ mixing, we illustrate the NP contributions to ${\mathcal C}_{(\phi, \eta', \pi, \rho, \omega, f_0)K_S}$ and ${\mathcal S}_{(\phi, \eta', \pi, \rho, \omega, f_0)K_S}$ in Fig. \[figure2\]. In this case, the third parameter ($B_{dd}^R$) is also involved. For the channel $B_d\to \pi K_S$, we take a strategy different from that used in [@Barger:2009eq; @Barger:2009qs], in which it was assumed that the NP enters the hadronic decays of neutral $B_d$ meson only through electroweak penguins. In that case, the NP effects in the $B_d\to \pi K_S$ channel can be resolved into a factor $qe^{i\phi}$ [@Buras:2003dj]; the constraints on this factor from a $\chi^2$ fit of $B\to \pi K_S$ and $B \to \pi\pi$ data have been studied in [@Fleischer:2008wb]. For our NUSSM model, the NP enters generically through QCD as well as electroweak penguins, and hence we treat this channel in the same way as the other $B_d$ decay channels. We also assume a $15\%$ uncertainty in the SM calculations for each of these modes and a $25\%$ uncertainty for the NP contributions. Here $15\%$ is a typical uncertainty level for the hadronic matrix elements of the SM FC operators (see e.g. [@Wirbel:1985ji]) that is needed to explain the experimental results for ${\mathcal C}_{\psi K_S}$ and ${\mathcal S}_{\psi K_S}$ in the SM  [@Barger:2009eq; @Barger:2009qs]. The difference of the uncertainty levels between the SM and NP calculations arises because the hadronic matrix elements of the FC operators in the SM are better understood than those of the NP operators. To see whether the anomalies in $B_s-\bar B_s$ mixing and the $B_d \to (\phi, \eta', \pi, \rho, \omega, f_0)K_S$ CP asymmetries can be simultaneously accommodated, we have carried out a correlated analysis within the $U(1)_I$ model. The distributions of $|B_{bs}^R|$, $\phi_{bs}^R$ and $B_{dd}^R$ constrained at different C.L. are illustrated in Fig. \[figure4\]. Indeed, there exist parameter regions for which the tension between the observations and the SM predictions are greatly relaxed. In Figs. \[figure2\] and \[figure4\], we require $0.005 < |B_{dd}^R| < 0.5$. Given that $|B_{dd}^R| \approx g' M_{Z_1}/(2g_Z M_{Z_2})$, we immediately find that $1\, {\rm TeV} < M_{Z_2} < 10\, {\rm TeV}$ for $g' \simeq g_Z$. Here $B_{dd}^R$ can be positive or negative, since it resolves a minus sign from the degeneracy of two solutions in $B_{bs}^R$ that is specified by a $\pi$ phase difference [@Barger:2009eq; @Barger:2009qs]. The red lines in Fig. \[figure4\] represent the parameter region discussed in our numerical example in which $|B_{dd}^R| \approx 0.02$. Indeed, we see that the anomalies in the hadronic $B_d$ meson decays can be explained simultaneously, given the $B_{bs}^R$ values required to fit the $B_s\to \bar B_s$ mixing data. Discussion and Conclusions ========================== In this paper, we have discussed a class of family non-universal $U(1)^\prime$ models based on non-standard $E_6$ embeddings of the SM that interchange the standard roles of the two ${\bf 5^*}$ representations present in the fundamental ${\bf 27}$ representation of $E_6$ for the third family. The NUSSM models in this class are simple and anomaly-free. They are not full $E_6$ grand unified theories, so the $U(1)^\prime$ breaking can occur at the TeV scale, resulting in a TeV-scale $Z^\prime$ gauge boson that can mediate FCNC in the $b\to s$ transitions. We analyzed a representative example of a NUSSM model (the $U(1)_I$ model), in which we described the low energy spectrum of the theory and determined the constraints on the family non-universal $Z^\prime$ couplings from the $B$ sector. NUSSM models such as the $U(1)_I$ model are characterized by a rich spectrum of states with masses at the electroweak to TeV scale. The $Z^\prime$-mediated FCNC in the $U(1)_I$ model can easily accommodate the observed discrepancies in the $b\to s$ transitions. Related observables such as $\tau\to \mu$ and $\tau \to e$ can also be studied in NUSSM models; we defer this to future work. Acknowledgments {#acknowledgments .unnumbered} =============== We thank Carlos E. M. Wagner for helpful discussions and the Aspen Center for Physics for hospitality in the preparation of this work. The work of L. E. is supported by the DOE grant No. DE-FG02-95ER40896 and the Wisconsin Alumni Research Foundation. The work of P. L. is supported by the IBM Einstein Fellowship and by the NSF grant PHY-0503584. The work of T. L. is supported by the Fermi-McCormick Fellowship and by the DOE grant No. DE-FG02- 90ER40560. Tree-level Mass-squared Matrix for Charged Higgs Bosons {#CH mass matrix} ======================================================= For charged Higgs bosons, the entries of its mass-squared matrix $M_{H^\pm}^2$ at tree level are given in the basis $\{h_d^-,h_u^{+*},H_d^-,H_u^{+*}\}$ by $$\begin{aligned} (M_{H^\pm}^2)_{11} &=& -{{G^2}\over 4} \left(|v_2|^2 - |v_1|^2+|V_2|^2 - |V_1|^2\right)+ \frac{g_2^2}{2}(|v_2|^2+ |V_2|^2-|V_1|^2) \nonumber \\&& -\frac{g'^2}{4}\left(-|v_1|^2 + |s_1|^2 + |V_1|^2 -|s_2|^2\right)+(|\lambda_1|^2+|\lambda_2|^2)|s_1|^2+m_{h_d}^2~,~\, \nonumber \\ (M_{H^\pm}^2)_{22} &=& {{G^2}\over 4} \left(|v_2|^2 - |v_1|^2+|V_2|^2 - |V_1|^2\right)+\frac{g_2^2}{2} (|v_1|^2+|V_1|^2-|V_2|^2)\nonumber \\&&+|\lambda_1|^2|s_1|^2+|\lambda_3|^2|s_2|^2+m_{h_u}^2 ~,~\, \nonumber \\ (M_{H^\pm}^2)_{33} &=& -{{G^2}\over 4} \left(|v_2|^2 - |v_1|^2+|V_2|^2 - |V_1|^2\right)+\frac{g_2^2}{2} (|v_2|^2+ |V_2|^2-|v_1|^2)\nonumber \\&& + \frac{g'^2}{4}\left(-|v_1|^2 + |s_1|^2 + |V_1|^2 -|s_2|^2\right)+(|\lambda_3|^2+|\lambda_4|^2)|s_2|^2+m_{H_d}^2 ~,~\, \nonumber \\ (M_{H^\pm}^2)_{44} &=& {{G^2}\over 4} \left(|v_2|^2 - |v_1|^2+|V_2|^2 - |V_1|^2\right)+ \frac{g_2^2}{2} (|v_1|^2+|V_1|^2-|v_2|^2)\nonumber \\&& +|\lambda_2|^2|s_1|^2+|\lambda_4|^2|s_2|^2+m_{H_u}^2 ~,~\, \nonumber \\ (M_{H^\pm}^2)_{12} &=&(M_{H^\pm}^2)_{21}^* =\frac{g_2^2}{2} v_1^*v_2^* - \lambda_1(\lambda_1^*v_2^*v_1^*+\lambda_2^*V_2^*v_1^*+\lambda_5^* s_2^*)-A_{\lambda_1} \lambda_1 s_1 ~,~\, \nonumber \\ (M_{H^\pm}^2)_{13} &=& (M_{H^\pm}^2)_{31}^* =\frac{g_2^2}{2} v_1^*V_1 + (\lambda_1\lambda_3^*+\lambda_2\lambda_4^*) s_1 s_2^* ~,~\, \nonumber \\ (M_{H^\pm}^2)_{14} &=& (M_{H^\pm}^2)_{41}^* =\frac{g_2^2}{2} v_1^*V_2^* - \lambda_2(\lambda_1^*v_2^*v_1^*+\lambda_2^*V_2^*v_1^*+\lambda_5^* s_2^*)-A_{\lambda_2} \lambda_2 s_1 ~,~\, \nonumber \\ (M_{H^\pm}^2)_{23} &=& (M_{H^\pm}^2)_{32}^* =\frac{g_2^2}{2} v_2V_1 - \lambda_3^*(\lambda_3h_u^{0}V_1+\lambda_4V_2V_1+\lambda_5 s_1) -A_{\lambda_3}^* \lambda_3^* s_2^* ~,~\, \nonumber\\ (M_{H^\pm}^2)_{24} &=& (M_{H^\pm}^2)_{42}^* = \frac{g_2^2}{2} v_2V_2^* + \lambda_1^*\lambda_2 |s_1|^2 + \lambda_3^*\lambda_4 |s_2|^2 ~,~\, \nonumber \\ (M_{H^\pm}^2)_{34} &=& (M_{H^\pm}^2)_{43}^* = \frac{g_2^2}{2} V_1^*V_2^* -\lambda_4(\lambda_3^*v_2^*V_1^*+\lambda_4^*V_2^*V_1^*+\lambda_5^* s_1^*) -A_{\lambda_4} \lambda_4 s_2 ~.~\, \nonumber \end{aligned}$$ These entries can be applied to both cases with and without CP violation. Parameters {#Parameters} ========== The parameters used in our numerical analysis are summarized below: [**(1) QCD and EW Parameters**]{} $G_F = 1.16639 \times 10^{-5}$ GeV$^{-2}$, $\Lambda_{\overline{MS}}^{(5)} = 225$ MeV,\ $M_W = 80.42$ GeV, $\sin^2\theta_W = 0.23$,\ $\eta_{2B} = 0.55$, $J_5 = 1.627$,\ $\alpha_s(M_Z) = 0.118$, $\alpha_{em} = 1/128$,\ $\lambda = 0.2252$, $A = 0.8117$,\ $\bar{\rho} = 0.145$, $\bar{\eta} = 0.339$,\ $R_b = \sqrt{\rho^2 + \eta^2} = 0.378$.\ [**(2) Masses, Decay Constants, Hadronic Form Factors and Lifetimes**]{} $M_{{\pi}^{\pm}} =0.139$ GeV, $M_{{\pi}^{0}} = 0.135$ GeV,\ $M_{K} = 0.498$ GeV, $M_{B} = 5.279$ GeV,\ $M_\phi = 1.02$ GeV, $M_\psi = 2.097$ GeV,\ $M_{\eta^\prime} = 0.958$ GeV, $M_{\omega} = 0.783$ GeV,\ $M_{\rho} = 0.776$ GeV, $M_\eta = 0.548$ GeV\ $M_{f^0} = 0.980$ GeV,\ $X_{\eta} = 0.57$, $Y_{\eta} = 0.82$,\ $m_u (\mu = 4.2 ~{\rm GeV}) = 1.86$ MeV, $m_d (\mu = 4.2 ~{\rm GeV})= 4.22$ MeV,\ $m_s (\mu = 4.2 ~{\rm GeV}) = 80$ MeV, $m_c (\mu = 4.2 ~{\rm GeV}) = 0.901$ GeV,\ $m_b(\mu = 4.2 ~{\rm GeV}) = 4.2$ GeV, $m_t (\mu = M_Z) = 171.7$ GeV,\ $f_{\phi} = 237$ MeV, $f_{B} = 190$ MeV,\ $f_{\pi} = 130$ MeV, $f_{K} = 160$ MeV,\ $f_{\psi} = 410$ MeV, $f_{\omega} = 200$ MeV,\ $f_\rho = 209$ MeV, $f_{f^0} = 180$ MeV,\ $F_0^{B\pi} (0) = 0.330$, $F_0^{BK} (0) = 0.391$,\ $F_1^{BK} (0) = 0.379$, $A_0^{B\omega} (0) = 0.280$,\ $F_0^{B f} (0) = 0.250$, $F_0^{f K} (0) = 0.030$,\ $A_0^{B\rho} = 0.280$, $f_{B_s} \sqrt{{\hat B}_{B_s}} = 0.262$\ $\tau_{B^0}=1.530$ ps, $\tau_{B^-}=1.65$ ps,\ $M_{B_s} = 5.37$ GeV, $\tau_{B_s} = 1.47$ ps, [99]{} P. Langacker, Rev. Mod. Phys. [**81**]{}, 1199 (2009) \[arXiv:0801.1345 \[hep-ph\]\]. 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Hence, a detailed numerical analysis would be needed to ascertain whether the neutralino LSP satisfies the dark matter constraints. As this is tangential to the main purpose of our paper, we do not address it here. [^7]: In the $U(1)'_S$ model, in which all SM fermions are charged under the $U(1)'$ symmetry, the $Z'$-induced corrections to the Wilson coefficients take a more general form (see e.g. [@Barger:2009eq; @Barger:2009qs]). We will not discuss them here.

--- abstract: 'We present astrochemical photo-dissociation region models in which cosmic ray attenuation has been fully coupled to the chemical evolution of the gas. We model the astrochemical impact of cosmic rays, including those accelerated by protostellar accretion shocks, on molecular clouds hosting protoclusters. Our models with embedded protostars reproduce observed ionization rates. We study the imprint of cosmic ray attenuation on ions for models with different surface cosmic ray spectra and different star formation efficiencies. We find that abundances, particularly ions, are sensitive to the treatment of cosmic rays. We show the column densities of ions are under predicted by the “classic” treatment of cosmic rays by an order of magnitude. We also test two common chemistry approximations used to infer ionization rates. We conclude that the approximation based on the H$_3^+$ abundance under predicts the ionization rate except in regions where the cosmic rays dominate the chemistry. Our models suggest the chemistry in dense gas will be significantly impacted by the increased ionization rates, leading to a reduction in molecules such as NH$_3$ and causing H$_2$-rich gas to become \[C II\] bright.' author: - 'Brandt A.L. Gaches' - 'Stella S.R. Offner' - 'Thomas G. Bisbas' bibliography: - 'crchem1.bib' title: 'The Astrochemical Impact of Cosmic Rays in Protoclusters I: Molecular Cloud Chemistry' --- Introduction {#sec:intro} ============ Molecular cloud dynamics and chemistry are sensitive to the ionization fraction. The chemistry of molecular clouds is dominated by ion-neutral reactions [@watson1976] and thus controlled by the ionization fraction. The gas (kinetic) temperature of a typical molecular cloud with an average H-nucleus number density of $n\approx10^3\,{\rm cm}^{-3}$ is approximately 10 K for cosmic-ray ionization rates $\zeta\lesssim10^{-16}\,{\rm s}^{-1}$ [@bisbas2015; @bisbas2017], thus rendering neutral-neutral reactions inefficient. Ionization in molecular clouds is produced in three difference ways: UV radiation, cosmic rays (CRs), and X-Ray radiation. Ultra-violet radiation, from external O- and B-type stars and internal protostars, does not penetrate very far into the cloud due to absorption by dust. However, cosmic rays, which are relativistic charged particles, travel much further into molecular clouds and dominate the ionization fraction when $A_V \geq 5\,{\rm mag}$ [@mckee1989; @strong2007; @grenier2015]. CR-driven chemistry is initiated by ionized molecular hydrogen, ${\rm H}_2^+$ [@dalgarno2006]. The ion-neutral chemistry rapidly follows: $${\rm CR + H_2 \rightarrow H_2^+ + e^- + CR'}$$ $${\rm H_2^+ + H_2 \rightarrow H_3^+ + H,}$$ where CR$'$ is the same particle as CR but with a lower energy. The ejected electron from the first reaction can have an energy greater than the ionization potential of H$_2$ and thus cause further ionization. Once H$_3^+$ forms, more complex chemistry follows, thereby creating a large array of hydrogenated ions: $${\rm X + H_3^+ \rightarrow [XH]^+ + H_2}.$$ Both HCO$^+$ and N$_2$H$^+$, important molecules used to map the dense gas in molecular clouds, form this way with X being CO and N$_2$, respectively. These species are also used to constrain the cosmic ray ionization rate (CRIR) [i.e., @caselli1998; @ceccarelli2014]. OH$^+$ and H$_n$O$^+$ are also formed this way through H$_3^+$ and H$^+$ [@hollenbach2012]. In addition, at low column densities ($A_V<1\,{\rm mag}$), which is typical of the boundaries of molecular clouds), the non-thermal motions between ions and neutrals may overcome the energy barrier of the reaction $${\rm C^+ + H_2 \rightarrow CH^+ + H,}$$ leading to an enhancement of the CO column density [@federman1996; @visser2009] and a shift of the H[i]{}-to-H$_2$ transition to higher $A_V$ [@bisbas2019]. The ionization fraction controls the coupling of the magnetic fields to the gas, influencing non-ideal magnetohydrodynamic (MHD) effects such as ambipolar diffusion [@mckee2007]. These non-ideal effects can play a signficant role in the evolution in the cores and disks of protostars. On galactic scales, numerical simulations have shown that CRs can help drive large outflows and winds out of the galaxy [e.g., @girichidis2016]. Our study focuses on the impact of CRs on Giant Molecular Cloud scales which is typically not resolved fully in such simulations. There have been a plethora of studies modeling the impact of CRs on chemistry and thermal balance [i.e., @caselli1998; @bell2006; @meijerink2011; @bayet2011; @clark2013; @bisbas2015]. However, in these studies, and the vast majority of astrochemical models, the CRIR is held constant throughout the cloud, despite the recognition that CRs are attenuated and modulated while traveling through molecular gas [@schlickeiser2002; @padovani2009; @schlickeiser2016; @padovani2018]. Galactic-CRs, thought to be accelerated in supernova remnants or active galactic nuclei, are affected by hadronic and Coulombic energy losses and screening mechanisms that reduce the flux with increasing column density [@strong2001; @moskalenko2005; @evoli2017]. The modulation of CRs has not previously been included within astrochemical models of molecular clouds due to the difficulty in calculating the attenuation and subsequent decrease in the CRIR [@wakelam2013; @cleeves2014]. Given that CRs are thought to be attenuated, it is expected that the ionization rate should decline within molecular clouds. However, recent observations do not universally show a lower ionization rate. [@favre2017] inferred the CRIR towards 9 protostars and found a CRIR consistent with the rate inferred for galactic CRs. The OMC-2 FIR 4 protocluster, hosting a bright protostar, is observed to have a CRIR 1000 higher than the expected rate from galactic CRs [@ceccarelli2014; @fontani2017; @favre2018]. [@gaches2018b] show that this system can be modelled assuming the central source is accelerating protons and electrons within the accretion shocks on the protostar’s surface. In general, accreting, embedded protostars may accelerate enough CRs to cancel the effect of the attenuation of external CRs at high column densities, producing a nearly constant ionization rate throughout the cloud [@padovani2016; @gaches2018b]. Typical accretion shocks and shocks generated by protostellar jets satisfy the physical conditions necessary to accelerate protons and electrons [@padovani2016; @gaches2018b]. Accretion shocks in particular are a promising source of CRs since they are strong, with velocities exceeding 100 km/s and temperatures of millions of degrees Kelvin [@hartmann2016]. [@gaches2018b] calculated the spectrum of accelerated protons in protostellar accretion shocks and the attenuation through the natal core assuming that the CRs free-stream outwards. These models suggest that clusters of a few hundred protostars accelerate enough CRs into the surrounding cloud to exceed the ionization rate from Galactic CRs. In this study, we explore the effects of protostellar CRs on molecular cloud chemistry by employing the model of [@gaches2018b]. We implement an approximation for CR attenuation into the astrochemistry code [3d-pdr]{} [@bisbas2012] to account for CR ionization rate gradients. We investigate the signatures of a spatially varying ionization rate. We further explore the impact of protostellar CR sources and their observable signatures. The layout of the paper is as follows. In §\[sec:methods\] we present the CR and protostellar models and describe the implementation of CR attenuation into [3d-pdr]{}. We discuss our results in §\[sec:res\]. Finally, in §\[sec:disc\] we create observational predictions and compare them to observations. Methods {#sec:methods} ======= Protocluster Model {#sec:proto_model} ------------------ We generate protoclusters following the method of [@gaches2018a], where the model cluster is parameterized by the number of stars and gas surface density, N$_*$ and $\Sigma_{\rm cl}$, respectively. These parameters are connected to the star formation efficiency $\varepsilon_g = M_*/M_{\rm gas}$, where M$_{\rm gas}$ is related to $\Sigma_{\rm cl}$ following [@mckee2003] $\Sigma_{\rm cl} = \frac{M_{\rm gas}}{\pi R^2}$, where the cloud radius, $R$, is determined from the density distribution (See §\[sec:dens\]). We model protoclusters with surface densities in the range $0.1 \leq \Sigma_{\rm cl} \leq 10$ g cm$^{-2}$ and with a number of stars in the range $10^2 \leq N_* \leq 10^4$. In this parameter space, we always consider $\varepsilon_g \leq 25\%$. We generate $N_{\rm cl} = 20$ protoclusters for each point in the parameter space and adopt the average CR spectrum for the chemistry modelling. We use the Tapered Turbulent Core (TTC) accretion history model [@mckee2003; @offner2011], where the protostellar core is supported by turbulent pressure and accretion is tapered to produce smaller accretion rates as the protostellar mass, $m$, approaches the final mass, $m_f$. [@gaches2018a] showed the TTC model is able to reproduce the bolometric luminosities of observed local protoclusters. Cosmic Ray Model {#sec:cr} ---------------- We brielfy summarize the CR acceleration and propagation model in [@gaches2018b] and refer the reader to that paper for more details. We assume CRs are accelerated in the accretion shock near the protostellar surface. Accreting gas is thought to flow along magnetic field lines in collimated flows with the shock at the termination of the flow. We assume the shock velocities are comparable to the free-fall velocity at the stellar surface. Following [@hartmann2016], we assume fully ionized strong shocks with the shock front perpendicular to the magnetic field lines. We adopt a mean molecular weight $\mu_I = 0.6$ and a filling fraction of accretion columns on the surface, $f = 0.1$. We calculate the CR spectrum under the Diffusive Shock Acceleration (DSA) limit, also known as first-order Fermi acceleration [e.g., reviewed in @drury1983; @kirk1994; @melrose2009]. Under DSA, the CR momentum distribution is a power-law, $f(p) \propto p^{-q}$, where $q$ is related to the shock properties. We attenuate the CR spectrum through the protostellar core following [@padovani2009]. [@padovani2018] presented updated attenuation models for surface densities up to 3000 g cm$^{-2}$, but the results remain unchanged for the surface of concern in this work. The core surface density and radius for a turbulence-supported core are [@mckee2003]: $$\begin{aligned} \Sigma_{\rm core} &= 1.22\Sigma_{\rm cl} = 0.122 {\rm \, g \, cm^{-2}} \left (\frac{\Sigma_{\rm cl}}{{\rm 0.1 \, g \, cm^{-2}}} \right )\\ N({\rm H_2})_{\rm core} &= \frac{\Sigma_{\rm core}}{\mu_M m_H} \\ &= 3\times 10^{22} {\rm \, cm^{-2}\, } \left ( \frac{\Sigma_{\rm core}}{0.122 {\rm \, g \, cm^{-2}}} \right ) \left ( \frac{2.4}{\mu} \right ) \nonumber\\ R_{\rm core} &= 0.057 \Sigma_{\rm cl}^{-\frac{1}{2}} \left ( \frac{m_f}{30 ~M_{\odot}} \right )^{\frac{1}{2}} {~~~\rm pc} \\ &= 0.104 {\rm \, pc} \left ( \frac{\Sigma_{\rm cl}}{0.1 {\rm \, g \, cm^{-2}}} \right )^{-\frac{1}{2}} \left ( \frac{m_f}{10 {\rm \, M_{\odot}}}\right )^{\frac{1}{2}} \nonumber ,\end{aligned}$$ where $\mu_M = 2.4$ is the mean molecular weight for a molecular gas. We calculate the total protocluster CRIR by summing over the $N_*$ attenuated CR spectra. Density Structure {#sec:dens} ----------------- We assume the molecular cloud density is described by an inverse power law $$n(r) = n_0 \left ( \frac{R}{r} \right )^2,$$ where $R$ is the cloud radius and $n_0$ is the number density with an inner radius of 0.1 pc. The $r^{-2}$ dependence matches the solution for isothermal collapse [@shu1977]. We take $n_0 = 100$ cm$^{-3}$, corresponding to a gas regime in which the cloud is expected to be mostly molecular under typical conditions. The radius is set by constraining the total surface density by $\Sigma_{\rm cl}$ as defined: $$\int_{R_c}^R n(r) dr = \frac{\Sigma_{\rm cl}}{\mu_M m_H},$$ where $R_c$ is the inner radius delineating the transition between the molecular cloud and protostellar core. The turbulent-linewidth, $\sigma$, of a cloud with density profile $n(r) \propto r^{-2}$ and a virial $\alpha$ parameter is [@bertoldi1992]: $$\sigma = \left ( \frac{G M^2 \alpha}{3 \mu_M m_H \bar{n} V R} \right )^{\frac{1}{2}} ,$$ where $\bar{n}$ is the volume-averaged density from $n(r)$, $G$ is the gravitational constant, and $V = \frac{4}{3} \pi R^3$ is the volume of the molecular cloud. We take $\alpha = 2$ for our fiducial model [@heyer2015]. Chemistry with Cosmic Ray Attenuation ------------------------------------- We use a modified version of the [3d-pdr]{} astrochemistry code[^1] introduced in [@bisbas2012]. [3d-pdr]{} solves the chemical abundance and thermal balance in one-, two- and three-dimensions for arbitrary gas distributions. The code can be applied to arbitrary three dimensional gas distributions, such as post-processing simulations [@offner2013; @offner2014; @bisbas2018]. Here, we use the code in one dimension to model a large parameter space. We adopt the [@mcelroy2013] [umist12]{} chemical network containing 215 species and approximately 3,000 reactions. We assume the gas is initially composed of molecular H$_2$ with the rest being atomic with abundances from [@sembach2000] shown in Table \[tab:abund\]. Cooling is included from line emission, which is mainly due to carbon monoxide at low temperatures and forbidden lines of \[OI\],\[CI\] and \[CII\] at higher temperatures. Heating is due to dissipation of turbulence, photoelectric heating of dust from far-ultraviolet emission, H$_2$ fluorescence and CR heating of gas. We use a temperature floor of 10 K. Previously, [3d-pdr]{} included CRs via a single global CRIR parameter. See [@bisbas2012] for more technical details. [cc]{} H & 1.0\ He & 0.1\ C & 1.41$\times10^{-4}$\ N & 7.59$\times10^{-5}$\ O & 3.16$\times10^{-4}$\ S & 1.17$\times10^{-5}$\ Si & 1.51$\times10^{-5}$\ Mg & 1.45$\times10^{-5}$\ Fe & 1.62$\times10^{-5}$ We modify [3d-pdr]{} to account for CR attenuation through the cloud. Currently, our implementation is restricted to one-dimensional models where we assume spherical symmetry. [3d-pdr]{} calculates the CRIR from $N_{\rm SRC}$ CR sources. The user provides a CR spectrum for any number of sources and whether it is external (incident at the surface) or internal (originating at the cloud center). In 1D, the fluxes are defined on either surface of the domain. The flux due to external sources is attenuated while the point sources are assumed to radiate isotropically; both are attenuated and spatially diluted. The spectra are attenuated after every update of the molecular abundances to keep the amount of H$_2$ for interaction losses self-consistent. [3d-pdr]{} stores the initial CR flux in $N_{\rm ene}$ bins and self-consistently calculates $\zeta$ from all sources across the domain. Point sources require a user-set radial scaling factor, $r_s$, and a transport parameter, $a$. For our model results, we set $r_s = R_C$ to represent the core radius. Point source CR spectra, $j(E, r)$, are attenuated by the H$_2$ column density [@padovani2009] and diluted by the radial distance following $$j(r) \propto \left ( \frac{r_s}{r + r_s}\right )^a,$$ to approximate the effects of transport. Solving the transport equations for Galactic transport problems has been done with specialized codes, such as [Galprop]{} [@moskalenko1998] and [Dragon2]{} [@evoli2017]. Fully solving the steady-state transport equations are beyond the scope of this work but will be investigated in the future. In our study, we include two CR flux sources. First, we include the internal protostellar clusters discussed above. We set the radial scaling $r_s = R_C = 0.1$ pc, which is approximately the size of a protostellar core. Second, we include an external isotropic CR flux to model interstellar CRs. We follow [@ivlev2015] and parameterize the external flux as $$j_{\rm ext} = C \frac{E^{\alpha}}{\left ( E + E_0 \right )^{\beta}} {~~~\rm (particles ~eV^{-1} ~cm^{-2} ~s^{-1} ~sr^{-1})}.$$ We use their “low” model ($\mathcal{L}$), with $C = 2.4\times10^{15}$, $E_0 = 650$ MeV, $\alpha = 0.1$ and $\beta = 2.8$ and their “high” model ($\mathcal{H}$), with $C = 2.4\times10^{15}$, $E_0 = 650$ MeV, $\alpha = -0.8$ and $\beta = 1.9$. The $\mathcal{L}$ model is a direct extrapolation of the Voyager-1 data [@stone2013], while the $\mathcal{H}$ is a maximal model to correct for any possible effects of the solar magnetic field. The CRIR is calculated by integrating the spectrum multiplied by the H$_2$ cross section: $$\zeta_p = 2 \pi \int j(E) \sigma_{i}(E) dE ,$$ where the factor of $2\pi$ accounts for irradiating the 1-D surface on one side and $\sigma_i(E)$ is the H$_2$ ionization cross section with relativistic corrections [@krause2015]. The code allows for an arbitrary number of energy bins, $N_{\rm bins}$, for input CR spectrum. We compared the CRIR for bin sizes ranging from N$_{\rm bin} = 4$ to $1000$ and found that N$_{\rm bins} > 40$ only produces changes in the CRIR at the 1% level. We do not fully solve for primary or secondary electrons. Therefore, we multiply the proton CRIR by $\frac{5}{3}$ to account for the electron population [@dalgarno1958; @takayanagi1973]. Our fiducial parameters for the study are shown in Table \[tab:params\]. Table \[tab:models\] shows the full suite of models we adopt. The model names describe the included physics: L/H denotes using the $\mathcal{L}$/$\mathcal{H}$ (Low/High) external spectrum, NI denotes no internal sources, DI denotes internal sources with $a = 1$ (diffusive transport), RI denotes internal sources with $a = 2$ (rectilinear transport) and NA denotes no internal sources or CR attenuation. We study the impact of these parameters in Section \[sec:res\]. [ccc]{} $\mu_{\rm I}$ & Reduced gas mass & 0.6 (ionized)\ $\mu_{\rm M}$ & Reduced gas mass & 2.4 (neutral molecular)\ $n_0$ & Density at edge of cloud & $10^2$ cm$^{-3}$\ $N_{\rm src}$ & Number of CR Sources & 2\ $N_{\rm bins}$ & Number of CR spectrum bins & 40\ a & CR transport parameter & 1\ $r_s$ & Scaling radius for CR flux & 0.1 pc\ $\alpha$ & Cloud virial parameter & 2\ [c|cccc]{} LDI\[model:fid\] & $r^{-1}$ & $\checkmark$ & $\mathcal{L}$ & $\checkmark$\ LRI\[model:rec\] & $r^{-2}$ & $\checkmark$ & $\mathcal{L}$ & $\checkmark$\ LNI\[model:ni\] & ... & ... & $\mathcal{L}$ & $\checkmark$\ LNA\[model:na\] & ... & ... & $\mathcal{L}$ & ...\ HNI\[model:hni\] & ... & ... & $\mathcal{H}$ & $\checkmark$\ HDI\[model:hdi\] & $r^{-1}$ & $\checkmark$ & $\mathcal{H}$ & $\checkmark$\ Results {#sec:res} ======= Cosmic Ray Spectrum {#sec:crsp} ------------------- Our modified [3d-pdr]{} code requires as an input the flux of CR protons for any number of sources. As a result, the CR proton flux and CRIR throughout the spatial domain become outputs rather than inputs. Figure \[fig:crspecs\] shows an example CR spectrum for a molecular cloud with $\Sigma_{\rm cl} = 0.75$ g cm$^{-2}$ and $N_* = $ 750 using the LDI model. The CR proton flux increases inside the cloud because of the embedded sources. The double peaked shape of the spectrum is due to peaks in the loss function from ionization and Coulomb losses. The inset shows the CRIR as a function of the position within the cloud. In this model, the ionization rate climbs nearly two orders of magnitude throughout the cloud with increasing proximity to the protostellar cluster. Cosmic Ray Ionization Rate Models {#sec:crir} --------------------------------- A number of prescriptions have been used to calculate the CRIR from observed column densities of various tracer species [@caselli1998; @indriolo2012]. The inclusion of CR attenuation allows us to directly test the accuracy of these approximations. Our astrochemical models provide the abundances throughout the cloud and the local CRIR in-situ. We test two different prescriptions that are typically used for diffuse and dense gas, respectively, from [@indriolo2012]. The first, and simplest, denoted as “Simple Electron Balance” (SEB), estimates the CRIR using only the abundance of H$_3^+$ and $e^-$: $$\label{eq:ebal} \zeta = k_e n({\rm e^-}) \frac{n({\rm H_2})}{n({\rm H_3^+})},$$ where $k_e$ is the H$_3^+$ electron-recombination rate and $n({\rm e^-})$, $n({\rm H_2})$ and $n({\rm H_3^+})$ are the densities of electrons, molecular hydrogen and H$_3^+$, respectively. The second approximation includes the destruction of H$_3^+$ with CO and O, which we denote the “Reduced Analytic” (RA) model: $$\label{eq:simpanaly} \zeta = \frac{x({\rm H_3^+})}{x({\rm H_2})} n_H \left [ k_e x({\rm e^-}) + k_{\rm CO} x({\rm CO}) + k_O x({\rm O}) \right ] \times \left [ 1 + \frac{2 k_3 x({\rm e^-})}{k_2 f({\rm H_2})} + \frac{2 k_4}{k_2} \left ( \frac{1}{f({\rm H_2})} - 1 \right ) \right ],$$ where $k_i$ are the reaction rate coefficients for the reactions in Table from [@indriolo2012] (repeated in Table \[tab:react\] below), $x_i$ is the abundance of species i with respect to total Hydrogen nuclei and $f({\rm H_2}) = 2n({\rm H_2})/n_H$ is the molecular hydrogen fraction. Figure \[fig:zeta\] shows the calculated CRIR using the full model and the approximations in Equations \[eq:ebal\] and \[eq:simpanaly\] as a function of the H$_3^+$ abundance. We show the cases of four different CR models: LNA, LNI, LDI and HDI. The first model, LNA, is of particular importance since it represents the simplest one-dimensional PDR model. Observations typically assume 0D distribution, such that the ratio of column densities is equal to the ratio of the abundances. This makes a tacit assumption that the ionization rate is the same throughout the domain. We find that both approximations produce a large range of CRIRs – even when the input CRIR is fixed due to other effects impacting the chemistry, notably the influence of the external FUV radiation. [*The SEB and RA approximation models systematically underestimate the CRIR and produce an artificial spread in the inferred CRIR.*]{} When internal sources are included, we find that both approximations infer the CRIR reasonably well. When there are no internal sources, both approximations underestimate the CRIR by up to an order of magnitude and in general do not represent any real spread in the CRIR accurately. [ccc]{} \[tab:react\] H$_2^+$ + H$_2$ $\rightarrow$ H$_3^+$ + H & $k_2 = 2.09 \times 10^{-9}$ & [@theard1974]\ H$_2^+$ + e$^-$ $\rightarrow$ H + H & $k_3 = 1.6\times10^{-8} (T/300)^{-0.43}$ & [@mitchell1990]\ H$_2^+$ + H $\rightarrow$ H$_2$ + H$^+$ & $k_4 = 6.4\times10^{-10}$ & [@karpas1979]\ H$_3^+$ + e$^-$ $\rightarrow$ products & $k_5 = k_e = -1.3\times10^{-8} + 1.27 \times 10^{-6} T_e^{-0.48}$ & [@mccall2004]\ H$_3^+$ + CO $\rightarrow$ H$_2$ + HCO$^+$ & $k_6 = 1.36\times10^{-9}(T/300)^{-0.142} \exp{3.41/T}$ & [@klippenstein2010]\ H$_3^+$ + CO $\rightarrow$ H$_2$ + HCO$^+$ & $k_7 = 8.49\times10^{-10}(T/300)^{0.0661}\exp{-5.21/T}$ & [@klippenstein2010]\ H$_3^+$ + O $\rightarrow$ H$_2$ + OH$^+$ & $k_8 = k_O = 1.14\time10^{-9}(T/300)^{-0.156}\exp{-1.41/T}$ & [@klippenstein2010]\ ![\[fig:crspecs\] Proton cosmic ray spectrum with line color indicating position within the cloud for $\Sigma_{\rm cl} = 0.75$ g cm$^{-2}$ and N$_*$ = 750 using the LDI CR model. Inset: Cosmic ray ionization rate versus position, $x$, into the cloud, where $x = 0$ is the cloud surface.](fig1.pdf){width="50.00000%"} {width="85.00000%"} ---------------------------------------- {width="85.00000%"} {width="85.00000%"} {width="85.00000%"} Impact of Cosmic Ray Sources on Cloud Chemistry {#sec:chem} ----------------------------------------------- We examine in detail two different CR models: the traditional LNA and the LDI model. Figure \[fig:colVirNA\] shows the column densities of different species and density averaged temperature and CRIR for the LNA model as a function of $\Sigma_{\rm cl}$ and $N_*$. The total column density of each species increases with the gas surface density, $\Sigma_{\rm cl}$, and hence the total gas mass. Furthermore, we find across the whole parameter space that N(CO) $>$ N(C) $>$ N(C$^+$). This qualitative behavior is to be expected with no internal sources. Figure \[fig:profVirNA\] shows the abundance profiles for the LNA model as a function of $A_V$ into the cloud, $\Sigma_{\rm cl}$ and $N_*$. Since there are no embedded sources in this model, there is no difference between models of different $N_*$. The abundance profiles for C$^+$, C and CO exhibit the expected “layered” behavior [@draine2011]: C$^+$ is confined to the surface, C exists in a thin, warm layer and CO asymptotically approaches an abundance of \[CO/H$_2$\] $\approx 10^{-4}$. Similarly the abundance of NH$_3$ steadily increases into the cloud. The abundance ratio ${\rm [HCO^+/N_2H^+]}$ is sometimes used to infer the CRIR under the assumption the two molecules are co-spatial [i.e., @ceccarelli2014]. We find that, while they share some local maxima, they are not completely co-spatial [in agreement with]{} the turbulent cloud study of [@gaches2015].Moreover, observations show that while they are not entirely co-spatial, there is overlap in the emission regions [i.e., @jorgensen2004; @ceccarelli2014; @storm2016; @favre2017; @pety2017; @pound2018]. In particular, we show HCO$^+$ can exist at much lower $A_V$ than N$_2$H$^+$. Due to similar critical densities however, the two molecules thermalize at nearly the same densities. Figure \[fig:colVir\] shows the column densities across the parameter space for the LDI model. Here we find a very different behavior compared to the LNA model shown in Figure \[fig:colVirNA\], where the differences are especially pronounced for the more diffuse gas tracers. The column densities are no longer strictly functions of $\Sigma_{\rm cl}$ but depend on $N_*$. For large, massive star-forming regions (upper right corner in each panel), the gas becomes CO deficient and C rich while the bulk of gas remains molecular. Similarly, there is a slight increase in the column density of HCO$^+$ and N$_2$H$^+$ due to the increase in ionization. The qualitative trends exhibited by C$^+$, C, and marginally by HCO$^+$ and N$_2$H$^+$, follow that of the density-averaged CRIR, $\langle \zeta_\rho \rangle$. The effect of an embedded protocluster is also visible in the abundance profiles. Figure \[fig:profVir\] shows the abundance profiles for the LDI model. We find that CO only approaches abundances of $10^{-4}$ for clusters with little embedded star formation.For smaller mass clouds (i.e., smaller $\Sigma_{\rm cl}$), the C$^+$, C and CO abundance remains fairly unchanged compared to the LNA model. In the most massive clouds, the amount of CO at $A_V \leq 1$ is enhanced by an order of magnitude and reduced by an order of magnitude at $A_V \geq 5$. We further see a reduction in N$_2$H$^+$ at mid-$A_V$ with an enhancement of HCO$^+$. Likewise the gas temperature exceeds $T > 30$ K for most of the clouds with $\Sigma_{\rm cl} > 0.25$ g cm$^{-2}$. As $A_V \rightarrow 10^3$, the differences between the molecular ion abundances is much less due to the greatly increased density compared to the surface of the cloud. The abundance of H$_2$ in the dense gas is unaffected by the increased CRIR. We now statistically quantify the impact of different CR models on the six different molecules: C$^+$, C, CO, N$_2$H$^+$, HCO$^+$ and H$_3^+$. We investigate the H$_3^+$ column density because it is the simplest molecule that can be used to constrain the CRIR [@dalgarno2006]. We calculate the column density logarithmic difference: $$\Delta_s = \log \frac{N_{s,i}}{N_{s, {\rm LNA}}},$$ with $s$ representing the different species, and $i$ the different CR models exluding the LNA model. Figure \[fig:violins\] shows violin plots representing the probability distribution of $\Delta_s$ using all clouds in the $(\Sigma_{\rm cl}-N_*)$ space. In all cases, CO is never enhanced but rather depleted. This is because the maximum abundance \[CO\] = 10$^{-4}$ is set by the C/O ratio. Our models generally increase the local CRIR, thereby dissociating the CO and reducing its abundance. We find very little difference between the LNI and LNA for all molecules except for N$_2$H$^+$ and HCO$^+$, which exhibit a 25% linear dispersion. This is caused by the impact of higher ionization rates towards the surface of the clouds. The HNI model,which has the highest overall CRIR at the surface, shows a clear offset for the atomic and ionic species and a slight deficit for CO. In models LRI, LDI and HDI there is a significant dispersion in the column density difference, $\Delta_s$, in all species. Figure \[fig:violins\] demonstrates that considerable care must be taken when modeling observed column densities of atomic or ionic species: the possible error, $\Delta_s$, in the modeled column densities may be off by an order of magnitude depending on the transport of the cosmic rays and the amount of ongoing embedded star formation. The CRIR is not the only factor that leads to the creation of molecular ions, as typically assumed in observational studies. The abundances are influenced by the FUV flux, which is also enhanced by a central protocluster [@gaches2018a]. {width="\textwidth"} Abundance Ratio Diagnostics for the Cosmic Ray Ionization Rate -------------------------------------------------------------- Line and abundance ratios of various tracers are often used to constrain the CRIR in dense gas. The species are typically assumed to be co-spatial (although as demonstrated in Section \[sec:chem\] that is not typically the case). We examine two different ratio diagnostics: global diagnostics using column densities and local diagnostics using the local abundance ratios and CRIRs. Figure \[fig:ratsI\] shows three different column density ratios for the LNA, LNI and HNI models: ${\rm [HCO^+/N_2H^+]}$, ${\rm [CO/C^+]}$ and ${\rm [C/C^+]}$. We find that the column density ratios in these cases increase monotonically with $\Sigma_{\rm cl}$. The ratio of ${\rm [HCO^+/N_2H^+]}$ is nearly constant, changing by less than a factor of two across two dex of $\Sigma_{\rm cl}$. The HNI case shows a slightly different behavior with a slight local minimum in ${\rm [HCO^+/N_2H^+]}$ at $\Sigma_{\rm cl} = 6$ g cm$^{-2}$. The trends in these models are not due to changes in the CRIR but rather in the total amount of gas column. The \[CO/C$^+$\] ratio shows a buildup of CO compared to C$^+$. This is to be expected in an externally irradiated model: C$^+$ remains consistently on the surface, while the amount of CO continues to build with $\Sigma_{\rm cl}$ with a proportional increase in the amount of dense gas. Similarly, the \[C/C$^+$\] remains fairly constant since these species exist only in limited areas of $A_V$. Figure \[fig:ratsS\] shows the same column density ratios for three models including the embedded protoclusters: LRI, LDI and HDI. Here the trends are significantly more complicated. The ${\rm [HCO^+/N_2H^+]}$ ratio still only varies by a factor of two throughout the parameter space, but it exhibits more complex behavior. The ratio decreases with $\Sigma_{\rm cl}$ and rises with $N_*$ up to some maximum, with an additional increase in ${\rm [HCO^+/N_2H^+]}$ for N$_* \approx 10^4$ for the LDI and HDI model. To understand this, we can look at the abundance profiles of the LDI model in detail in Figure \[fig:profVir\]. The abundance of HCO$^+$ increases with both with $\Sigma_{\rm cl}$ and N$_*$ with the abundance profile flattening as a function of $A_V$ for $\Sigma_{\rm cl} > 1$ g cm$^{-2}$. For N$_2$H$^+$ the trends are separated by an A$_V$ threshold at $A_V = 1$. At $A_V < 1$, the abundance of N$_2$H$^+$ increases like HCO$^+$, with $\Sigma_{\rm cl}$ and N$_*$. For $1 < A_V < 100$, the abundance of N$_2$H$^+$ is sensitive primarily to N$_*$. In high ionization environments, CO will be destroyed in the creation of HCO$^+$ due to interactions with H$_3^+$. These environments will also produce N$_2$H$^+$ which destroys CO to create HCO$^+$. This is likely the main driving cause in the abundance profiles: there is a reduction of CO and N$_2$H$^+$ in the dense more ionized gas, and an systematic increase in HCO$^+$. The ${\rm [CO/C^+]}$ ratio increases monotonically across two orders of magnitude towards high $\Sigma_{\rm cl}$ and low $N_*$: cold gas is less ionized (lower right corner), so the amount of CO increases with respect to C$^+$. ${\rm [C/C^+]}$ shows a different trend compared to ${\rm [CO/C^+]}$. High ionization rates, in both the LDI and HDI models, have an increased ${\rm [C/C^+]}$ in lower mass clouds hosting smaller clusters and a decreased ${\rm [C/C^+]}$ at high $\Sigma_{\rm cl}$ compared to the LRI model. The ${\rm [C/C^+]}$ ratio is nearly flat across the $\Sigma_{\rm cl} - N_*$ parameter space in the HDI model. Clouds with fewer CRs and more gas shielding to the incident the FUV radiation have more C compared to C$^+$. Observational measurements of $\zeta$ in dense gas typically use astrochemical modeling and local abundance ratios (See §\[sec:compcrir\]). Figure \[fig:zetaLocal\] plots the CRIR for models with $5\% \leq \varepsilon \leq 25\%$ as a function of different abundance ratios. A good CRIR indicator should exhibit a monotonic trend in response to changes in the CR flux. The LNI model does not exhibit much change in the CRIR, so the local trends depend on density and radiative effects. In the HNI model, only the ${\rm [HCO^+/CO]}$ ratio exhibits a monotonic trend. The models with sources show completely different abundance ratios because the dense gas is warmer and the ionization rates are higher. In all of these cases, the ratios are monotonic for $\Sigma_{\rm cl} \gtrsim 1$ g cm$^{-2}$. For $\Sigma_{\rm cl} \lesssim 1$ g cm$^{-2}$ each exhibits a similar trend as in the NI model subsets. This demonstrates that these diagnostics only constrain regimes where the CRIR influences the chemistry more than radiative or other heating processes. {width="\textwidth"} ----------------------------------------- {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} ----------------------------------------- {width="\textwidth"} {width="\textwidth"} {width="90.00000%"} ----------------------------------------- {width="90.00000%"} {width="90.00000%"} {width="90.00000%"} Discussion {#sec:disc} ========== Model Assumptions and Caveats {#sec:caveat} ----------------------------- Our models require a variety of assumptions. First and most importantly, the models are one-dimensional and we assume protostars are clustered in the center. In reality, protostars are distributed throughout molecular clouds. Furthermore, the density distribution of molecular clouds is set by turbulence and is not a purely radial distribution. However, our results will hold qualitatively for the molecular gas around young, dense embedded clusters in molecular clouds, such as the central cluster in $\rho$ Oph. Second, our chemical network does not include any gas-grain chemistry or freeze-out [see @mcelroy2013]. Therefore, we over-predict the CO abundance in regions where $n > 10^4$ cm$^{-3}$ and $T \lessapprox 30$ K [@vandishoeck2014]. Comparing this criteria to the temperature profiles in Figure \[fig:profVir\] shows the models with $\Sigma_{\rm cl} < 0.5$ g cm$^{-2}$ and where $1 \leq A_V \leq 10$ are below the freeze-out temperature. When embedded sources are included, the densest gas heats to temperatures $> 50$ K. These temperatures lead to desorption from the grains, producing gas-phase CO: any CO-ice that formed before star formation occured would be evaporated back into the gas phase [@jorgensen2013; @jorgensen2015]. We do not fully solve the cosmic ray transport equations or the acceleration dynamics of protons in protostellar accretion shocks. We use analytic approximations to describe the acceleration of CRs at the protostellar shock and the transport out of both the parent core and cloud. [@gaches2018b] explore the changes in the CRIR for different transport regimes, shock efficiencies and magnetic fields. Differences in the protostellar magnetic field changes the maximum energy of the accelerated CRs but has little effect on the CRIR. However, the CRIR scales nearly linearly with the shock efficiency. Our results assume CR transport in the rectilinear regime through the core. More diffusive transport would produce higher temperatures at the surface of the core than observed. The details of the transport depend both on the magnetic field morphology and on the coupling between the particles and the field. In molecular clouds, turbulence is much stronger than in the cores allowing CRs to diffuse across magnetic field lines rather than streaming along them [@schlickeiser2002]. Conversely, if the particles are well-coupled their trajectories would follow the field lines, potentially producing asymmetries in the CR flux. The directionality imposed by the protostellar outflow could cause CR beaming in the outflow direction or simply advect the particles along with the outflow gas [@rodgers-lee2017]. We assume CRs transport from their parent cores through the clouds by parameterizing the radial scaling by either diffusive ($1/r$) or free-streaming ($1/r^2$). We do not fully solve the transport equations, which has yet to be done for CRs propagating out of molecular clouds from internal sources. Comparison to Observed CRIRs {#sec:compcrir} ---------------------------- ![\[fig:zetaav\] Cosmic ray ionization rate, $\zeta$, versus $A_V$ for the six different models in Table \[tab:models\]. The filled curves represent the $1\sigma$ spread from the models covering the ($\Sigma_{\rm cl}$ - $N_*$) parameter space. Squares represent diffuse gas measurements from [@indriolo2012] and [@indriolo2015]. Diamonds represent dense gas measurements from [@caselli1998]. Crosses represent observations towards high-mass protostars from [@boisanger1996; @vandertak2000].](fig11.pdf){width="50.00000%"} Figure \[fig:zetaav\] shows the results from the different PDR models in Table \[tab:models\] compared to four different observational surveys covering a range of $1 \leq A_V \leq 10^3$. The CRIR is one of the trickier astrochemical parameters to constrain from observations. Unfortunately, no universal method is applicable to all clouds conditions. Historically, there have been two main methods: absorption measurements of simple ions, such as H$_3^+$ or OH$^+$, in the infrared or molecular line observations using key molecules in neutral-ion pathways along with astrochemical modeling. H$_3^+$ is typically thought to be among the best tracers of the CRIR due to its simple chemistry. However, H$_3^+$ is only observed in infrared absorption, limiting its use to sight lines with bright background sources. [@indriolo2012] and [@indriolo2015] used H$_3^+$, H$_2$O$^+$ and OH$^+$ absorption to trace the CRIR in diffuse gas with $A_V < 1$ and found the CRIR in low $A_V$ gas varies between $10^{-16}$ -$10^{-14}$ s$^{-1}$. The gas at low A$_V$ is particularly sensitive to external influences, motivating the need to model the chemistry with external CR spectra derived from examining the local galactic environment. The grouping of points at low $A_V$ with high CRIR ($\zeta \geq 10^{-14}$ s$^{-1}$) are clouds in sightlines towards the galactic center and thus in environments with extreme external particle irradiation. [@caselli1998] used a combination of HCO$^+$, DCO$^+$ and CO together with analytic chemistry approximations to infer the CRIR in 24 dense cores. Their observations exhibit a nearly bi-model distribution: some are clustered at $\zeta \lessapprox 10^{-17}\,{\rm s}^{-1}$, while the majority are at $\zeta \approx 10^{-16}\,{\rm s}^{-1}$. They infer the ionization rates using the abundance ratios of \[DCO$^+$/HCO$^+$\] and \[HCO$^+$\]/\[CO\] under 0D spatial assumptions and a reduced analytic chemical network. Finally, we include the CRIRs from the [@vandertak2000] survey towards single high-mass protostars with the central protostar being massive enough to provide a bright background source for H$_3^+$ absorption. They find CRIRs scattered from $10^{-17}$ to 10$^{-16}$ using an assumed H$_3^+$ abundance and density distribution. They find the observed H$_3^+$ column density increases with cloud distance, which can be explained by contamination from low-density clouds along the line of sight. Our model results show good agreement with the inferred CRIRs from [@indriolo2012], [@indriolo2015] and [@caselli1998]. We find the LDI model is able to replicate the spread in the CRIR. There are two main controlling factors for the CRIR in the clouds: the number of embedded sources and the cloud environment. Embedded sources create a natural dispersion in $\zeta$ for different molecular cloud masses and star formation efficiencies. Without internal sources, there is no spread in our modelled CRIR as a function of column density. In order to represent the observations, the external CRIR must be increased instead for different regions. Local sources of CRs, such as nearby OB associations or supernova, contribute significantly to the CR flux at the cloud boundary. As the external CR flux is increased, the impact of attenuation also increases due to the rapid reduction in low energy CRs. Figure \[fig:zetaav\] shows that the impact of attenuation is different between the HNI and LNI models. For the LNI model, $\zeta$ changes by less than an order of magnitude across 4 orders of magnitude in $A_V$. Conversely, the HNI model CRIR decreases by 2 orders of magnitude due to an overall reduction in MeV-scale CRs. The HNI model over predicts the CRIRs measured in diffuse gas to CRIRs measured near high-mass protostars, excluding the galactic center sightlines. However, the $\mathcal{H}$ spectrum is the maximal CR spectrum from Voyager-1 observations [@stone2013; @ivlev2015]. The LNI and LNA models under-predict the observed CRIR for all but a few sight-lines. Thus, Figure \[fig:zetaav\] demonstrates that it is essential to consider the cloud environment and properly treat the CR physics and cloud density distributions. Models without attenuation only represent the CRIR within narrow ranges of $A_V$ and not in the cloud interiors. Figure \[fig:zetaav\] also underscores that the low energy CR spectrum, which if often adopted in astrochemical modelling, is a poor fit to the majority of the observations. Challenges for Deriving the CRIR from Chemical Diagnostics {#sec:constrain} ---------------------------------------------------------- There have been numerous attempts to find chemical diagnostics that are strong tracers of the CRIR [@caselli1998; @neufeld2010; @neufeld2017; @indriolo2007; @indriolo2012; @indriolo2015; @albertsson2018]. Some of these, such as \[DCO$^+$\]/\[HCO$^+$\], cannot be modelled with the current [3d-pdr]{} version due to the lack of deuterium and isotopic chemistry. Most probes of the CRIR are based on the local abundance, which is difficult to directly ascertain from observations. The use of column density ratios typically assumes the line emission observed between species is co-spatial. Figures \[fig:ratsI\] and \[fig:ratsS\] examine effects of CR physics on the ${[\rm HCO^+/N_2H^+]}$, ${[\rm CO/C^+]}$ and ${\rm [C/C^+]}$ column density ratio diagnostics. However, we find that none of these ratios are monotonically sensitive to the average CRIR, shown for the LDI model in Figure \[fig:colVir\]. The \[CO/C$^+$\] column density ratio is anti-correlated with the density-averaged CRIR because the amount of CO declines while the amount of C$^+$ increases in large, massive clusters. Local abundance ratios are used for fitting observations with astrochemical models which assume the ratio of the column densities is equal to the ratio of the abundances [@indriolo2012]. These are constrained through the use of 0-D spatial models, where a single density, temperature and extinction are evolved over time. However, this ignores the physical structure of clouds, which have non-uniform density, temperature, FUV and, as we show here, CRIR distributions [@clark2012; @offner2013; @gaches2015; @glover2015; @seifried2017]. As Figure \[fig:zetaLocal\] shows, in models without internal sources, none of the abundance ratios are strong diagnostics. Furthermore, the range in the CRIR is small despite some large changes in each of the ratios. For models with internal sources, the CRs from embedded sources dominate the chemistry throughout the cloud. Here, we find the ratios of C/C$^+$ and HCO$^+$/CO are mostly monotonic with the CRIR. However, there is significant variation with $\Sigma_{\rm cl}$ and thus with the gas mass. The results signify that more careful physical and chemical modeling needs to be done to accurately constrain the CRIR in high-surface density, star-forming regions. The ratios are only a good diagnostic in regions where CRs dominate the thermo-chemistry. Recently, [@lepetit2016] used H$_3^+$ absorption to infer the CRIR and physical conditions in the CMZ. They used a similar relation to Equation \[eq:ebal\]: $$N({\rm H_3^+}) = 0.96\frac{\zeta L}{k_e}\frac{f}{2x_e},$$ where $k_e$, $f$, and $x_e$ have the same definition as in Equations \[eq:ebal\] and \[eq:simpanaly\] and $L$ is the size of the cloud. They fit observed H$_3^+$ column densities with PDR models as a function of the gas density, $n_H$, and the size of the cloud, $L$. Most sightlines are well fit using their method by clouds with densities $10 \leq n_H \leq 100$ cm$^{-3}$ corresponding to $5 \lesssim L \lesssim 100$ pc. Our models suggest that these length scales would incur CR screening effects which would change the CRIR. Similarly, for the high density clouds, the energy losses will deplete MeV CRs and reduce the CRIR. In these cases, there is a further degeneracy in the $n_H - L$ plane resulting in an average decrease in the CRIR. The reduction would systematically produce model fits with lower densities to correct for the lower CRIR. [@rimmer2012] used a similar hybrid approach, adopting a prescription for $\zeta(N)$ ad-hoc with the [Meudon]{} PDR code [@lepetit2006] to model the Horsehead Nebula. They found their high $\zeta(N_H)$ model improved agreement over standard constant CRIR PDR prescriptions. However, their treatment of $\zeta(N)$ is static and fixed in time. The decoupling of CR attenuation and chemistry is only a good approximation if the abundance of neutral Hydrogen (H, H$_2$) does not change much in time, ensuring that the CR spectrum is constant in time. The new approach presented here will allow $\zeta(N)$ to be connected to the chemical time evolution. Impact of Cosmic Ray Feedback on Cloud Chemistry {#sec:chemImpact} ------------------------------------------------ The [*Herschel*]{} Galactic Observations of Terahertz C+ (GOT C+) [@pineda2013] survey mapped \[C II\] 158 $\mu$m emission over the whole galactic disk, providing the best constraint on where \[C II\] emission originates. [@pineda2013] found that nearly half of the \[C II\] emission originates from dense photon dominated regions with about another quarter of the emission from CO-dark H$_2$ gas. [@clark2019] performed synthetic observations of young simulated molecular clouds and found the majority of their \[C II\] emission originates from atomic-Hydrogen dominated gas. This discrepancy was explained by the time evolution of molecular gas due to star formation and feedback. The results presented here provide an complementary explanation for the \[C II\]-bright molecular gas. When protoclusters become active, they accelerate CRs into the densest regions of molecular clouds. Figure \[fig:profVir\] shows that high-mass protoclusters will lead to \[C II\]-bright H$_2$ dominated gas since CRs i) increase the gas temperature to values closer to the \[C II\] excitation temperature of 91.2 K, ii) increase the abundance of C$^+$ in dense gas due to the destruction of CO and iii) do not significantly alter the abundance of H$_2$. In local star-forming regions, the lowest inversion transitions of ammonia, NH$_3$, have been widely used to map the dense gas cores within molecular clouds [i.e., @goodman1993; @jijina1999; @rosolowsky2008; @wienen2012]. [Ammonia remains optically thin, and while it does suffer from depletion, it’s formation is enhanced in regions where CO freezes out [@caselli2012] (although this effect is not included in our models)]{}. However, this also makes ammonia much more susceptible to local variations in the FUV radiation field, temperature and CRIR. Recently, the Green Bank Ammonia Survey (GAS) mapped all the Gould Belt clouds with $A_V > 7 \, {\rm mag}$ [@friesen2017]. The DR1 data show the line-of-sight averaged abundance, $X({\rm NH_3}) = N({\rm NH_3})/N({\rm H_2})$, exhibits a spread through molecular clouds. The spread could be caused by the porosity of molecular clouds allowing more FUV radiation into regions of dense gas. However, our 1D models also exhibit a variation in this abundance measurement for the models with internal sources (LRI, LDI and HDI). Figure \[fig:profVir\] shows that ammonia is depleted in clusters exhibiting more embedded star formation by a couple of orders of magnitude. The abundance within the dense gas goes from 10$^{-8}$ in small clusters to 10$^{-10}$ in the largest. Furthermore, the gas also heats up leading to stronger emission in higher transitions, such as NH$_3$(3,3). [@redaelli2017] examined the NH$_3$ GAS map of the Barnard 59 clump in more detail. They found that the abundance appears to drop in gas around the central central 0/1 protostar. The dust temperature shows a clear increase around the same source with a slight increase in the ammonia excitation temperature. Impact of Cosmic Ray Feedback on Chemistry in Dense Cores --------------------------------------------------------- Protostars are observed to be dimmer than classic collapse models predict, i.e., the “Protostellar Luminosity Problem" [@dunham2014]. One possible solution is that accretion is strongly episodic [@kenyon90; @vorobyov09; @offner2011]. Although our models assume steady-state accretion, we can infer the impact of large bursts of accretion on cloud chemistry. An accretion burst leads to a stronger accretion shock, which in turn produces higher energy CRs and a higher CRIR. The CRIR increase in the dense gas then raises the temperature. The higher temperatures, whether caused by radiative or CR heating, lead to several different chemical effects. First, molecules frozen onto dust grains will evaporate, both by thermal desorption [@oberg2016] and CR-induced desorption [@hasegawa1993], into the gas phase. Second, the increase of the CRIR will increase the ionization fraction leading to a chain of ion-neutral reactions following H$_3^+$. Finally, the elevated radiative and CR flux may be strong enough to destroy some molecules. [@jorgensen2015] and [@frimann2016] showed that episodic accretion can cause the sublimation of CO-ice and explain the excess C$^{18}$O emission observed near protostars. Intuitively, a burst a CRs will lead to a reaction chain: H$_3^+$ is created, thereby leading to the destruction of CO to form HCO$^+$. However, the increase in CRs will provide a large population of free electrons which recombine with HCO$^+$ to form CO. HCO$^+$ also interacts with water and other dipole neutrals (in the case of water, the reaction leads to the formation of CO and protonated water). HCO$^+$ is observed to be depleted near protostars that have undergone episodic accretion [@jorgensen2013]. Ices sublimated by an accretion burst will cause a more active gas-phase chemistry and lead to an increase in carbon-chain molecules in molecular clouds as well as increase gas-phase CO in the dense gas where it would otherwise freeze-out. Overall, the addition of CRs magnifies the effect of an accretion burst. Temperatures increase beyond that expected from radiative heating alone. This suggests that a smaller change in accretion rate may be needed to produce the observed chemical changes. Implications for Comparing Data and Models ------------------------------------------ Synthetic observations of hydro-dynamic simulations are a vital tool for comparing theoretical predictions to observations. The synthetic observations may treat the chemistry in different ways: from assuming a constant abundance of some molecule to post-processing simulations with an astrochemical code or using the reduced-network chemistry from the hydrodynamic simulation [see review by @haworth2018]. These synthetic observations are used to gauge how well the simulations correspond to the observed universe. As such, it is paramount to ensure that all astrochemical parameters are as accurate as possible. Radiation-MHD simulations can provide the density and temperature at every point [e.g., @Offner2009]. Simulations also now often include FUV radiation and optical depth calculations using packages such as [Fervent]{} [@baczynski2015], [TreeRay]{} (Wünsch et al., in preparation, used in [@haid2019]) and [Harm$^2$]{} [@rosen2017]. These methods can provide the FUV radiation and/or optical depth into the cloud. Our results, here, show that the H$_2$ optical depth into the cloud should be considered when calculating the appropriate CRIR for post-processing. Typically, the CRIR is held constant throughout the entire simulation domain, which will lead to systematic differences in the simulation line emission. Conclusions =========== We implement cosmic-ray attenuation in the public astrochemistry code [3d-pdr]{}. The implementation uses the H$_2$ column density from the chemistry to attenuate the CR spectra. We couple the code to the protostellar CR models from [@gaches2018b], which solve for the total attenuated protocluster CR spectrum as a function of the cloud surface density, $\Sigma_{\rm cl}$, and number of constituent protostars, N$_*$. We present one-dimensional astrochemical models for molecular clouds with a wide range of $\Sigma_{\rm cl}$ and $N_*$. We compare the abundance distributions for a low external CR spectrum, representing an extrapolation of the Voyager-1 data, and a high external CR spectrum, representing a maximal correction for solar influences. Our model results show that CRs originating from the accretion shocks of protostars affect the chemistry of the surrounding molecular cloud. We conclude the following: - Models with no sources or attenuation cannot explain observed CRIRs. Models with no internal sources but a higher ($\mathcal{H}$) external spectrum (HNI) match the observed CRIRs, although it may under-predict the CRIRs inferred for high-mass protostars. We find that a model using the commonly adopted spectrum with internal sources (LDI) matches both the low and mid $A_V$ observations of $\zeta$ and the observed spread. - CRs accelerated by protostellar accretion shocks significantly alter the Carbon chemistry in star-forming clouds. The amount of neutral and ionized Carbon increases in the dense gas as the number of protostars increases. Models with embedded sources (LDI, LRI, HDI) increase the amount of C, HCO$^+$ and NH$_3$ at lower $A_V$ and decrease the abundance of CO and NH$_3$ at higher $A_V$. Overall, models including internal sources (LDI, LRI and HDI) exhibit a higher abundance of HCO$^+$ and H$_3^+$ with $\Sigma_{\rm cl}$ and $N_*$. - Approximations that use H$_3^+$ and C-based tracers to estimate the CRIR systematically under-predict the CRIR unless CRs are the dominant source of ionization. The Reduced Analytic Approximation, which uses the abundances of H$_3^+$, CO and O,always produces more accurate values of the CRIR, highlighting the importance of obtaining accurate Oxygen and Carbon Monoxide abundances within molecular clouds. - Ions are systematically under produced using the canonical CRIR while CO is over produced. Internal sources created a dispersion in the distribution of column densities by driving more active ion-neutral chemistry deep within molecular clouds. - Models using the low external CR spectrum ($\mathcal{L}$) and/or no internal sources of CRs under estimate the H$_3^+$ column density by a order of magnitude or more. - Internally-accelerated CRs will naturally lead to molecular gas which become CO-deficient but \[C II\]-bright, particularly for high surface density molecular clouds hosting large clusters. - Including CR attenuation in PDR models will help break the denegeracy in astrochemical modeling between the density, CRIR and FUV radiation. As protoclusters grow in constituent nubmers, the impact on the chemistry is amplified, greatly so if CRs diffuse out of molecular clouds rather than stream. Comparison to observed CRIRs suggest the external CR spectrum, attenuation and internal sources are important for modelling the chemistry of molecular clouds. However, the current uncertainties are large due to lack of observational data that can simultaneously constrain the density, FUV radiation and CRIR on molecular cloud scales. Observations to constrain the CRIR within dense gas necessitate multi-line data, to independently determine the temperature as in e.g., [@ceccarelli2014], and multi-species data, to act as astrochemical diagnostics as in e.g., [@caselli1998]. The [3d-pdr]{} CR attenuation [Fortran]{} module can be included in any [Fortran]{} astrochemistry code. SSRO and BALG acknowledge support from the National Science Foundation (NSF) grant AST-1510021. SSRO was also supported by NSF CAREER grant AST-1650486. TGB acknowledges funding by the German Science Foundation (DFG) via the Collaborative Research Center SFB 956 “Conditions and impact of star formation”. The authors thank helpful discussions with Neal Evans and Nick Indriolo and the anonymous referee for their useful comments. The calculations performed for this work were done on the Massachusetts Green High Performance Computing Center (MGHPCC) in Holyoke, Massachusetts supported by the University of Massachusetts, Boston University, Harvard University, MIT, Northeastern University and the Commonwealth of Massachusetts. [^1]: The code can be downloaded from <https://uclchem.github.io>, including the new modifications presented in this paper.

--- abstract: 'The XMM-OM instrument extends the spectral coverage of the XMM-Newton observatory into the ultraviolet and optical range. It provides imaging and time-resolved data on targets simultaneously with observations in the EPIC and RGS. It also has the ability to track stars in its field of view, thus providing an improved post-facto aspect solution for the spacecraft. An overview of the XMM-OM and its operation is given, together with current information on the performance of the instrument.' author: - 'K. O. Mason, A. Breeveld, R. Much, M. Carter, F. A. Cordova, M. S. Cropper, J. Fordham, H. Huckle, C. Ho, H. Kawakami, J. Kennea, T. Kennedy, J. Mittaz, D. Pandel, W. C. Priedhorsky, T. Sasseen, R. Shirey, P. Smith, J.-M. Vreux' title: 'The XMM-Newton Optical/UV Monitor Telescope' --- Introduction ============ The Optical/UV Monitor Telescope (XMM-OM) is a standalone instrument that is mounted on the mirror support platform of XMM-Newton (Jansen et al. 2001) alongside the X-ray mirror modules. It provides coverage between 170 nm and 650 nm of the central 17 arc minute square region of the X-ray field of view (FOV), permitting routine multiwavelength observations of XMM targets simultaneously in the X-ray and ultraviolet/optical bands. Because of the low sky background in space, XMM-OM is able to achieve impressive imaging sensitivity compared to a similar instrument on the ground, and can detect a $B=23.5$ magnitude A-type star in a 1000 s integration in “white” light (6 sigma). It is equipped with a set of broadband filters for colour discrimination. The instrument also has grisms for low-resolution spectroscopy, and an image expander (Magnifier) for improved spatial resolution of sources. Fast timing data can be obtained on sources of interest simultaneously with image data over a larger field. In the following sections we give an overview of the instrument followed by an account of its operation in orbit and the instrument characteristics. Instrument overview =================== The XMM-OM consists of a Telescope Module and a separate Digital Electronics Module, of which there are two identical units for redundancy (see Fig. 1). The Telescope Module contains the telescope optics and detectors, the detector processing electronics and power supply. There are two distinct detector chains, again for redundancy. The Digital Electronics Module houses the Instrument Control Unit, which handles communications with the spacecraft and commanding of the instrument, and the Data Processing Unit, which pre-processes the data from the instrument before it is telemetered to the ground. (8.8,10) (10,0) Optics ------ The XMM-OM uses a Ritchey Chrétien telescope design modified by field flattening optics built into the detector window. The f/2 primary mirror has a 0.3 m diameter and feeds a hyperboloid secondary which modifies the f-ratio to 12.7. A $45$[$^{\circ}$]{} flat mirror located behind the primary can be rotated to address one of the two redundant detector chains. In each chain there is a filter wheel and detector system. The filter wheel has 11 apertures, one of which is blanked off to serve as a shutter, preventing light from reaching the detector. Another seven filter locations house lenticular filters, six of which constitute a set of broad band filters for colour discrimination in the UV and optical between 180 nm and 580 nm (see Table 2 for a list of filters and their wavelength bands). The seventh is a “white light” filter which transmits light over the full range of the detector to give maximum sensitivity to point sources. The remaining filter positions contain two grisms, one optimised for the UV and the other for the optical range, and a 4 field expander (or Magnifier) to provide high spatial resolution in a 380–650 nm band of the central portion of the (FOV). Detector -------- The detector is a microchannelplate-intensified CCD (Fordham et al. 1992). Incoming photons are converted into photoelectrons in an S20 photocathode deposited on the inside of the detector window. The photoelectrons are proximity focussed onto a microchannelplate stack, which amplifies the signal by a factor of a million, before the resulting electrons are converted back into photons by a P46 phosphor screen. Light from the phosphor screen is passed through a fibre taper which compensates for the difference in physical size between the microchannelplate stack and the fast-scan CCD used to detect the photons. The resulting photon splash on the CCD covers several neighbouring CCD pixels (with a FWHM of approximately 1.1 CCD pixels, if fitted with a Gaussian). The splash is centroided, using a 33 CCD pixel subarray to yield the position of the incoming photon to a fraction of a CCD pixel (Kawakami et al. 1994). An active area of 256256 CCD pixels is used, and incoming photon events are centroided to 1/8th of a CCD pixel to yield 20482048 pixels on the sky, each 0.4765 arc seconds square. In this paper, to avoid confusion, while CCD pixels (256256 in FOV) will be referred to explicitly, a pixel refers to a centroided pixel (20482048 in FOV). As described later, images are normally taken with pixels binned 22 or at full sampling. The CCD is read out rapidly (every 11 ms if the full CCD format is being used) to maximise the coincidence threshold (see sect. 5.2). Telescope mechanical configuration ---------------------------------- The XMM-OM telescope module consists of a stray light baffle and a primary and secondary mirror assembly, followed by the detector module, detector processing electronics and telescope module power supply unit. The separation of the primary and secondary mirrors is critical to achieving the image quality of the telescope. The separation is maintained to a level of 2 by invar support rods that connect the secondary spider to the primary mirror mount. Heat generated by the detector electronics is transferred to the baffle by heat pipes spaced azimuthally around the telescope, and radiated into space. In this way the telescope module is maintained in an isothermal condition, at a similar temperature to the mirror support platform. This minimizes changes in the primary/secondary mirror separation due to thermal stresses in the invar rods. Fine focussing of the telescope is achieved through two sets of commandable heaters. One set of heaters is mounted on the invar support rods. When these heaters are activated, they cause the rods to expand, separating the primary and secondary mirrors. A second set of heaters on the secondary mirror support brings the secondary mirror closer to the primary when activated. The total range of fine focus adjustment available is $\pm10\mu$m. The filter wheel is powered by stepper motor, which drives the wheel in one direction only. The filters are arranged taking into account the need to distribute the more massive elements (grisms, Magnifier) uniformly across the wheel. Digital Electronics Module -------------------------- There are two identical Digital Electronics Modules (DEM) serving respectively the two redundant detector chains. These units are mounted on the mirror support platform, separate from the telescope module. Each DEM contains an Instrument Control Unit (ICU) and a Digital Processing Unit (DPU). The ICU commands the XMM-OM and handles communications between the XMM-OM and the spacecraft. The DPU is an image processing computer that digests the raw data from the instrument and applies a non-destructive compression algorithm before the data are telemetered to the ground via the ICU. The DPU supports two main science data collection modes, which can be used simultaneously. In Fast Mode, data from a small region of the detector are assembled into time bins. In Image Mode, data from a large region are extracted to create an image. These modes are described in more detail in the next sect. The DPU autonomously selects up to 10 guide stars from the full XMM-OM image and monitors their position in detector coordinates at intervals that are typically set in the range 10–20 seconds, referred to as a tracking frame. These data provide a record of the drift of the spacecraft during the observation accurate to $\sim 0.1$ arc second. The drift data are used within the DPU to correct Image Mode data for spacecraft drift (see sect. 5.5). Observing with XMM-OM ===================== Specifying windows ------------------ The full FOV of XMM-OM is a square 1717 arc minutes, covering the central portion of the X-ray FOV. Within this field the observer can define a number of data collection windows around targets or fields of interest. Up to five different Science Windows can be defined with the restriction that their boundaries may not overlap. However, one window can be completely contained within another. Because of constraints on the telemetry rate available, it is not possible to transmit the full data on every photon that XMM-OM detects. Instead a choice has to be made between image coverage and time resolution. Thus two types of Science Window can be defined, referred to as Image Mode and Fast Mode. A maximum of two of the five available science windows can be Fast Mode. *Image Mode emphasizes spatial coverage at the expense of timing information. Images can be taken at the full sampling of the instrument or binned by a factor of 2 or 4, to yield a resolution element on the sky of approximately 0.5, 1.0 or 2.0 arc seconds (a factor of four finer for the Magnifier). The maximum total size of the Science Windows is determined by the memory available in the DPU. A single Image Mode window binned by a factor of 22 can be up to 976960 detector pixels, which results in a 488480 binned pixel image being stored in the DPU. At full sampling (with no binning) the window can be up to 652652 pixels. Any drift in the pointing direction of the spacecraft is corrected in the image by tracking guide stars (section 5.5).* *Fast Mode emphasizes timing information at the expense of spatial coverage. The maximum total number of pixels that can be specified for a Fast Mode window is 512. Thus the maximum size of an approximately square window would be 2223 pixels ( = 506 total). Note that there is no binning within a Fast Mode window. The pixel locations of individual photons within the window are recorded and assigned a time tag, which has a user-specified integration time of between 100 ms and the tracking frame duration (10–20 s). No tracking correction is applied to Fast Mode data. This can be applied on the ground, from the drift history supplied by XMM-OM.* To simplify observation set-up, two standard observing sequences of five exposures have been created that together cover the whole XMM-OM FOV at one arc second sampling while at the same time monitoring a central target at full spatial sampling (0.5 arcsec). In the first variant, each of the five exposures contains an unbinned Image Mode window centred on the prime instrument’s boresight (the position of the main target), and a second Image Mode window, binned by 22 pixels, that is defined in each of the set of five exposures so as to form a mosaic of the entire field (see Fig. 2). The second variant is exactly the same as the first except that a Fast Mode window is added around the prime instrument’s boresight position. (8.8,8) (0,0) The length of an XMM-OM Image Mode only exposure can be set in the range 800–5000 s. However it should be noted that there is an approximately 300 second overhead associated with each individual exposure. The maximum length of an exposure that contains a single Fast Mode window is 4400 s, or 2200 s if there are two Fast Mode windows. At the time of writing a further mode is being commissioned which allows the full field to be imaged at 1 arcsec sampling in one go, at the expense of tracking information and correction. This is made possible by the impressive stability of the XMM-Newton spacecraft compared to pre-launch expectation. Window coordinates can be specified either in detector pixels, or in sky coordinates. To facilitate the latter, the XMM-OM performs a short V-band observation at the start of each pointing. The DPU compares the image with the positions of uploaded field stars to calibrate the absolute pointing of the OM. Filter selection ---------------- The XMM-OM filter wheel rotates in one direction only and, to conserve the total number of wheel rotations over the expected lifetime of XMM-Newton, the number of filter wheel rotations per pointing is limited to one (unless there are very strong scientific arguments for more). Thus filter observations have to be executed in a particular order during a given target pointing. The filter elements are listed in the order they occur in the filter wheel in Table 2. The instrument is slewed with the blocked filter in place, and thereafter a field acquisition exposure is performed in V. The same telescope focus setting is used for all the filters except for the Magnifier (see sect. 5.6.1), where the optimum focus is different (the image quality is the most sensitive to focus position when using the Magnifier). The XMM-OM instrument is optimised for the detection of faint sources. If the source count-rate is too high the response of the detector is non-linear. This “coincidence loss” occurs when the probability of more than one photon splash being detected on a given CCD pixel within the same CCD readout frame becomes significant. Coincidence loss is discussed in more detail in sect. 5.2. If a source is predicted to exceed the coincidence threshold for a given filter, then a different filter with lower throughput can be selected. Alternatively a grism can be selected which disperses the available light over many pixels. The XMM-OM detectors can also be permanently damaged by exposure to a source that is too bright, reducing both the quantum efficiency of the photocathode and the gain of the channelplates. This is a cumulative effect dependent on the total number of photons seen over the lifetime of the instrument at a particular location on the detector. The deterioration is therefore more severe for longer observations of a bright source. For this reason limits are imposed on the maximum brightness of stars in the FOV (see Table 1) and apply to any star in the FOV irrespective of whether it is within a science window or not. Even more stringent limits are applied to the central region of the detector that will usually contain the target of interest. In the event that there is a star in (or near to) the FOV that violates the brightness constraints, a different filter, which has lower photon throughput, can be selected. Also, if the bandpass is appropriate, the Magnifier can be used to exclude bright stars further than a few arc minutes from the field centre. The grisms (one optimised for the UV, the other for the optical) form a dispersed first order image on the detector, together with a zeroth order image that is displaced in the dispersion direction. The counts in the zeroth order image of field stars determines the brightness limits used for observing with the grisms. OM performance ============== (8.8,10) (0,10) The first light observation for XMM-OM took place on 2000 January 11. Since then the various engineering and science data taking modes of the instrument have been commissioned including full-field image engineering mode (not generally available for science observations because of the very large telemetry overhead required to transmit the data to the ground), and the Image and Fast science modes. The telescope focus has been optimised using the heater-based fine focus control, and the gain of the image intensifier has been optimised. The performance of the DPU in tracking image motion due to spacecraft drift has also been verified and distortion maps derived to relate XMM-OM detector coordinates to the sky. Photometric calibrations have been derived for all filter elements, but work continues on colour equations and to tie these more accurately into standard systems. Similarly, preliminary throughput and wavelength calibrations have been derived for the Grisms. To illustrate the capabilities of XMM-OM, we show in Figure 3 images of part of the Lockman Hole field in the White Light filter, and in five of the six colour filters (the remaining filter, UVM2, was not used during this observation). The images contains an $R=18.1$ magnitude AGN identified in the ROSAT observation of the field, and referred to as R32 by Schmidt et al.  (1998). The AGN is clearly UV bright. The XMM-OM detects approximately 12from the AGN in White Light, while the count rate in the colour filters ranges from a high of 2.9 in U, down to about 0.25 in the UV filter UVW2. To illustrate the spectral capability of XMM-OM, we show in Figure 4 the extracted spectrum of the DA white dwarf standard BPM16274. The Balmer absorption lines can be clearly discerned. (8.8,7) (0,0) Analysis issues =============== Throughput ---------- An initial estimate of the zero points of the various XMM-OM broadband filters (i.e. the magnitude which yields 1 count per second; Table 2.) was derived from calibration observations of two white dwarfs. - The OM response throughput was determined based on measurements of the spectrophotometric standards BPM16274 and LBB227. - Using the OM throughput model an OM exposure of Vega was simulated. - The zeropoints in U-, B-, V-filter were fixed in a way that the brightness of Vega matches the literature values. - In the UV-filters the brightness of Vega was set to 0.025 mag; the average [*U*]{} magnitude of Vega in the literature being the most appropriate for the UV filters, and to 0.03 mag (the average Vega [*V*]{} magnitude) for the broadband Magnifier and the White Light filters. The calculated OM zeropoints are written into the relevant XMM-CCF file, to be used by the XMM Science Analysis System (SAS) (Watson et al. 2001). Updates of the zeropoint definitions as well as more precise colour transformations (Royer et al. 2000) to the standard UBV system are expected once the results of a dedicated ground based photometric observation programme become available. In the framework of this programme several OM calibration fields are anchored to high quality secondary photometric standards deep fields established by ESO. Early results of this ground observations are expected in October 2000. From analysis of the Lockman Hole field, avoiding the central 1 arc minute of the FOV where the background is enhanced (section 5.6.4) the limiting magnitude after 1000 s is calculated to be 21.0 in [*V*]{}, 22.0 in [*B*]{} and 21.5 in [*U*]{} (6 sigma). The limiting magnitude for the White Light filter is very dependant on the spectral type of the star, because the bandpass is so broad. However, for an A0 star we estimate that the 6 sigma limiting magnitude above background is $\sim$23.5. Coincidence loss and deadtime ----------------------------- Coincidence loss is observed whenever the count rate is such that more than one photon arrives in the same place within a given readout frame. Losses become significant for a point source at a count rate of about $10$ (for 10% coincidence) when the full CCD chip is being readout (i.e. about 2.5 magnitudes brighter than the zero points listed in Table 2). A factor of approximately two improvement can be achieved by restricting the area of the CCD used, since this reduces the time required to readout the chip. The coincidence loss can be approximated by $$ph_{\rm in}= { \log{(1-{cts}_{\rm detected}*T)} \over T_{\rm ft} - T } \label{equation1}$$ where\ ------------------------ ---------------------------------------- $ph_{\rm in}$ infalling photon rate per second ${cts}_{\rm detected}$ measured count rate per second $T$ CCD frametime in units of seconds ${T}_{\rm ft}$ frametransfer time in units of seconds ------------------------ ---------------------------------------- Equation 1 applies strictly in the case of a perfectly point-like source. In practice a real stellar profile has wings, and the formula will break down at very high rates when coincidence among photons in the wings of the profile becomes significant. The CCD deadtime depends on the size and shape of the science window used but can be calculated accurately. The deadtime correction should be applied by the SAS after any coincidence loss corrections. Flat fields ----------- An LED can be used to illuminate the detector by backscatter of the photons from the blocked filter. These images are not completely flat due to the illumination pattern of the LED, the gross shape of which could be removed by comparing with sky flats. However, using the LED allows a large number of events to be collected in every pixel to give sufficiently high statistics for pixel to pixel sensitivity to be measured and the relative measurement of any variation of the detector response on a fine scale. The LED brightness is adjustable and is currently operated at a level that produces $3.25\exp{-3}$  per binned (22) pixel. So far, flat fields have been obtained to the level of 400 counts per binned (22) pixel allowing an accuracy of 5% in the sensitivity measurement. A CCF file in the SAS currently represents the accuracy of flat fields obtained before mid June, which is at the 10% level. Once sufficient flat fields have been obtained for a 2–3% sensitivity the relevant CCF file will be updated. Background ---------- The background count rate in the OM is dominated by the zodiacal light in the optical. In the far UV the intrinsic detector background becomes important. Images are regularly taken with the blocked filter and no LED illumination to measure the detector dark counts. The mean OM dark count rate is $2.56\exp{-4}$  per pixel. The variation in dark count rate across the detector is $\pm 9\%$ and shows mainly a radial dependence, being highest in an annulus at about 8 arcmin radius and lowest in the centre. When the spacecraft is pointing at a very bright star, the dark rate is noticeably increased (e.g. up to 65% higher for the $V=0$ star Capella) despite the blocked filter. Excluding those dark frames taken during Capella ($V=0$) and Zeta Puppis ($V=2$) observations, the counts per dark frame vary by only $\pm 7\%$ and show no trend of change with time. Tracking performance -------------------- The positions of selected guide stars in the XMM-OM FOV are measured each 10–20 second tracking frame, and an X-Y offset applied to image mode data obtained during the tracking frame before they are added to the master image in the DPU memory. The tracking offsets are computed in pixels irrespective of the binning parameter chosen. Using this “Shift and Add” technique, the final image is corrected on timescales greater than a few tens of seconds and on spatial scales down to $\sim 0.5$ pixels, for drift in the pointing direction of the spacecraft. The performance of tracking can be verified by comparing the PSFs of stars taken during Fast Mode (at high time resolution and with no tracking) with those data taken using Image Mode when tracking is enabled. This analysis has shown that XMM-OM tracking is performing as expected. Analysis of OM tracking histories show that the spacecraft drift is less than 0.5 pixels for approximately 75% of all frames taken, and therefore require no shift and add correction (a shift of one pixel will be made if the guide stars are calculated to have drifted more than $\pm 0.5$ pixels from their reference positions). Of the remaining, the corrections due to drift are rarely more than 2 pixels in any one direction. Tracking is turned off automatically when no suitable guide stars are found, which is usually due to poor statistics. This can occur in observations of very sparse fields (rare) or when using the UVM2 and UVW2 filters, where throughput is lower than in the optical bands. However, given the pointing stability of XMM-Newton and the intrinsically poorer resolution of the detector in the UV (section 5.6), this does not normally lead to any significant degradation in the PSF for non-magnified data. Image quality ------------- ### Point Spread Function After launch the measured PSFs in the V-filter had FWHM widths broader than expected from preflight measurements. The focus was therefore adjusted using the control heaters as discussed in section 2.3. Fig. 5 shows the gradual change in the PSF with the heater setting. As can be seen from the figure, the optimum setting for the Magnifier is clearly at $-100$% i.e. at the minimum separation of the primary and secondary mirrors, whereas for the V-filter it is above 70%. A value of 100% (maximum separation of mirrors) was chosen for subsequent measurements in all filters, except for the Magnifier where $-100$% is selected. To allow for the thermal settling time involved in a change of focus, twenty minutes of additional overhead time is inserted before and after a sequence of Magnifier exposures. (8.8,5.5) (0,0) The PSFs contain a contribution from the telescope optics and from the detector. They can be assumed to be radially symmetric in shape, with an approximately gaussian central peak and extended wings. The width of the PSF increases with photon energy because of the detector component, from 3.1 pixels (1.5 arcsecs) FWHM in the V band to $\sim 6$ pixels (3 arcsecs) pixels in the UV filters (see Fig. 6). (8.8,6) (0,0) ### Distortion The XMM-OM optics, filters and (primarily) the detector system result in a certain amount of image distortion. It is mainly in the form of barrel distortion, and if not corrected can result in shifts from the expected position of up to 20 arcsecs. By comparing the expected position with the measured position for a large number of stars in the FOV a distortion map has been derived. The preliminary V-filter analysis was performed on the LMC pointing and is based on 230 sources. A 3rd order polynomial was fitted to the deviations assuming that there is no error at the centre of the FOV (i.e. at address (1024.5,1024.5)). This polynomial can be used to correct source positions measured in other fields, and currently gives a positional RMS accuracy of 1.0 arcsec (1.9 pixels) in the V-filter (see Fig. 7; astrometry relative to stars of known position over restricted regions of the field can of course be more accurate than this). Using higher orders of the polynomial does not increase the accuracy and is detrimental particularly for sources at the edges of the FOV. Using functions other than polynomials has not yet been investigated, but may lead to an improvement to the correction for sources near the edges of the FOV. Distortion maps using the 3C273 field have been derived for the other filters, but are not yet to such high accuracy. Further work will either use fields with more sources in the FOV or combine data from several observations. The preliminary distortion maps have been entered into the appropriate CCF files and can be used in conjunction with the SAS. They are also used on board to automatically position windows on the detector that are specified in sky coordinates. This is important for small windows such as those used in Fast Mode. (8.8,6) (0,0) ### Modulo-8 pattern As discussed earlier, the XMM-OM detector functions by centroiding a photon splash to within a fraction (1/8th) of a physical CCD pixel. This calculation is performed in real-time by the detector electronics, and therefore has to be fast. It is done by means of a lookup table whose parameters are computed onboard once per revolution, based on a short image taken with the internal flood LED lamp, and periodically updated. The lookup table parameters are the mean values derived from a selected part of the active area of the detector (usually the central region). They do not take into account small variations in the shape of the photon splash over the detector face and as such are an approximation to the optimum value at a given location on the detector. The result of imperfections in the lookup table is that the size of the pixels is not equal on the sky. When displayed with a normal image display routine, therefore, uncorrected XMM-OM images can exhibit a faint modulation in the apparent background level repeating every eight pixels, corresponding to every physical CCD pixel (see Fig. 8). SAS tasks that, for example, search for sources in XMM-OM images take the variation in pixel size into account and compute the local 88 pattern post facto based on the measured image. Similarly the raw image can be resampled for display purposes. The SAS routine does not lose or gain counts, but resamples them according to the true pixel sizes. The detector centroiding process also breaks down if more than one photon splash overlaps on a given CCD frame. Thus an 88 pattern is often seen around bright stars (see Fig. 8a), or when two bright stars occur close together on an image. ### Scattered light (10,12.0) (0,0) (0,6.5) Artifacts can appear in XMM-OM images due to light being scattered within the detector. These have two causes: internal reflection of light within the detector window and reflection of off-axis starlight and background light from part of the detector housing. The first of these causes a faint, out of focus ghost image of a bright star displaced in the radial direction away from the primary image due the curvature of the detector window (Fig. 8a). The second effect is due to light reflecting off a chamfer in the detector window housing. Bright stars that happen to fall in a narrow annulus 12.1 to 13 arc minutes off axis shine on the reflective ring and form extended loops of emission radiating from the centre of the detector (Fig. 8b). Similarly there is an enhanced “ring” of emission near the centre of the detector due to diffuse background light falling on the ring (Fig. 8b). The reflectivity of the ring, and of the detector window, reduces with increasing photon energy. Therefore these features are less prominent when using the UV filters. Conclusion ========== The first stage of commissioning and calibrating XMM-OM has been completed. The instrument is fulfilling its role of extending the spectral coverage of XMM-Newton into the ultraviolet and optical band, allowing routine observations of targets simultaneously with EPIC and RGS. Specifically, the instrument has successfully been demonstrated to provide wide field simultaneous imaging with the X-ray camera, simultaneous timing studies, and boresight information to arcsecond accuracy. A number of results illustrating the scientific potential of XMM-OM are contained within this volume. We would like to thank all the people who have contributed to the instrument; in its building, testing and operation in orbit, as well as those who have analysed the calibration data. The author list only contains a small fraction of those people involved. XMM-OM was built by a consortium led by the Principal Investigator, Prof. K.O.Mason, and comprising, in the UK, the Mullard Space Science Laboratory and Department of Physics and Astronomy, University College London; in the USA, University of California Santa Barbara, Los Alamos National Laboratory, & Sandia National Laboratory; and in Belgium, the Centre Spatial Liege & the University of Liege. JMV acknowledges support from the SSTC-Belgium under contract P4/05 and by the PRODEX XMM-OM Project. The U.S. investigators acknowledge support from NASA contract NAS5-97119. The UK contribution was supported by the PPARC. Based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and the USA (NASA) Fordham, J. L. A., Bone, D. A., Norton, R. J., Read, P. D. 1992, Proc. ESA Symp. on Photon Counting Detectors for Space Instrumentation, ESA SP-356, 103 Jansen, F., Lumb, D., Altieri, B. et al. 2001, A&A, 365 (this issue) Kawakami, H., Bone, D., Fordham, J.L.A., Michel, R. 1994, Nucl. Instrum. & Meth. A 348, 707 Royer, P., Manfroid, J., Gosset, E., Vreux J.-M. 2000, A&AS, 145, 351. Schmidt, M., Hasinger, G., Gunn, J. et al. 1998, A&A, 329, 495-503 Watson, M. G., Augeres, J.-L., Ballet, J. et al. 2001, A&A, 365 (this issue)

--- abstract: 'Given a disk $O$ in the plane called the objective, we want to find $n$ small disks $P_1,\ldots,P_n$ called the pupils such that $\bigcup_{i,j=1}^nP_i\ominus P_j\supseteq O$, where $\ominus$ denotes the Minkowski difference operator, while minimizing the number of pupils, the sum of the radii or the total area of the pupils. This problem is motivated by the construction of very large telescopes from several smaller ones by so-called Optical Aperture Synthesis. In this paper, we provide exact, approximate and heuristic solutions to several variations of the problem.' author: - | Trung Nguyen$^1$[^1], Jean-Daniel Boissonnat$^1$,\ Fréderic Falzon$^2$ and Christian Knauer$^3$\ \ [$^1$Geometrica project, INRIA Sophia Antipolis, France]{}\ [$^2$Research department, Alcatel Alenia Space, France]{}\ [$^3$Institut für Informatik, Freie Universität Berlin, Germany]{} bibliography: - 'bibfile.bib' title: 'A disk-covering problem with application in optical interferometry' --- Introduction {#sec:intro} ============ The diameter of the pupil of a telescope is proportional to its resolution power. A simple calculus shows that we would need a telescope having a diameter of approximately $20m$ to observe the Earth from a high orbit [@NBBFT06]. Needless to say, such an instrument would not be adapted to the observation from space. In order not to build too large pupils, Optical Aperture Synthesis is adopted to synthesize (very) large pupils by interferometrically combining several smaller pupils [@disrupt05] (see Fig. \[fig:instr\]). The auto-correlation support (ACS) of a system of pupils denotes all the observable spatial frequency domain. The underlying problem can be stated in geometric terms as follows. Given an objective $O$ supposed to be a disk, design a set of disks $\mathcal{P}=\{P_1,\ldots,P_n\}$ such that its ACS $\mathcal{D}$ covers entirely the objective while minimizing some cost function. Here $\mathcal{D} = \bigcup_{i,j=1}^n(P_i\ominus P_j)$ where $\ominus$ denotes the Minkowski difference operator. The cost function may include the number of pupils, the sum of the radii or the total area of the pupils, etc. This problem is a variant of the disk-covering problem. To the best of our knowledge, the variant we consider is new and the interferometry problem has not been considered before from a geometric perspective. This paper is a follow-up of our initial investigation [@NBBFT06]. The reader interested in the general disk-covering problem or some other variants can refer to [@alt2006mcc; @CB05; @booth2003cac]. \[fig:instr\] ![[Examples of using Optical Aperture Synthesis to synthesize large pupils [@disrupt05]]{}[]{data-label="fig:SOO-const"}](soo-ex.ps "fig:"){height="3cm"} ![[Examples of using Optical Aperture Synthesis to synthesize large pupils [@disrupt05]]{}[]{data-label="fig:SOO-const"}](multi2.ps "fig:"){height="3cm"} The outline of this paper is as follows. In section \[sec:apollonius\_diagram\], we introduce Apollonius diagrams (additively weighted Voronoi diagrams) which play a central role in our study, and use them to decide whether the objective is covered. Section \[sec:three\_pupils\] deals with the case of three pupils for which we provide an optimal solution. We describe in section \[sec:approximation\_number\_pupils\] a constant-factor approximation algorithm for the case where the pupils are restricted to have the same radius. In section \[sec:fixed-center\_problem\], we consider the centers of the pupils to be given and provide efficient algorithms to minimize the sum of the radii or the total area of the pupils under the constraint that the ACS covers the objective. Finally, section \[sec:fixed-radius\_problem\] considers the problem where the radii of the pupils are known but their positions are unknown. Apollonius diagrams and the decision problem {#sec:apollonius_diagram} ============================================ \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Proposition]{} \[thm\][Fact]{} Apollonius diagrams (aka Additively weighted Voronoi diagrams) -------------------------------------------------------------- Let $\mathcal{D}=\{ D_1,\ldots ,D_N\}$ be a set of $N$ disks in the plane. We denote by $c_i$ the center of $D_i$ and by $\rho_i$ its radius. Let $\|.\|$ denote the Euclidean distance and $\partial S$ denote the boundary of a subset of points $S$. The [*distance*]{} of a point $x$ to the circle $\partial D_i$ is defined as $$\delta_i(x)=\|x-c_i\|-\rho_i.$$ For a point $x$, $\delta_i(x)$ is $<0,0,>0$ depending whether $x$ lies inside, on the boundary of, or outside $D_i$. The [*Apollonius cell*]{} of $D_i$ consists of the points whose distance to $\partial D_i$ is less than or equal to their distance to any other circle of $\mathcal{D}$: $$A_i=\{x\in\mathbb{R}^2\mid \delta_i(x)\leq \delta_j(x), j=1,\ldots,N \}.$$ ![[An Apollonius diagram of 8 disks in the Euclidean plane. The black disk has no cell.]{}[]{data-label="fig:power_diagram"}](apo_graph.ps){height="4cm"} Unlike the case of points, it is possible that a disk may have an empty cell. This happens when the disk is inside another disk. The one-dimensional connected sets of points that belong to exactly two Apollonius cells are called [*Apollonius edges*]{}, while points that belong to at least three Apollonius cells are called [*Apollonius vertices*]{}. The collection of the cells, edges and vertices forms the [*Apollonius diagram*]{} of $\mathcal{D}$, denoted by $Apo(\mathcal{D})$ (see Fig. \[fig:power\_diagram\]). The Apollonius diagram $Apo(\mathcal{D})$ can be computed in time $O(N\log N)$ which is worst-case optimal [@KY02], and robust and efficient implementations exist [@cgal]. More information on Apollonius diagrams can be found in [@BWY06; @KY02]. We start by stating some properties of Apollonius diagrams. Let $B_{ij}$ define the bisector of two disks $D_i$ and $D_j$ $$B_{ij}=\{x\in\mathbb{R}^2\mid \delta_i(x) = \delta_j(x) \}.$$ \[lem:bisector\_is\_linear\] The restriction of $\delta_i$ and $\delta_j$ to $B_{ij}$ are unimodal functions. More precisely, these functions decrease linearly to a minimum and then increase linearly. Consider two disks $D_i$ and $D_j$ with radii $\rho_i, \rho_j$ and centers, w.l.o.g., $c_i = (-c, 0)$ and $c_j = (c, 0)$. The bisector of $D_i$ and $D_j$ is a sheet of the hyperbola whose equation is $$\frac{x^2}{a^2} - \frac{y^2}{c^2 - a^2} = 1,$$ where $a = |\rho_j - \rho_i|/2$. Then the distance of a point with abscissa $x$ on the hyperbola to $c_i$ is a linear function of $x$: $d = \pm(ex + a)$, where $e = \frac{c}{a}$ is the eccentricity of the hyperbola and sign $\pm$ is positive if $\rho_i\leq \rho_j$ and negative otherwise. \[cor:cover\_cell\_by\_minidisk\] Any arc $pq$ contained in the edge of a cell $A_i$ is included in the smallest disk of center $c_i$ that contains $p$ and $q$. Since the distance function to $D_i$ of the points on arc $pq$ is unimodal by Lemma \[lem:bisector\_is\_linear\], it reaches a maximum at $p$ or $q$. Hence any disk with center $c_i$ that contains $p$ and $q$ covers the whole arc. The Apollonius cell $A_i$ is included in the disk centered at $c_i$ that contains the set of its vertices. If $A_i$ is unbounded we are done. Otherwise, as $A_i$ is star-shaped [@BWY06], it is included in a disk if its edges are. Applying Corollary \[cor:cover\_cell\_by\_minidisk\] to all edges of $A_i$ concludes our proof. Let $\delta_\mathcal{D}(x)$ denote the smallest distance of $x$ to the disks of $\mathcal{D}$, i.e., $\delta_\mathcal{D}(x) \leq \delta_i(x)$ for any $1\leq i\leq N$ and equality holds iff $x\in A_i$. We see that $\delta_\mathcal{D}(x)\leq 0$ when $x$ lies inside the union of the disks of $\mathcal{D}$. The decision problem -------------------- ![[A system of three pupils (left) and its ACS (right), the objective is represented by a thick circle.]{}[]{data-label="fig:autocorrelation"}](fig2.eps){height="4cm"} Let $\mathcal{P}=\{P_1,\ldots,P_n\}$ be a set of $n$ disks called the [*pupils*]{} and $O$ be a disk of radius $R$ centered at the origin called the [*objective*]{}. The ACS of $\mathcal{P}$ is $\mathcal{D}=\bigcup_{i,j=1}^n(P_i\ominus P_j)$. The decision problem consists in determining whether $O$ is covered by $\mathcal{D}$. Let $c_i$ and $\rho_i$ denote the center and the radius of pupil $P_i$ and let $D_{ij}=P_i\ominus P_j$. It is not difficult to see that $D_{ij}$ is a disk with center $c_{ij}=c_i-c_j$ and radius $\rho_{ij}=\rho_i+\rho_j$. Moreover, $\mathcal{D}=\bigcup_{i,j=1}^n D_{ij}$ (see Fig. \[fig:autocorrelation\]). If the radius $\rho_i$ of some pupil $P_i$ is greater than half the objective’s radius $R$, $D_{ii}$ covers $O$. We assume in the sequel that the pupils all have a radius at most $\frac{R}{2}$ which implies that all disks of $\mathcal{D}$ have radii smaller than $R$. We write $A_{ij}$ for the cell of $D_{ij}$ in the Apollonius diagram of $\mathcal{D}$. Let $V_{ij}$ denote the set of vertices of $A_{ij}$ inside $O$ and the intersection points of $\partial A_{ij}$ with $\partial O$. We denote by $N=n^2$ the number of the disks of $\mathcal{D}$. It can be argued that the cardinality of all $V_{ij}$ is $O(N)$. The following shows a necessary and sufficient condition for covering $O$ by $\bigcup_{i,j=1}^n D_{ij}$ (see Fig. \[fig:alpha\_pupils\]). ![[: A set of three pupils whose ACS does not cover the objective. The x-marks correspond the vertices of $V_{ij}$ of which some lie outside the union of disks. [Right]{}: The set of pupils with the same position but radii enlarged by $\alpha^*$ as computed by Algorithm \[alg:same\_alpha\]. All vertices of $V_{ij}$ are inside $\mathcal{D}$ and the objective is covered.]{}[]{data-label="fig:alpha_pupils"}](config1p.ps "fig:"){height="1.635cm"}![[: A set of three pupils whose ACS does not cover the objective. The x-marks correspond the vertices of $V_{ij}$ of which some lie outside the union of disks. [Right]{}: The set of pupils with the same position but radii enlarged by $\alpha^*$ as computed by Algorithm \[alg:same\_alpha\]. All vertices of $V_{ij}$ are inside $\mathcal{D}$ and the objective is covered.]{}[]{data-label="fig:alpha_pupils"}](config1d.ps "fig:"){height="3.27cm"} ![[: A set of three pupils whose ACS does not cover the objective. The x-marks correspond the vertices of $V_{ij}$ of which some lie outside the union of disks. [Right]{}: The set of pupils with the same position but radii enlarged by $\alpha^*$ as computed by Algorithm \[alg:same\_alpha\]. All vertices of $V_{ij}$ are inside $\mathcal{D}$ and the objective is covered.]{}[]{data-label="fig:alpha_pupils"}](config2p.ps "fig:"){height="1.75cm"}![[: A set of three pupils whose ACS does not cover the objective. The x-marks correspond the vertices of $V_{ij}$ of which some lie outside the union of disks. [Right]{}: The set of pupils with the same position but radii enlarged by $\alpha^*$ as computed by Algorithm \[alg:same\_alpha\]. All vertices of $V_{ij}$ are inside $\mathcal{D}$ and the objective is covered.]{}[]{data-label="fig:alpha_pupils"}](config2d.ps "fig:"){height="3.5cm"} \[lem:decision\_problem\] $O\subseteq\mathcal{D}$ iff $V_{ij}\subseteq D_{ij}$ for all $i,j=1,\ldots,n$. First we argue that $O\subseteq\mathcal{D}$ iff $A_{ij}\cap O\subseteq D_{ij}$ for all $i,j=1,\ldots,n$. Since the set of $A_{ij}$ forms a decomposition of the plane, $A_{ij}\cap O\subseteq D_{ij}, i,j=1,\ldots,n$, implies that $O\subseteq \bigcup_{i,j=1}^n D_{ij}=\mathcal{D}$. Conversely, suppose that $O\subseteq\mathcal{D}$ and $p\in A_{ij}\cap O$, we will show that $p\in D_{ij}$. Indeed, $p\in O\subseteq \mathcal{D}$ implies $\delta_\mathcal{D}(p)\leq 0$. Together with $p\in A_{ij}$, we conclude $\delta_{ij}(p)=\delta_\mathcal{D}(p)\leq 0$ which implies that $p$ is inside $D_{ij}$. We show next that $A_{ij}\cap O\subseteq D_{ij}$ is equivalent to $V_{ij}\subseteq D_{ij}$ by proving that a disk $\Delta$ centered at $c_{ij}$ covering $V_{ij}$ covers also $A_{ij}\cap O$. We first observe that the edges of $A_{ij}$ with both endpoints in $O$ are covered by $\Delta$ by Corollary \[cor:cover\_cell\_by\_minidisk\]. It remains to verify that the intersection points of $\partial A_{ij}$ with $\partial O$ and the arcs linking them are also in $O$. Consider two such points $p$ and $q$ consecutive along the boundary of $O$. Call $p_1p_2$ and $q_1q_2$ the two Apollonius edges that intersect $\partial O$ at $p$ and $q$ respectively. Suppose $p_1,q_1\in O$ and $p_2, q_2\notin O$, which implies that $p_1, p, q, q_1$ belong to $V_{ij}$. Since $p_1$ and $p$ lie on edge $p_1p_2$, and $q$ and $q_1$ are contained in $q_1q_2$, $\Delta$ will cover the arcs $p_1p$ and $qq_1$ by Corollary \[cor:cover\_cell\_by\_minidisk\]. It thus remains to show that the circular arc $pq$ of $O$ is included in $\Delta$, which is true since $p, q\in D_{ij}$ whose radius has been assumed to be smaller than the radius of $O$. The following simple result is important in sections \[sec:fixed-center\_problem\] and \[sec:fixed-radius\_problem\]. \[cor:disks\_cover\_cells\] Given a configuration of pupils with the corresponding sets $D_{ij}$ and $V_{ij}$. We move/resize the pupils such that each new disk $D^{'}_{ij}$ includes $V_{ij}$. Then, $O$ is covered by $\bigcup_{i,j=1}^n D^{'}_{ij}$. Since $V_{ij}\subseteq D^{'}_{ij}$ is equivalent to $A_{ij}\cap O\subseteq D^{'}_{ij}$ (see the proof of Lemma \[lem:decision\_problem\]) and the sets $A_{ij}\cap O$ cover $O$, the objective is covered by $\bigcup_{i,j=1}^n D^{'}_{ij}$. Lemma \[lem:decision\_problem\] gives us a simple $O(N\log N)$-time algorithm that solves the decision problem. It still works when we replace Apollonius diagrams by power diagrams. The reason of using the formers will be seen in section \[sec:fixed-center\_problem\]. Problem with three pupils {#sec:three_pupils} ========================= A configuration of pupils is called valid if its ACS covers the objective. In this section, we want to minimize the sum $\rho_1+\rho_2+\rho_3$ among the valid configurations. Let denote by $l$ the line passing through $c_{23}$ and $c_{32}$. Since the disks and the objective are symmetric about the origin, it suffices to consider only one half-plane bounded by $l$. Among the valid configurations, those in which one radius is half of the objective’s radius $R$ and the other two are zero are optimal. It is straightforward to see that such configurations are valid. Consider now a configuration in which $\rho_1+\rho_2+\rho_3 < R/2$. We will prove that it cannot be a valid configuration. Indeed, suppose w.l.o.g. $P_1$ has the largest radius among three pupils. Then $D_{11}$ is the largest disk among $D_{11}, D_{22}$ and $D_{33}$ and its radius $2\rho_1$ is smaller than $R$. Let $p_1, q_1, q_2, p_2$ be the intersection points from left to right of $\partial O$ and $\partial D_{11}$ with $l$ (see Fig. \[fig:three\_pupils\]). If segment $\overline{p_1q_1}$ is covered by $D_{23}$, then the diameter of $D_{23}$ is at least the length of $\overline{p_1q_1}$, i.e., $2(\rho_2+\rho_3) \geq R - 2\rho_1$ which implies $\rho_1+\rho_2+\rho_3 \geq R/2$ (a contradiction). The case where $\overline{p_2q_2}$ is covered by $D_{23}$ is symmetrical. We can therefore assume that $D_{23}$ does not cover $\overline{p_1q_1}$ nor $\overline{p_2q_2}$, and, by symmetry, the same holds for $D_{32}$. Without loss of generality, we can assume that $D_{12}$ contains $p_1$ or $q_1$ and that $D_{13}$ contains $p_2$ or $q_2$. We denote by $c$ the midpoint of the arc $p_1p_2$ of $\partial O$. The distance of $c$ to $p_1, q_1, p_2, q_2$ is at least $\sqrt{R^2+(2\rho_1)^2} > R$. Then $c$ is not included in neither $D_{12}$ nor $D_{13}$ whose diameters are smaller than $R$. It is not included in $D_{23}$ and $D_{32}$ either since the distance from $c$ to $c_{23}$ and $c_{32}$ is at least $R$ and the radii of $D_{23}$ and $D_{32}$ are less than $R$. Hence, the configuration is not valid. ![[A configuration of three pupils and the upper part of its ACS]{}[]{data-label="fig:three_pupils"}](3pupils.eps){height="3.5cm"} It is interesting to see from the above lemma that configurations of three pupils consisting of a pupil of radius $R/2$ and two points are optimal, whatever the position of the pupils may be. An $8\sqrt{2}-$approximation to the smallest number of the pupils of the same radius {#sec:approximation_number_pupils} ==================================================================================== In this section, we restrict to the case $\rho_1=\ldots=\rho_n=\rho/2$, then the disks $D_{ij}$ have the same radius $\rho$. We want to find an upper bound for $n$ to cover an objective of radius $R$. As the number of disks is $n^2$, a lower bound $\lceil R/\rho\rceil$ is easily obtained. Let $p$ be any prime number, we start by stating a basic property of $p$ \[fac:prime\_property\] Let $k,l\in\mathbb{Z}$ such that $\gcd(p,k) = 1$, there exists an integer $0\leq i<p$ satisfying $ik \equiv l \pmod p$. \[thm:covering\_set\]$\{x_i-x_j\mid i,j=0,\ldots,4p-1\}\supseteq\{x\in\mathbb{Z}, |x| < p^2\}$ where $$\begin{aligned} x_k &=& kp + (\frac{k(k + 1)}{2} \bmod p)\\ x_{k+2p} &=& x_k + p,\end{aligned}$$ for $k=0\ldots,2p-1$. Let $x$ be an arbitrary integer between 0 and $p^2-1$, then $x$ can be written as $kp + l$ for some $0\leq k,l < p$. Let $X_i=x_{k+i}-x_{i}$ for $i=0,\ldots,p-1$, we observe that $$\begin{aligned} \label{exp:difference_of_centers} (k - 1)p < X_i < (k + 1)p.\end{aligned}$$ $$X_i \equiv X_0 + ik \pmod p$$ By Fact \[fac:prime\_property\] there exists some $0\leq i<p$ such that $X_i\equiv l\pmod p$. Hence together with (\[exp:difference\_of\_centers\]) the difference of either $x_{k+i}$ or $x_{k+i+2p}$ with $x_i$ will be $x$. The only case where Fact \[fac:prime\_property\] does not apply is when $k = 0$. In this case choose $k = 1$ instead and easily see that the set $\{x_{i+1}-x_{i+2p}\}\cup\{x_{i+1+2p}-x_{i+2p}\}$ generates all integers $1,\ldots,p-1$ and hence contains $x$. The above set should not be confused with Golomb ruler [@D02] and the set defined by Erdös and Turán [@ET41] since in the latter sets, the differences between any pair of distinct elements must be unique but do not generally cover all points $1,\ldots,p^2$. Suppose, w.l.o.g., radius of the disks $\rho = \frac{1}{\sqrt{2}}$ and $R = p^2$ for some prime $p$. Let $\mathcal{S}=\{x\in\mathbb{Z}^2\mid \|x\|_\infty < p^2\}$. We see that the disks of radius $\frac{1}{\sqrt{2}}$ whose centers cover $\mathcal{S}$ are sufficient to cover completely the objective. In other words, we want to find $n$ centers of pupils $c_i\in\mathbb{Z}^2$ such that $$\{c_i-c_j\mid 1\leq i,j\leq n\} \supseteq \mathcal{S}$$ \[cor:upper\_bound\] $\lceil 8\sqrt{2}R/\rho\rceil$ pupils of radius $\rho$ are sufficient to cover an objective of radius $R$. The set of pupils is constructed as follows: $c_i=(x_{\lfloor\frac{i}{4p}\rfloor}, x_{i\bmod 4p})$ for $i = 0,\ldots, 16p^2-1$. By applying Theorem \[thm:covering\_set\] first for $x$-coordinate and then for $y$-coordinate, we see that these $16p^2$ pupils are able to cover any element of $\mathcal{S}$ thus the objective of radius $R$. As $R = p^2$ and $\rho = \frac{1}{\sqrt{2}}$, we yield the upper bound. The following is an immediate consequence of Corollary \[cor:upper\_bound\] and the lower bound observed earlier. There is an $8\sqrt{2}-$approximation algorithm to cover the objective of radius with the smallest number of pupils of the same radius. The fixed-center problem {#sec:fixed-center_problem} ======================== In sections \[subsec:same\_alpha\] and \[subsec:different\_alphas\], the centers of the pupils are fixed and we present two heuristic algorithms for optimizing the radii among the valid configurations. Both algorithms are based on the fact that the circle of center $c_{ij}$ and radius $\rho_{ij}+\delta_{ij}(p)$ passes through $p$. Then we provide an approximation algorithm with a given error bound and compare it with the heuristic algorithms. We end up the section with a method to maximize the objective while keeping fixed the radii as well as the positions of the pupils. A simple optimization problem {#subsec:same_alpha} ----------------------------- If we increase each of the radii of the pupils by a real number $\alpha/2$, the radii of the disks $D_{ij}$ then increase by $\alpha$ and $Apo(\mathcal{D})$ remains unchanged. Hence there exists a minimal value of $\alpha$, denoted $\alpha^*$, for which the objective is covered by the union of the new (enlarged) disks. The following shows that $\alpha^*$ can be computed exactly in $O(N\log N)$ time (see Algorithm \[alg:same\_alpha\]). We recall that $V_{ij}$ is the set of vertices of $A_{ij}$ inside $O$ and the intersection points of $\partial A_{ij}$ with $\partial O$. \[lem:alpha\_star\] $$\begin{aligned} \alpha^* = \max_{ij}\max_{p\in V_{ij}}\delta_{ij}(p)\end{aligned}$$ It is easy to see that $\max_{ij}\max_{p\in V_{ij}}\delta_{ij}(p)$ is the minimal value of $\alpha$ for which $V_{ij}\subseteq D_{ij}$. The result follows from Lemma 4. $\alpha^* \gets -\infty$ compute $Apo(\mathcal{D})$ and $V_{ij}$ $\alpha^*\gets \max(\alpha^*, \delta_{ij}(x))$ $\alpha^*$ Minimizing the sum of the radii of the pupils {#subsec:different_alphas} --------------------------------------------- We consider now the more difficult problem of optimizing the sum of the radii of the pupils and propose a heuristic solution that turns out to perform well in practice. Instead of increasing the radii of the $P_i$ by a same amount as in the previous subsection, we consider them as $n$ variables. Algorithm \[alg:different\_alphas\] below proceeds in two main steps. First, we compute minimal quantities, denoted $\alpha_{ij}$, by which the radii of the $D_{ij}$ must be enlarged/reduced so as to satisfy Lemma \[lem:decision\_problem\] (lines 3–9). This step is similar to Algorithm \[alg:same\_alpha\]. Thanks to the fact that the already visited $\alpha_{ij}$ necessarily increase, the initial $V_{ij}$ will be covered upon termination by the disks $D^{'}_{ij}$ (which are $D_{ij}$ augmented by $\alpha_{ij}$). The objective is then covered by $\bigcup_{i,j=1}^n D^{'}_{ij}$ according to Corollary \[cor:disks\_cover\_cells\]. Finally, we want to minimize the sum of the radii of the $P^{*}_i$ under the constraint that $\rho^{*}_i+\rho^{*}_j$ must be at least the radius of $D^{'}_{ij}$ (line 10): $$\begin{aligned} \textrm{min} && \sum_{i=1}^n\rho^{*}_i \label{*}\\ \textrm{s.t.} && \rho^{*}_i+\rho^{*}_j\geq(\rho_i+\rho_j)+\alpha_{ij},\hspace{1.5cm}i,j=1,\ldots,n\hspace{1cm}(*)\\ && \rho^{*}_i \geq 0, \hspace{4.6cm} i=1,\ldots,n.\end{aligned}$$ Here, $\rho_i$ are the radii of the initial pupils $P_i$ and hence known. This is a linear program whose feasible set is non-empty and bounded. Thus, there exists an optimal solution. $\varepsilon \gets$ any small positive constant $\alpha_{ij} \gets -\infty$, $i,j=1,\ldots,n$ compute $Apo(\mathcal{D})$ and $V_{ij}$ $\alpha_{ij}\gets \max(\alpha_{ij}, \delta_{ij}(x))$ compute $\{\rho^{*}_i\}_{i=1,\ldots,n}$ by solving the linear program (\*) $err \gets \sum_{i=1}^n \rho_i - \sum_{i=1}^n\rho^{*}_i$ $\rho_i \gets \rho^{*}_i, i = 1,\ldots,n$ $err < \varepsilon$ except for the first iteration $\{\rho^{*}_i\}_{i=1,\ldots,n}$ Note that we need to update the Apollonius diagram since the pupils’ radii change after each iteration of the [**repeat**]{} loop. Algorithm \[alg:different\_alphas\] always terminates. The initial $V_{ij}$ is included in $D^{'}_{ij}$ by the construction of $\alpha_{ij}$. According to Corollary \[cor:disks\_cover\_cells\], $O$ is therefore covered by $\bigcup_{ij}D_{ij}$ after the first iteration. Hence, we may assume that the objective is covered. In this case, Lemma \[lem:alpha\_star\] implies that no $\alpha_{ij}$ is positive which shows that, at each step, $\rho^{*}_i+\rho^{*}_j \leq \rho_i+\rho_j$ and hence $\sum_{i=1}^n\rho^{*}_i\leq\sum_{i=1}^n\rho_i$. Since $\sum_{i=1}^n\rho^{*}_i$ is positive, Algorithm \[alg:different\_alphas\] necessarily terminates after a finite number of iterations. \ [**Minimizing the total area of the pupils:**]{} Replacing the objective function $\sum_{i=1}^n\rho^{*}_i$ in (\*) with $\pi\sum_{i=1}^n\rho^{*2}_i$ yields a quadratic program which minimizes the total area of the pupils. \ [**Additional constraints:**]{} In addition to covering the objective, we can also bound the radii of the pupils and forbid any overlap among the pupils. This can be done by adding the following constraints to the linear program (\*) $\rho^{*}_i+\rho^{*}_j\leq\|c_i-c_j\|, \hfill 1\leq i<j\leq n,$ $min\_radius \leq\rho_i\leq max\_radius, \hfill i=1,\ldots,n$. Algorithm \[alg:different\_alphas\] has been implemented and appears to work well in practice. Fig. \[fig:fixed-center\_iterations\] compares the results of Algorithms \[alg:same\_alpha\], \[alg:different\_alphas\] with the optimal solution computed by the following exhaustive search method. \ [**Exhaustive search algorithm:**]{} If the radii of the pupils are assumed to be integer multiples of a small number $\theta$, then the exhaustive search methods can be applied and the optimal solution in the continuous case must be at least the solution found by these methods minus $n\theta$. We hence have an approximation algorithm within a given error bound. Maximizing the objective ------------------------ Now we keep the pupils fixed (radii and positions) and maximize the radius of the objective under the constraint that it is covered by the union of the disks. \[pro:intersection\_point\] If an edge $pq$ of $A_{ij}$ cuts $\partial D_{ij}$ at a point $x\neq p$ and $q$, then there is a point $x'$ on $pq$ that is close to $x$ and not contained in $\mathcal{D}$. From the fact that $\delta_{ij}(.)$ is a unimodal function and $\delta_{ij}(x) = \delta_\mathcal{D}(x) = 0$. The following corollary, whose proof is referred to the full version of the paper, computes the maximal radius $R^*$ of the objective for which it is covered by $\mathcal{D}$. If $D_{ii}\subseteq A_{ii}$ for some $i=1,\ldots,n$ then $R^* = 2\rho_i$. Otherwise, $$R^* = \min_{ij} \min_{x\in \partial A_{ij}\cap \partial D_{ij}}\| x \|$$ The fixed-radius problem {#sec:fixed-radius_problem} ======================== a)![[]{data-label="fig:fixed-center_iterations"}](config3p.ps "fig:"){height="1.5cm"}![[]{data-label="fig:fixed-center_iterations"}](config3d.ps "fig:"){height="3.2cm"} b)![[]{data-label="fig:fixed-center_iterations"}](config4p.ps "fig:"){height="1.435cm"}![[]{data-label="fig:fixed-center_iterations"}](config4d.ps "fig:"){height="3.06cm"}\ c)![[]{data-label="fig:fixed-center_iterations"}](config5p.ps "fig:"){height="1.26cm"}![[]{data-label="fig:fixed-center_iterations"}](config5d.ps "fig:"){height="2.69cm"} d)![[]{data-label="fig:fixed-center_iterations"}](config6p.ps "fig:"){height="1.295cm"}![[]{data-label="fig:fixed-center_iterations"}](config6d.ps "fig:"){height="2.76cm"} e)![[]{data-label="fig:fixed-center_iterations"}](config7p.ps "fig:"){height="1.23cm"}![[]{data-label="fig:fixed-center_iterations"}](config7d.ps "fig:"){height="2.62cm"} a)![[]{data-label="fig:fixed-radius_iterations"}](config8p.ps "fig:"){height="1.065cm"}![[]{data-label="fig:fixed-radius_iterations"}](config8d.ps "fig:"){height="2.93cm"} b)![[]{data-label="fig:fixed-radius_iterations"}](config9p.ps "fig:"){height="1.75cm"}![[]{data-label="fig:fixed-radius_iterations"}](config9d.ps "fig:"){height="3cm"}\ c)![[]{data-label="fig:fixed-radius_iterations"}](config10p.ps "fig:"){height="1.54cm"}![[]{data-label="fig:fixed-radius_iterations"}](config10d.ps "fig:"){height="3.08cm"} d)![[]{data-label="fig:fixed-radius_iterations"}](config11p.ps "fig:"){height="1.75cm"}![[]{data-label="fig:fixed-radius_iterations"}](config11d.ps "fig:"){height="3.5cm"} a)![[]{data-label="fig:combine_algorithms"}](config12p.ps "fig:"){height="1.6cm"}![[]{data-label="fig:combine_algorithms"}](config12d.ps "fig:"){height="3.2cm"} b)![[]{data-label="fig:combine_algorithms"}](config13p.ps "fig:"){height="1.59cm"}![[]{data-label="fig:combine_algorithms"}](config13d.ps "fig:"){height="3.18cm"} c)![[]{data-label="fig:combine_algorithms"}](config14p.ps "fig:"){height="1.4cm"}![[]{data-label="fig:combine_algorithms"}](config14d.ps "fig:"){height="2.80cm"} In this section we fix the radii and propose a heuristic algorithm for moving the set of pupils so that its ACS covers the objective. Our algorithm is based on Corollary \[cor:disks\_cover\_cells\]. More precisely, we want to capture the point sets $V_{ij}$ by the disks $D_{ij}$. Given a set of points $P$ and a disk, the optimal center position for the disk to cover $P$ is the point that minimizes the maximal distance to any point of $P$ $$\label{eqn:smallest_enclosing_disk} \min_{p\in P} \max \|x - p\|.$$ This is the so-called smallest enclosing disk problem and a linear algorithm to compute exactly the disk center can be found in [@BKOS00]. Unfortunately, function (\[eqn:smallest\_enclosing\_disk\]) being non-differentiable makes it hard to apply to our problem. Another approach is to minimize the sum of the squared distance from the center to each point of $P$ $$\min{\sum_{p\in P}}\|x - p\|^2.$$ This function is convex and attains its minimum at the barycenter of $P$. Our algorithm works as follows. We begin with a given configuration of pupils, compute the set $V_{ij}$ and move the pupils to minimize the following function $$%\label{eq:optimization_program} \textrm{min} \sum_{i,j=1}^n\sum_{p\in V_{ij}}\|(c^{*}_i-c^{*}_j)-p\|^2$$ Here the centers $c^{*}_i$ of the pupils are variables and we recall that $c^{*}_i-c^{*}_j$ becomes the center of disk $D^{*}_{ij}$. The objective function being the sum of convex functions, is thus convex. We can update the sets $V_{ij}$ and iterate the algorithm until we obtain the desired result. As shown in Fig. \[fig:fixed-radius\_iterations\], the initial configuration is not critical. The algorithm can also be used as a preprocessing step to improve Algorithm \[alg:different\_alphas\] (see Figs. \[fig:fixed-center\_iterations\]e and \[fig:combine\_algorithms\]). [**ACKNOWLEDGMENT.**]{} We thank Helmut Alt, Günter Rote and Mariette Yvinec for helpful discussions and careful proofreading of early drafts of this paper. [^1]: The work of the author is supported by Alcatel Alenia Space and INRIA and was partly carried out while he was visiting the Freie Universität Berlin.

--- abstract: 'In this paper, we present an online two-level vehicle trajectory prediction framework for urban autonomous driving where there are complex contextual factors, such as lane geometries, road constructions, traffic regulations and moving agents. Our method combines high-level policy anticipation with low-level context reasoning. We leverage a long short-term memory (LSTM) network to anticipate the vehicle’s driving policy (e.g., forward, yield, turn left, turn right, etc.) using its sequential history observations. The policy is then used to guide a low-level optimization-based context reasoning process. We show that it is essential to incorporate the prior policy anticipation due to the multimodal nature of the future trajectory. Moreover, contrary to existing regression-based trajectory prediction methods, our optimization-based reasoning process can cope with complex contextual factors. The final output of the two-level reasoning process is a continuous trajectory that automatically adapts to different traffic configurations and accurately predicts future vehicle motions. The performance of the proposed framework is analyzed and validated in an emerging autonomous driving simulation platform (CARLA).' author: - 'Wenchao Ding and Shaojie Shen [^1]' bibliography: - 'paper.bib' title: | **Online Vehicle Trajectory Prediction using Policy Anticipation Network\ and Optimization-based Context Reasoning** --- Introduction {#sec:introduction} ============ In recent years, there has been growing interest in building fully autonomous vehicles. Our requirement of such vehicles is to have accurate anticipation over other traffic participants so that their planned motions are neither too aggressive nor too conservative. To achieve this goal, autonomous vehicles are expected to reason about the behavior and intentions of surrounding vehicles and subsequently predicts future trajectories of these vehicles. Given an urban driving environment where there are complex latent factors such as lane geometries, traffic regulations, road constructions and dynamical agents, the complexity of the prediction problem is high. Under such a scenario, there are two challenges to be addressed. First, given the complex environment, it is essential to consider the multimodal nature of the future trajectory [@lee2017desire]. For example, at the intersection as depicted in Fig. \[fig:motivation\_example\], there are two distinct choices, moving forward and turning left, which result in totally different future trajectories. Second, the prediction method must be highly flexible and able to easily adapt to the complex contextual factors. Many handcrafted prediction models, such as [@agamennoni2012estimation; @laugier2011probabilistic; @lefevre2012evaluating; @havlak2014discrete], may lack flexibility and require refactoring when a new contextual factor is introduced. Meanwhile, other methods, especially the popular RNN-based models [@kim2017probabilistic; @alahi2016social], treat the trajectory prediction as a pure regression problem in spite of the multimodal nature of the future trajectory. We are therefore motivated to develop a flexible trajectory prediction framework which can easily adapt to various complex urban environments while incorporating high-level intentions to enhance the prediction accuracy. In this paper, we propose an *online* two-level vehicle trajectory prediction framework. We develop a policy anticipation network using a long short-term memory (LSTM) network to anticipate high-level policies of vehicles (such as moving forward, yielding, turning and lane changing) based on sequential past observations. Given the high-level policy, we propose an optimization-based context reasoning process in which the complex contextual information is naturally encoded in a multi-layer cost map structure. A policy interpreter is set up to bridge the high-level and low-level reasoning by transforming the policy to a trajectory initial guess of the non-linear optimization. The policy anticipation network is used to capture the intention and guide the trajectory prediction process. Our optimization-based context reasoning process can easily adapt to different traffic configurations by transforming different factors into a unified notation of cost. The motivation for modeling trajectory prediction as an optimization problem is that human drivers internally balance their maneuvers in terms of the “cost”. For example, driving through red lights or breaking speed limits would risk receiving penalties, and human drivers have an inborn ability to balance various kinds of costs during driving. The optimization-based reasoning process can be easily extended by adding another cost term to the unified cost map structure. The idea of modeling drivers as optimizing agents is not new [@wolf2008artificial; @abbeel2004apprenticeship; @bahram2016combined; @sadigh2016planning], especially in the field of imitating human driving behaviors using inverse reinforcement learning (IRL). However, from the prediction perspective, the multimodal nature of the future trajectory [@lee2017desire; @deo2018convolutional] is not well modeled by the optimization process. For example, the non-linear optimization process may converge to either of the two possible intentions in Fig. \[fig:motivation\_example\]. To this end, we propose the policy anticipation network, which guides the optimization process to the anticipated high-level intention. Note that our optimization-based context reasoning can also incorporate the IRL technique for weight tuning, which is left as important future work. We summarize the contributions of this paper as follows: - An online two-level trajectory prediction framework which incorporates the multimodal nature of future trajectories. - A highly flexible optimization-based context reasoning process which incorporates a multi-layer cost map structure to encode various contextual factors. - Integration of the vehicle trajectory prediction framework and presentation of the results on accuracy, efficiency, and flexibility in various traffic configurations. The related literature is reviewed in Sect. \[sec:related\_works\]. A system overview is given in Sect. \[sec:overview\]. The main methodology is presented in Sect. \[sec:policy\] and Sect. \[sec:optimization\]. The implementation details and experimental results are provided in Sect. \[sec:implementation\] and Sect. \[sec:results\]. Conclusions and future work are given in Sect. \[sec:conclusion\]. Related Works {#sec:related_works} ============= The problem of vehicle trajectory prediction has been actively studied in the literature. As concluded in [@lefevre2014survey], there are three levels of prediction models, namely, physics-based, maneuver-based and interaction-aware motion models. Physics-based motion models use dynamic and kinematic vehicle models to propagate future states [@ammoun2009real; @brannstrom2010model]. However, the prediction results only hold for the very short-term (less than one second). Maneuver-based motion models are more advanced in the sense that the model may forecast relatively complex maneuvers, such as lane change and turns at intersections, by revealing the maneuver pattern. Many of the works on this level present a probabilistic framework to account for the uncertainty and variation of the motion patterns, such as Gaussian processes (GPs) [@tran2014online; @laugier2011probabilistic], Monte Carlo sampling [@eidehall2008statistical], Gaussian mixture models (GMMs) [@havlak2014discrete] and hidden Markov models [@aoude2012driver]. However, they typically assume vehicles are independent entities and fail to model interactions within the context and with other agents. Interaction-aware models, on the other hand, take the driving context and vehicle interactions into account, and most of them, such as [@gindele2015learning; @lefevre2012evaluating] and [@agamennoni2012estimation], are based on dynamic Bayesian networks (DBNs). Though these methods are context-aware, they require refactoring the models when considering a new contextual factor. Our method belongs to the interaction-aware level. Compared to the DBN-based prediction methods, our method is more flexible and can be easily adapted to different traffic configurations. It is notable that recurrent neural networks (RNNs) and their variants, such as LSTM networks, have recently been applied to predict or track moving targets, as in [@kim2017probabilistic; @khosroshahi2016surround] and [@ondruska2016deep]. Our policy anticipation network shares a similar structure with [@khosroshahi2016surround]. But the fundamental difference is that the network in [@khosroshahi2016surround] is only used to analyze the maneuver pattern at an intersection and cannot actively predict the future trajectories. Many learning-based end-to-end trajectory prediction models [@kim2017probabilistic; @alahi2016social; @deo2018convolutional] lack the ability to encode the contextual information. In [@lee2017desire], Lee *et al.* suggest combining IRL with an environment feature map to learn the interaction with contextual factors. However, this requires a large amount of training data to generalize due to the high complexity of the model. Also, it is hard to learn the interaction in some rare driving situations, such as red light offences. System Overview {#sec:overview} =============== The overview of our vehicle trajectory prediction framework is shown in Fig. \[fig:framework\]. During the high-level reasoning, the sequential state observations are fed to the policy anticipation network, which provides the future policy that a vehicle is likely to execute. Together with the map information, the policy can be properly interpreted in the driving context and a reference prediction is generated and fed to the optimization-based context reasoning process. The optimization process renders various environment observations and encodes them into the multi-layer cost map structure. A non-linear optimization process is then conducted to generate the predicted vehicle trajectory. Policy Anticipation and Interpretation {#sec:policy} ====================================== Problem formulation ------------------- We assume that the vehicle is equipped with a detector that provides the pose estimation $\mathbf{p}^k_i = (x_i,y_i,\theta_i, v_i)$ of a neighboring vehicle with ID $k$ at different time-instants, where $x_i$ and $y_i$ denote global coordinates at frame $i$, $\theta_i$ denotes the vehicle orientation in the 2-D plane, and $v_i$ denotes the body velocity. We accumulate observations from different time-instants inside a sliding window with a total window size of $T_{\text{obs}}$. And the network predicts vehicle $k$’s future policy in a look-ahead window from $T_{\text{obs}}$ to $T_{\text{pred}}$. The annotated labels include *forward*, *yield*, *turn left*, *turn right*, *lane change left*, and *lane change right*, and the labels can be easily extended when considering complex lane geometries. Network structure ----------------- Our policy anticipation network is based on an RNN encoder structure [@cho2014rnnecdc]. We refer interested readers to [@cho2014rnnecdc] and [@lee2017desire] for the detailed structure. Note that the output layer is modified to a softmax layer to provide the likelihood for all the policy labels. The probability distribution is used in the interpretation of the policy in Sec. \[sec:policy\_interpretation\]. We adopt negative log-likelihood (NLL) loss for this classification problem. Policy interpretation {#sec:policy_interpretation} --------------------- The policy interpretation module combines the policy anticipation results with a local map, so that the optimization-based context reasoning can start with a reasonable initial guess. As shown in Fig. \[fig:policy\_interp\], with different initial guesses (turning left or forward in this case), the optimization will be devoted to finding a solution in a totally different local solution space. Specifically, we use the likelihood provided by the policy anticipation network as follows: 1) we prune the infeasible anticipations (turning right in this example); 2) we take the policy of the maximum likelihood, and 3) we generate an initial trajectory prediction by extracting reference points corresponding to the selected policy. The initial guess is fed to the optimization-based context reasoning for further processing. In the future, instead of using deterministic reasoning based on one selected policy, we plan to use a probabilistic interpretation process. Optimization-based Context Reasoning {#sec:optimization} ==================================== Cost map structure {#sec:cost_map_design} ------------------ In this section, we present the cost map structure, which encodes the whole driving context. We specify different kinds of costs by separating them into different layers with distinct physical meanings, for the sake of illustration. A toy example of the multi-layer cost map is given in Fig. \[fig:multi\_layer\]. We adopt a four-layer cost map design in which we encode the cost induced by the lane geometry and static obstacles into the *static layer*, the cost induced by the moving objectives (MO) into the *MO layer*, the cost induced by traffic regulations into the *context layer*, and the cost induced by the vehicles’ nonholonomic constraints into the *nonholonomic layer*. Cost functions -------------- We adopt a discrete notation of the vehicle trajectory [@ziegler2014trajectory] where a continuous trajectory $\mathbf{x}(t) = \left( x(t), y(t) \right){\mbox{${}^{\text{T}}$}}$ is represented by a series of rear axle center points $\mathbf{x}_i = \left( x_i, y_i \right){\mbox{${}^{\text{T}}$}}$ in a global coordinate system. Namely, the predicted trajectory is approximated by $N$ points $\mathbf{x}_i = \mathbf{x}(t_i)$, which are sampled at equidistant times $t_i = t_0 + ih, 0\leq i < N$ of sampling step width $h$. The dynamics of the trajectory $\mathbf{x}(t)$ can be expressed as a function of its time derivatives, which are the finite differences of the sampling points. The orientation and curvature of the trajectory can be expressed by its time derivatives [@ziegler2014trajectory]. Following these notations, we introduce the cost functions $f(\mathbf{x})$. 1. *Lane geometry*: Ideally, the point $\mathbf{x}$ that exceeds the solid-lane boundary should receive a repulsive force (cost) $f_g(\mathbf{x}) $ pointing into the travelable lanes. For broken-lane boundaries, we pose the cost of the same structure, but the magnitude is much smaller to allow for lane changing. We present a bi-directional signed distance field (bi-SDF) to describe the corresponding cost characteristics: $$f_g( \mathbf{x} ) = \begin{cases} \alpha_g (- d_b(\mathbf{x}) + \tau_b) ^2 &\text{if $ d_b(\mathbf{x}) \leq 0$} \\ \alpha_g ( d_b(\mathbf{x}) - \tau_b) ^2 &\text{if $0 < d_b(\mathbf{x})\leq \tau_b$}\\ 0 &\text{if $\tau < d_b (\mathbf{x})$}, \end{cases}$$ where $d_b (\mathbf{x})$ measures the distance to the nearest solid-lane boundary, $\tau_b$ is the distance threshold, and $\alpha_g$ is the cost magnitude. Note that $d_b(\mathbf{x}) > 0$ means the in-boundary area, while $d_b(\mathbf{x})<0$ represents that the point exceeds the boundary and needs to be pushed back to the travelable lanes. Different from the traditional SDF, which does not define the gradient when $d_b (\mathbf{x}) < 0$, we slightly extend the definition so that the point outside of the boundary will receive a force pointing inside the lane. The benefit of extending $d_b (\mathbf{x}) < 0$ is that the optimization process is less likely to get stuck in the infeasible out-of-boundary area. 2. *Static obstacles/ driveable area*: The cost $f_s(\mathbf{x})$ induced by static obstacles shares a similar form to $f_g(\mathbf{x})$, and these two costs are categorized into the static layer. The distance measure to static obstacles $d_s(\mathbf{x})$ is also extended to allow a negative distance. 3. *Moving obstacles*: To take interaction with other agents into account, we introduce a cost $f^j_d(\mathbf{x}_i)$ for $t_i$ if the position $\mathbf{x}_i$ of the predicted vehicle is within a distance threshold $\tau_d$ of the prediction $\mathbf{x}_{\text{pred},i}^{j} $ of another agent $j \in \mathcal{J}$, where $\mathcal{J}$ denotes the set of all the interacting agents. The practical method of acquiring $\mathbf{x}_{\text{pred},i}^{j} $ is introduced in Sec. \[sec:optimization\_procedure\]. The MO cost at time $t_i$ is given by $$f^j_d(\mathbf{x}_i) = \alpha_d (d_o(\mathbf{x}_i, j) - \tau_o)^2 \mathbb{1}_{d_o(\mathbf{x}_i,j) < \tau_o},$$ where $f^j_d(\mathbf{x}_i)$ is specified by the quadratic error between the distance $d_o(\mathbf{x}_i,j)$ to the moving agent $j$ and $\tau_0$ if the distance threshold $\tau_0$ is reached. 4. *Red lights*: We argue that red lights should not be enforced as hard constraints since in a real-world driving scenario there exist red light offences. To capture the real intention of other drivers under traffic control, we introduce a red light repulsive force $r(\mathbf{x})$. The repulsive force is supposed to produce larger resistance for vehicles travelling at higher velocity. It is notable that if a vehicle refuses to brake and tries to go through a red light, as shown in Fig. \[fig:through\_red\], the cost $f_r(\mathbf{x})$ will not dominate the optimization process and the abnormal behavior is captured. The overall cost $f_r(\mathbf{x})$ can be expressed by the dot product of the velocity $\dot{\mathbf{x}} $ and the repulsive force $r (\mathbf{x})$ as follows: $$\begin{aligned} f_r(\mathbf{x}) & = \norm{ \dot{\mathbf{x}} \cdot r (\mathbf{x}) }^2 \\ & = \begin{cases} \alpha_r (d_r(\mathbf{x})-\tau_r ) ^2 \norm{\dot{\mathbf{x}} \cdot {\boldsymbol{\hat{\textbf{r}}}}(\mathbf{x})}^2 &\text{if $0 < d_r(\mathbf{x})\leq \tau_r$}\\ 0 &\text{if $\tau_r < d_r (\mathbf{x})$}, \end{cases} \end{aligned}$$ where $\alpha_r$ is the cost magnitude, ${\boldsymbol{\hat{\textbf{r}}}}(\mathbf{x})$ denotes the unit direction of the force $r(\mathbf{x})$, $d_r (\mathbf{x})$ denotes the distance to the red light, and $\tau_r$ is the distance threshold below which the force $r(\mathbf{x})$ will take effect. 5. *Speed limits*: Like red lights, speed limits should not be encoded in hard constraints when taking speed limit offences into account. We introduce the cost $f_v(\mathbf{x})$, which is induced by the speed limit and should also allow the vehicle to stop in the case of a traffic jam. As a result, we model $f_v(\mathbf{x})$ as the quadratic error between the predicted velocity $\dot{\mathbf{x}}$ and a desired velocity $\mathbf{v}_{\text{des}}$. The magnitude of the desired velocity $\norm{ \mathbf{v}_{\text{des}}} $ is determined by the minimum between two factors, namely, the speed limit $v_{\max}$ and the velocity trend $v_{\text{trend}}$. Specifically, $v_{\text{trend}}$ is obtained by conducting velocity fitting for the historical velocity observations in $T_{\text{obs}}$, which captures the acceleration and deceleration trend of the predicted vehicle and is close to zero in the case of a traffic jam. The direction of the desired velocity ${\boldsymbol{\hat{\textbf{v}}}}_{\text{des}}$ conforms to the lane geometry. Mathematically, we have $\norm{\mathbf{v}_{\text{des}}} = \min {(v_{\max}, v_{\text{trend}}) } $ and $f_v(\mathbf{x}) = \norm{\dot{\mathbf{x}} - \mathbf{v}_{\text{des}} }^2$. 6. *Nonholonomic constraints*: The predicted trajectory should obey the limits of the vehicle motion model. Due to the steering geometry of the vehicle, the curvature should be bounded by the maximum curvature allowed. However, when taking abnormal operations, such as skidding, into account, the hard curvature constraint should also be modeled by the feasibility cost $f_{\kappa}(\mathbf{x})$ as follows: $$f_{\kappa}(\mathbf{x}) =\alpha_{\kappa} (\kappa(\mathbf{x}) - \kappa_{\max})^2 \mathbb{1}_{\kappa(\mathbf{x}) > \kappa_{\max}},$$ where the cost takes effect when the curvature exceeds the limit $\kappa_{\max}$ and $\alpha_{\kappa}$ is the cost magnitude. Similarly, due to the friction limit of tires and throttle limit of vehicles, the maximum acceleration of vehicles cannot exceed a limit $a_{\max}$. We model the acceleration feasibility cost $f_{a}(\mathbf{x})$ as follows: $$f_{a}(\mathbf{x}) =\alpha_{a} (\ddot{\mathbf{x}} - a_{\max})^2\mathbb{1}_{\ddot{\mathbf{x}} > a_{\max}},$$ where the maximum acceleration is denoted by $a_{\max}$ and cost magnitude is denoted by $\alpha_{a}$. The motivation for using quadratic functions with barriers for the cost functions is that 1) they tolerate a mild deviation from the best driving practices, and 2) they penalize abnormal behaviors while still allowing their existence. Non-linear optmization {#sec:optimization_detail} ---------------------- Based on the cost functions, we now introduce the optimization formulation. At a top level, the predicted trajectory is generated by minimizing $J(\mathbf{x}(t))$, which is the integral of the overall loss $ L(\mathbf{x}(t))$ over time $T$, i.e., $J(\mathbf{x}(t)) = \int_{t_0}^{t_0 + T} L(\mathbf{x}(t))$, which can be approximated by finite summation in the discrete case as follows: $$\label{eq:non-linear-opt} \begin{aligned} J(\mathbf{x}(t)) = &\sum_{i=0}^{N-1} \big( w_g f_g(\mathbf{x}_i) + w_s f_s(\mathbf{x}_i)+w_d\sum_{j\in \mathcal{J}} f^j_d(\mathbf{x}_i) \\ +& w_r f_r(\mathbf{x}_i) + w_v f_v(\mathbf{x}_i)+w_{\kappa} f_{\kappa}(\mathbf{x}_i) + w_a f_a(\mathbf{x}_i) \big) h. \end{aligned}$$ The weights of different costs represent the tradeoff among different contextual factors. We tune the weights so that predicted trajectories match a human prior for different traffic configurations. As mentioned in Sec. \[sec:introduction\], the optimization process can incorporate IRL for automatic weight tuning, which is important future work. Implementation Details {#sec:implementation} ====================== Simulation environment {#sec:simulation_environment} ---------------------- We adopt an open-source urban autonomous driving simulator named CARLA [@Dosovitskiy17]. In this section, we present our environment setup. For a scene containing $n$ vehicles, the first $n-2$ vehicles (agent vehicles) are controlled by the autopilot module provided by CARLA, the $n-1$-th vehicle (player vehicle) is controlled by a human player and the $n$-th vehicle is an observer vehicle which is supposed to closely follow the player vehicle, sense the environment, and predict the trajectory of the player vehicle. We focus on predicting the trajectories for the player vehicle since it reflects real human intentions. Another reason is that the agent vehicles do not have complex maneuver patterns due to the fixed handwritten logic of the autopilot module. Hence, when presenting the experimental results (Sec. \[sec:results\]) we will focus on illustrating the prediction results for the player vehicle, to give a clean and informative visualization. Data collection and network training ------------------------------------ We collect the training data for the policy anticipation network from CARLA by driving the player vehicle ourselves using a Logitech G29 racing wheel. During the driving, we follow the traffic rules most of the time and conduct different maneuver patterns, but we also commit intentional traffic rule offences, as in Fig. \[fig:through\_red\], to examine how our prediction module will respond. Moreover, we add virtual road construction sites, as in Fig. \[fig:multi\_layer\], and respond to them during driving using the feedback from our visualization system. The collected data is $21,260$ frames in total. $T_{\text{obs}}$ and $T_{\text{pred}}$ are both set to 40 frames (4s). The policy label can be determined by examining the statistics on the steering angle and acceleration in the $T_\text{pred}$ in an unsupervised way. One problem with the data collected from CARLA is that the current version[^2] only includes two-lane roads with traffic moving in opposite directions, which means that lane change behavior cannot be effectively incorporated. In the future, we will collect data from more complex environments to enrich the dataset. Non-linear optimization procedure {#sec:optimization_procedure} --------------------------------- The non-linear optimization formulation (\[eq:non-linear-opt\]) is implemented in Ceres [@ceres-solver] since the objectives can be rewritten into non-linear least squares. If more complex objectives are involved, non-linear solvers such as NLOPT [@Johnson2011] can be used. The maximum number of iterations is set to $20$. Recall that the prediction for a certain vehicle can depend on the prediction of other vehicles due to the moving obstacle cost term $f^j_d(\mathbf{x})$. In practice, we use the prediction results from the last prediction round to calculate $f^j_d(\mathbf{x})$. Results {#sec:results} ======= Prediction accuracy {#sec:result_accuray} ------------------- We adopt the root mean square error (RMSE) between the predicted coordinates and the true coordinates as the error metric. We are concerned with how the RMSE error statistics change with respect to the look-ahead time, especially when the look-ahead time is large. To this end, we plot the mean and variance of the RMSE loss with respect to the look-ahead time, as shown in Fig. \[fig:runtime\_sim\]. We compare our method with the following two methods: - *Naive fitting method*. The future trajectory is generated using least mean square polynomial regression with an acceleration regulator. This method can capture the trend but cannot incorporate the driving context. - *RNN encoder-decoder trajectory regression*. This method uses an RNN to encode the past maneuver history and directly outputs the future trajectory through the RNN decoder. This structure is popular, and is adopted in [@lee2017desire] and [@deo2018convolutional]. Since the source code of [@lee2017desire] is not officially available and [@deo2018convolutional] is mainly tested in a highway dataset, we adopt the RNN encoder-decoder part in [@lee2017desire] according to the available implementation details [@lee2017sup]. We conduct the experiments in the form of case studies to show that our proposed framework can easily adapt to various traffic configurations, as elaborated in Sec. \[sec:simulated\_cases\]. [0.245]{} [0.245]{} [0.245]{} [0.245]{} \[fig:runtime\_sim\] Testing in different traffic configurations {#sec:simulated_cases} ------------------------------------------- To verify that our proposed method can automatically adapt to different traffic configurations and take various latent factors into account, we design five test cases: driving along a curved road, heading towards a pedestrian who is crossing the road, passing through an intersection with road construction, heading towards a red light with road construction, and committing a red light offence. To give a clean visualization, we focus on the prediction for the vehicle being driven by us, namely, the vehicle with ID $0$. 1. *Curved road*: This case is used to verify the capability of reasoning about lane geometries. As illustrated in Fig. \[fig:simu\_curvy\_road\], both baseline methods can capture the motion trend. However, because they are unaware of the lane geometries, they take a long time to conform to the shape of the road. On the other hand, our proposed method produces a reasonable prediction immediately. Quantitatively, our method achieves $44\%$ accuracy improvement for the ending frame in $T_{\text{pred}}$. From the instantaneous error statistics, i.e., the average error for the whole predicted trajectory, we observe that the maximum instantaneous error is reduced from $6.3$ m to $3$ m. This testing case verifies the effectiveness of optimization-based context reasoning. 2. *Intersection with road construction*: This case is used to illustrate the importance of high-level reasoning, which the two baseline methods lack. As shown in Fig. \[fig:simu\_int\], neither baseline methods can effectively capture the turning left intention and both converge slowly. The benefit of incorporating high-level behavior anticipation is validated by an accuracy gain of $56\%$ for the ending frame and a lower instantaneous error during the intersection entrance. The results verify that it is essential to incorporate the high-level intention. 3. *Red light with road construction*: This case is taken as one example of the non-linear optimization (Fig. \[fig:multi\_layer\]). The statistics are provided in Fig.\[fig:simu\_light\], which confirms the necessity of modeling contextual factors. 4. *Heading towards a pedestrian*: This case is used to illustrate the ability to reason about other moving agents. As shown in Fig. \[fig:simu\_dno\], the predicted vehicle is moving at high speed, but a pedestrian is crossing the road ahead of the vehicle. Sudden braking of the vehicle should be the reaction. As shown in Fig. \[fig:simu\_dno\], the two baselines are still giving out forward trajectories, while our method expects hard braking by modeling the interactions between agents. The instantaneous error shows that our method predicts the braking intention beforehand. 5. *Red light offence*: This case is used to show how the proposed method responds to abnormal driving behavior, and is elaborated in Fig. \[fig:through\_red\]. Run-time efficiency ------------------- In this section, we test the run-time efficiency. We collect $3886$ rounds of predictions and record the time consumption of the three parts of the system, namely, network inference, cost map rendering, and non-linear optimization. The experiment is conducted on a desktop computer equipped with an Intel I7-8700K CPU and an NVIDIA GTX 1080-Ti graphics card for network training and inference. As we can see from Tab. \[tab:runtime\_analysis\], the network inference (on GPU) consumes $2.9$ ms on average since the network structure is not complex. It takes an average computing time of $16.2$ ms to render a 4-layer $40\times40$ 2-D cost map (CPU implementation). The non-linear optimization is efficient, with an average time consumption of $3.4$ ms. In total, our prediction system typically consumes $22.6$ ms to complete one round of prediction, and a large part of that time is consumed in the cost map rendering. Conclusion and Future work {#sec:conclusion} ========================== In this paper, we propose an online two-level vehicle trajectory prediction framework which utilizes a policy anticipation network for high-level policy reasoning and a non-linear optimization process for low-level context reasoning. We highlight the flexibility of the proposed framework, and provide various test cases, including normal operations and abnormal driving behavior, in urban environments. In the future, we will explore using IRL [@abbeel2004apprenticeship] to acquire the weights from data. Modeling interaction in prediction is another direction we are actively exploring [@ding2019int]. [^1]: This work was supported by the Hong Kong PhD Fellowship Scheme (HKPFS). All authors are with the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong, China. [wdingae@ust.hk, eeshaojie@ust.hk]{} [^2]: CARLA release 0.7.0 is used for all the experiments.

--- abstract: 'The spectra of two early B-type supergiant stars in the Sculptor spiral galaxy NGC 300 are analysed by means of non-LTE line blanketed unified model atmospheres, aimed at determining their chemical composition and the fundamental stellar and wind parameters. For the first time a detailed chemical abundance pattern (He, C, N, O, Mg and Si) is obtained for a B-type supergiant beyond the Local Group. The derived stellar properties are consistent with those of other Local Group B-type supergiants of similar types and metallicities. One of the stars shows a near solar metallicity while the other one resembles more a SMC B supergiant. The effects of the lower metallicity can be detected in the derived wind momentum.' author: - 'Miguel Alejandro Urbaneja, Artemio Herrero[^1] , Fabio Bresolin, Rolf-Peter Kudritzki[^2] , Wolfgang Gieren[^3]  and Joachim Puls[^4]' title: 'Quantitative spectral analysis of early B-type supergiants in the Sculptor galaxy NGC 300[^5]' --- Introduction ============ The 8-10 meter class telescopes and their new generation instruments make it possible to extend the quantitative stellar spectroscopy beyond the Local Group. Early B-type supergiant stars are ideal targets for detailed spectroscopy even at low resolution (R$\sim$1000). Their blue spectra are rich in metal features which allows us the analysis of chemical species like C, N, O, Si and Mg. Although our knowledge of the evolution of massive stars still has open questions, most of the recent works indicate that the blue luminous supergiants do not show any contamination of their oxygen surface abundances during the early stages of their evolution, neither the O-types [@villamariz2002], nor the B-types [@smartt1997; @monteverde2000; @smartt2002], nor the A-types [@venn1995; @takeda1998; @przybilla2002], which enables a direct comparison between the stellar oxygen abundances and the ones derived from regions. This has become extremely important, especially in the extragalactic field where oxygen is used as the primary metallicity indicator, due to the fact that at high metallicity (larger than approx. 0.5 solar) strong line methods must be used, for which the choice of the calibration strongly influences the derived abundances [@kewley2002; @pilyugin2002]. In addition to chemical abundance studies, blue luminous stars have strong radiatively driven mass outflows which can provide us with information on extragalatic distances by means of the Wind Momentum - Luminosity Relationship, WLR [@kudritzki2000 and references therein]. Recently, and within a wide program aimed at the spectroscopy study of luminous blue stars beyond the Local Group, first steps have been done for A-type supergiants in NGC 3621 [6.7 Mpc away, @bresolin2001]. Quantitative spectroscopy has been shown to be possible for A-type supergiants [@bresolin2002a] and Wolf-Rayet stars [@bresolin2002b] in NGC 300, 2.02 Mpc away in the Sculptor group. Here we report the first quantitative analysis of B-type supergiants (hereafter B-Sg) out of the Local Group, presenting the detailed chemical pattern along with the stellar parameters and the wind properties. The technique will be applied in a forthcoming paper to a large set of early B-Sg located at several galactocentric distances in order to derive radial abundance gradients of the $\alpha$-elements. Combined with the results of a similar study of A-type supergiants it will provide a wealth of information on the chemical evolution of the host galaxy NGC 300. Observations ============ The stars are part of a spectroscopic survey of photometrically selected blue luminous supergiants in the Sculptor galaxy NGC 300, obtained at the VLT with the FORS multiobject spectrograph, and described in detail by @bresolin2002a, which presents a spectral catalog of 70 luminous blue supergiants in the blue region ($\sim$ 4000 - 5000 Å). The selected stars are identified as B-12 and A-9 in that spectral catalog (see their Table 2 and finding charts). In September 2001 the spectra of the H$\alpha$ region were obtained in order to measure the mass-loss rates, which provide us with a complete coverage of the 3800 - 7200 Å wavelength range at R$\sim$1000 resolution. The reader is referred to @bresolin2002a for a detailed description of the observations and reduction process, as well as for the photometry and the spectral classification of the stars. Spectral analysis ================= The spectra of early B-Sg are dominated by the lines, followed by /, /, / and , in addition to H and lines. At high resolution it is possible to detect some other metal lines of , / and but, due to their intrinsic weakness, these lines do not have any influence in the analysis at low resolution and could hardly be used to fix the abundance of such elements. Fig. \[fig1\] shows the high resolution - high S/N ratio (R$\sim$15000, SNR$\sim$350) blue spectrum of the Galactic supergiant HD14956 (B1.5Ia), and the same spectrum degraded to the resolution of the NGC 300 data, R$\sim$1000 (labeled as [*\#d*]{} in the figure). We have also included the identification of the more important lines. As can be seen, only a few strong lines remain isolated at that low resolution, therefore the analysis must be based on the comparison of the observed spectra to a set of model atmospheres that include a vast number of lines in the calculation of the emergent fluxes. We have taken into account more than two hundreds metal lines in the 3800 - 6000 Å wavelength range. It is important to include extense metal line lists because of the fact that some spectral features are formed by the contribution of several chemical species (e.g. the strong blend of O, N and C at $\sim$ 4650 Å). We have excluded some strong isolated lines because our atomic models do not consider the levels involved in these transitions. Nevertheless, these lines are isolated and have no influence on the results. Even considering the noise effects in the lower resolution FORS spectra (displayed also in Fig. \[fig1\]), strong metal features can still be detected and used for a detailed chemical abundance analysis. In particular a wealth of information can be extracted from the selected regions at 4070, 4320, 4420 (), 4550 - 4570 (), 4600 - 4660 (, , and ) and 5010 ( and ). Atmosphere models ----------------- We use the newest version of the FASTWIND code [first presented by @santolayarey1997] which solves the radiation transfer in a moving media by means of suitable approximations which simplify the numerical treatment of the problem but without affecting the physical significance of the results. The atmospheric structure is treated in a consistent way, assuming a $\beta$-velocity law in the wind, ensuring a smooth transition between the “photosphere” and the “wind”; the temperature structure is approximated by means of [*non-LTE Hopf functions*]{} carefully chosen to ensure the flux conservation better than 2 % at any depth point; rate equations are solved in the co-moving frame scheme, with the coupling between the radiation field and the rate equations solved using local ALOs [following @puls1991]. This new version includes the effects of the [*line blanketing*]{}. The reader is referred to Puls et al. (2003, in preparation) for a detailed description. We have analysed two Galactic stars, 10 Lac (O9V) and HD209975 (O9.5Ib) in order to compare our results with the ones obtained with other codes. In the case of 10 Lac, our results agree with the recent ones by @herrero2002 [see their comparison to the results by Hubeny et al. 1998]. The derived parameters for HD209975 are consistent with the results by @villamariz2002 which used plane-parallel model with line blocking. A model is prescribed by the effective temperature $T_{eff}$, the surface gravity [*log g*]{}, the stellar radius $R_*$ (all these three quantities are defined at $\tau_{Ross.} = 2/3$), the mass-loss rate $\dot{M}$, the wind terminal velocity $v_\infty$, the $\beta$ exponent of the wind velocity law, the He abundance $Y_{He}$, the microturbulent velocity $v_{turb}$ and, in the case of B-type stars, the [*Si*]{} abundance. The $T_{eff}$ is well determined from the triplet and the blends of (with at 4090 Å and with / at 4120 Å), and the surface gravity from the Balmer hydrogen lines, provided that the mass-loss rate information is extracted from the H$\alpha$ profile. An important issue concerns the wind terminal velocity, that must be adopted from a spectral type - v$_\infty$ empirical calibration [@haser1995; @kudritzki2000]. The assumed terminal velocity affects the derived $\dot{M}$ and the [*log g*]{}. But, with the joined information from H$\alpha$ and H$\beta$, the mass-loss rate and v$_\infty$ can be constrained to yield reasonable uncertainties in [*log g*]{}. The stellar radius is derived interactively from the absolute magnitude, deduced from the apparent magnitude after adopting a distance modulus [$\mu = 26.53$, @freedman2001], and the model emergent flux [@kudritzki1999], which also provides the reddening by the comparison of the synthetic colors with the observed ones. Results ------- Best-fitting models are displayed in Fig. \[fig1\] and the results summarized in Tab. \[tabla1\]. The derived $\beta$ values are consistent with those obtained by @kudritzki1999 for Galactic B-Sg, with lower values excluded by the arise of emission wings in the synthetic H$\alpha$ profiles. We estimated an uncertainty of $\pm$0.25 in $\beta$. In the case of B-12 only the higher Balmer lines have been considered in the surface gravity determination, as the cores of H$\gamma$ and H$\beta$ are particularly affected by the sky substraction. As it has been quoted, the O and N abundances are very well constrained because of the large number of features from these species. The presence of a lot of weak metal lines in the 4600 - 4700 Å wavelength range makes the selection of the continuum level in this area difficult, good S/N ratio is also needed to disentangle between a real feature and the noise effects. Final abundance uncertainties are estimated to be $\pm\ 0.2$ [*dex*]{} from model comparisons (see Fig. \[fig4\]). We define the mean metallicity as the sum of the $\alpha$-elements abundances, $X_{Si} + X_{Mg} + X_{O}$ and refer it to the Sun abundances by @grevesse1998; at the early stages of massive star evolution, the O surface abundance is not affected by the CNO cycle, which means that the abundance of the $\alpha$-elements is a direct measurement of the ZAMS metallicity of the star. The results for B-12, located close to the galactic center, resembles the abundance patterns of the early B-Sg in the solar neighborhood, having a solar metallicity within the uncertainties of the analysis. On the other hand A-9, in the outskirts of the galaxy, has clearly a lower metallicity, around 0.3 $Z_\odot$. This is in agreement with the results for A-8, a B9-A0 supergiant close to A-9, by @bresolin2002a [see the Fig. 2]. These authors find a mean metallicity of 0.2 $Z_\odot$ for A-8. We must emphasize that both the model atmospheres and the metallicity indicators are different, but the results agree extremely well. The metallicity and the spatial location of both stars in NGC 300 points to a M33-like radial metallicity gradient. The CNO abundances indicate a different degree of chemical evolution, while B-12 displays a normal CNO spectrum, A-9 shows indications of strong N enrichment. Synthetic magnitudes and colors (see Tab. \[tabla3\]) are consistent with almost no reddening for both stars, except the observed $(V-I)$ for B-12 that seems to be anomalous, probably reflecting the presence of the region. Fig. \[fig2\] shows the location of the stars on the Hertzprung-Russel diagram, along with theoretical stellar tracks without rotation at solar metallicity from @schaller1992. We have also added the location of the Galactic stars 10 Lac, HD209975 and HD14956 as a reference. Comparing the wind momentum of both NGC 300 stars with the results for Galactic supergiants (Fig. \[fig5\]), B-12 agrees well with the results by Herrero et al. (2002) for O-type supergiants in the Galactic association Cyg OB2, as does HD14956. Note, however that the Herrero et al. (2002) stars are considerably hotter than the ones considered here. The wind momentum of A-9 is also compatible with the WLR of Galactic early B-Sg as derived by @kudritzki1999. With respect to this relationship, however, B-12 (being an early B-type supergiant as well) shows an enhanced wind momentum rate, which might be related to clumping effects in the wind that would produce an overestimation of the mass-loss rate. The failure of our models to reproduce the blue absorption of H$\alpha$ for B-12, in parallel with an H$\gamma$ core which is too strongly refilled might then be explained by this effect, at least in part, and not only by the rather problematic sky substraction outlined above. The location of A-9, compared to HD14956, reflects the lower metal content of the NGC 300 supergiant. It must be considered here that we have adopted the same $v_\infty$ for both stars, HD14956 and A-9, while a lower value for A-9 could be expected due to its lower metallicity [@kudritzki2000]. In any case the effect of the lower wind terminal velocity would reduce even more the wind momentum of A-9, reinforcing the difference with respect to the Galactic B1.5Ia. Recently @kudritzki2003 have proposed a new extragalactic distance indicator, the “Flux-weighted - Luminosity Relationship (FGLR)”. The results for both NGC 300 B-type supergiants, B-12 and A-9, follow this relationship, within the observed scatter (see the Fig. 2 of the latter reference). We are gratefull to L. J. Corral for making us available the spectrum of HD14956. MAU thanks F. Najarro for providing the routines for the computation of the synthetic magnitudes. AH and MAU thank the Spanish MCyT for a support under proyect PNAYA2001-0436, partially funded with FEDER funds from the EU. WG gratefully acknowledges financial support for this work from the Chilean Center for Astrophysics FONDAP 15010003. Bresolin, F., Kudritzki, R.-P., Méndez, R. H., & Przybilla, N. 2001, , 548, 149 Bresolin, F., Gieren, W., Kudritzki, R.-P., Pietrzyńki, G., & Przybilla, N. 2002a, , 567, 277 Bresolin, F., Kudritzki, R.-P., Najarro, F., Gieren, W. & Pietrzyńki, G. 2002b, , 577, L107 Freedman, W. L., et al. 2001, , 553, 47 Grevesse, N. & Sauval, A. J. 1998, Space Sci. Rev., 85, 161 Haser, S. M. 1995, Ph.D. thesis, Ludwing-Maximillians Univ., Munich Herrero, A., Puls, J., & Najarro, F. 2002, , in press Hubeny, I., Heap, S. R., & Lanz, T. 1998, ASP Conf. Series Vol 131, 108 Kewley, L. J. & Dopita, M. A. 2002, , 142, 35 Kudritzki, R.-P., et al. 1999, , 350, 970 Kudritzki, R.-P. & Puls, J. 2000, , 38, 613 Kudritzki, R.-P., Bresolin, F., & Przybilla, N. 2003, , 582, 83 Monteverde, M. I., Herrero, A., & Lennon, D. J. 2000, , 545, 813 Pilyugin, L. S. 2002, preprint (astro-ph/0211319) Przybilla, N. 2002, Ph.D. thesis, Ludwing-Maximillians Univ., Munich Puls, J. 1991, , 248, 581 Santolaya-Rey, E., Puls, J., & Herrero, A. 1997, , 323, 488 Schaller, G., Schaerer, D., Meynet, G., & Maeder, G. 1992, , 96, 269 Smartt, S. J., Dufton, P. L., & Lennon, D. J. 1997, , 326, 763 Smartt, S. J., Lennon, D. J., Kudrityzki, R.-P., Rosales, F., Ryans, R. S. I., & Wright, N. 2002, , 979, 991 Takeda, Y. & Takada-Hidai, M. 1998, , 50, 629 Venn, K. A. 1995, , 449, 839 Villamariz, M. R., Herrero, A., Becker, S. R., & Butler, K. 2002, , 388, 940 Vink, J. S., de Koter, A., & Lamers, H. J. G. L. M. 2000, , 362, 295 [lccccccccccccccrc]{} B-12 & 24.0$\pm$1.0 & 2.60$\pm$0.15 & 43.5$\pm$1.5 & 0.10 & 20. & 1500. & 3.00$\pm$0.50 & 1.50 & 7.45 & 8.65 & 7.50 & 7.50 & 8.00 & 1.00 & 0.00 & 5.75$\pm$0.10\ A-9 & 21.0$\pm$1.0 & 2.50$\pm$0.15 & 32.0$\pm$1.0 & 0.10 & 15. & 800. & 0.25$\pm$0.07 & 2.00 & 7.10 & 8.30 & 7.20 & 8.00 & 7.60 & 0.30 & -0.50 & 5.24$\pm$0.11\ [lccccccccccc]{} B-12 & 19.30 & -0.18 & 0.00 & & -7.29 & -0.17 & -0.23 & -2.33 & & 0.00 & 0.23\ A-9 & 20.23 & -0.17 &…& & -6.36 & -0.16 & -0.20 & -1.97 & & 0.00 &…\ [^1]: Instituto de Astrofísica de Canarias, Vía Láctea S/N, E-38200 La Laguna, Canary Islands, Spain, maup@ll.iac.es, ahd@ll.iac.es; Dpto. de Astrofísica, Universidad de La Laguna, Avda. Astrofísico Francisco Sanchéz, E-38271 La Laguna, Canary Islands, Spain, ahd@ll.iac.es [^2]: Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, Hawaii 96822, bresolin@ifa.hawaii.edu, kud@ifa.hawaii.edu [^3]: Universidad de Concepción, Departamento de Física, Casilla 160-C, Concepción, Chile, wgieren@coma.cfm.udec.cl [^4]: Universitäts-Sternwarte München, Scheinerstr. 1, D-81679 München, Germany, uh101aw@usm.uni-muenchen.de [^5]: Based on observations obtained at the ESO Very Large Telescope

--- abstract: 'Auger recombination is a non-radiative process, where the recombination energy of an electron-hole pair is transferred to a third charge carrier. It is a common effect in colloidal quantum dots that quenches the radiative emission with an Auger recombination time below nanoseconds. In self-assembled QDs, the Auger recombination has been observed with a much longer recombination time in the order of microseconds. Here, we use two-color laser excitation on the exciton and trion transition in resonance fluorescence on a single self-assembled quantum dot to monitor in real-time every quantum event of the Auger process. Full counting statistics on the random telegraph signal give access to the cumulants and demonstrate the tunability of the Fano factor from a Poissonian to a sub-Poissonian distribution by Auger-mediated electron emission from the dot. Therefore, the Auger process can be used to tune optically the charge carrier occupation of the dot by the incident laser intensity; independently from the electron tunneling from the reservoir by the gate voltage. Our findings are not only highly relevant for the understanding of the Auger process, it also demonstrates the perspective of the Auger effect for controlling precisely the charge state in a quantum system by optical means.' author: - 'P. Lochner' - 'A. Kurzmann' - 'J. Kerski' - 'P. Stegmann' - 'J. König' - 'A. D. Wieck' - 'A. Ludwig' - 'A. Lorke' - 'M. Geller' title: 'Real-time detection of every Auger recombination in a self-assembled quantum dot' --- Keywords: Quantum dots, Resonance fluorescence, Auger recombination, Full counting statistics, Random telegraph signal\ The excitonic transitions in self-assembled quantum dots (QDs) [@Bimberg1999; @Petroff2001] realize perfectly a two-level system in a solid-state environment. These transitions can be used to generate single photon sources [@Michler2000; @Yuan2001] with high photon indistinguishability[@Santori2002; @Matthiesen2013], an important prerequisite to use quantum dots as building blocks in (optical) quantum information and communication technologies[@Kimble2008; @Ladd2010]. Moreover, self-assembled QDs are still one of the best model systems to study in an artificial atom the carrier dynamics[@Kurzmann2016b; @Geller2019], the spin- and angular-momentum properties[@Bayer2000; @Vamivakas2009] and charge carrier interactions[@Labud2014]. One important effect of carrier interactions is the Auger process: An electron-hole pair recombines and instead of emitting a photon, the recombination energy is transferred to a third charge carrier, which is then energetically ejected from the QD[@Kharchenko1996; @Efros1997; @Fisher2005; @Jha2009]. This is a common effect, mostly studied in colloidal QDs, where it quenches the radiative emission with recombination times in the order of picoseconds to nanoseconds[@Vaxenburg2015; @Klimov2000; @Park2014]. This limits the efficiency of optical devices containing QDs like LEDs[@Caruge2008; @Cho2009] or single photon sources[@Brokmann2004; @Michler2000a; @Lounis2000]. In self-assembled QDs, Auger recombination was speculated to be absent, and only recently, it was directly observed in optical measurements on a single self-assembled QD coupled to a charge reservoir with recombination times in the order of microseconds[@Kurzmann2016]. As a single Auger process is a quantum event, it is unpredictable and only the statistical evaluation of many processes gives access to the physical information of the recombination process[@Levitov1996; @Blanter2000]. The most in-depth evaluation - the so-called full counting statistics - becomes possible when each single quantum event in a time trace is recorded. Such real-time detection in optical experiments on a single self-assembled QD have until now only been shown for the statistical process of electron tunneling between the QD and a charge reservoir, where tunneling and spin-flip rates could be tuned by the applied electric and magnetic field[@Kurzmann2019]. Here, Auger recombination in a single self-assembled QD is investigated by optical real-time measurements of the random telegraph signal. With the technique of two-laser excitation, we are able to detect every single quantum event of the Auger recombination. These events take place in the single QD, leaving the quantum dot empty until single-electron tunneling into the QD from the charge reservoir takes place again. This reservoir is coupled to the QD with a small tunneling rate in the order of ms$^{-1}$. The laser intensity, exciting the trion transition, precisely controls the electron emission by the Auger recombination and, hence, the average occupation with an electron. It also tunes the Fano factor from a Poissonian to a sub-Poissonian distribution, which we observe in analyzing the random telegraph signal by methods of full counting statistics. The investigated sample was grown by molecular beam epitaxy (MBE) with a single layer of self-assembled In(Ga)As QDs embedded in a p-i-n diode (see Supporting Information for details). A highly n-doped GaAs layer acts as charge reservoir, which is coupled to the QDs via a tunneling barrier, while a highly p-doped GaAs layer defines an epitaxial gate[@Ludwig2017]. An applied gate voltage $V_\text{G}$ shifts energetically the QD states with respect to the Fermi energy in the electron reservoir and controls the charge state of the dots by electron tunneling through the tunneling barrier. The sample is integrated into a confocal microscope setup within a bath cryostat at 4.2K for resonant fluorescence (RF) measurements (see Methods). {width="1\columnwidth"} Figure \[1\] shows the RF of the neutral exciton (X^0^) and the negatively charged exciton, called trion (X^-^). A RF measurement as function of gate voltage in Figure \[1\]**b** shows the fine-structure split exciton[@Hoegele2004] with an average linewidth of about 1.8$\upmu$eV at low excitation intensity ($1.6\cdot10^{-3}\,\upmu$W/$\upmu$m$^2$). Please note, that this measurement was recorded at a laser energy where the exciton gets into resonance at negative gate voltages because here, the measurement conditions were the best. The quantum-confined Stark effect shifts the exciton resonance X^0^ for higher gate voltages to higher frequencies up to 325.760THz, seen in Figure \[1\]**a**. This quadratic Stark shift of the two exciton transitions[@Li2000] is indicated by two white lines. At a voltage of about 0.375V (dashed vertical line in Fig. \[1\]**a**), the electron ground state in the dot is in resonance with the Fermi energy in the charge reservoir. An electron tunnels into the QD and the exciton transition vanishes while the trion transition can be excited at lower frequencies from 324.5095THz to 324.5115THz. The spectrum of the exciton (blue dots) and the trion transition (red dots) under two-laser excitation is shown in Figure \[1\]**c**. The trion transition is measured at a laser frequency of 324.511THz (corresponding to the red line, “Laser 1” in Fig. \[1\]**a**) and a laser excitation intensity of $8\cdot10^{-6}\,\upmu$W/$\upmu$m$^2$ at a gate voltage of 0.515V. The exciton spectrum in Figure \[1\]**c** was obtained simultaneously by a second laser 2 on the exciton transition (blue line in Fig. \[1\]**a** at 325.7622THz) with a laser excitation intensity of $1.6\cdot10^{-3}\,\upmu$W/$\upmu$m$^2$, as the Auger recombination with rate $\gamma_\text{a}$ leads to an empty QD until an electron tunnels into the dot from the reservoir with rate $\gamma_\text{In}=\gamma_\text{In}^0+\gamma_\text{In}^\text{X}$. This rate comprises the tunneling into the empty dot $\gamma_\text{In}^0$ and the tunneling into the dot charged with an exciton $\gamma_\text{In}^\text{X}$[@Seidl2005] (see Fig. \[1\]**d** for a schematic representation). This has been explained previously in Kurzmann et al.[@Kurzmann2016] with the important conclusion that the intensity ratio between trion/exciton intensity in equilibrium measurements is given by the ratio between Auger/tunneling rate $\gamma_\text{a}/\gamma_\text{In}$. As the tunneling rate $\gamma_\text{In}$ in the sample used here is in the range of ms$^{-1}$, the Auger rate $\gamma_\text{a}$ exceeds the tunneling rate by more than two orders of magnitude (see below). As a consequence, the intensity of the trion transition in equilibrium is by more than two orders of magnitude smaller than the exciton transition. {width="0.5\columnwidth"} The interplay between electron tunneling and optical-driven Auger recombination can be studied in more detail by a real-time random telegraph signal of the resonance fluorescence. In these measurements, the time stamp of every detected RF photon is recorded, see Figure \[2\], enabling the evaluation by full counting statistics. As the intensity of the trion is very weak, the random telegraph signal has been investigated in a two-color excitation scheme. The bright exciton transition with count rates exceeding 10MCounts/s (see Supporting Information) is used as an optical detector for the telegraph signal of the Auger recombination. In this two-color laser excitation scheme, the “exciton off” signal corresponds to the “trion on” signal and vice versa[@Kurzmann2016]. Hence, the trion statistics can directly be determined from the “inverse” exciton signal. The intensity of the exciton excitation laser 2 is held constant at $1.6\cdot 10^{-3}\,\upmu$W/$\upmu$m$^2$. This intensity is far below the saturation of the RF signal of the exciton (see Supporting Information) and avoids the photon-induced electron capture at high excitation intensities[@Kurzmann2016a]. However, this laser intensity yields count rates above 200kcounts/s (see Fig. \[1\]**b**), sufficiently-high for recording single quantum events in a real-time measurement[@Kurzmann2019]. While the intensity of the exciton detection laser 2 is kept constant, the laser intensity of the trion excitation laser 1 is increased from $1.6\cdot10^{-7}\,\upmu$W/$\upmu$m$^2$ up to $1.6\cdot10^{-5}\,\upmu$W/$\upmu$m$^2$. For every trion laser intensity, the time-resolved RF signal is recorded for 15 minutes using a fast (350ps) avalanche photo diode and a bin time of 100$\upmu$s. Figure \[2\] shows parts of three different time traces at three different trion laser 1 intensities. As the exciton laser 2 intensity always exceeds the trion laser 1 intensity by at least nearly two orders of magnitude, the small amount of RF counts from the trion can be neglected. As a consequence, the detected RF signal of the exciton is directly related to the Auger recombination: An Auger recombination empties the dot and the exciton transition detects an empty dot (no trion transition possible) with a count rate of about 25 counts per bin time (100$\upmu$s). After a time $\tau_\text{On}$, an electron tunnels into the QD in Figure \[2\] and the exciton RF signal quenches for a charged dot (trion transition possible) until, after a time $\tau_\text{Off}$, another Auger recombination happens. Increasing the trion laser intensity from $8\cdot10^{-7}$ up to $1.6\cdot10^{-5}\,\upmu$W/$\upmu$m$^2$ in Figure \[2\] increases the probability of an electron emission with rate $\gamma_\text{E}= n \gamma_\text{a}$ by an Auger process, as the probability for occupation of the dot with a trion $n$ increases with increasing laser 1 intensity. Therefore, the exciton transition is observed most frequently for the highest trion laser intensity. This can be observed in Figure \[2\], where the optical random telegraph signal is compared for three different trion excitation intensities. A threshold between exciton “on” and “off” is set for the following statistical evaluation[@Gustavsson2009; @Gustavsson2006]. All exciton RF intensities smaller than this threshold (dashed red line at 7counts/0.1ms in Fig. \[2\]) are counted as “exciton off” (white areas), all intensities above the threshold are counted as “exciton on” (blue areas). {width="\columnwidth"} From these time-resolved RF data sets, the Auger and tunneling rates can be determined by analysing the probability distributions of the “off”-times $\tau_\text{Off}$ and the “on”-times $\tau_\text{On}$ for every 15 minutes long data set[@Gustavsson2009]. A representative distribution at a trion laser intensity of $8\cdot10^{-7}\,\upmu$W/$\upmu$m$^2$ can be seen in Figure \[3\]**a**. An exponential fit to the “on”-times (blue line in Fig. \[3\]**a**) yields the tunneling rate $\gamma_\text{In}$ into the QD, while an exponential fit to the “off”-times (red line in Fig. \[3\]**a**) yields the emission rate $\gamma_\text{E}=n\gamma_\text{a}$ for this specific trion laser 1 intensity. In the example in Figure \[3\]**a**, we find $\gamma_\text{In}=0.80\,$ms$^{-1}$ and $\gamma_\text{E}=0.074\,$ms$^{-1}$. As discussed above, the probability for emitting an electron by an Auger recombination process increases with the occupation probability of the QD with a trion $n$. The occupation probability with a trion $n$ depends on the laser 1 excitation intensity and has been determined from a pulsed measurement of the trion RF intensity, where the highest trion intensity corresponds to an occupation probability of $n=$ 0.5[@Loudon2000] (see Supporting Information for more details). Figure \[3\]**b** shows the expected linear dependence of the electron emission rate $\gamma_\text{E}=n \gamma_\text{a}$ on the occupation probability of the QD with a trion $n$; tuning the emission rate $\gamma_\text{E}$ from almost zero to more than $\gamma_\text{E}= 2$ ms$^{-1}$. The Auger rate is the proportional factor $\gamma_\text{a}$=1.7$\upmu\text{s}^{-1}$ (red data points) and in good agreement with the value obtained before for a different QD with slightly different size[@Kurzmann2016]. The tunneling rate $\gamma_\text{In}$ remains approximately constant at a mean value of 0.74ms$^{-1}$ (blue data points in Fig. \[3\]**b**). This is in agreement with the probability for an electron to tunnel into the empty QD at a constant gate voltage: it is independent on the trion laser intensity. That means, we are able to use the Auger recombination to tune optically the electron emission rate independently from the gate voltage, influencing the emission rate without changing the rate for capturing an electron into the QD (here by the tunneling rate $\gamma_\text{In}$). An independent tuning of electron emission and capture rate is usually not possible for a QD that is tunnel-coupled to one charge reservoir. Changing the coupling strength or Fermi energy by a gate voltage always changes both rates for tunneling into and out of the dot simultaneously. Using the standard methods of full counting statistics[@Gustavsson2009; @Flindt2009] in the following, first of all the asymmetry $a=\frac{\gamma_\text{In}-\gamma_\text{E}}{\gamma_\text{In}+\gamma_\text{E}}$ between the tunneling $\gamma_\text{In}$ and emission rate $\gamma_\text{E}$ has been evaluated. The asymmetry in Figure \[3\]**c** can be tuned by the trion excitation laser intensity from -1 up to 0.55 at a maximum laser intensity of $1.6\cdot10^{-5}\,\upmu$W/$\upmu$m$^2$. Important to mention here: At high trion laser intensities above $1.6\cdot10^{-5}\,\upmu$W/$\upmu$m$^2$, the electron emission by Auger recombination after an tunneling event from the reservoir happens much faster than the bin time of 0.1ms. Therefore, the RF intensity within the bin time is not falling below the threshold and these events are not detected, i.e. the maximum bandwidth of 10kHz (given by the bin time) of the optical detection scheme distorts the statistical analysis at trion laser intensities above $1.6\cdot10^{-5}\,\upmu$W/$\upmu$m$^2$. Below this laser intensity, every single Auger recombination event is detected in the real-time telegraph signal. ![**Probability distribution and cumulants of the time-resolved RF random telegraph signal.** Panel **a** and **b** show the probability $P(N)$ for a number $N$ of Auger events in a bin time of 200ms (blue bars) and the Poissonian distribution related to the mean value of the probability $P(N)$ (red curve). At a trion excitation intensity (laser 1) of $3\cdot10^{-7}\,\upmu$W/$\upmu$m$^2$ (panel **a**), which corresponds to an asymmetry close to -1, the probability $P(N)$ is close to a Poissonian distribution. At a trion excitation intensity of $6\cdot10^{-6}\,\upmu$W/$\upmu$m$^2$ (panel **b**), which corresponds to an asymmetry close to 0, the probability $P(N)$ is sub-Poissonian. Panel **c** shows the second (blue) and third (red) normalized cumulant as a function of the asymmetry. Symbols are measured values, lines are calculated curves for a two-state system[@Gustavsson2006].[]{data-label="4"}](Figure4.pdf){width="\columnwidth"} Finally, full counting statistics[@Gustavsson2009; @Fricke2007; @Gorman2017] is performed on the telegraph signal: Every 15-min long telegraph signal is divided into sections with length $t_0$. The number $N$ of Auger events within the time interval $t_0$ is counted. Figure \[4\]**a** and **b** show two examples for the corresponding probability distributions $P(N)$ in the limit of large $t_0$ (0.2s). At an asymmetry close to -1 (a trion laser intensity of $3\cdot 10^{-7}\,\upmu$W/$\upmu$m$^2$, Fig. \[4\]**a**), the probability is close to a Poissonian distribution. At an asymmetry of about 0 (laser intensity of $6\cdot 10^{-6}\,\upmu$W/$\upmu$m$^2$, Fig. \[4\]**b**), the probability distribution is sub-Poissonian, indicating a relation between Auger recombination and electron tunneling. The Auger recombination emits an electron after an electron has tunneled from the reservoir into the dot. Vice versa, the electron can only tunnel after the Auger recombination has emptied the QD. From the probability distributions, the cumulants $C_m(t_0)=\partial_z^m \ln \mathcal{M}(z,t_0)|_{z=0}$ can be derived with the generating function $\mathcal{M}(z,t_0)=\sum_{N}e^{zN}P(N)$[@Gustavsson2009]. The first cumulant $C_1$ corresponds to the mean value, the second cumulant $C_2$ is the variance and the third one describes the skewness of the distribution. The second and third normalized cumulant in the limit of large $t_0$ (20ms and 5ms, respectively) can be seen as data points in Figure \[4\]**c**. For a two-state system, theory predicts for these normalized cumulants in the long-time limit $C_2/C_1=(1+a^2)/2$ (also called “Fano factor”) and $C_3/C_1=(1+3a^4)/4$[@Gustavsson2006], shown as lines in Figure \[4\]**c**. The data for the second and third normalized cumulant coincide perfectly with the calculated curves. We can conclude from the statistical analysis that the QD behaves like a two-state system, where one state is the QD charged with one electron (or a trion after optical excitation) and the other state is the empty dot (or charged with an exciton). The QD charged with one electron cannot be distinguished from the dot containing a trion (same for empty dot and exciton) as the optical transition times in the order of nanoseconds are orders of magnitude faster than the tunneling and emission time by the Auger recombination [@Zrenner2002]. The statistical analysis demonstrates the influence of the Auger recombination on the cumulants, especially on the Fano factor, which can be tune from $F=1$ to $F=0.5$ by increasing the incident laser intensity on the trion transition. In summary, we performed real-time RF random telegraph measurements and studied full counting statistics of the Auger effect in a single self-assembled QD. With this technique, we were able to measure every single Auger recombination as a quantum jump from a charged to an uncharged QD; followed by single-electron tunneling. The full counting statistics gives access to the normalized cumulants and demonstrates the tunability of the Fano factor from Possonian to sub-Poissonian distribution by the incident laser intensity on the trion transition. Comparison with theoretical prediction shows that the empty and charged QD with the Auger recombination and tunneling follows a dynamical two-state system. For future quantum state preparation, the Auger process can be used to control optically the charge state in a quantum system by optical means. Methods ======= As the same measurement technique is used, this methods section follows the Supplemental information of Kurzmann et al.[@Kurzmann2019]. Optical measurements -------------------- Resonant optical excitation and collection of the fluorescence light is used to detect the optical response of the single self-assembled QD, where the resonance condition is achieved by applying a specific gate voltage between the gate electrode and the Ohmic back contact. The QD sample is mounted on a piezo-controlled stage under an objective lens with a numerical aperture of $NA=0.65$, giving a focal spot size of about 1$\upmu$m diameter. All experiments are carried out in a liquid He confocal dark-field microscope at 4.2K with a tunable diode laser for excitation and an avalanche photodiode (APD) for fluorescence detection. The resonant laser excitation and fluorescence detection is aligned along the same path with a microscope head that contains a 90:10 beam splitter and two polarizers. Cross-polarization enables a suppression of the spurious laser scattering into the detection path by a factor of more than $10^7$. The counts of the APD (dead time of 21.5ns) were binned by a QuTau time-to-digital converter with a temporal resolution of 81ps. This work was supported by the German Research Foundation (DFG) within the Collaborative Research Centre (SFB) 1242, Project No. 278162697 (TP A01), and the individual research grant No. GE2141/5-1. 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--- address: - 'Universität Essen, FB6 Mathematik, 45117 Essen, Germany' - 'University of Michigan, Ann Arbor, MI 48109, USA' author: - Manuel Blickle - Robert Lazarsfeld bibliography: - 'MultiplierNotes.bib' title: | An Informal introduction to\ multiplier ideals --- [^1] Introduction ============ Given a smooth complex variety $X$ and an ideal (or ideal sheaf) ${{\mathfrak{a}}}$ on $X$, one can attach to ${{\mathfrak{a}}}$ a collection of *multiplier ideals* ${{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}^c})}$ depending on a rational weighting parameter $c > 0$. These ideals, and the vanishing theorems they satisfy, have found many applications in recent years. In the global setting they have been used to study pluricanonical and other linear series on a projective variety ([@Demailly93c], [@Angehrn-Siu95a], [@Siu98a], [@Ein-Lazarsfeld97a], [@ELNull], [@Demailly99b]). More recently they have led to the discovery of some surprising uniform results in local algebra ([@ELS1], [@ELS2], [@ELSV]). The purpose of these lectures is to give an easy-going and gentle introduction to the algebraically-oriented local side of the theory. Multiplier ideals can be approached (and historically emerged) from three different viewpoints. In commutative algebra they were introduced and studied by Lipman [@lip.adj] in connection with the Briançon-Skoda theorem.[^2] On the analytic side of the field, Nadel [@Nadel90] attached a multiplier ideal to any plurisubharmonic function, and proved a Kodaira-type vanishing theorem for them.[^3] This machine was developed and applied with great success by Demailly, Siu and others. Algebro-geometrically, the foundations were laid in passing by Esnault and Viehweg in connection with their work involving the Kawamata-Viehweg vanishing theorem. More systematic developments of the geometric theory were subsequently undertaken by Ein, Kawamata and the second author. We will take the geometric approach here. The present notes follow closely a short course on multiplier ideals given by the second author at the Introductory Workshop for the Commutative Algebra Program at the MSRI in September 2002[^4]. The three main lectures were supplemented with a presentation by the first author on multiplier ideals associated to monomial ideals (which appears here in §3). We have tried to preserve in this write-up the informal tone of these talks: thus we emphasize simplicity over generality in statements of results, and we present very few proofs. Our primary hope is to give the reader a feeling for what multiplier ideals are and how they are used. For a detailed development of the theory from an algebro-geometric perspective we refer to Part Three of the forthcoming book [@PAG]. The analytic picture is covered in Demailly’s lectures [@Dem.Mult]. We conclude this Introduction by fixing the set-up in which we work and giving a brief preview of what is to come. Throughout these notes, $X$ denotes a smooth affine variety over an algebraically closed field $k$ of characteristic zero and $R = k[X]$ is the coordinate ring of $X$, so that $X = {{\operatorname{Spec}}}R$. We consider a non-zero ideal ${{\mathfrak{a}}}\subseteq k[X]$ (or equivalently a sheaf of ideals ${\mathfrak{a}}\subseteq {{\ensuremath{{\mathcal{O}}}}}_X$). Given a rational number $c \geq 0$ our plan is to define and study the multiplier ideal $${{\ensuremath{{\mathcal{J}}}}}(c \cdot {\mathfrak{a}})\ =\ {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c) \ \subseteq \ k[X].$$ As we proceed, there are two ideas to keep in mind. The first is that ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c)$ measures in a somewhat subtle manner the singularities of the divisor of a typical function $f$ in ${\mathfrak{a}}$: for fixed $c$, “nastier" singularities are reflected by “deeper" multiplier ideals. Secondly, ${{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}^c})}$ enjoys remarkable formal properties arising from the Kawamata-Viehweg-Nadel Vanishing theorem. One can view the power of multiplier ideals as arising from the confluence of these facts. The theory of multiplier ideals described here has striking parallels with the theory of tight closure developed by Hochster and Huneke in positive characteristic. Many of the uniform local results that can be established geometrically via multiplier ideals can also be proven (in more general algebraic settings) via tight closure. For some time the actual connections between the two theories were not well understood. However very recent work [@HaraYosh], [@Takagi.MultTest] of Hara-Yoshida and Takagi has generalized tight closure theory to define a so called test ideal $\tau({\mathfrak{a}})$, which corresponds to the multiplier ideal ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}})$ under reduction to positive characteristic. This provides a first big step towards identifying concretely the links between these theories. Concerning the organization of these notes, we start in §2 by giving the basic definition and several examples. Multiplier ideals of monomial ideals are discussed in detail in §3. Invariants arising from multiplier ideals, with some applications to uniform Artin-Rees numbers, are taken up in §4. Section 5 is devoted to a discussion of some basic results about multiplier ideals, notably Skoda’s theorem and the restriction and subaddivity theorems. We consider asymptotic constructions in §6, with applications to uniform bounds for symbolic powers following [@ELS1]. We are grateful to Karen Smith for suggestions concerning these notes. Definition and Examples {#sec.defex} ======================= As just stated, $X$ is a smooth affine variety of dimension $n$ over an algebraically closed field of characteristic zero, and we fix an ideal ${{\mathfrak{a}}}\subseteq k[X]$ in the coordinate ring of $X$. Very little is lost by focusing on the case $X = {{\ensuremath{\mathbb{C}}}}^n$ of affine $n$-space over the complex numbers ${{\ensuremath{\mathbb{C}}}}$, so that ${{\mathfrak{a}}}\subseteq {{\ensuremath{\mathbb{C}}}}[x_1, \ldots, x_n]$ is an ideal in the polynomial ring in $n$ variables. Log resolution of an ideal -------------------------- The starting point is to realize the ideal ${{\mathfrak{a}}}$ geometrically. A *log resolution* of an ideal sheaf ${\mathfrak{a}}\subseteq {{\ensuremath{{\mathcal{O}}}}}_X$ is a proper, birational map $\mu: Y {\xrightarrow{\ \ }}X$ whose exceptional locus is a divisor $E = \text{Exceptional}(\mu)$ such that 1. $Y$ is non-singular. 2. ${\mathfrak{a}}\cdot {{\ensuremath{{\mathcal{O}}}}}_{Y} =\mu^{-1}{\mathfrak{a}}= {{\ensuremath{{\mathcal{O}}}}}_{Y}(-F)$ with $F=\sum r_iE_i$ an effective divisor. 3. $F+E$ has simple normal crossing support. Recall that a (Weil) divisor $D=\sum \alpha_i D_i$ has simple normal crossing support if each of its irreducible components $D_i$ is smooth, and if locally analytically one has coordinates $x_1,\ldots,x_n$ of $Y$ such that ${{\operatorname{Supp}}}D=\sum D_i$ is defined by $x_1\cdot\ldots\cdot x_a$ for some $a$ between $1$ and $n$. In other words, all the irreducible components of $D$ are smooth and intersect transversally. The existence of a log resolution for any sheaf of ideals in any variety over a field of characteristic zero is essentially Hironaka’s celebrated result on resolution of singularities [@Hironaka.ResSing]. Nowadays there are more elementary constructions of such resolutions, for instance [@Bierstone-Milman97], [@EncVill.Desing] or [@Paranjape]. Let $X={{\ensuremath{\mathbb{A}}}}^2={{\operatorname{Spec}}}k[x,y]$ and ${\mathfrak{a}}= (x^2,y^2)$. Blowing up the origin in ${{\ensuremath{\mathbb{A}}}}^2$ yields $$Y = {\mathit{Bl}}_0({{\ensuremath{\mathbb{A}}}}^2) {\xrightarrow{\ \mu\ }} {{\ensuremath{\mathbb{A}}}}^2=X.$$ Clearly, $Y$ is nonsingular. Computing on the chart for which the blowup $\mu$ is a map from ${{\ensuremath{\mathbb{A}}}}^2 {\xrightarrow{\ \ }}{{\ensuremath{\mathbb{A}}}}^2$ given by $(u,v) \mapsto (u,uv)$ shows that ${\mathfrak{a}}\cdot {{\ensuremath{{\mathcal{O}}}}}_{Y} = {{\ensuremath{{\mathcal{O}}}}}_{Y}(-2E)$. On the described chart we have ${\mathfrak{a}}\cdot {{\ensuremath{{\mathcal{O}}}}}_Y = (u^2,u^2v^2)=(u^2)$ and $(u=0)$ is the equation of the exceptional divisor. This resolution is illustrated in Figure \[Resolve.Double.Pt\], where we have drawn schematically the curves in ${{\ensuremath{\mathbb{A}}}}^2$ defined by typical $k$-linear combinations of generators of ${{\mathfrak{a}}}$, and the proper transforms of these curves on $Y$. Note that these proper transforms do not meet: this reflects the fact that ${{\mathfrak{a}}}$ has become principal on $Y$. ![Log resolution of $(x^2, y^2)$[]{data-label="Resolve.Double.Pt"}](ResolveDoubPt) Now let ${\mathfrak{a}}= (x^3,y^2)$. In this case a log resolution is constructed by the familiar sequence of three blowings-up used to resolve a cuspidal curve (Figure \[Resolve.Cusp\]). Here we have ${\mathfrak{a}}\cdot {{\ensuremath{{\mathcal{O}}}}}_{Y} = {{\ensuremath{{\mathcal{O}}}}}_{Y}(-2E_1-3E_2-6E_3)$ where $E_i$ is the exceptional divisor of the $i$th blowup. 5 pt ![Log resolution of $(x^3,y^2)$.[]{data-label="Resolve.Cusp"}](ResolveCusp "fig:") 10 pt These examples illustrate the principle that a log resolution of an ideal ${\mathfrak{a}}$ is very close to being the same as a resolution of singularities of a divisor of a general function in ${\mathfrak{a}}$. Definition of multiplier ideals ------------------------------- Besides a log resolution of $\mu:Y {\xrightarrow{\ \ }}X$ of the ideal ${\mathfrak{a}}$, the other ingredient for defining the multiplier ideal is the relative canonical divisor $$K_{Y/X}=K_{Y}-\mu^*K_X= {{\operatorname{div}}}(\det ({\operatorname{Jac}} \mu)).$$ It is unique as a divisor (and not just as a divisor class) if one requires its support to be contained in the exceptional locus of $\mu$. Alternatively, $K_{Y/X}$ is the effective divisor defined by the vanishing of the determinant of the Jacobian of $\mu$. The canonical divisor $K_X$ is just the class corresponding to the canonical line bundle $\omega_X$. If $X$ is smooth, $\omega_X$ is just the sheaf of top differential forms $\Omega^n_X$ on $X$. Extremely useful for basic computations of multiplier ideals is the following proposition, see [@Ha], Exercise II.8.5. \[prop.CanBlow\] Let $Y={\mathit{Bl}}_Z X$ where $Z$ is a smooth subvariety of the smooth variety $X$ of codimension $c$. Then the relative canonical divisor $K_{Y/X}$ is $(c-1)E$, $E$ being the exceptional divisor of the blowup. Now we can give a provisional definition of the multiplier ideal of an ideal ${{\mathfrak{a}}}$: it coincides in our setting with Lipman’s construction in [@lip.adj]. Let ${\mathfrak{a}}\subseteq k[X]$ be an ideal. Fix a log resolution $\mu: Y {\xrightarrow{\ \ }}X$ of ${\mathfrak{a}}$ such that ${\mathfrak{a}}\cdot{{\ensuremath{{\mathcal{O}}}}}_{Y}={{\ensuremath{{\mathcal{O}}}}}_{Y}(-F)$, where $F=\sum r_iE_i$, and $K_{Y/X}=\sum b_iE_i$. The *multiplier ideal* of ${\mathfrak{a}}$ is $$\begin{split} {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}) \ &=\ \mu_*{{\ensuremath{{\mathcal{O}}}}}_{Y}(K_{Y/X} - F) \\ &= \ \big \{ \, h \in k[X] \mid {{\operatorname{div}}}(\mu^*h)+K_{Y/X} - F \, \geq \, 0 \, \big \} \\ &=\ \big\{\, h \in k[X] \mid {{\operatorname{ord}}}_{E_i}(\mu^* h) \geq r_i - b_i \text{ for all $i$}\, \big\}. \end{split}$$ (We will observe later that this is independent of the choice of resolution.) The definition may seem at first blush a little mysterious. One way to motivate it is to note that ${{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}})}$ is the push-forward of a bundle which is very natural from the viewpoint of vanishing theorems. In fact, the bundle ${{\ensuremath{{\mathcal{O}}}}}_Y(-F)$ appearing above is (close to being) ample for the map $\mu$. Therefore $K_{Y/X} - F$ has the shape to which Kodaira-type vanishing results will apply. In any event, the definition will justify itself before long through the properties of the ideals so defined. Use the fact that $\mu_*\omega_Y=\omega_X$ to show that ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}})$ is indeed an ideal in $k[X]$. \[ex.intclo\] Show that the integral closure ${\overline{{\mathfrak{a}}}}$ of ${\mathfrak{a}}$ is equal to $\mu_* {{\ensuremath{{\mathcal{O}}}}}_{Y}(-F)$. Use this to conclude that ${\mathfrak{a}}\subseteq {\overline{{\mathfrak{a}}}} \subseteq {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}) = {\overline{{{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}})}}$. (Recall that the integral closure of an ideal ${\mathfrak{a}}$ consists of all elements $f$ such that $v(f) \geq v(a)$ for all valuations $v$ of ${{\ensuremath{{\mathcal{O}}}}}_X$.) \[Mult.Ideal.Int.Closure.Ideal\] Verify that for ideals ${\mathfrak{a}}\subseteq {\mathfrak{b}}$ one has ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}) \subseteq {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{b}})$. Use this and the previous exercise to show that ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}})={{\ensuremath{{\mathcal{J}}}}}({\overline{{\mathfrak{a}}}})$. The above definition of the multiplier ideal is not general enough for the most interesting applications. As it turns out, allowing an additional rational (or real) parameter $c$ considerably increases the power of the theory. Note that a log resolution of an ideal ${\mathfrak{a}}$ is at the same time a log resolution of any integer power ${\mathfrak{a}}^n$ of that ideal. Thus we extend the last definition, using the same log resolution for every $c \geq 0$: \[Gen.Mult.Ideal.Def\] For every rational number $c\geq 0$, the *multiplier ideal* of the ideal ${\mathfrak{a}}$ with exponent (or coefficient) $c$ is $$\begin{split} {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c)\ = \ {{\ensuremath{{\mathcal{J}}}}}(c\cdot {\mathfrak{a}}) \ &= \ \mu_*{{\ensuremath{{\mathcal{O}}}}}_{Y}(K_{Y/X}-{\lfloor c\cdot F \rfloor}) \\ &= \ \big \{ h \in k[X] \, |\, {{\operatorname{ord}}}_{E_i}(\mu^*h) \, \geq \, {\lfloor cr_i \rfloor}-b_i \text{ for all $i$}\, \big \} \end{split}$$ where $\mu : Y {\xrightarrow{\ \ }}X$ is a log resolution of ${\mathfrak{a}}$ such that ${\mathfrak{a}}\cdot {{\ensuremath{{\mathcal{O}}}}}_Y = {{\ensuremath{{\mathcal{O}}}}}_Y(-F)$. Note that we do not assign any meaning to ${\mathfrak{a}}^c$ itself, only to ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c)$.[^5] The round-down operation ${\lfloor \cdot \rfloor}$ applied to a ${{\ensuremath{\mathbb{Q}}}}$-divisor $D = \sum a_iD_i$ for distinct prime divisors $D_i$ is just rounding down the coefficients. That is, ${\lfloor D \rfloor} = \sum {\lfloor a_i \rfloor}D_i$. The round up ${\lceil D \rceil}=-{\lfloor -D \rfloor}$ is defined analogously. Show that rounding does not in general commute with restriction or pullback. \[MI.Max.Idea.Ex\] Let ${\mathfrak{m}}$ be the maximal ideal of a point $x \in X$. Show that $${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{m}}^c)=\begin{cases} {\mathfrak{m}}^{{\lfloor c \rfloor}+1-n} & \text{for $c\geq n=\dim X$.} \\ {{\ensuremath{{\mathcal{O}}}}}_X & \text{otherwise.} \end{cases}$$ Let ${\mathfrak{a}}= (x^2,y^2) \subseteq k[x,y]$. For the log resolution of ${\mathfrak{a}}$ as calculated above we have $K_{Y/X}=E$. Therefore, $${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c) = \mu_*({{\ensuremath{{\mathcal{O}}}}}_{Y}(E-{\lfloor 2c \rfloor}E))= (x,y)^{{\lfloor 2c \rfloor}-1}$$ (In view of Exercise \[Mult.Ideal.Int.Closure.Ideal\], this is a special case of Exercise \[MI.Max.Idea.Ex\].) Let ${\mathfrak{a}}= (x^2,y^3)$. In this case we computed a log resolution with $F= 2E_1+3E_2+6E_3$. Using the basic formula for the relative canonical divisor of a blowup along a smooth center, one computes $K_{Y/X}= E_1 + 2 E_2 +4 E_3$. Therefore, $$\begin{split} {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c) &=\ \mu_*\big({{\ensuremath{{\mathcal{O}}}}}_{Y}(E_1 + 2 E_2 +4 E_3-{\lfloor c(2E_1+3E_2+6E_3) \rfloor})) \\ &= \mu_*({{\ensuremath{{\mathcal{O}}}}}_{Y}((1-{\lfloor 2c \rfloor})E_1+(2-{\lfloor 3c \rfloor})E_2+(4-{\lfloor 6c \rfloor})E_3)). \end{split}$$ This computation shows that for $c < 5/6$ the multiplier ideal is trivial, [*i.e.*, ]{}${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c)={{\ensuremath{{\mathcal{O}}}}}_X$. Furthermore, ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^{\frac{5}{6}})=(x,y)$. The next coefficient for which the multiplier ideal changes is $c=1$. This behavior of multiplier ideals to be piecewise constant with discrete jumps is true in general and will be discussed in more detail later. \[Smooth ideals\] \[Smooth.Ideal.Exercise\] Suppose that ${\mathfrak{q}}\subseteq k[X]$ is the ideal of a smooth subvariety $Z \subseteq X$ of pure codimension $e$. Then $${{{\ensuremath{{\mathcal{J}}}}}({{\mathfrak{q}}^\ell})} \ = \ {\mathfrak{q}}^{\ell + 1 - e}.$$ (Blowing up $X$ along $Z$ yields a log resolution of ${\mathfrak{q}}$.) The case of fractional exponents is similar. Two basic properties -------------------- The definitions of the previous subsection are justified by the fact that they lead to two very basic results. The first point is that the ideal ${{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}^c})}$ constructed in Definition \[Gen.Mult.Ideal.Def\] is actually independent of the choice of resolution. If $X_1 {\xrightarrow{\ \mu_1\ }} X$ and $X_2 {\xrightarrow{\ \mu_2\ }} X$ are two log resolutions of the ideal ${\mathfrak{a}}\subseteq {{\ensuremath{{\mathcal{O}}}}}_X$ such that ${\mathfrak{a}}{{\ensuremath{{\mathcal{O}}}}}_{X_i} = {{\ensuremath{{\mathcal{O}}}}}_{X_i}(-F_i)$, then $${\mu_1}_*\big({{\ensuremath{{\mathcal{O}}}}}_{X_1}(K_{X_1/X}-{\lfloor c\cdot F_1 \rfloor}\big)\ = \ {\mu_2}_*\big({{\ensuremath{{\mathcal{O}}}}}_{X_2}(K_{X_2/X}-{\lfloor c\cdot F_2 \rfloor}\big).$$ As one would expect, the proof involves dominating $\mu_1$ and $\mu_2$ by a third resolution. It is in the course of this argument that it becomes important to know that $F_1$ and $F_2$ have normal crossing support, see [@PAG Chapter 9]. By contrast, give an example to show that if $c$ is non-integral, then the ideal $\mu_*(-{\lfloor cF \rfloor})$ may indeed depend on the log resolution $\mu$. The second fundamental fact is a vanishing theorem for the sheaves computing multiplier ideals. \[Local Vanishing Theorem\] \[Local.Vanishing.Thm\] Consider an ideal ${{\mathfrak{a}}}\subseteq k[X]$ as above, and let $\mu: Y {\xrightarrow{\ \ }}X$ be a log resolution of ${\mathfrak{a}}$ with ${\mathfrak{a}}\cdot {{\ensuremath{{\mathcal{O}}}}}_Y = {{\ensuremath{{\mathcal{O}}}}}_Y(-F)$. Then $$R^i\mu_*{{\ensuremath{{\mathcal{O}}}}}_Y(K_{Y/X}-{\lfloor cF \rfloor}) = 0$$ for all $i > 0$ and $c > 0$. This leads one to expect that the multiplier ideal, being the zeroth derived image of ${{\ensuremath{{\mathcal{O}}}}}_Y(K_{Y/X}-{\lfloor cF \rfloor})$ under $\mu_*$, will display particularly good cohomological properties. Theorem \[Local.Vanishing.Thm\] is a special case of the Kawamata-Viehweg vanishing theorem for a mapping, see [@PAG Chapter 9] . It is the essential fact underlying all the applications of multiplier ideals appearing in these notes. When $c$ is a natural number, the result can be seen as a slight generalization of the classical Grauert-Riemenschneider Vanishing Theorem. However as we shall see it is precisely the possibility of working with non-integral $c$ that opens the door to applications of a non-classical nature. Analytic construction of multiplier ideals {#Analytic.Interp.Subsectn} ------------------------------------------ We sketch briefly the analytic construction of multiplier ideals. Let $X$ be a smooth complex affine variety, and ${{\mathfrak{a}}}\subseteq {{\ensuremath{\mathbb{C}}}}[X]$ an ideal. Choose generators $g_1,\ldots,g_p \in {{\mathfrak{a}}}$. Then $${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c)^{\text{an}}\ =_{\text{locally}} \ \Big \{ \, h \text{ holomorphic }\Big | \,\, \frac{|h|^2}{\big(\sum |g_i|^{2}\big)^c} \text{ is locally integrable} \, \Big \}.$$ In other words, the analytic ideal associated to ${{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}^c})}$ arises as a sheaf of “multipliers". See [@Demailly99b (5.9)] or [@PAG Chapter 9.3.D] for the proof. In brief the idea is to show that both the algebraic and the analytic definitions lead to ideals that transform the same way under birational maps. This reduces one to the situation where ${{\mathfrak{a}}}$ is the principal ideal generated by a single monomial in local coordinates. Here the stated equality can be checked by an explicit calculation. Multiplier ideals via tight closure ----------------------------------- As already hinted at in the introduction there is an intriguing parallel between effective results in local algebra obtained via multiplier ideals on the one hand and tight closure methods on the other. Almost all the results we will discuss in these notes are of this kind: there are tight closure versions of the Briançon-Skoda theorem, the uniform Artin-Rees lemma and even of the result on symbolic powers we present as an application of the asymptotic multiplier ideals in Section \[sec.symb\].[^6] There is little understanding for why such different techniques (characteristic zero, analytic in origin vs. positive characteristic) seem to be tailor made to prove the same results. Recently, Hara-Yoshida and Takagi strengthened this parallel by constructing in [@HaraYosh] and [@Takagi.MultTest; @TakWat; @HaTak; @Tak.Inv] multiplier-like ideals using techniques modelled after tight closure theory. Their construction builds on earlier work of Smith [@Smith.MultTest] and Hara [@Hara.GeomIntTest], who had established a connection between the multiplier ideal associated to the unit ideal $(1)$ on certain singular varieties with the so-called test ideal in tight closure. The setting of the work of Hara and Yoshida is a regular[^7] local ring $R$ of positive characteristic $p$. For simplicity one might again assume $R$ is the local ring of a point in ${{\ensuremath{\mathbb{A}}}}^n$. Just as with multiplier ideals, one assigns to an ideal ${\mathfrak{a}}\subseteq R$ and a rational parameter $c \geq 0$, the *test ideal* $$\tau({\mathfrak{a}}^c) = \{\, h \in R\,|\, h I^{*{\mathfrak{a}}^c} \subseteq I \text{ for all ideals } I\,\}.$$ Here $I^{*{\mathfrak{a}}^c}$ denotes the ${\mathfrak{a}}^c$–tight closure of an ideal, specifically introduced for the purpose of constructing these test ideals $\tau({\mathfrak{a}}^c)$.[^8] The properties the test ideals enjoy are strikingly similar to those of the multiplier ideal in characteristic zero: For example the Restriction Theorem (Theorem \[thm.restr\]) and Subadditivity (Theorem \[thm.subadd\]) hold. What makes the test ideal a true analog of the multiplier ideal is that under the process of reduction to positive characteristic the multiplier ideal ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c)$ corresponds to the test ideal $\tau({\mathfrak{a}}^c)$, or more precisely to the test ideal of the reduction mod $p$ of ${\mathfrak{a}}^c$. The multiplier ideal of monomial ideals {#sec.nonomial} ======================================= Even though multiplier ideals enjoy extremely good formal properties, they are very hard to compute in general. An important exception is the class of monomial ideals, whose multiplier ideals are described by a simple combinatorial formula, established by Howald [@Howald.MultMon]. By way of illustration we discuss this result in detail. To state the result let ${\mathfrak{a}}\subseteq k[x_1,\ldots,x_n]$ be a nonomial ideal, that is an ideal generated by monomials of the form $x^m=x_1^{m_1}\cdot\ldots\cdot x_n^{m_n}$ for $m \in {{\ensuremath{\mathbb{Z}}}}^n \subseteq {{\ensuremath{\mathbb{R}}}}^n$. In this way we can identify a monomial ideal ${\mathfrak{a}}$ of $k[x_1,\ldots,x_n]$ with the set of exponents (contained in ${{\ensuremath{\mathbb{Z}}}}^n$) of the monomials in ${\mathfrak{a}}$. The convex hull of this set in ${{\ensuremath{\mathbb{R}}}}^n={{\ensuremath{\mathbb{Z}}}}^n {\otimes}{{\ensuremath{\mathbb{R}}}}$ is called the *Newton polytope* of ${\mathfrak{a}}$ and it is denoted by ${{\operatorname{Newt}}}({\mathfrak{a}})$. Now Howald’s result states: \[thm.mon\] Let ${\mathfrak{a}}\subseteq k[x_1,\ldots,x_n]$ be a monomial ideal. Then for every $c>0$, $${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c) = \langle x^m \,|\, m+(1,\ldots,1) \in \text{ interior of }c \cdot {{\operatorname{Newt}}}({\mathfrak{a}}) \rangle$$ For example, the picture of the Newton polytope of the monomial ideal ${\mathfrak{a}}= (x^4,xy^2,y^4)$ in Figure \[fig.NPx4\] shows, using Howald’s result, that ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}})=(x^2,xy,y^2)$. Note that even though $(0,1)+(1,1)$ lies in the Newton polytope ${{\operatorname{Newt}}}({\mathfrak{a}})$ it does not lie in the interior. Therefore, the monomial $y$ corresponding to $(0,1)$ does *not* lie in the multiplier ideal ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}})$. But for all $c<1$, clearly $y \in {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c)$. \[fig.NPx4\]  To pave the way for clean proofs we need to formalize our setup slightly and recall some results from toric geometry. Basic notions from toric geometry. ---------------------------------- Note that $k[X]=k[x_1,\ldots,x_n]$ carries a natural ${{\ensuremath{\mathbb{Z}}}}^n$-grading by giving a monomial $x^m=x_1^{m_1}\cdot\ldots\cdot x_n^{m_n}$ degree $m \in {{\ensuremath{\mathbb{Z}}}}^n$. Equivalently we note that the $n$-dimensional torus $$T^n \ =\ {{\operatorname{Spec}}}k[x_1^{\pm 1},\ldots,x_n^{\pm 1}] \ \cong \ (k^*)^n$$ acts on $k[X]$ via $\lambda \cdot x^m = \lambda^m x^m$ for $\lambda \in (k^*)^n$. In terms of the varieties this means that $X={{\ensuremath{\mathbb{A}}}}^n$ contains the torus $T^n$ as a dense open subset, and the action of $T^n$ on itself naturally extends to an action of $T^n$ on all of $X$. Under this action, the torus fixed ($={{\ensuremath{\mathbb{Z}}}}^n$-graded) ideals are precisely the monomial ideals. We denote the lattice ${{\ensuremath{\mathbb{Z}}}}^n$ in which the grading takes place by $M$. It is just the lattice of the exponents of the Laurent monomials of $k[T^n]$. As indicated above, the Newton polytope ${{\operatorname{Newt}}}({\mathfrak{a}})$ of a monomial ideal ${\mathfrak{a}}$ is just the convex hull in $M_{{\ensuremath{\mathbb{R}}}}=M {\otimes}_{{\ensuremath{\mathbb{Z}}}}{{\ensuremath{\mathbb{R}}}}$ of the set $\{\, m \in M\, |\, x^m \in {\mathfrak{a}}\,\}$. The Newton polytope of a principal ideal $(x^v)$ is just the positive orthant in $M_{{\ensuremath{\mathbb{R}}}}$ shifted by $v$. In general, the Newton polytope of any ideal is an unbounded region contained in the first orthant. With every point $v$ the Newton polytope also contains the first orthant shifted by $v$. \[ex.intclomon\] Let ${\mathfrak{a}}$ be a monomial ideal in $k[x_1,\ldots,x_n]$. Then the lattice points (viewed as exponents) in the Newton polytope ${{\operatorname{Newt}}}({\mathfrak{a}})$ of ${\mathfrak{a}}$ define an ideal ${\overline{{\mathfrak{a}}}} \supseteq {\mathfrak{a}}$. Show that ${\overline{{\mathfrak{a}}}}$ is the integral closure of ${\mathfrak{a}}$ (see [@Fulton.Toric]). The property of $X={{\ensuremath{\mathbb{A}}}}^n$ to contain the torus $T^n$ as a dense open set such that the action of $T^n$ on itself extends to an action on $X$ as just described is the definition of a *toric variety*. The language of toric varieties is the most natural to phrase, prove (and generalize, see [@Bli.MultToric]) Howald’s result. To set this up completely would take us somewhat afield, so we choose to take a more direct approach using only a bare minimum of toric geometry. A first fact we have to take without proof from the theory of toric varieties is that log resolutions of torus fixed ideals of $k[X]$ exist in the category of toric varieties.[^9] Let ${\mathfrak{a}}\subseteq k[x_1,\ldots,x_n]$ be a monomial ideal. Then there is a log resolution $\mu: Y {\xrightarrow{\ \ }}X$ of ${\mathfrak{a}}$ such that $\mu$ is a map of toric varieties and consequently ${\mathfrak{a}}\cdot {{\ensuremath{{\mathcal{O}}}}}_Y = {{\ensuremath{{\mathcal{O}}}}}_Y(-F)$ is such that $F$ is fixed by the torus action on $Y$. This follows from the theory of toric varieties. First one takes the normalized blowup of ${\mathfrak{a}}$, which is a (possibly singular) toric variety since ${\mathfrak{a}}$ was a torus invariant ideal. Then one torically resolves the singularities of the resulting variety as described in [@Fulton.Toric]. Note that this is a much easier task than resolution of singularities in general. It comes down to a purely combinatorial procedure. An alternative proof could use Encinas and Villamayor’s [@EncVill.Desing] equivariant resolution of singularities. They give an algorithmic procedure of constructing a log resolution of ${\mathfrak{a}}$ such that the torus action is preserved — that is by only blowing up along torus fixed centers. ### Toric Divisors A toric variety $X$ has a finite set of torus fixed prime (Weil) divisors. Indeed, since an arbitrary torus fixed prime divisor cannot meet the torus ($T^n$ acts transitive on itself and is dense in $X$) it has to lie in the boundary $Y-T^n$, which is a variety of dimension $\leq n-1$ and thus can only contain finitely many components of dimension $n-1$. Furthermore, these torus fixed prime divisors $E_1,\ldots,E_r$ generate the lattice of all torus fixed divisors which we shall denote by $L^X$. We denote the sum of all torus invariant prime divisors $E_1+\ldots+E_r$ by $1_X$. The torus invariant rational functions of a toric variety are just the Laurent monomials $x_1^{m_1}\cdot\ldots\cdot x_n^{m_n} \in k[T^n]$. For the toric variety $X={{\ensuremath{\mathbb{A}}}}^n$ one clearly has the identification of $M$, the lattice of exponents, with $L^X$ by sending $m$ to ${{\operatorname{div}}}x^m$. In general this map will not be surjective and its image is precisely the set of torus invariant Cartier divisors. We note the following easy lemma which will nevertheless play an important role in our proof of Theorem \[thm.mon\]. It makes precise the idea that a log resolution of a monomial ideal ${\mathfrak{a}}$ corresponds to turning its Newton polytope ${{\operatorname{Newt}}}({\mathfrak{a}}) \subseteq M_{{\ensuremath{\mathbb{R}}}}$ into a translate of the first orthant in $L^X_{{\ensuremath{\mathbb{R}}}}$. \[lem.techmon\] Let $\mu: Y {\xrightarrow{\ \ }}X={{\operatorname{Spec}}}k[x_1,\ldots,x_n]$ be a toric resolution of the monomial ideal ${\mathfrak{a}}\subseteq k[x_1,\ldots,x_n]$ such that ${\mathfrak{a}}\cdot {{\ensuremath{{\mathcal{O}}}}}_Y = {{\ensuremath{{\mathcal{O}}}}}_Y(-F)$. Then, for $m \in M$ we have $$c \cdot m \in c'{{\operatorname{Newt}}}({\mathfrak{a}}) \iff c\cdot\mu^* {{\operatorname{div}}}x^m \geq c' \cdot F$$ for all rational $c,c' > 0$. We first show the case $c=c'=1$. Assume that $m \in {{\operatorname{Newt}}}({\mathfrak{a}})$. By Exercise \[ex.intclomon\], this is equivalent to $x^m \in {\overline{{\mathfrak{a}}}}$, the integral closure of ${\mathfrak{a}}$. Since, by Exercise \[ex.intclo\], ${\overline{{\mathfrak{a}}}} = \mu_*{{\ensuremath{{\mathcal{O}}}}}_Y(-F)$ it follows that $x^m \in {\overline{{\mathfrak{a}}}}$ if and only if $\mu^*x^m \in {{\ensuremath{{\mathcal{O}}}}}_Y(-F)$. This, finally, is equivalent to $\mu^*({{\operatorname{div}}}x^m) \geq F$. For the general case express $c$ and $c'$ as integer fractions. Clearing denominators and noticing that for an integer $a$ one has $a {{\operatorname{Newt}}}({\mathfrak{a}})={{\operatorname{Newt}}}({\mathfrak{a}}^a)$ one reduces to the previous case. ### Canonical divisor As the final ingredient for computing the multiplier ideal we need an understanding of the canonical divisor (class) of a toric variety. \[lem.can\] Let $X$ be a (smooth) toric variety and let $E_1,\ldots,E_r$ denote the collection of all torus invariant prime Weil divisors. Then the canonical divisor is $K_Y=-\sum E_i=-1_X$. We leave the proof as an exercise or alternatively refer to [@Fulton.Toric] or [@Danilov78] for this basic result. We verify it for $X={{\ensuremath{\mathbb{A}}}}^n$. Then $E_i=(x_i=0)$ for $i=1,\ldots,n$ are the torus invariant divisors and $K_X$ is represented by the divisor of the $T^n$-invariant rational $n$-form $\tfrac{dx_1}{x_1}\wedge\ldots\wedge\tfrac{dx_n}{x_n}$, which is $-(E_1+\ldots+E_n)$. As a consequence of the last lemma we get the following lemma. \[lem.mon1X\] Let $\mu: Y {\xrightarrow{\ \ }}X={{\ensuremath{\mathbb{A}}}}^n$ be a birational map of (smooth) toric varieties. Then $K_{Y/X}=\mu^* 1_X - 1_Y$ and the support of $\mu^* 1_X$ is equal to the support of $1_Y$. As the strict transform of a torus invariant divisor on $X$ is a torus invariant divisor on $Y$ it follows that $\mu^* 1_X - 1_Y$ is supported on the exceptional locus of $\mu$. Since $-1_X$ represents the canonical class $K_X$ and respectively for $Y$, the first assertion follows from the definition of $K_{Y/X}$. Since $\mu^* 1_X$ is torus invariant clearly its support is included in $1_Y$. Since $\mu$ is an isomorphism over the torus $T^n \subseteq X$ it follows that $\mu^{-1}(1_X) \supseteq 1_Y$ which implies the second assertion. This exercise shows how to avoid taking Lemma \[lem.can\] on faith but instead using a result of Russel Goward [@Goward.PrincMon] which states that a log resolution of a monomial ideal can be obtained by a sequence of monomial blowups. A *monomial blowup* $Y={\mathit{Bl}}_Z(Y)$ of ${{\ensuremath{\mathbb{A}}}}^n$ is the blowing up of ${{\ensuremath{\mathbb{A}}}}^n$ at the intersection $Z$ of some of the coordinate hyperplanes $E_i=(x_i=0)$ of ${{\ensuremath{\mathbb{A}}}}^n$. For such monomial blowup $\mu: Y={\mathit{Bl}}_Z(X) {\xrightarrow{\ \ }}X \cong {{\ensuremath{\mathbb{A}}}}^n$ show that $Y$ is a smooth toric variety which is canonically covered by ${{\operatorname{codim}}}(Z,X)$ many ${{\ensuremath{\mathbb{A}}}}^n$ patches. Show that $1_Y=E_1+\ldots+E_n+E$ where $E$ is the exceptional divisor of $\mu$. Via a direct calculation verify the assertions of the last two lemmata for $Y$. Since a monomial blowup is canonically covered by affine spaces one can repeat the process and obtains the notion of a *sequence of monomial blowups*. Using Goward’s result show directly that a monomial ideal has a toric log resolution $\mu: Y {\xrightarrow{\ \ }}{{\ensuremath{\mathbb{A}}}}$ with the properties as in Lemma \[lem.mon1X\]. Proof of Theorem \[thm.mon\] ---------------------------- By the existence of a toric (or equivariant) log resolution of a monomial ideal ${\mathfrak{a}}$ it follows immediately that the multiplier ideal ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c)$ is also generated by monomials. Thus, in order to determine ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c)$ it is enough to decide which monomials $x^m$ lie in ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c)$. With our preparations this now an easy task. As usual we denote ${{\operatorname{Spec}}}k[x_1,\ldots,x_n]$ by $X$ and let $\mu:Y {\xrightarrow{\ \ }}X$ be a toric log resolution of ${\mathfrak{a}}$ such that ${\mathfrak{a}}\cdot {{\ensuremath{{\mathcal{O}}}}}_Y = {{\ensuremath{{\mathcal{O}}}}}_Y(-F)$. Abusing notation by identifying ${{\operatorname{div}}}(x_1\cdot\ldots\cdot x_n) = 1_X \in L^X$ with $(1,\ldots,1) \in M$, the condition of the theorem that $m + 1_X$ is in the interior of the Newton polytope $c\cdot{{\operatorname{Newt}}}({\mathfrak{a}})$ is equivalent to $$m + 1_X - {\varepsilon}1_X \in c{{\operatorname{Newt}}}({\mathfrak{a}})$$ for small enough rational ${\varepsilon}> 0$. By Lemma \[lem.techmon\] this holds if and only if $$\mu^* {{\operatorname{div}}}g + \mu^* 1_X - {\varepsilon}\mu^*1_X \geq c\,F.$$ Using the formula $K_{Y/X}=\mu^*1_X- 1_Y$ from Lemma \[lem.can\] this is furthermore equivalent to $$\mu^*{{\operatorname{div}}}g + K_{Y/X}+{\lfloor 1_{Y} - {\varepsilon}\mu^*1_X -c\,F \rfloor} \geq 0$$ for sufficiently small ${\varepsilon}> 0$. Since by Lemma \[lem.mon1X\], $\mu^*1_X$ is effective with the same support as $1_{Y}$ it follows that all coefficients appearing in $1_{Y} -{\varepsilon}\mu^*1_X$ are very close to but strictly smaller than 1 for small ${\varepsilon}> 0$. Therefore, ${\lfloor 1_{Y} - {\varepsilon}\mu^*1_X -cF \rfloor} = {\lceil -cF \rceil} = -{\lfloor cF \rfloor}$. Thus we can finish our chain of equivalences with $$\mu^* {{\operatorname{div}}}g \geq -K_{Y/X} + {\lfloor cF \rfloor}$$ which says nothing but that $g \in {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c)$. This formula for the multiplier ideal of a monomial ideal will be applied in the next section to concretely compute certain invariants arising from multiplier ideals. Invariants arising from multiplier ideals and applications {#sec.invariants} ========================================================== We keep the notation of a smooth affine variety $X$ over an algebraically closed field of characteristic zero, and an ideal ${\mathfrak{a}}\subseteq k[X]$. In this section we use multiplier ideals to attach some invariants to ${{\mathfrak{a}}}$, and we study their influence on some algebraic questions. The log canonical threshold --------------------------- If $c > 0$ is very small, then ${{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}^c})} = k[X]$. For large $c$, on the other hand, the multiplier ideal ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c)$ is clearly nontrivial. This leads one to define: The *log canonical threshold* of ${{\mathfrak{a}}}$ is the number $${{\operatorname{lct}}}({\mathfrak{a}}) = {{\operatorname{lct}}}(X,{\mathfrak{a}}) = \inf\{\, c > 0\, |\, {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c) \neq {{\ensuremath{{\mathcal{O}}}}}_X \, \}.$$ The following exercise shows that ${{\operatorname{lct}}}({{\mathfrak{a}}})$ is a rational number, and that the infimum appearing in the definition is actually a minimum. Consequently, the log canonical threshold is just the smallest $c > 0$ such that ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c)$ is nontrivial. \[LCT.via.discrepencies\] As usual, fixing notation of a log resolution $\mu: Y {\xrightarrow{\ \ }}X$ with ${\mathfrak{a}}\cdot{{\ensuremath{{\mathcal{O}}}}}_{Y}=\sum r_iE_i$ and $K_{Y/X}=\sum b_iE_i$, show that ${{\operatorname{lct}}}(X,{\mathfrak{a}})= \min \{\, \frac{b_i+1}{r_i}\,\}$. Recall the notions from singularity theory [@Kollar.Sings.of.Pairs] in which a pair $(X,{\mathfrak{a}}^{c})$ is called *log terminal* if and only if $b_i - cr_i + 1 > 0$ for all $i$. It is called *log canonical* if and only if $b_i -cr_i + 1\geq 0$ for all $i$. The last exercise also shows that $(X,{\mathfrak{a}}^c)$ is log terminal if and only if the multiplier ideal ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c)$ is trivial. Continuing previous examples we observe that ${{\operatorname{lct}}}((x^2,y^2))=1$ and ${{\operatorname{lct}}}((x^2,y^3))=\frac{5}{6}$. The formula for the multiplier ideal of a monomial ideal ${\mathfrak{a}}$ on $X = {{\operatorname{Spec}}}k[x_1,\ldots,x_n]$ shows that ${{\ensuremath{{\mathcal{J}}}}}(a^c)$ is trivial if and only if $1_X=(1,\ldots,1)$ is in the interior of the Newton polytope $c\,{{\operatorname{Newt}}}({\mathfrak{a}})$. This allows to compute the log canonical threshold of ${\mathfrak{a}}$: ${{\operatorname{lct}}}({\mathfrak{a}})$ is the largest $t>0$ such that $1_X \in t \cdot {{\operatorname{Newt}}}({\mathfrak{a}})$. As a special case of the previous example, take $${\mathfrak{a}}\ =\ (x_1^{a_1},\ldots,x_n^{a_n}).$$ Then the Newton polytope is the subset of the first orthant consisting of points $(v_1,\ldots,v_n)$ satisfying $\sum \frac{v_i}{a_i} \geq 1$. Therefore $1_X \in t\cdot {{\operatorname{Newt}}}({\mathfrak{a}})$ if and only if $\sum \frac{1}{a_i} \geq t$. In particular, ${{\operatorname{lct}}}({\mathfrak{a}}) = \sum \frac{1}{a_i}$. Jumping numbers --------------- The log-canonical threshold measures the triviality or non-triviality of a multiplier ideal. By using the full algebraic structure of these ideals, it is natural to see this threshold as merely the first of a sequence of invariants. These so-called jumping numbers were first considered (at least implicitly) in work of Libgober, Loeser and Vaquié ([@Libgober], [@Loeser.Vaquie]). They are studied more systematically in the paper [@ELSV]. We start with a lemma: \[prop.jump\] For ${\mathfrak{a}}\subseteq {{\ensuremath{{\mathcal{O}}}}}_X$, there is an increasing discrete sequence of rational numbers $$0 \ = \ \xi_0\ <\ \xi_1\ < \ \xi_2 \ <\ \ldots$$ such that ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c)$ is constant for $\xi_i\leq c < \xi_{i+1}$ and ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^{\xi_i}) \varsupsetneq{{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^{\xi_{i+1}})$. We leave the (easy) proof to the reader. The $\xi_i = \xi_i({{\mathfrak{a}}})$ are called the *jumping numbers* or *jumping coefficients* of ${\mathfrak{a}}$. Referring to the log resolution $\mu$ appearing in Example \[LCT.via.discrepencies\], note that the only candidates for jumping numbers are those $c$ such that $cr_i$ is an integer for some $i$. Clearly the first jumping number $\xi_1({\mathfrak{a}})$ is the log canonical threshold ${{\operatorname{lct}}}({\mathfrak{a}})$. Let ${{\mathfrak{a}}}\subseteq k[x_1, \ldots, x_n]$ be a monomial ideal. For the multiplier ideal ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c)$ to jump at $c=\xi$, it is equivalent that some monomial, say $x^v$, is in ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^\xi)$ but not in ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^{\xi-{\varepsilon}})$ for all ${\varepsilon}> 0$. Thus, the largest $\xi>0$ such that $v+(1,\ldots,1) \in \xi {{\operatorname{Newt}}}({\mathfrak{a}})$ is a jumping number. Doing this construction for all $v \in {{\ensuremath{\mathbb{N}}}}^n$ one obtains all jumping numbers of ${\mathfrak{a}}$ (this uses the fact that the multiplier ideal of a monomial ideal is a monomial ideal). Consider again ${\mathfrak{a}}= (x_1^{a_1},\ldots,x_n^{a_n})$. Then the jumping numbers of ${{\mathfrak{a}}}$ are precisely the rational numbers of the form $$\tfrac{v_1+1}{a_1}+\ldots+\tfrac{v_n+1}{a_n}$$ where $(v_1,\ldots,v_n)$ ranges over ${{\ensuremath{\mathbb{N}}}}^n$. Note however that different vectors $(v_1, \ldots,v_n)$ may give the same jumping number. It is instructive to picture the jumping numbers of an ideal graphically. Figure \[Jumping.Nos.Picture\], taken from [@ELSV], shows the jumping numbers of the two ideals $(x^9, y^{10})$ and $(x^3, y^{30})$: the exponents are chosen so that the two ideals have the same Samuel multiplicity, and so that the pictured jumping coefficients occur “with multiplicity one" (in a sense whose meaning we leave to the reader). Jumping length -------------- Jumping numbers give rise to an additional invariant in the case of principal ideals. \[Jump.Nos.Poly.Lemma\] Let $f\in k[X]$ be a non-zero function. Then ${{\ensuremath{{\mathcal{J}}}}}(f) = (f)$ but for $c<1$ one has $(f) \varsubsetneq {{\ensuremath{{\mathcal{J}}}}}(f^c)$. In other words, $\xi = 1$ is a jumping number of the principal ideal $(f)$. Deferring the proof for a moment, we note that the Lemma means that $\xi_{\ell}(f) = 1$ for some index $\ell$. We define $\ell = \ell(f)$ to be the *jumping length* of $f$. Thus $\ell(f)$ counts the number of jumping coefficients of $(f)$ that are $\le 1$. Let $f = x^4 + y^3 \in {{\ensuremath{\mathbb{C}}}}[x,y]$. One can show that $f$ is sufficiently generic so that ${{\ensuremath{{\mathcal{J}}}}}(f^c) = {{\ensuremath{{\mathcal{J}}}}}((x^4,y^3)^c)$ provided that $c < 1$. Therefore the first few jumping numbers of $f$ are $$0\ < \ {{\operatorname{lct}}}(f)\ = \ \tfrac{1}{4} + \tfrac{1}{3}\ <\ \tfrac{2}{4} \ +\ \tfrac{1}{3}\ <\ \tfrac{1}{4} + \tfrac{2}{3} \ <\ 1,$$ and $\ell(f) = 4$. \[Proof of Lemma \[Jump.Nos.Poly.Lemma\]\] Let $\mu: Y {\xrightarrow{\ \ }}X$ be a log resolution of $(f)$ and denote the integral divisor $(f=0)$ by $D = \sum a_i D_i$. Clearly, ${\mathfrak{a}}\cdot {{\ensuremath{{\mathcal{O}}}}}_Y = {{\ensuremath{{\mathcal{O}}}}}_Y(-\mu^*D)$ and $\mu^*D$ is also an integral divisor. Thus $$\begin{split} {{\ensuremath{{\mathcal{J}}}}}(f)&=\mu_*{{\ensuremath{{\mathcal{O}}}}}_Y(K_{Y/X}-\mu^*D) \\ &= \mu_*({{\ensuremath{{\mathcal{O}}}}}_Y(K_{Y/X}){\otimes}\mu^*{{\ensuremath{{\mathcal{O}}}}}_X(-D)) \\ &={{\ensuremath{{\mathcal{O}}}}}_X {\otimes}{{\ensuremath{{\mathcal{O}}}}}_X(-D)\\ &=(f). \end{split}$$ On the other hand, choose a general point $x \in D_i$ on any of the components of $D = \text{div}(f) = \sum a_i D_i$. Then $\mu$ is an isomorphism over $x$ and consequently $$\text{ord}_{D_i} \big ( {{{\ensuremath{{\mathcal{J}}}}}({ f^c})} \big ) \ < \ a_i \ \text{ for } 0 < c < 1.$$ Therefore ${{{\ensuremath{{\mathcal{J}}}}}({(f)^c})} \subsetneqq (f)$ whenever $c < 1$. Finally, we note that the jumping length can be related to other invariants of the singularities of $f$: \[Tyurina.No.Bound\] Assume the hypersurface defined by the vanishing of $f$ has at worst an isolated singularity at $x \in X$. Then $$\ell(f) \leq \tau(f,x) + 1,$$ where $\tau(f,x)$ is the Tjurina number of $f$ at $x$, defined as the colength in ${{\ensuremath{{\mathcal{O}}}}}_{x,X}$ of $(f,\frac{\partial f}{\partial z_1},\ldots,\frac{\partial f}{\partial z_n})$ for $z_1,\ldots,z_n$ parameters around $x$. Application to uniform Artin-Rees numbers ----------------------------------------- We next discuss a result relating jumping lengths to uniform Artin-Rees numbers of a principal ideal. To set the stage, recall the statement of the Artin-Rees lemma in a simple setting: Let ${\mathfrak{b}}$ be an ideal and $f$ an element of $k[X]$. There exists an integer $k = k(f,{\mathfrak{b}})$ such that $${\mathfrak{b}}^m \cap (f) \ \subseteq \ {\mathfrak{b}}^{m-k}\cdot(f)$$ for all $m \geq k$. In other words, if $fg \in {\mathfrak{b}}^m$ then $g \in {\mathfrak{b}}^{m-k}$. Classically, $k$ is allowed to depend both on ${{\mathfrak{b}}}$ and $f$. However in his paper [@Huneke.UniformBounds], Huneke showed that in fact there is a single integer $k=k(f)$ which works simultaneously for all ideals ${\mathfrak{b}}$.[^10] Any such $k$ is called a *uniform Artin-Rees number* of $f$. The next result shows that the jumping length gives an effective estimate (of moderate size!) for uniform Artin-Rees numbers. \[thm.ArtReesNum\] As above, write $\ell(f)$ for the jumping length of $f$. Then the integer $k=\ell(f)\cdot \dim X$ is a uniform Artin-Rees number of $f$. If $f$ defines a smooth hypersurface, its jumping length is $1$ and it follows that $n=\dim X$ is a uniform Artin-Rees number in this case. (In fact, Huneke showed that $n - 1$ also works in this case.) If $f$ defines a hypersurface with only an isolated singular point $x\in X$, it follows from Proposition \[Tyurina.No.Bound\] and the Theorem that $k = n \cdot \big(\tau(f,x) + 1) \big)$ is a uniform Artin-Rees number. (One can show using the next Lemma and some observations of Huneke that in fact $k = \tau(f,x) + n$ also works: see [@ELSV §3].) The essential input to Theorem \[thm.ArtReesNum\] is a statement involving consecutive jumping coefficients: \[Lemma.UAR.Pr.Ideals\] Consider two consecutive jumping numbers $$\xi \, =\, \xi_i(f) \ <\ \xi_{i+1}(f) \, =\, \xi'$$ of $f$, and let ${\mathfrak{b}}\subseteq k[X]$ be any ideal. Then given a natural number $m>n=\dim X$, one has $${\mathfrak{b}}^m\cdot{{\ensuremath{{\mathcal{J}}}}}(f^{\xi})\, \cap \, {{\ensuremath{{\mathcal{J}}}}}(f^{\xi'}) \ \subseteq\ {\mathfrak{b}}^{m-n}\cdot{{\ensuremath{{\mathcal{J}}}}}(f^{\xi'}).$$ We will deduce this from Skoda’s theorem in the next section. In the meantime, we observe that it leads immediately to the \[Proof of Theorem \[thm.ArtReesNum\]\] We apply the previous Lemma repetitively to successive jumping numbers in the chain of multiplier ideals $$k[X] \ = \ {{\ensuremath{{\mathcal{J}}}}}(f^0) \ \varsupsetneq {{\ensuremath{{\mathcal{J}}}}}(f^{\xi_1}) \ \varsupsetneq\ {{\ensuremath{{\mathcal{J}}}}}(f^{\xi_2})\ \varsupsetneq \ \ldots \ \varsupsetneq \ {{\ensuremath{{\mathcal{J}}}}}(f^{\xi_\ell}) = {{\ensuremath{{\mathcal{J}}}}}(f) = (f).$$ After further intersection with $(f)$ one finds: $$\begin{aligned} {\mathfrak{b}}^m \cap (f) \ &\subseteq \ {\mathfrak{b}}^{m-n} \cdot {{\ensuremath{{\mathcal{J}}}}}(f^{\xi_1}) \cap (f) \\ &\subseteq {\mathfrak{b}}^{m-2n}\cdot {{\ensuremath{{\mathcal{J}}}}}(f^{\xi_2}) \cap (f) \\ & \quad \ldots \\ &\subseteq {\mathfrak{b}}^{m-\ell n}(f),\end{aligned}$$as required. When ${{\mathfrak{a}}}= (f )$ is a principal ideal, the jumping numbers of $f$ are related to other invariants appearing in the literature. In particular, if $f$ has an isolated singularity then (suitable translates of) the jumping coefficients appear in the Hodge-theoretically defined *spectrum* of $f$. See [@ELSV §5] for precise statements and references. Further Local Properties Of Multiplier Ideals ============================================= In this section we discuss some results involving the local behavior of multiplier ideals. We start with Skoda’s theorem and some variants. Then we discuss the restriction and subadditivity theorems, which will be used in the next section. Skoda’s theorem --------------- An important (and early) example of a uniform result in local algebra was established by Skoda and Briançon [@BS.74] using analytic results of Skoda [@Skoda72]. In our language, Skoda’s result is this: \[thm.version1\] Consider any ideal ${\mathfrak{b}}\subseteq k[X]$ with $X$ smooth of dimension $n$. Then for all $m \geq n$ $${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{b}}^m) \ = \ {\mathfrak{b}}\cdot {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{b}}^{m-1})\ =\ \ldots\ =\ {\mathfrak{b}}^{m+1-n}\cdot {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{b}}^{n-1}).$$ As Hochster noted in his lectures, the statement in [@Skoda72] has a more analytic flavor. In fact, using the analytic interpretation of multiplier ideals (§\[Analytic.Interp.Subsectn\]) one sees that (the analytic analogue of) Theorem \[thm.version1\] is essentially equivalent to the following statement. > Suppose that ${\mathfrak{b}}$ is generated by $(g_1,\ldots,g_t)$, and that $f$ is a holomorphic function such that $$\int > \frac{{\left\vertf\right\vert}^2}{(\sum > {\left\vertg_i\right\vert}^{2})^{m}} \ < \ \infty \notag$$ for some $m \ge n = \dim X$. Then locally there exist holomorphic functions $h_i$ such that $ f=\sum h_ig_i $, and moreover each of the $h_i$ satisfies the local integrability condition $\int > \frac{\vert h_i > \vert^2}{(\sum \vert g_i \vert^2)^{m-1}} < > \infty$. (The hypothesis expresses the membership of $f$ in ${{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{b}}}^m})}^{\text{an}}$ and the conclusion writes $f$ as belonging to ${{\mathfrak{b}}}^{\text{an}} \cdot {{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{b}}}^{m-1}})}^{\text{an}}$.) As a corollary of Skoda’s theorem, one obtains the classical theorem of Briançon-Skoda. With the notation as before, $${\overline{{\mathfrak{b}}^m}}\ \subseteq\ {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{b}}^m) \ \subseteq\ {\mathfrak{b}}^{m+1-n}$$ where ${\overline{\phantom{{\mathfrak{b}}}}}$ denotes the integral closure and $n = \dim X$. The argument follows ideas of Teissier and Lipman. We choose generators $g_1,\ldots,g_k$ for the ideal ${\mathfrak{b}}$ and fix a log resolution $\mu: Y {\xrightarrow{\ \ }}X$ of ${\mathfrak{b}}$ with ${\mathfrak{b}}\cdot {{\ensuremath{{\mathcal{O}}}}}_{Y} = {{\ensuremath{{\mathcal{O}}}}}_{Y}(-F)$. Write $g^{\prime}_i=\mu^*(g_i) \in \Gamma(Y,{{\ensuremath{{\mathcal{O}}}}}_{Y}(-F))$ to define the surjective map $$\label{eqn.FirstMapKoszul} \oplus_{i=0}^{k} {{\ensuremath{{\mathcal{O}}}}}_{Y} {\xrightarrow{\ \ }}{{\ensuremath{{\mathcal{O}}}}}_{Y}(-F)$$ by sending $(x_1,\ldots,x_k)$ to $\sum x_ig^{\prime}_i$. Tensoring this map with ${{\ensuremath{{\mathcal{O}}}}}_{Y}(K_{Y/X}-(m-1)F)$ yields the surjection $$\oplus_{i=1}^k {{\ensuremath{{\mathcal{O}}}}}_{Y}(K_{Y/X}-(m-1)F) {\xrightarrow{\ {\varphi}\ }} {{\ensuremath{{\mathcal{O}}}}}_{Y}(K_{Y/X}-mF).$$ Further applying $\mu_*$ we get the map $\oplus_{i=0}^k {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{b}}^{m-1}) {\xrightarrow{\ \mu_*{\varphi}\ }} {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{b}}^m)$ which again sends a tuple $(y_1, \ldots, y_k)$ to $\sum y_ig_i$. Therefore, the image of $\mu_*({\varphi})$ is $${\operatorname{Image}}(\mu_*{\varphi}) \ = \ {\mathfrak{b}}{{\ensuremath{{\mathcal{J}}}}}({\mathfrak{b}}^{m-1}) \ \subseteq\ {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{b}}^m).$$ What remains to show is that $\mu_*{\varphi}$ is surjective. For this consider the Koszul complex on the $g_i'$ on $Y$ which resolves the map in (\[eqn.FirstMapKoszul\]). $$\begin{split} 0 {\xrightarrow{\ \ }}{{\ensuremath{{\mathcal{O}}}}}_Y((k-1)F) &{\xrightarrow{\ \ }}\oplus^{k} {{\ensuremath{{\mathcal{O}}}}}_Y((k-2)F) {\xrightarrow{\ \ }}\ldots \\ \ldots & {\xrightarrow{\ \ }}\oplus^{\binom{k}{2}} {{\ensuremath{{\mathcal{O}}}}}_Y(F) {\xrightarrow{\ \ }}\oplus^k {{\ensuremath{{\mathcal{O}}}}}_Y {\xrightarrow{\ \ }}{{\ensuremath{{\mathcal{O}}}}}_Y(-F) {\xrightarrow{\ \ }}0. \end{split}$$ As above, tensor through by ${{\ensuremath{{\mathcal{O}}}}}_Y(K_{Y/X}-(m-1)F)$ to get a resolution of ${\varphi}$. Local vanishing (Theorem \[Local.Vanishing.Thm\]) applies to the $m \geq n = \dim X$ terms on the right. Chasing through the sequence while taking direct images then gives the required surjectivity. See [@PAG Chapter 9] or [@ELNull] for details. It will be useful to have a variant involving several ideals and fractional coefficients. For this we extend slightly the definition of multiplier ideals. ### Mixed multiplier ideals Fix a sequence of ideals ${\mathfrak{a}}_1,\ldots,{\mathfrak{a}}_t$ and positive rational numbers $c_1, \ldots, c_t$. Then we define the multiplier ideal $${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}_1^{c_1}\cdot\ldots\cdot{\mathfrak{a}}_t^{c_t})$$ starting with a log resolution $\mu: Y {\xrightarrow{\ \ }}X$ of the product ${\mathfrak{a}}_1 \cdot \ldots \cdot {\mathfrak{a}}_t$. Since this is at the same time also a log resolution of each ${\mathfrak{a}}_i$ write ${\mathfrak{a}}_i\cdot {{\ensuremath{{\mathcal{O}}}}}_{Y} = {{\ensuremath{{\mathcal{O}}}}}_{Y}(-F_i)$ for simple normal crossing divisors $F_i$. With the notation as indicated, the mixed multiplier ideal is $${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}_1^{c_1}\cdot\ldots\cdot{\mathfrak{a}}_t^{c_t} ) \ = \ \mu_*({{\ensuremath{{\mathcal{O}}}}}_{Y}(K_{Y/X}-{\lfloor c_1F_1+\ldots+c_tF_t \rfloor})).$$ As before, this definition is independent of the chosen log resolution. Note that once again we do not attempt to assign any meaning to the expression ${\mathfrak{a}}_1^{c_1}\cdot\ldots\cdot{\mathfrak{a}}_t^{c_t}$ in the argument of ${{\ensuremath{{\mathcal{J}}}}}$. This expression is meaningful a priori whenever all $c_i$ are positive integers and our definition is consistent with this prior meaning. With this generalization of the concept of multiplier ideals we get the following variant of Skoda’s theorem. \[Skoda.Thm.II\] For every integer $c \ge n = \dim X$ and any $d > 0$ one has $${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}_1^{c}\cdot{\mathfrak{a}}_2^{d}) \ =\ {\mathfrak{a}}_1^{c-(n-1)}{{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}_1^{n-1}\cdot{\mathfrak{a}}_2^{d}).$$ The proof of this result is only a technical complication of the proof of the First Version we discussed above. We refer to [@PAG Chapter 9] for details. We conclude by using Skoda’s Theorem to prove (a slight generalization of) the Lemma \[Lemma.UAR.Pr.Ideals\] underlying the results on uniform Artin-Rees numbers in the previous section. \[lem.ArtResUsed\] Let ${\mathfrak{a}}\subseteq k[X]$ be an ideal and let $\xi < \xi'$ be consecutive jumping numbers of ${\mathfrak{a}}$. Then for $m > n$ we have $${\mathfrak{b}}^m\cdot {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^{\xi})\ \cap \ {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^{\xi'}) \ \subseteq \ {\mathfrak{b}}^{m-n}\cdot {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^{\xi'})$$ for all ideals ${\mathfrak{b}}\subseteq k[X]$. We first claim that $${\mathfrak{b}}^m{{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^{\xi}) \, \cap \, {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^{\xi'}) \ \subseteq \ {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{b}}^{m-1} \cdot {\mathfrak{a}}^{\xi'}).$$ This is shown via a simple computation. In fact, to begin with one can replace $\xi$ by $c \in [\xi, \xi')$ arbitrarily close to $\xi'$ since this does not change the statement. Let $\mu:Y {\xrightarrow{\ \ }}X$ be a common log resolution of ${\mathfrak{a}}$ and ${\mathfrak{b}}$ such that ${\mathfrak{a}}\cdot{{\ensuremath{{\mathcal{O}}}}}_Y = {{\ensuremath{{\mathcal{O}}}}}_Y(-A)$ and ${\mathfrak{b}}\cdot{{\ensuremath{{\mathcal{O}}}}}_Y = {{\ensuremath{{\mathcal{O}}}}}_Y(-B)$. Let $E$ be a prime divisor on $Y$ and denote by $a$, $b$ and $e$ the coefficient of $E$ in $A$, $B$ and $K_{Y/X}$, respectively. Then $f$ is in the left-hand side if and only if $${{\operatorname{ord}}}_E f \ \geq\ \max(-e+mb+{\lfloor ca \rfloor}\, ,\, -e+{\lfloor \xi'a \rfloor}).$$ If $b=0$ this implies that ${{\operatorname{ord}}}_E f \geq -e+(m-1)b+{\lfloor \xi'a \rfloor}$. If $b\neq 0$ then $b$ is a positive integer $\geq 1$. Since $c$ is arbitrarily close to $\xi'$ we get ${\lfloor \xi' a \rfloor} - b \leq {\lfloor \xi' a \rfloor} - 1 \leq {\lfloor ca \rfloor}$. Adding $-e+mb$ it follows that also in this case ${{\operatorname{ord}}}_E f \geq -e+(m-1)b+{\lfloor \xi'a \rfloor}$. Since this holds for all $E$ it follows that $f\in{{\ensuremath{{\mathcal{J}}}}}({\mathfrak{b}}^{m-1} \cdot {\mathfrak{a}}^{\xi'})$. Now, using Theorem \[Skoda.Thm.II\] we deduce $${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{b}}^{m-1}\cdot{\mathfrak{a}}^{\xi'}) \ \subseteq\ {\mathfrak{b}}^{m-n}{{\ensuremath{{\mathcal{J}}}}}({\mathfrak{b}}^{n-1}\cdot{\mathfrak{a}}^{\xi'}) \ \subseteq \ {\mathfrak{b}}^{m-n}{{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^{\xi'}).$$ Putting all the inclusions together, the Lemma follows. \[Periodicityu.Jump.Nos\] Let ${\mathfrak{a}}\subseteq k[X]$ be an ideal. Starting at $\dim X-1$, the jumping numbers are periodic with period 1. That is, $\xi \geq \dim X-1$ is a jumping number if and only if $\xi+1$ is a jumping number. Restriction theorem ------------------- The next result deals with restrictions of multiplier ideals. Consider a smooth subvariety $Y \subseteq X$ and an ideal ${{\mathfrak{b}}}\subseteq k[X]$ which does not vanish on $Y$. There are then two ways to get an ideal on $Y$. First, one can compute the multiplier ideal ${{{\ensuremath{{\mathcal{J}}}}}({X, {{\mathfrak{b}}}^c})}$ on $X$ and then restrict it to $Y$. On the other hand, one can also restrict ${{\mathfrak{b}}}$ to $Y$ and then compute the multiplier ideal on $Y$ of this restricted ideal. The Restriction Theorem – which is arguably the most important local property of multiplier ideals – states that there is always an inclusion among these ideals on $Y$. \[thm.restr\] Let $Y \subseteq X$ be a smooth subvariety of $X$ and ${\mathfrak{b}}$ an ideal of $k[X]$ such that $Y$ is not contained in the zero locus of ${\mathfrak{b}}$. Then $${{\ensuremath{{\mathcal{J}}}}}(Y,({\mathfrak{b}}\cdot k[Y])^c)\ \subseteq\ {{\ensuremath{{\mathcal{J}}}}}(X,{\mathfrak{b}}^c)\cdot k[Y].$$ One can think of the Theorem as reflecting the principle that singularities can only get worse under restriction. In the present setting, the result is due to Esnault and Viehweg [@EV Proposition 7.5] When $Y$ is a hypersurface, the statement is proved using the local vanishing theorem \[Local.Vanishing.Thm\]. Since in any event a smooth subvariety is a local complete intersection, the general case then follows from this. Give an example where strict inclusion holds in the Theorem. Subadditivity theorem --------------------- We conclude with a result due to Demailly, Ein and the second author [@DEL] concerning the multiplicative behavior of multiplier ideals. This subadditivity theorem will be used in the next section to obtain some uniform bounds on symbolic powers of ideals. \[thm.subadd\] Let ${\mathfrak{a}}$ and ${\mathfrak{b}}$ be ideals in $k[X]$. Then for all $c,d > 0$ one has $${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c\cdot {\mathfrak{b}}^d) \ \subseteq\ {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c)\cdot {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{b}}^d).$$ In particular, for every positive integer $m$, ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^{cm})\subseteq {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^c)^m$. The idea of the proof is to pull back the data to the product $X \times X$ and then to restrict to the diagonal $\Delta$. Specifically, assume for simplicity that $c=d=1$, and consider the product $$\xymatrix{ & {X\times X} \ar^{{{\operatorname{p}}}_1}[dl]\ar_{{{\operatorname{p}}}_2}[dr]&\\ X &&X }$$ along with its projections as indicated. For log resolutions $\mu_1$ and $\mu_2$ of ${\mathfrak{a}}$ and ${\mathfrak{b}}$ respectively one can verify that $\mu_1 \times \mu_2$ is a log resolution of the ideal ${{\operatorname{p}}}_1^{-1}({\mathfrak{a}}) \cdot {{\operatorname{p}}}_2^{-1}({\mathfrak{b}})$ on $X \times X$. Using this one shows that $${{\ensuremath{{\mathcal{J}}}}}(X\times X,{{\operatorname{p}}}_1^{-1}({\mathfrak{a}}) \cdot {{\operatorname{p}}}_2^{-1}({\mathfrak{b}}))\ = \ {{\operatorname{p}}}_1^{-1}{{\ensuremath{{\mathcal{J}}}}}(X,{\mathfrak{a}})\cdot{{\operatorname{p}}}_2^{-1}{{\ensuremath{{\mathcal{J}}}}}(X,{\mathfrak{b}}).$$ Now let $\Delta \subseteq X\times X$ be the diagonal. Apply the Restriction Theorem \[thm.restr\] with $Y=\Delta$ to conclude $$\begin{split} {{\ensuremath{{\mathcal{J}}}}}(X,{\mathfrak{a}}\cdot{\mathfrak{b}}) \ &= \ {{\ensuremath{{\mathcal{J}}}}}(\Delta,{{\operatorname{p}}}_1^{-1}({\mathfrak{a}}) \cdot {{\operatorname{p}}}_2^{-1}({\mathfrak{b}})\cdot {{\ensuremath{{\mathcal{O}}}}}_{\Delta}) \\ &\subseteq \ {{\ensuremath{{\mathcal{J}}}}}(X \times X, {{\operatorname{p}}}_1^{-1}({\mathfrak{a}}) \cdot {{\operatorname{p}}}_2^{-1}({\mathfrak{b}}))\cdot{{\ensuremath{{\mathcal{O}}}}}_{\Delta} \\ &= \ {{\ensuremath{{\mathcal{J}}}}}(X,{\mathfrak{a}}) \cdot {{\ensuremath{{\mathcal{J}}}}}(X,{\mathfrak{b}}), \end{split}$$ as required. Asymptotic Constructions ======================== There are many natural situations in geometry and algebra where one is forced to confront rings or algebras that fail to be finitely generated. For example, if $D$ is a non-ample divisor on a projective variety $V$, then the section ring $R(V,D) = \oplus \Gamma(V, {{\ensuremath{{\mathcal{O}}}}}_V(mD))$ is typically not finitely generated. Or likewise, if ${\mathfrak{q}}$ is a radical ideal in some ring, the symbolic blow-up algebra $\oplus {\mathfrak{q}}^{(m)}$ likewise fails to be finitely generated in general. It is nonetheless possible to extend the theory of multiplier ideals to such settings. It turns out that there is finiteness built into the resulting multiplier ideals that may not be present in the underlying geometry or algebra. This has led to some of the most interesting applications of the theory. In the geometric setting, the asymptotic constructions have been known for some time, but it was only with Siu’s work [@Siu98a] on deformation-invariance of plurigenera that their power became clear. Here we focus on an algebraic formulation of the theory from [@ELS1]. As before, we work with a smooth affine variety $X$ defined over an algebraically closed field $k$ of characteristic zero. Graded systems of ideals ------------------------ We start by defining certain collections of ideals, to which we will later attach multiplier ideals. A *graded system* or *graded family of ideals* is a family ${{\mathfrak{a}}}_{\bullet} = {\left\{{\mathfrak{a}}_k\right\}}_{k \in {{\ensuremath{\mathbb{N}}}}}$ of ideals in $k[X]$ such that $${{\mathfrak{a}}}_{\ell} \cdot {\mathfrak{a}}_m \ \subseteq {\mathfrak{a}}_{\ell+m}$$ for all $\ell,m \geq 1$. To avoid trivialities, we also assume that ${{\mathfrak{a}}}_k \ne (0) $ for $k \gg 1$. The condition in the definition means that the direct sum $$R({{\mathfrak{a}}}_{\bullet}) \ \overset{\text{def}}{=} \ k[X] \, \oplus {{\mathfrak{a}}}_1 \oplus {{\mathfrak{a}}}_2 \oplus \ldots$$ naturally carries a graded $k[X]$-algebra structure and $R({\mathfrak{a}}_\bullet)$ is called the *Rees algebra* of ${\mathfrak{a}}_\bullet$. In the interesting situations $R({{\mathfrak{a}}}_{\bullet})$ is not finitely generated, and it is here that the constructions of the present section give something new. One can view graded systems as local objects displaying complexities similar to those that arise from linear series on a projective variety $V$.[^11] We give several examples of graded systems. 1. Let ${\mathfrak{b}}\subseteq k[X]$ be a fixed ideal, and set ${\mathfrak{a}}_k={\mathfrak{b}}^k$. One should view the resulting as a trivial example. 2. Let $Z \subseteq X$ be a reduced subvariety defined by the radical ideal ${\mathfrak{q}}$. The symbolic powers $${\mathfrak{q}}^{(k)} \ {\stackrel{\scriptscriptstyle {\operatorname{def}}}{=}}\ {\left\{f \in k[X] \,|\, {{\operatorname{ord}}}_z f \geq k,\ z\in Z \text{ generic }\right\}}$$ form a graded system.[^12] 3. Let $<$ be a term order on $k[x_1,\ldots,x_n]$ and ${\mathfrak{b}}$ be an ideal. Then $${\mathfrak{a}}_k \ {\stackrel{\scriptscriptstyle {\operatorname{def}}}{=}}\ {\operatorname{in}}_<({\mathfrak{b}}^k)$$ defines a graded system of monomial ideals, where ${\operatorname{in}}_<({\mathfrak{b}}^k)$ denotes the initial ideal with respect to the given term order. \[Valuation ideals\] \[Valuation.GSI.Ex\] Let $\nu$ be a ${{\ensuremath{\mathbb{R}}}}$-valued valuation centered on $k[X]$. Then the valuation ideals $${\mathfrak{a}}_k \ {\stackrel{\scriptscriptstyle {\operatorname{def}}}{=}}\ {\left\{f \in k[X] \,|\, \nu(f)\geq k\right\}}$$ form a graded family. Special cases of this construction are interesting even when $X = {{\ensuremath{\mathbb{A}}}}^2_{{\ensuremath{\mathbb{C}}}}$. 1. Let $\eta: Y {\xrightarrow{\ \ }}{{\ensuremath{\mathbb{A}}}}^2$ be a birational map with $Y$ also smooth and let $E \subseteq Y$ be a prime divisor. Define the valuation $\nu(f) {\stackrel{\scriptscriptstyle {\operatorname{def}}}{=}}{{\operatorname{ord}}}_E(f)$. Then $${\mathfrak{a}}_k {\stackrel{\scriptscriptstyle {\operatorname{def}}}{=}}\mu_* {{\ensuremath{{\mathcal{O}}}}}_Y(-kE) = {\left\{f \in {{\ensuremath{{\mathcal{O}}}}}_X\,|\, \nu(f)={{\operatorname{ord}}}_E(f) \geq k\right\}}.$$ 2. In ${{\ensuremath{\mathbb{C}}}}[x,y]$ put $\nu(x)=1$ and $\nu(y)=\frac{1}{\sqrt{2}}$. Then one gets a valuation by weighted degree. Here ${\mathfrak{a}}_k$ is the monomial ideal generated by the monomials $x^iy^j$ such that $i+\tfrac{j}{\sqrt{2}} \geq k$. 3. Given $f \in {{\ensuremath{\mathbb{C}}}}[x,y]$ define $\nu(f)={{\operatorname{ord}}}_z(f(z,e^z-1))$. This yields a valuation giving rise to the graded system $${\mathfrak{a}}_k \ {\stackrel{\scriptscriptstyle {\operatorname{def}}}{=}}\ (x^k,y-P_{k-1}(x))$$ where $P_{k-1}(x)$ is the $(k-1)$st Taylor polynomial of $e^x-1$. Note that the general element in ${{\mathfrak{a}}}_k$ defines a smooth curve in the plane. Except for Example \[First.Ex.GSI\](i), all these constructions give graded families ${{\mathfrak{a}}}_{\bullet}$ for which the corresponding Rees algebra need not be finitely generated. Asymptotic multiplier ideals ---------------------------- We now attach multiplier ideals ${{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}{_{\bullet}}^c})}$ to a graded family ${{\mathfrak{a}}}{_{\bullet}}$ of ideals. The starting point is: \[Def.AMI.Lemma\] Let ${\mathfrak{a}}_\bullet$ be a graded system of ideals on $X$, and fix a rational number $c > 0$. Then for $p \gg 0$ the multiplier ideals ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^{c/p}_{ p})$ all coincide. \[Def.AMI\] Let ${\mathfrak{a}}_\bullet={\left\{{\mathfrak{a}}_k\right\}}_{k \in {{\ensuremath{\mathbb{N}}}}}$ be a graded system of ideals on $X$. Given $c > 0$ we define the *asymptotic multiplier ideal* of ${{\mathfrak{a}}}{_{\bullet}}$ with exponent $c$ to be the common ideal $${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}_\bullet^c) \ {\stackrel{\scriptscriptstyle {\operatorname{def}}}{=}}\ {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^{c/p}_{ p})$$ for any sufficiently big $p\gg 0$.[^13] We first claim that one has an inclusion of multiplier ideals ${{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^{c/p}_{ p}) \subseteq {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^{c/pq}_{ pq})$ for all $p,q \geq 0$. Granting this, it follows from the Noetherian condition that the collection of ideals ${\left\{{{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^{c/p}_{p})\right\}}_{p\geq 0}$ has a unique maximal element. This proves the lemma at least for sufficiently divisible $p$. (The statement for all $p \gg 0$ requires a little more work; see [@PAG Chapter 11].) To verify the claim let $\mu: X^\prime {\longrightarrow}X$ be a common log resolution of ${\mathfrak{a}}_{ p }$ and ${\mathfrak{a}}_{ pq}$ with ${\mathfrak{a}}_{ p} \cdot {{\ensuremath{{\mathcal{O}}}}}_Y = {{\ensuremath{{\mathcal{O}}}}}_Y(-F_{ p})$ and ${\mathfrak{a}}_{ pq} \cdot {{\ensuremath{{\mathcal{O}}}}}_Y = {{\ensuremath{{\mathcal{O}}}}}_Y(-F_{ pq})$. Since the ${{\mathfrak{a}}}_k$ form a graded system one has ${\mathfrak{a}}_{ p}^q \subseteq {\mathfrak{a}}_{ pq}$ and therefore $-cqF_{ p}\leq-cF_{ pq}$. Thus $$\mu_*{{\ensuremath{{\mathcal{O}}}}}_Y(K_{Y/X}-{\lfloor \textstyle{\frac{cq}{pq}}F_ { p} \rfloor}) \ \subseteq \ \mu_*{{\ensuremath{{\mathcal{O}}}}}_Y(K_{Y/X}-{\lfloor \frac{c}{pq}F_{ pq} \rfloor})$$ as claimed. Lemma \[Def.AMI.Lemma\] shows that any information captured by the multiplier ideals ${{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}_{ p}^{c/p}})}$ is present already for any one sufficiently large index $p$. It is in this sense that multiplier ideals have some finiteness built in that may not be present in the underlying graded system ${{\mathfrak{a}}}{_{\bullet}}$. \[Ex.AMI.computations\] We return to the graded systems in Example \[Valuation.GSI.Ex\] coming from valuations on ${{\ensuremath{\mathbb{A}}}}^2$. 1. Here ${{\mathfrak{a}}}_k $ is the monomial ideal generated by $x^i y^j$ with $i + \tfrac{j}{\sqrt 2} \ge k$, and ${{\ensuremath{{\mathcal{J}}}}}({{\mathfrak{a}}}{_{\bullet}}^c)$ is the monomial ideal generated by all $x^iy^j$ with $$(i+1) + \frac{(j+1)}{\sqrt{2}} \ > \ c .$$ (Compare with Theorem \[thm.mon\].) 2. Now take the valuation $\nu(f) = {{\operatorname{ord}}}_z f(z,e^z-1)$. Then $${{\ensuremath{{\mathcal{J}}}}}({{\mathfrak{a}}}{_{\bullet}}^c) \ = \ {{\ensuremath{\mathbb{C}}}}[x,y]$$ for all $c > 0$. (Use the fact that each ${\mathfrak{a}}_k$ contains a smooth curve.) Growth of graded systems ------------------------ We now use the Subadditivity Theorem \[thm.subadd\] to prove a result from [@ELS1] concerning the multiplicative behavior of graded families of ideals: \[thm.asympIncl\] Let ${\mathfrak{a}}_\bullet$ be a graded system of ideals and fix any $\ell \in {{\ensuremath{\mathbb{N}}}}$. Then $${{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}{_{\bullet}}^\ell})} \ = \ {{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}_{\ell p}^{1/p}})} \ \ \text{for} \ \ p \gg 0.$$ Moreover for every $m \in {{\ensuremath{\mathbb{N}}}}$ one has: $$\label{Sub.AMI.Eqn} {\mathfrak{a}}_\ell^m \ \subseteq \ {\mathfrak{a}}_{\ell m} \subseteq \ {{\ensuremath{{\mathcal{J}}}}}({{\mathfrak{a}}}{_{\bullet}}^{\ell m}) \ \subseteq \ {{\ensuremath{{\mathcal{J}}}}}({{\mathfrak{a}}}{_{\bullet}}^\ell)^m.$$ In particular, if ${{\ensuremath{{\mathcal{J}}}}}({{\mathfrak{a}}}{_{\bullet}}^\ell) \subseteq {\mathfrak{b}}$ for some natural number $\ell$ and ideal ${\mathfrak{b}}$, then ${\mathfrak{a}}_{\ell m} \subseteq {\mathfrak{b}}^m$ for all $m$. The crucial point here is the containment ${{\ensuremath{{\mathcal{J}}}}}({{\mathfrak{a}}}{_{\bullet}}^{\ell m}) \subseteq {{\ensuremath{{\mathcal{J}}}}}({{\mathfrak{a}}}{_{\bullet}}^\ell)^m$: it shows that passing to multiplier ideals “reverses" the inclusion ${{\mathfrak{a}}}_\ell^m \subseteq {{\mathfrak{a}}}_{\ell m}$. \[Proof of Theorem \[thm.asympIncl\]\] For the first statement, observe that if $p \gg 0$ then $${{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}{_{\bullet}}^\ell})} \ = \ {{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}_p^{\ell/p}})} \ = \ {{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}_{ \ell p}^{\ell/ \ell p}})} \ = \ {{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}^{1/p}_{\ell p}})},$$ where the second equality is obtained by taking $ \ell p$ in place of $p$ as the large index in Lemma \[Def.AMI.Lemma\]. For the containment $ {{\mathfrak{a}}}_{\ell m} \subseteq {{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}{_{\bullet}}^{\ell m}})}$ it is then enough to prove that ${{\mathfrak{a}}}_{\ell m} \subseteq {{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}_{\ell m p}^{1/p}})}.$ But we have ${{\mathfrak{a}}}_{\ell m} \subseteq {{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}_{\ell m}})}$ thanks to Exercise \[ex.intclo\], while the inclusion ${{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}_{\ell m}})} \subseteq {{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}^{1/p}_{\ell m p}})}$ was established during the proof of \[Def.AMI.Lemma\]. It remains only to prove that ${{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}{_{\bullet}}^{\ell m}})} \subseteq {{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}{_{\bullet}}^\ell})}^m$. To this end, fix $p \gg 0$. Then by the definition of asymptotic multiplier ideals and the Subadditivity Theorem one has $$\begin{aligned} {{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}{_{\bullet}}^{\ell m}})} \ &= \ {{\ensuremath{{\mathcal{J}}}}}({\mathfrak{a}}^{\ell m/p}_{p}) \\ &\subseteq \ {{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}_{ p}^{\ell/p}})}^m \\ &= \ {{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}{_{\bullet}}^\ell})}^m, \end{aligned}$$ as required. The Theorem gives another explanation of the fact that the multiplier ideals associated to the graded system ${{\mathfrak{a}}}{_{\bullet}}$ from Example \[Valuation.GSI.Ex\].(iii) are trivial. In fact, in this example the colength of ${{\mathfrak{a}}}_k$ in ${{\ensuremath{\mathbb{C}}}}[X]$ grows linearly in $k$. It follows from Theorem \[thm.asympIncl\] that then ${{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}{_{\bullet}}^\ell})} = (1)$ for all $\ell$. \[Triv.GSI.AMI\] Let ${{\mathfrak{a}}}_k = {{\mathfrak{b}}}^k$ be the trivial graded family consisting of powers of a fixed ideal. Then ${{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}{_{\bullet}}^c})} = {{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{b}}}^c})}$ for all $c > 0$. So we do not get anything new in this case. A comparison theorem for symbolic powers {#sec.symb} ---------------------------------------- As a quick but surprising application of Theorem \[thm.asympIncl\] we discuss a result due to Ein, Smith and the second author from [@ELS1] concerning symbolic powers of radical ideals. Consider a reduced subvariety $Z \subseteq X$ defined by a radical ideal ${\mathfrak{q}}\subseteq k[X]$. Recall from Example \[First.Ex.GSI\](ii) that one can define the symbolic powers ${\mathfrak{q}}^{(k)}$ of ${\mathfrak{q}}$ to be $${\mathfrak{q}}^{(k)} \ {\stackrel{\scriptscriptstyle {\operatorname{def}}}{=}}\ {\left\{f \in {{\ensuremath{{\mathcal{O}}}}}_X \,|\, {{\operatorname{ord}}}_z f \geq k,\ z\in Z\right\}}.$$ Thus evidently ${\mathfrak{q}}^k \subseteq {\mathfrak{q}}^{(k)}$, and equality holds if $Z$ is smooth. However if $Z$ is singular then in general the inclusion is strict: Take $Z \subseteq {{\ensuremath{\mathbb{C}}}}^3$ to be the union of the three coordinate axes, defined by the ideal $${\mathfrak{q}}\ = \ ( \, xy \, , \, yz \, , \, xz \,) \ \subseteq \ {{\ensuremath{\mathbb{C}}}}[x,y,z].$$ Then $xyz \, \in \, {\mathfrak{q}}^{(2)}$ since evidently the union of the three coordinate planes has multiplicity $2$ at a general point of $Z$. But ${\mathfrak{q}}^2$ is generated by monomials of degree $4$, thus cannot contain $xyz$, which is of degree $3$. Swanson [@Swanson00a] proved (in a much more general setting) that there exists an integer $k = k(Z)$ such that $${\mathfrak{q}}^{(km)} \ \subseteq \ {\mathfrak{q}}^m$$ for all $m \ge 0$. At first glance, one might be tempted to suppose that for very singular $Z$ the coefficient $k(Z)$ will have to become quite large. The main result of [@ELS1] shows that this isn’t the case, and that in fact one can take $k(Z) = {{\operatorname{codim}}}Z$: \[Symb.Pow.Thm\] Assume that every irreducible component of $Z$ has codimension $\le e$ in $X$. Then $${\mathfrak{q}}^{(e m)} \ \subseteq {\mathfrak{q}}^m \ \text{ for all } \ m \ge 0.$$ In particular, ${\mathfrak{q}}^{(m \cdot \dim X)} \subseteq {\mathfrak{q}}^m$ for all radical ideals ${\mathfrak{q}}\subseteq k[X]$ and all $m \ge 0$. \[Points in the plane\] Let $T \subseteq {{\ensuremath{\mathbb{P}}}}^2$ be a finite set (considered as a reduced scheme), and let $I \subseteq S = {{\ensuremath{\mathbb{C}}}}[x,y,z]$ be the homogeneous ideal of $T$. Suppose that $f \in S$ is a homogeneous form which has multiplicity $\ge 2m$ at each of the points of $T$. Then $f \in I^m$. (Apply Theorem \[Symb.Pow.Thm\] to the homogeneous ideal $I$ of $T$.) In spite of the classical nature of this statement, we do not know a direct elementary proof. \[Proof of Theorem \[Symb.Pow.Thm\]\] Applying Theorem \[thm.asympIncl\] to the graded system ${{\mathfrak{a}}}_k = {\mathfrak{q}}^{(k)}$, it suffices to show that $${{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}{_{\bullet}}^e})} \ \subseteq \ {\mathfrak{q}}. \tag{*}$$ Since ${\mathfrak{q}}$ is radical, it suffices to test the inclusion (\*) at a general point of $Z$. Therefore we can assume that $Z$ is smooth, in which case ${\mathfrak{q}}^{(k)} = {\mathfrak{q}}^k$. Now Exercises \[Smooth.Ideal.Exercise\] and \[Triv.GSI.AMI\] apply. Using their theory of tight closure, Hochster and Huneke [@HH.ComOrdSymbPow] have extended Theorem \[Symb.Pow.Thm\] to arbitrary regular Noetherian rings containing a field. Theorem \[thm.asympIncl\] is applied in [@ELS2] to study the multiplicative behavior of Abyhankar valuations centered at a smooth point of a complex variety. Working with the asymptotic multiplier ideals ${{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}{_{\bullet}}^c})}$ one can define the log-canonical threshold and jumping coefficients of a graded system ${{\mathfrak{a}}}{_{\bullet}}$ much as in §4. However now these numbers need no longer be rational, the periodicy of jumping numbers (Exercise \[Periodicityu.Jump.Nos\]) may fail, and in fact the collection of jumping coefficients of ${{\mathfrak{a}}}{_{\bullet}}$ can contain accumulation points. See [@ELSV §5]. [^1]: Research of the second author partially supported by NSF Grant DMS 0139713 [^2]: Lipman used the term “adjoint ideals", but this has come to refer to a different construction. [^3]: In fact, the “multiplier" in the name refers to their analytic construction (see §\[Analytic.Interp.Subsectn\]) [^4]: Handwritten notes and the lectures on streaming video are available at\ http://www.msri.org/publications/video/index05.html [^5]: There is a way to define the integral closure of an ideal ${\mathfrak{a}}^c$, for $c \geq 0$ rational, such that it is consistent with the definition of the multiplier ideal. For $c=p/q$ with positive integers $p$ and $q$, set $f \in {\mathfrak{a}}^{p/q}$ if and only if $f^q \in {\overline{{\mathfrak{a}}^p}}$, where the bar denotes the integral closure. [^6]: The tight closure analogues of these result can be found in [@HH.BrianSkoda], [@Huneke.UniformBounds] and [@HH.ComOrdSymbPow], respectively. [^7]: One feature of their theory is that there is no reference to resolutions of singularities. As a consequence no restriction on the singularity of $R$ arises, whereas for multiplier ideals at least some sort of ${{\ensuremath{\mathbb{Q}}}}$–Gorenstein assumption is needed. [^8]: Similarly as for tight closure, $x \in I^{*{\mathfrak{a}}^c}$ if there is a $h \neq 0$ such that for all $q=p^e$ one has $hx^q{\mathfrak{a}}^{{\lceil qc \rceil}} \subseteq I^{[q]}$. Note that $I^{[q]}$ denotes the ideal generated by all $q$th powers of the elements of $I$, whereas ${\mathfrak{a}}^{{\lceil qc \rceil}}$ is the usual ${\lceil qc \rceil}$th power of ${\mathfrak{a}}$. [^9]: To be precise, a toric variety comes with the datum of the torus embedding $T^n \subseteq X$. Maps of toric varieties are such that they preserve the torus action. [^10]: We stress that both the classical Artin-Rees Lemma and Huneke’s theorem are valid in a much more general setting. [^11]: If $D$ is an effective divisor on $V$, the base ideals ${{\mathfrak{b}}}_k = {{\mathfrak{b}}}(|kD|) \subseteq {{\ensuremath{{\mathcal{O}}}}}_V$ form a graded family of ideal sheaves on $V$: this is the prototypical example. [^12]: When $Z$ is reducible, we ask that the condition hold at a general point of each component. The fact that this is equivalent to the usual algebraic definition is a theorem of Zariski and Nagata: see [@Eisenbud Chapter 3]. [^13]: In [@ELS1] and early versions of [@PAG], one only dealt with the ideals ${{{\ensuremath{{\mathcal{J}}}}}({{{\mathfrak{a}}}{_{\bullet}}^{\ell}})} $ for integral $\ell$, which were written ${{{\ensuremath{{\mathcal{J}}}}}({\Vert {{\mathfrak{a}}}_\ell \Vert})}$.

--- abstract: | The goal of branch length estimation in phylogenetic inference is to estimate the divergence time between a set of sequences based on compositional differences between them. A number of software is currently available facilitating branch lengths estimation for homogeneous and stationary evolutionary models. Homogeneity of the evolutionary process imposes fixed rates of evolution throughout the tree. In complex data problems this assumption is likely to put the results of the analyses in question. In this work we propose an algorithm for parameter and branch lengths inference in the discrete-time Markov processes on trees. This broad class of nonhomogeneous models comprises the general Markov model and all its submodels, including both stationary and nonstationary models. Here, we adapted the well-known Expectation-Maximization algorithm and present a detailed performance study of this approach for a selection of nonhomogeneous evolutionary models. We conducted an extensive performance assessment on multiple sequence alignments simulated under a variety of settings. We demonstrated high accuracy of the tool in parameter estimation and branch lengths recovery, proving the method to be a valuable tool for phylogenetic inference in real life problems. ${\texttt{Empar}}$ is an open-source C++ implementation of the methods introduced in this paper and is the first tool designed to handle nonhomogeneous data. [: nucleotide substitution models; branch lengths; maximum-likelihood; expectation-maximization algorithm.]{} author: - 'A. M. Kedzierska$^{1,2}$' - 'M. Casanellas$^{2,**}$' title: '${\texttt{Empar}}$: EM-based algorithm for parameter estimation of Markov models on trees.' --- {#section .unnumbered} Assuming that an evolutionary process can be represented in a phylogenetic tree, the tips of the tree are assigned operational taxonomic units (OTUs) whose composition is known. Here, the OTUs are thought of as the DNA sequences of either a single or distinct taxa. Internal vertices represent ancestral sequences and inferring the branch lengths of the tree provides information about the speciation time. Choice of the evolutionary model and the method of inference have a direct impact on the accuracy and consistency of the results [@SulSwo97; @Fel78; @BruHal99; @Pen94; @HueHil93; @Schwartz2010]. We assume that the sites of a multiple sequence alignment (MSA) are independent and identically distributed (i.i.d. hypothesis of all sites undergoing the same process without an effect on each other), the evolution of a set of OTUs along a phylogenetic tree ${\tau}$ can be modeled by the evolution of a single character under a hidden Markov process on ${\tau}$. Markovian evolutionary processes assign a conditional substitution (transition) matrix to every edge of ${\tau}$. Most current software packages are based on the continuous-time Markov processes where the transition matrix associated to an edge $e$ is given in the form $\exp(Q^e t_e)$, where $Q^ee$ is an instantaneous mutation rate matrix. Although in some cases the rate matrices are allowed to vary between different lineages (cf. [@Galtier1998],[@YY99]), it is not uncommon to equate them to a *homogeneous* rate matrix $Q$, which is constant for every lineage in ${\tau}$. Relaxing the homogeneity assumption is an important step towards increased reliability of inference (see [@mitoch]). In this work, we consider a class of processes more general than the homogeneous ones: the discrete-time Markov processes. If ${\tau}$ is rooted, these models are given by a root distribution $\pi,$ and a set of transition matrices $A^e$ (e.g. chap. 8 of @Semple2003). The transition matrices $A^e$ can freely vary for distinct edges and are not assumed to be of exponential form, thus are highly applicable in the analyses of non-homogeneous data. Among these models we find the general Markov model (${\mathtt{GMM}}$) and all its submodels, e.g. discrete-time versions of the Jukes-Cantor model (denoted as ${\mathtt{JC69}^{\ast}}$), Kimura two-parameters (${\mathtt{K80}^{\ast}}$) and Kimura 3-parameters models (${\mathtt{K81}^{\ast}}$), and the strand symmetric model ${\mathtt{SSM}}$. Though the discrete-time models provide a more realistic fit to the data [@YY99; @Ripplinger2008; @Ripplinger2010], their complexity requires a solid inferential framework for accurate parameter estimation. In continuous-time models, *maximum-likelihood estimation* (MLE) was found to outperform Bayesian methods [@Schwartz2010]. The most popular programs of phylogenetic inference (PAML [@Yang1997], PHYLIP [@Felsenstein1989], PAUP\* [@PAUP]) are restricted to the homogeneous models. Though more realistic, the use of nonhomogeneous models in phylogenetic inference is not yet an established practice. Recently, [@Jayaswal2011] proposed two new non-homogeneous models. With the objective of testing stationarity, homogeneity and inferring the proportion of invariable sites, the authors propose an iterative procedure based on the *Expectation Maximization* (${\mathtt{EM}}$) algorithm to estimate parameters of the non-homogeneous models (cf. [@barryhartigan87]). The ${\mathtt{EM}}$ algorithm was formally introduced by [@Dempster1977] (cf. @Hartley1958). It is a popular tool to handle incomplete data problems or problems that can be posed as such (e.g. missing data problems, models with latent variables, mixture or cluster learning). This iterative procedure globally optimizes all the parameters conditional on the estimates of the hidden data and computes the maximum likelihood estimate in the scenarios, where, unlike in the fully-observed model, the analytic solution to the likelihood equations are rendered intractable. An exhaustive list of references and applications can be found in [@Tanner1996], and more recently in [@Ambroise1998]. Here, we extend on the work of [@Jayaswal2011] and present ${\texttt{Empar}}$, a MLE method based on the ${\mathtt{EM}}$ algorithm which allows for estimating the parameters of the (discrete-time) Markov evolutionary models. ${\texttt{Empar}}$ is an implementation suitable for phylogenetic trees on any number of leaves and currently includes the following evolutionary models: ${\mathtt{JC69}^{\ast}}$, ${\mathtt{K80}^{\ast}}$, ${\mathtt{K81}^{\ast}}$, ${\mathtt{SSM}}$ and ${\mathtt{GMM}}.$ We test the proposed method on simulated data and analyze the accuracy of the parameter and branch length recovery. The tests are conducted in a settings analogue to that of [@Schwartz2010] and evaluate the performance of ${\texttt{Empar}}$ on the four and six-taxon trees with several sets of branch lengths, ${\mathtt{JC69}^{\ast}}$ and ${\mathtt{K81}^{\ast}}$ models under varying alignment lengths. We present an in-depth theoretical study, investigating the dependence of the performance on factors such as model complexity, size of the tree, positioning of the branches, data and total tree lengths. Our findings suggest that the method is a reliable tool for parameter inference of small sets of taxa, best results obtained for shorter branches. The algorithm underlying ${\texttt{Empar}}$ was implemented in C++ and is freely available to download at <http://genome.crg.es/cgi-bin/phylo_mod_sel/AlgEmpar.pl>. METHODS {#methods .unnumbered} ======= Models {#models .unnumbered} ------ We fix a set of $n$ taxa labeling the leaves of a rooted tree ${\tau}$. We denote by $N({\tau})$ the set of all nodes of ${\tau}$, the set of leaves as $L({\tau})$, the set of interior nodes as $Int({\tau}),$ and the set of edges as $E({\tau}).$ We are given a DNA multiple sequence alignment (MSA) associated to the taxa in ${\tau}$ and a discrete-time Markov process on ${\tau}$ associated to an evolutionary by a model ${\mathcal{M}}$, where the nodes in${\tau}$ are discrete random variables with values in the set of nucleotides $\{{\mathtt{A}},{\mathtt{C}},{\mathtt{G}},{\mathtt{T}}\}$. We assume that all sites in the alignment are i.i.d. and model evolution per site as follows: for each edge $e$ of ${\tau}$ we collect the conditional probabilities $P(y|x,e)$ (nucleotide $x$ being replaced by $y$ at the descendant node of $e$) in a transition matrix $A^e=(P(y|x,e))_{x,y}$; $\pi=(\pi_{{\mathtt{A}}},\pi_{{\mathtt{C}}},\pi_{{\mathtt{G}}},\pi_{{\mathtt{T}}})$ is the distribution of nucleotides at the root $r$ of ${\tau}$ and $\xi=\{\pi, (A^e)_{e}\}$ the set of continuous parameters of ${\mathcal{M}}$ on ${\tau}$. We denote by $X$ the set of $4^n$ possible patterns at the leaves and $Y$ the set of $4^{|Int({\tau})|}$ possible patterns at the interior nodes of ${\tau}.$ In what follows, the joint probability of observing $\textbf{x}=(x_l)_{l \in L({\tau})} \in X$ at the leaves and nucleotides $\textbf{y}=(y_v)_{v\in Int({\tau})} \in Y$ at the interior nodes in ${\tau}$ is calculated as $$p_{\textbf{x},\textbf{y}}(\xi)=\pi_{y_r}\prod_{v \in N({\tau})\setminus\{r\} }A^{e_{an(v),v}}_{y_{an(v)},y_{v}}$$ where $an(v)$ denotes a parent node of node $v,$ $e_{an(v),v}$ is an edge from $an(v)$ to $v$ (note: if $v$ is a leaf, then $y_v=x_v$). In the *complete model* the states at the interior nodes are observed and the joint distribution is computed as above. On the other hand, the *observed model* assumes the variables at the interior nodes to be latent. In the latter case, the probability of observing $\textbf{x}=(x_l)_{l \in L({\tau})}$ at the leaves of ${\tau}$ can be expressed as $$p_{\textbf{x}}(\xi)=\sum_{\textbf{y}=(y_v)_{v\in Int({\tau})} \in Y}p_{\textbf{x},\textbf{y}}(\xi).$$ Restricting the shape of the transition matrices $A^e$ leads to different evolutionary models such as ${\mathtt{JC69}^{\ast}}$, ${\mathtt{K80}^{\ast}}$, ${\mathtt{K81}^{\ast}}$, ${\mathtt{SSM}}$ (see @CFK, @arBook, and @AllmanRhodes_chapter4 for references and background on the discrete-time models). The first three are the discrete-time versions of the widely used continuous-time JC69 [@Jukes1969], K80 [@Kimura1980] and K81 [@Kimura1981] models. The Strand Symmetric model ${\mathtt{SSM}}$ (@CS) is a discrete-time generalization to the `HKY` model [@Hasegawa1985] with equal distribution of the pairs of bases (`A,T`) and (`C,G`) at each node of the tree. It reflects the double-stranded nature of DNA and was found to be well-suited for long stretches of data [@Yap2004]. Lastly, the general Markov model ${\mathtt{GMM}}$ (@Allman2003 [@Steel1994a]) is free of restrictions on the entries of $A^e$, non-stationary, and can be thought as a non-homogeneous version of the general time reversible model ([@GTR]). Expectation-Maximization algorithm {#expectation-maximization-algorithm .unnumbered} ---------------------------------- An algebraic approach to the Expectation-Maximization (${\mathtt{EM}}$) algorithm was first introduced in [@Pachter2005]. In this work, we adapted this approach to the context of phylogenetic trees. Let $D$ denote a MSA recorded into a vector of $4^{|{{L({\tau})}}|}$ counts of patterns $u_D=(u_\textbf{x})_{\textbf{x}\in X},$ where each $u_\textbf{x}$ stands for the counts of a particular configuration of nucleotides **x** at the leaves, observed as columns in the alignment. We are interested in maximizing the likelihood function: $$\mathcal{L}_{obs}(\xi ; u_D)=\prod_{\textbf{x}\in X}p_{\textbf{x}}(\xi)^{u_{\textbf{x}}}$$ (up to a constant). Let $U_{cD}=(u_{\textbf{x},\textbf{y}})_{\textbf{x}\in X,\textbf{y}\in Y}$ be an array of counts for the complete model, where $u_{\textbf{x},\textbf{y}}$ is the number of times $\textbf{x}$ was observed at the leaves and $\textbf{y}$ at the interior nodes. The likelihood for the complete model has a multinomial form $$\label{eq:likelEM} \mathcal{L}_c(\xi ; U_{cD})=\prod_{\textbf{x}\in X,\textbf{y}\in Y}p_{\textbf{x},\textbf{y}}(\xi)^{u_{\textbf{x},\textbf{y}}}=\prod_{\textbf{x}\in X,\textbf{y}\in Y}(\pi_{y_r}\prod_{v \in N({\tau})\setminus\{r\} }A^{e_{an(v),v}}_{y_{an(v)},y_{v}})^{u_{\textbf{x},\textbf{y}}}$$ (up to a constant), which is guaranteed to have a global maximum given by a model-specific explicit formula (see ${\textit{Supp. mat.}}$ A). ${\mathtt{EM}}$ algorithm iteratively alternates between the expectation (${\emph{E-step}}$) and maximization step (${\emph{M-step}}$). In the ${\emph{E-step}}$ the algorithm uses the tree topology, current estimates of parameters and the observed data $u_D$ to assign a posterior probability to each of the possible $4^{|{{L({\tau})}}|}$ patterns in $X$ and the expected counts of the complete model, $u_{cD}$. This step can be efficiently performed using the peeling algorithm of [@Felsenstein2004]. In the ${\emph{M-step}}$ the updated MLE of the parameters are obtained by maximizing the likelihood of the complete model . The procedure is depicted in Fig. 1. ![Expectation-maximization algorithm.[]{data-label="fig:em_procedure"}](plots/code-crop){width="90.00000%"} The likelihood is guaranteed to increase at each iteration of this process (e.g. [@Wu], [@Husmeier2005]). Moreover, for a compact set of parameters the algorithm converges to a critical point of the likelihood function. Although the output of the algorithm is not guaranteed to be a global maximum, multiple starting points are used for optimal solution. Statistical tests {#statistical-tests .unnumbered} ----------------- The substitution matrices are assumed stochastic. The number $d$ of free parameters for transition matrices in ${\mathtt{JC69}^{\ast}}$, ${\mathtt{K80}^{\ast}}$, ${\mathtt{K81}^{\ast}}$, ${\mathtt{SSM}}$ and ${\mathtt{GMM}}$ models is 1, 2, 3, 6, and 12 respectively. The root distribution under the ${\mathtt{SSM}}$ (respectively ${\mathtt{GMM}}$) model has 2 (resp. 3) free parameters, and is uniform for the remaining models considered here. For clarity of exposition, hereon the reference to the root distribution will be omitted; however, the formulas can be easily modified to include the root. We let $\xi^e$ be the vector of free parameters defined as the off-diagonal elements of a transition matrix $A$ associated to a edge $e$ ($\xi^e_1=A_{1,2}$, $\xi^e_2$ is the next –from left to right, top down– off-diagonal entry distinct from $\xi^e_1$, etc). The procedure is repeated until $\xi^e_d$ is reached. Let $\xi=(\xi_i^e)_{i=1,\ldots,d; e\in{E({\tau})}}$ denote the vector of free parameters for an evolutionary model ${\mathcal{M}}$ as above and let $\hat{\xi}$ be its MLE. Under certain regularity conditions [@Zacks1971 Chap. 5], $\hat{\xi}$ exists, is consistent, efficient and asymptotically normal with mean $\xi$ and the covariance matrix given by the inverse of the Fisher information matrix [@Rao1973; @Efron1978]. The entries of the $d|{E({\tau})}|\times d|{E({\tau})}|$ Fisher information matrix ${\textbf{I}}$ over free parameters are given by: $${\textbf{I}}(\xi^e_k, \xi^{e'}_{k'}) = -\mathbf{E}\left(\frac{\partial^2 \log{\mathcal{L}}_{obs}(\xi; u_D)}{\partial \xi^{e}_{k}\partial \xi^{e'}_{k'}}\right)$$ (see ${\textit{Supp. mat.}}$ B for details). The Wald statistics for testing the null hypothesis $\xi_i^e=\hat{\xi}_i^e$, $e\in{E({\tau})}, i=1,\ldots,d$, is $$\label{eq:null_distr} (\hat{\xi^e}-\xi^e)^T {\textbf{I}}^e(\hat{\xi^e}-\xi^e)\sim\chi^2_{d},$$ where ${\textbf{I}}^e$ denotes the $d\times d$ slice of ${\textbf{I}}$ corresponding to the free parameters of $e\in{E({\tau})}$. The $p-value$ can thus be easily calculated by looking at the tails of the corresponding $\chi^2$ distribution. We tested the validity of the test statistics in our data by simulating a variety of MSAs under the complete model and compared it to the theoretical distribution . Figure I in the ${\textit{Supp. mat.}}$ C shows high fit and proves that the setting is appropriate. Variances of the free parameters of the model and the full (observed) covariance matrix are saved in the output of ${\texttt{Empar}}$. These in turn can be used as the plug-in estimators in to calculate the $p-values$ and normal confidence intervals for the parameters. We denote by $V^{e}_{i,i}$ the $i^{th}$ diagonal entry of the matrix $({\textbf{I}}^e)^{-1}$ corresponding to the variance of the free parameter $\xi^e_i$, $i=1,\ldots,d$. For the models with $d>1$ (i.e. all but ${\mathtt{JC69}^{\ast}}$), the variances of the free parameters can be summarized in a combined form $cV^e$ for each edge $e$: $$\label{eq:comb_var} cV^{e}(\xi^e)=\frac{\sum^d_{j=1} \left(V^e_{j,j}+\left(\xi^e_j-\frac{\sum^d_{j=1}\xi^e_j}{d}\right)^2\right)}{d}.$$ Branch lengths {#branch-lengths .unnumbered} -------------- The evolutionary distance between two nodes in ${\tau}$ joined by an edge $e$ with substitution matrix $A^e$ is defined as the total number of substitutions per site along $e$. This quantity is referred to as the *branch length* of edge $e$ (or of matrix $A^e$) and, following on [@barryhartigan87trans], can be approximated by: $$l({{A}^{e}})=-\frac{1}{4}\log\det({{A}^{e}}).\label{eq:brlength}$$ We denote the *total length of the tree* ${\tau}$ by ${\mathsf{L}_{{\tau}}}$, ${\mathsf{L}_{{\tau}}}=\sum_{e\in|{E({\tau})}|}l({{A}^{e}}).$ Now, let $A$ and $A'$ be two invertible $4\times 4$ matrices such that the entries of $(A')^{-1}(A-A')$ are small. From , we get $$\begin{aligned} \label{eq:br_bound} |l(A)-l(A')|&=&\frac{1}{4}|\log\frac{\det(A)}{\det(A')}|=\frac{1}{4}|\log\det((A')^{-1}A)| \nonumber\\ &=&\frac{1}{4}|\log\det(\textbf{Id} + (A')^{-1}(A-A'))|\nonumber\\&\approx& \frac{1}{4}|\log(1+Tr((A')^{-1}(A-A')))|\nonumber\\ &\approx&\frac{1}{4}|Tr((A')^{-1}(A-A'))|\leq \frac{1}{4}4||(A')^{-1}(A'-A))||_{1} \nonumber\\ &\leq& ||(A')^{-1} ||_{1} ||A-A'||_{1},\end{aligned}$$ where $||.||_1$ is the maximum absolute column sum of the matrix. Therefore if $A'$ is a good approximation to $A$, then $l(A')$ is a good approximation to $l(A)$. In what follows, we use the statistical test above to show the accurate recovery of the parameters. By the above argument, we can conclude that the estimates of the branch lengths will also be accurate (see also Results section). Simulated data {#simulated-data .unnumbered} ============== Performance assessment of ${\texttt{Empar}}$ was conducted on the MSAs simulated on four and six-taxon trees following [@Schwartz2010]. In the case of four taxon trees we fixed an inner node as the root and considered three types of topologies: ${{\tau}_{\mathrm{balanced}}}^4$ corresponds to the “balanced” trees with all five branches equal; the inner branch in ${{\tau}_{1:2}}$ is half the length of the exterior branches; and ${{\tau}_{2:1}}$ denotes a topology with the inner branch double the length of the external ones (see Fig. 2). In ${{\tau}_{\mathrm{balanced}}}^4$ and ${{\tau}_{2:1}}$ we let the length $l_0$ of the inner branch vary from 0.01 to 1.4, where starting from 0.05 it increases in steps of 0.05; in ${{\tau}_{1:2}}$ we let $l_0$ vary in $(0,0.7).$ For 6-taxon trees we used only balanced trees ${{\tau}_{\mathrm{balanced}}}^6$ (see Fig. 2) with $l\in (0,0.7)$. ![Unrooted trees used for simulations: ${{\tau}_{\mathrm{balanced}}}^4$, ${{\tau}_{1:2}}$, ${{\tau}_{2:1}}$ and ${{\tau}_{\mathrm{balanced}}}^6$ (*from left to right*).[]{data-label="fig_trees"}](plots/tree-eq "fig:"){width="20.00000%"} ![Unrooted trees used for simulations: ${{\tau}_{\mathrm{balanced}}}^4$, ${{\tau}_{1:2}}$, ${{\tau}_{2:1}}$ and ${{\tau}_{\mathrm{balanced}}}^6$ (*from left to right*).[]{data-label="fig_trees"}](plots/tree-short "fig:"){width="20.00000%"} ![Unrooted trees used for simulations: ${{\tau}_{\mathrm{balanced}}}^4$, ${{\tau}_{1:2}}$, ${{\tau}_{2:1}}$ and ${{\tau}_{\mathrm{balanced}}}^6$ (*from left to right*).[]{data-label="fig_trees"}](plots/tree-long "fig:"){width="20.00000%"} ![Unrooted trees used for simulations: ${{\tau}_{\mathrm{balanced}}}^4$, ${{\tau}_{1:2}}$, ${{\tau}_{2:1}}$ and ${{\tau}_{\mathrm{balanced}}}^6$ (*from left to right*).[]{data-label="fig_trees"}](plots/tree-6t "fig:"){width="20.00000%"} We simulated multiple sequence alignments on trees with 4 and 6 leaves under ${\mathtt{JC69}^{\ast}}$ and ${\mathtt{K81}^{\ast}}$ models. We used the $\texttt{GenNon-H}$ package available from <http://genome.crg.es/cgi-bin/phylo_mod_sel/AlgGenNonH.pl>. In brief, based on an input phylogenetic tree with given branch lengths, $\texttt{GenNon-H}$ samples the substitution matrices corresponding to these lengths for all edges and uses them to generate the DNA MSAs following discrete-time Markov process on the tree. The output of this software is the alignment, the substitution matrices, root distribution (whenever non-stationary) and the variances of the continuous free parameters. We note that for the ${\mathtt{JC69}^{\ast}}$ model, there is a $1-1$ correspondence between the branch length and the free parameters of the substitution matrix. This does not hold for other models, were different substiution matrices may give the same branch length. We set the alignment length $L$ to $300nt, 500nt, 1,000nt$ and $10,000nt$ for 4-taxa and to $1,000nt$ or $10,000nt$ for 6-taxa. For the ${\mathtt{JC69}^{\ast}}$ and ${\mathtt{K81}^{\ast}}$ evolutionary models, a phylogenetic tree ${\tau}$ (with branch lengths), and a given alignment length, we run each analysis $1,000$ times. and estimated the parameters using ${\texttt{Empar}}$. All MSAs used for the tests are accessible at the ${\texttt{Empar}}$ webpage. Identifiability {#identifiability .unnumbered} --------------- It is known that in certain cases the substitution parameters are not identifiable (e.g. parameters at the edges adjacent to the root of valency 2). As shown in [@Allman2003], the ${\mathtt{GMM}}$ model and its submodels, are identifiable up to a permutation of rows. [@BHident] showed that incorrect order of rows in the matrices can lead to a negative determinant of the substitution matrix from which the branch lengths cannot be calculated. We expected this problem to arise in short data sets and large branch length, as those correspond to the substitution matrices with smaller diagonal value. For all the data sets used for tests, we calculated the percentage of cases among the 1,000 simulations for which the parameters estimated by the EM algorithm were permuted. This phenomenon was only observed in the data sets of $300nt$ and $1,000nt$. In the first case, the estimated matrices were permuted when the initial branch length was 0.55 or longer and corresponded to 0.005-0.023$\%$ of the cases; in the latter, for the branches of 0.6 or longer with at most 0.001$\%$ permuted matrices. Shorter branch lengths and longer alignments did not suffer from the above problem and recovered the underlying order in all of the cases. As shown by [@Chang96], the entries of the Diagonal Largest in Column (DLC) substitution matrices are identifiable. Namely, there exists a unique set of substitution matrices satisfying the DLC condition and a unique root distribution that leads to a given joint distribution at the leaves. In order to ensure the reliability of the results we designed a procedure that scans the tree in the search of the permutations that maximize the number of substitution matrices with larger diagonal entries. It is not possible to maximize it for all edges, thus the goal is to find the permutations giving more weights to the lower parts of the tree, starting with the nodes corresponding to the outer branches. Given a tree ${\tau}$, we choose an interior node to be the root, directing all edges outwards. For each interior node $x$, we apply a permutation $S(x)$ of $\{{\mathtt{A}},{\mathtt{C}},{\mathtt{G}},{\mathtt{T}}\}$ that maximizes the sum of diagonal entries of the matrices assigned to the outgoing edges of $x$. Permutations $S(x)$ are applied recursively to the subtrees of ${\tau}$, moving $x$ from the outer nodes towards the root. RESULTS AND DISCUSSION {#results-and-discussion .unnumbered} ====================== We present the results on the simulated data sets and discuss their dependence on the length of the alignments, the length of the branches and the depth of the branches in the tree–1 for the external branches and 2 for the internal branches [@Schwartz2010]. In cases with multiple branches of equal depth, we chose one of them at random. Results of statistical tests {#results-of-statistical-tests .unnumbered} ---------------------------- Each sample gave rise to a $p-value$ based on the $\chi^2_d$ test given by . The $p-values$ are a measure of strength of evidence against the null hypothesis: for both exceptionally small or large $p-values$ one can reject the null hypothesis. We recorded the proportion of samples for which the $p-value$ lied in the interval $(0.05,0.95)$. The results are shown Table \[tab:pval\_jc\_short\] for the ${\mathtt{JC69}^{\ast}}$ model on the ${{\tau}_{1:2}}$ tree (also see Tab. I-V in the ${\textit{Supp. mat.}}$ C). We observe that even for short alignments of 300nt the null hypothesis cannot be rejected in approximately 95% of the tests. [p[0.8cm]{}p[0.8cm]{}p[0.8cm]{}p[0.8cm]{}p[0.8cm]{}|p[0.8cm]{}p[0.8cm]{}p[0.8cm]{}p[0.8cm]{}]{} & &\ $\mathbf{l}$ $\mid$ L & & & & & & & &\ 0.01 & 0.971 & 0.972 & 0.968 & 0.946 & 0.972 & 0.949 & 0.868 & 0.958\ 0.05 & 0.947 & 0.951 & 0.947 & 0.948 & 0.974 & 0.943 & 0.953 & 0.952\ 0.10 & 0.949 & 0.953 & 0.964 & 0.952 & 0.952 & 0.948 & 0.948 & 0.955\ 0.15 & 0.952 & 0.954 & 0.958 & 0.938 & 0.946 & 0.953 & 0.940 & 0.947\ 0.20 & 0.957 & 0.944 & 0.944 & 0.954 & 0.949 & 0.965 & 0.944 & 0.954\ 0.25 & 0.957 & 0.955 & 0.955 & 0.956 & 0.945 & 0.939 & 0.955 & 0.936\ 0.30 & 0.957 & 0.943 & 0.945 & 0.955 & 0.943 & 0.946 & 0.941 & 0.948\ 0.35 & 0.952 & 0.943 & 0.958 & 0.958 & 0.948 & 0.943 & 0.950 & 0.960\ 0.40 & 0.955 & 0.946 & 0.947 & 0.957 & 0.951 & 0.951 & 0.936 & 0.944\ 0.45 & 0.949 & 0.944 & 0.944 & 0.947 & 0.948 & 0.955 & 0.958 & 0.958\ 0.50 & 0.948 & 0.935 & 0.942 & 0.941 & 0.929 & 0.949 & 0.954 & 0.946\ 0.55 & 0.954 & 0.949 & 0.946 & 0.957 & 0.936 & 0.944 & 0.944 & 0.952\ 0.60 & 0.940 & 0.942 & 0.937 & 0.953 & 0.944 & 0.934 & 0.948 & 0.955\ 0.65 & 0.940 & 0.934 & 0.955 & 0.952 & 0.938 & 0.938 & 0.945 & 0.948\ 0.70 & 0.944 & 0.936 & 0.942 & 0.946 & 0.917 & 0.940 & 0.944 & 0.948\ 0.75 & 0.922 & 0.932 & 0.947 & 0.934 & 0.922 & 0.932 & 0.943 & 0.950\ 0.80 & 0.909 & 0.932 & 0.926 & 0.957 & 0.957 & 0.928 & 0.943 & 0.941\ 0.85 & 0.912 & 0.912 & 0.932 & 0.948 & 0.968 & 0.930 & 0.936 & 0.947\ 0.90 & 0.870 & 0.885 & 0.919 & 0.951 & 0.980 & 0.918 & 0.929 & 0.953\ 0.95 & 0.852 & 0.888 & 0.939 & 0.951 & 0.981 & 0.965 & 0.908 & 0.944\ 1,00 & 0.824 & 0.866 & 0.893 & 0.935 & 0.982 & 0.981 & 0.896 & 0.933\ 1,05 & 0.816 & 0.853 & 0.889 & 0.930 & 0.980 & 0.981 & 0.898 & 0.937\ 1.10 & 0.806 & 0.852 & 0.891 & 0.921 & 0.990 & 0.995 & 0.925 & 0.945\ 1.15 & 0.784 & 0.812 & 0.867 & 0.938 & 0.980 & 0.987 & 0.982 & 0.951\ 1.20 & 0.797 & 0.785 & 0.823 & 0.923 & 0.986 & 0.986 & 0.984 & 0.942\ 1.25 & 0.786 & 0.803 & 0.824 & 0.938 & 0.983 & 0.981 & 0.984 & 0.941\ 1.30 & 0.789 & 0.793 & 0.800 & 0.894 & 0.981 & 0.976 & 0.992 & 0.925\ 1.35 & 0.755 & 0.787 & 0.786 & 0.893 & 0.973 & 0.991 & 0.989 & 0.912\ 1.40 & 0.761 & 0.789 & 0.785 & 0.864 & 0.970 & 0.974 & 0.994 & 0.879\ \[tab:pval\_jc\_short\] Error in transition matrices {#error-in-transition-matrices .unnumbered} ---------------------------- For a given branch, we quantified the divergence $D$ between the original and estimated parameters of its transition matrix $A$ using the induced $L_1$ norm: $||A-\hat{A}||_{1}$ (see ). The columns in the transition matrices of ${\mathtt{JC69}^{\ast}}$, $\mathtt{K80}^{\ast}$, and the ${\mathtt{K81}^{\ast}}$ are equal and the norm becomes: $$\label{eq:L2norm} D=\sum_{i=1}^4\mid A_{i,1}-\hat{A}_{i,1}\mid. $$ Figure 3 depicts the results for ${\mathtt{JC69}^{\ast}}$ and ${\mathtt{K81}^{\ast}}$ on the three 4-taxon phylogenies, different alignment lengths and depths of the branches. The shapes of the distribution of $D$ for both models are very similar. As expected, the performance is weaker for long branches and short alignments. A great improvement is observed with the increase in the alignment length, e.g. $10,000nt$ depicts very accurate estimates. The performance under the ${\mathtt{JC69}^{\ast}}$ model (Fig. 3a )is better than that of ${\mathtt{K81}^{\ast}}$ (Fig.3b) for shorter branch lengths. Parameter dispersion {#parameter-dispersion .unnumbered} -------------------- Figure 4 shows the variances of the estimated parameters for depth 1 and 2 branches on the ${{\tau}_{\mathrm{balanced}}}$, ${{\tau}_{1:2}}$, ${{\tau}_{2:1}}$ trees under the ${\mathtt{JC69}^{\ast}}$ model. The variances show an exponential increase, which is most significant in the ${{\tau}_{\mathrm{balanced}}}^4$ tree for both depths, and the ${{\tau}_{1:2}}$ for depth 2. The results for the depth 1 branch in ${{\tau}_{\mathrm{balanced}}}$ and ${{\tau}_{1:2}}$ are very similar. The smallest variance was observed for the depth 2 of ${{\tau}_{2:1}}$. For alignments of length $10,000nt$ on four taxa we can say that the method is quite accurate (see also Tab.VI-VIII in the ${\textit{Supp. mat.}}$ C). For the ${\mathtt{K81}^{\ast}}$ model we summarized the results on variances for each edge as the mean of combined variances of all samples (see formula ). The results are analogous to those of the ${\mathtt{JC69}^{\ast}}$ model, see Figure II in ${\textit{Supp. mat.}}$ C. As expected, the parameter estimates are less dispersed for shorter branches and longer alignments (see Tab. IX-XI in the ${\textit{Supp. mat.}}$ C). Error in the branch lengths {#error-in-the-branch-lengths .unnumbered} --------------------------- Using the formula  we calculated the actual difference $l_0-\hat{l}$ between the branch length $l_0$ computed from the original parameters $\xi$ and the branch length $\hat{l}$ computed using their MLEs $\hat{\xi^e}$. Negative values of this score imply overestimation of the branch length, while positive values indicate underestimation. The results are shown in Figures 5 and 6. In the case of ${\mathtt{JC69}^{\ast}}$ we observe that the method presented here does not tend to underestimate or overestimate the lengths for the depth 1 branches in all the 4-taxon trees ($l_0-\hat{l}$ is centered at 0 (see Fig. 5). The depth 2 branches have a tendency towards overestimation of the length for branches longer than (approximately) $0.45$ for ${{\tau}_{1:2}}$, $0.9$ for ${{\tau}_{2:1}}$, and $0.8$ for the ${{\tau}_{\mathrm{balanced}}}^4$ trees. In the latter case, lengths longer than $1.2$ for alignments up to $1,000$nt show opposite trend of underestimating the true lengths. The values were accurate when the alignment lengths were increased in the case of ${{\tau}_{1:2}}$ and ${{\tau}_{2:1}}$. On the other hand, for ${{\tau}_{\mathrm{balanced}}}^4$ the alignments of $10,000$nt resulted in overestimation. In the ${\mathtt{K81}^{\ast}}$ model the results are significantly more accurate (see Fig.  6). There is a trend of underestimation for branches longer than (approximately) $0.9$ for shorter alignments. That is especially noticeable for ${{\tau}_{\mathrm{balanced}}}^4$ and depth 1 branches of ${{\tau}_{1:2}}$. This trend diminishes with an increase in the alignment length. Overall, in the case of both models, the variance of the estimate is smaller for shorter lengths and both depth 1 and 2 branches of the ${{\tau}_{2:1}}$ tree. In addition, we calculated the tree length ${\mathsf{L}_{{\tau}}}$ (i.e. the sum of its branch lengths) from the estimated parameters and compared it to the theoretical result on the original branch length $l_0$: $4.5l_0$ for ${{\tau}_{1:2}}$ (where $l_0$ is a depth 1 branch), $3l_0$ for ${{\tau}_{2:1}}$ ($l_0$ for depth 2 branch) and $5l_0$ for ${{\tau}_{\mathrm{balanced}}}^4$. The rightmost columns of Figures 5 and 6 show the results for 4-taxon trees for the ${\mathtt{JC69}^{\ast}}$ and ${\mathtt{K81}^{\ast}}$ models respectively. The length of the tree is estimated accurately for all trees, the estimates being best for ${{\tau}_{2:1}}$. The variance is small and decreasing with an increase in the data length. As the sequences get longer, the distribution is centered around the true value. This is especially visible for the ${\mathtt{K81}^{\ast}}$ model (see Fig.  6). Results for larger trees {#results-for-larger-trees .unnumbered} ------------------------ We run the analysis on the 6-taxon balanced tree, ${{\tau}_{\mathrm{balanced}}}^6$, under the ${\mathtt{K81}^{\ast}}$ model, for alignment lengths of $1,000$nt and $10,000$nt and branch length ${\mathbf{l}}\in\{0.01, 0.1, 0.3, 0.5, 0.7, 0.9, 1.1, 1.3, 1.4\}$. The $p-values$ of the corresponding tests confirm that the performance of the algorithm is very satisfactory (see Tab. XII in the ${\textit{Supp. mat.}}$ C). We have seen in the 4-taxon study that the tree with equal branch lengths gave worse results than the unbalanced trees. Thus, we expect the results of the depth 2 branches to be similarly challenged in this case. Figure 7 depicts the estimated tree lengths. It can be seen that the estimates are accurate and the results improve for the alignments of $10,000$nt. As expected, the variance of the estimates increases with the increase in the length of the branch. By formula , long branches correspond to small values of the determinant of the transition matrix. Thus, statistical fluctuations in the parameter estimates have greater impact on the resulting length of the tree. Next, we calculated the difference between the original and estimated branch lengths. In Figure 8a we see that the depth 1 branches show some degree of underestimation of the length for lengths $1.1-1.4$ and alignments of $1,000$nt. In the case of $10,000$nt, the results improve and can be expected to show little bias for even longer data sets. Branches of depth 2 show higher degree of underestimation with improvement for longer data sets. The divergence of the original and estimated parameters for transition matrices given by formula is shown in Figure 8b. For branches of depth 1 and data of length $10,000$, the error is about 0.2. In the case of branches of depth 2, it is almost doubled for both alignment lengths. In both cases, branch lengths up to 0.5 give satisfactory results. The error of the estimates for longer branches seems to be approaching a plateau. Combined variance of the estimated parameters is much decreased for the $10,000$nt data sets in comparison with the $1,000$nt, and is smaller for the depth 1 branch (see Fig. 8c ). Again, the exponential shape of the plot can be attributed to the logarithm appearing in the formula . CONCLUSION {#conclusion .unnumbered} ========== In order to evaluate the performance of the method proposed here under various circumstances, we conducted many tests on simulated data sets. We observed that the performance of ${\texttt{Empar}}$ is most optimal for long alignments and short branch lengths. It is worth noting that even for short alignments of 300nt or 500nt on 4 taxa, the estimated parameters approximate closely the original parameters in $\approx 95\%$ of the cases as proved by the normality test of the MLE. Moreover, the branch lengths calculated based on the parameters estimated by ${\texttt{Empar}}$ were found very accurate already for short alignments. Though the measure of divergence $D$ for the parameters of transition matrices proposed here accumulates all errors in the entries of the transition matrix, alignment length of 10,000nt showed divergence values smaller than 0.1. In this paper, we provide the first implementation of a tool for inferring continuous parameters under the discrete-time models. The method allows for accurate estimation of branch lengths in non-homogeneous data. There are two limitations to applicability of the method. Firstly, the algorithm has an exponential computational time increase with the number of taxa. This is a restriction due to the fact that the algorithm computes large matrices of dimension that is exponential in the total number of nodes of a tree. Running time of ${\texttt{Empar}}$ on star trees with 3-8 nodes and equal branches of $0.5$ on Ubuntu 11.10, Intel Core i7 920 at 2.67 GHz with 6 Gb is given in Table \[tab:gennong\_times\]. Secondly, the memory usage of ${\texttt{Empar}}$ is approx. $8*4^{|N({\tau})|}$ and corresponds to the memory footprint of the matrix in the ${\mathtt{EM}}$ algorithm, e.g. for this matrix to fit in the memory of a 6Gb machine the bound on the number of nodes is $|N({\tau})|$ $\leq 14$. length $\mid$ n 3 4 5 6 7 8 ----------------- ------- ------- ------- ------- ------- ------ 1,000 0.004 0.02 0.033 0.222 1,049 7.14 10,000 0 0.011 0.043 0.171 1,044 6.95 : ${\texttt{Empar}}$ performance time– estimating the parameters of ${\mathtt{K81}^{\ast}}$ on star trees with equal branch lengths of $0.5$, varying number of leaves, $L({\tau})$, for the MSAs of $1,000$ and $10,000nt$. \[tab:gennong\_times\] We conclude that ${\texttt{Empar}}$ is a highly reliable method for estimating branch lengths of relatively small number of taxa and trees with short branch lengths (e.g. closely related species), and achieves high accuracy even when the results are based on short sequences. In particular, ${\texttt{Empar}}$ is a reliable method to compute quartets and to be used with quartet-based methods (see [@Berry99] and [@Berry2000]) on nonhomogeneous data. FUNDING {#funding .unnumbered} ======= Both authors were partially supported by Generalitat de Catalunya, 2009 SGR 1284. MC is partially supported by Ministerio de Educación y Ciencia MTM2009-14163-C02-02.We thank Roderic Guigó for generously providing funding for this project under grant BIO2011-26205 from the Ministerio de Educación y Ciencia (Spain). [45]{} natexlab\#1[\#1]{}url \#1[[\#1]{}]{} E S Allman and J A Rhodes. , 1860 (2):0 113–144, 2003. E S Allman and J A Rhodes. . Cambridge University Press, January 2004. ISBN 0-521-52586-1. E S Allman and J A Rhodes. Phylogenetic invariants. In O Gascuel and MA Steel, editors, [*Reconstructing Evolution*]{}. Oxford University Press, 2007. C Ambroise. . , 150 (1):0 154–156, January 1998. ISSN 0176-4268. D Barry and J A Hartigan. 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--- abstract: 'We propose driven dissipative Majorana platforms for the stabilization and manipulation of robust quantum states. For Majorana box setups, in the presence of environmental electromagnetic noise and with tunnel couplings to quantum dots, we show that the time evolution of the Majorana sector is governed by a Lindblad master equation over a wide parameter regime. For the single-box case, arbitrary pure states (‘dark states’) can be stabilized by adjusting suitable gate voltages. For devices with two tunnel-coupled boxes, we outline how to engineer dark spaces, i.e., manifolds of degenerate dark states, and how to stabilize fault-tolerant Bell states. The proposed Majorana-based dark space platforms rely on the constructive interplay of topological protection mechanisms and the autonomous quantum error correction capabilities of engineered driven dissipative systems. Once a working Majorana platform becomes available, only standard hardware requirements are needed to implement our ideas.' author: - 'Matthias Gau,$^{1,2}$ Reinhold Egger,$^{1}$ Alex Zazunov,$^{1}$ and Yuval Gefen$^{2}$' title: Towards dark space stabilization and manipulation in driven dissipative Majorana platforms --- \#1\#2[\#1|\#2]{}\#1[|\#1|]{}\#1[\#1]{}\#1[|\#1]{}\#1[\#1|]{} Introduction {#sec1} ============ It has been known for a long time that the dynamics of open quantum systems subject to external driving forces and coupled to environmental modes (‘heat bath’) can be described by master equations [@Weiss2007; @Breuer2006; @Gardiner2004]. For a Markovian bath, the memory time of the bath represents the shortest time scale of the problem. The master equation is then of Lindblad type [@Lindblad1976; @Lindblad1983], where a Hamiltonian describes the coherent time evolution of the system’s density matrix and a Lindbladian captures the dissipative dynamics. (We here use ‘Lindbladian’ for the dissipator terms in the master equations below.) The Lindblad equation is the most general Markovian master equation which preserves the trace and positive semi-definiteness of the density matrix. A major development over the past two decades has come from the realization that driven dissipative (DD) quantum systems can be stabilized in a pure quantum state by appropriate engineering of the driving fields and of the coupling to the dissipative environment [@Plenio1999; @Beige2000; @Plenio2002; @Diehl2008; @Kraus2008; @Diehl2010; @Diehl2011; @Bardyn2013; @Zanardi2014; @Albert2014; @Jacobs2014; @Albert2016; @Goldman2016; @Wiseman2010]. Such states are eigenstates of the corresponding Lindbladian with zero eigenvalue, i.e., the operation of the Lindbladian leaves them inert. We therefore will refer to these DD stabilized states as *dark states* in what follows. Rather than viewing the coupling to a dissipative environment as foe (e.g., leading to decoherence of quantum states and undermining the utilization of similar platforms for quantum information processing), the combined effect of drive and dissipation can thus be harnessed to engineer quantum-coherent pure states. Going beyond dark states, the stabilization of a *dark space* [@Iemini2015; @Iemini2016; @Santos2020] — a manifold spanned by multiple degenerate dark states — raises the prospects of employing such systems as viable platform for quantum information processing. Reference [@Touzard2018] reports on recent experimental results in this direction. Using trapped ions or superconducting qubits, the above ideas have already allowed for first qubit stabilization experiments [@Geerlings2013; @Lu2017; @Touzard2018], for the implementation of quantum simulators [@Barreiro2011; @Schindler2013], and for the generation of selected highly entangled multi-particle states [@Shankar2013; @Leghtas2013; @Reiter2016; @Liu2016]. Systems composed of many coupled qubits stabilized by DD mechanisms could eventually result in universal quantum computation platforms [@Verstraete2009; @Fujii2014], where fault tolerance is the consequence of autonomous error correction [@Terhal2015] due to the engineered dissipative environment, without the need for active feedback [@Wiseman2010; @Kerckhoff2010; @Murch2012; @Kapit2015; @Kapit2016]. Recent experimental progress on autonomous error correction in DD qubit systems has been described in Refs. [@Leghtas2013; @Liu2016; @Reiter2017; @Puri2019]. At present, reported fidelities in DD qubit setups (which by construction are stable in time) are typically below 90$\%$ for state stabilization, with significantly lower fidelities for single- or two-qubit gate operations. Another important and at first glance unrelated development towards the (so far elusive) goal of fault-tolerant universal quantum computation comes from the field of topological quantum computation [@Nayak2008]. By using topological quasiparticles [@Wen2017] for encoding and processing quantum information, the latter is nonlocally distributed in space and thereby protected against local environmental fluctuations. In general terms, for practically useful and scalable DD systems with multiple degenerate dark states, the coupling to the environment has to be carefully engineered such that it is blind to all system operators acting within the targeted dark space manifold [@Facchi2000]. It will thus be imperative to avoid residual (uncontrolled and unwanted) noise sources. In that regard, platforms harboring topological quasiparticles may offer a key advantage since they should come with a strongly reduced intrinsic sensitivity to residual environmental fluctuations as compared to conventional systems. The simplest candidate for topological quasiparticles is given by Majorana bound states (MBSs), which are localized zero-energy states in topological superconductors. For Majorana reviews, see Refs. [@Alicea2012; @Leijnse2012; @Beenakker2013; @Sarma2015; @Aguado2017; @Lutchyn2018; @Zhang2019a]. Topological codes relying on MBSs have so far been discussed in the context of active error correction [@Alicea2011; @Terhal2012; @Hyart2013; @Vijay2015; @Aasen2016; @Landau2016; @Plugge2016; @Plugge2017; @Karzig2017; @Litinski2017; @Wille2019], where periodically repeated stabilizer measurements are needed for fault tolerance. It remains an important challenge to devise feasible and scalable Majorana platforms exploiting passive error correction strategies, where DD mechanisms serve to continuously measure the system in a way that the desired highly entangled many-body quantum state becomes stabilized automatically, see, e.g., Ref. [@Herold2017]. While this ambitious goal is beyond the scope of our work, we here analyze related questions for DD systems with up to eight MBSs. For a mesoscopic floating (not grounded) topological superconductor harboring four MBSs, strong charging effects [@Fu2010] imply that the ground state is doubly degenerate under Coulomb valley conditions (see Sec. \[sec2a\] for details). Such a superconducting island is therefore a good candidate for a topologically protected Majorana qubit, named Majorana box qubit [@Plugge2017] or tetron [@Karzig2017]. Thanks to the nonlocal Majorana encoding of quantum information, such a qubit allows for unique addressability options via electron cotunneling when quantum dots (QDs) or normal leads are attached to the island by tunneling contacts, see also Refs. [@Gau2018; @Munk2019]. Majorana qubits have not yet been experimentally realized. However, the recent emergence of new Majorana platforms (see, e.g., Refs. [@Liu2018b; @Zhang2018b; @Wang2018b; @Sajadi2018; @Ghatak2018; @Murani2019]) in addition to the semiconductor nanowire platform mainly explored so far [@Lutchyn2018; @Zhang2019a] indicates that they may be available in the foreseeable future. We note that alternative Majorana qubit designs have been put forward, e.g., in Refs. [@Terhal2012; @Hyart2013; @Aasen2016]. Many of the ideas discussed below can be adapted to those setups as well. Motivation and goals of this work --------------------------------- We here show that once available, Majorana box devices yield highly attractive platforms for implementing DD protocols aimed at the realization of dark states and/or dark spaces. The driving field is applied to the tunnel link connecting a pair of QDs, and dissipation is due to environmental electromagnetic noise. To the best of our knowledge, apart from a distantly related proposal for the DD stabilization of Majorana-based quantum memories [@Bardyn2016], no studies of DD Majorana systems have appeared in the literature so far. We note that the DD engineering of MBSs in cold-atom based Kitaev chains [@Diehl2011; @Bardyn2013; @Goldman2016] differs from our ideas: We consider topological superconductors harboring native MBSs, and then subject the resulting Majorana systems to DD stabilization and manipulation protocols targeting dark states and/or dark spaces. Our unique platform enables us to employ QDs as external knobs to be used not only for state engineering but also for state manipulation. Our motivation for designing and studying novel DD stabilization and manipulation schemes using Majorana platforms rests on several arguments and expectations: 1. Since uncontrolled environmental effects are largely suppressed by topological protection mechanisms, one may reach higher fidelities than those reported so far for DD dark state or dark space implementions using conventional (topologically trivial) platforms. This point should be especially important for high-dimensional dark spaces, where residual noise effects could break the degeneracy of the dark states spanning the dark space manifold [@Facchi2000]. Such spaces are highly attractive candidates for implementing fault-tolerant quantum computing platforms. These topological protection elements are especially important for platforms where the Lindblad spectrum is not gapped. 2. It is known that for large-scale Majorana surface codes, where active feedback is needed for code stabilization, the fault-tolerance error threshold is much more benign than for conventional bosonic surface codes, see Refs. [@Vijay2015; @Plugge2016; @Fowler2012] and references therein. In particular, in Majorana surface codes no ancilla qubits are needed for stabilizer readout at all. We expect that our dark space constructions using MBS systems can allow for similar fault tolerance advantages over conventional dark space realizations. However, more work is needed to reach a quantitative conclusion on this point. 3. The DD stabilization and manipulation of Majorana-based dark states or dark spaces offers several practical advantages. In particular, the robustness of such states as quantified by the dissipative gap is expected to be superior to quantum states that are encoded without DD mechanisms in native Majorana devices, see Sec. \[sec3\]. Moreover, a small overlap between MBSs is often tolerable, without causing dephasing of dark states, cf. Sec. \[sec3e\]. 4. When steering a state into the dark space or manipulating a state within the dark space, one may need to maximize its purity, having in mind quantum information manipulation protocols. For this purpose, we may adiabatically switch on a suitable perturbation either to the Lindbladian dissipator or to the accompanying Hamiltonian, thereby breaking the degeneracy of the dark space. In this manner, one can revert to a specific pure dark state, manipulate this state, and subsequently adiabatically switch off this perturbation again. The DD Majorana platforms discussed below offer convenient tools to switch on and off such degeneracy-breaking perturbations. ![Schematic sketch of a driven dissipative Majorana box setup. The superconducting island harbors four Majorana operators $\gamma_\nu$, three of which are tunnel-coupled to two single-level quantum dots (QDs, in blue). The Majoranas could be realized as end states of two parallel topological superconductor nanowires (green) which are electrically connected by a superconducting bridge (orange) [@Plugge2017]. The tunnel links connecting QDs to MBSs are shown as dashed lines. The phases $\beta_j$ in Eq.  are also indicated. Due to the large box charging energy, transport between different QDs through the Majorana island proceeds only via cotunneling processes. These cotunneling processes can be inelastic, involving the emission or absorption of photons from the dissipative electromagnetic environment. In addition, a driving field can pump electrons via a tunnel link between the QDs (solid line). []{data-label="fig1"}](f1){width="0.9\columnwidth"} The dynamics of the Majorana degrees of freedom in a device such as the one depicted in Fig. \[fig1\] will here be discussed on several conceptual levels. We show that our DD protocols indeed give rise to master equations of Lindblad type. These equations contain both a Hamiltonian (governing the unitary part of the time evolution) and a Lindbladian (causing dissipative dynamics). By choosing suitable parameter values as discussed in Sec. \[sec3\], we demonstrate that an arbitrary dark state can be stabilized. In more complex two-box devices, see Sec. \[sec4\], the Lindbladian can be engineered to support a multi-dimensional dark space. As a generic initial state is driven towards the dark space, we show (see also Ref. [@ourprl]) how to optimize the purity, the fidelity (i.e., the overlap of the state with the target dark space), and the speed of approach. In our accompanying short paper [@ourprl], we provide a summary of our key ideas and apply them to show that in a two-box setup, one can stabilize and manipulate ‘dark qubit’ states. In effect, the topologically protected native Majorana qubit discussed in Refs. [@Plugge2017; @Karzig2017] (which exists in a single box) is thereby stabilized by adding another protection layer due to DD mechanisms (at the prize of adding a second box). The main benefit of applying DD strategies to a topologically nontrivial system comes from the insight that in the latter class of systems, one can implement unidirectional cotunneling processes in an elementary and practically useful manner. These cotunneling processes in turn directly determine the structure of the jump operators in the Lindblad equation. Overview -------- In order to guide the focused reader through this long article, we here provide a short overview summarizing the content of the subsequent sections. In addition, Table \[table1\] summarizes the key symbols and notations used throughout this paper. - In Sec. \[sec2\], we introduce the theoretical concepts and physical ingredients needed for the DD stabilization and manipulation of dark states using a single Majorana box, see Fig. \[fig1\], and we derive the dynamical equations. Our model is introduced in Sec. \[sec2a\], where the dissipation arises from environmental electromagnetic fluctuations and the drive is applied to a pair of QDs. We subsequently derive the Lindblad equation [@Weiss2007; @Breuer2006; @Gardiner2004; @Lindblad1976; @Lindblad1983] governing the time evolution of the combined QD-Majorana system in Sec. \[sec2b\], where we also present numerical results for the dynamics obtained from this Lindblad master equation. Remarkably, up to initial transient behaviors, one can describe the dynamics in the Majorana sector in terms of a reduced Lindblad equation, where the QD degrees of freedom have been traced out. We describe this step in Sec. \[sec2c\], along with a discussion of the conditions under which this reduced Lindblad equation applies. All of our subsequent results are obtained by employing this reduced Lindblad equation. - In Sec. \[sec3\] we then describe dark state stabilization protocols for the single-box device in Fig. \[fig1\]. We begin in Sec. \[sec3a\] with the case of Pauli operator eigenstates, followed by the stabilization of the so-called magic state in Sec. \[sec3b\]. In Sec. \[sec3c\], the role of increasing temperature on our stabilization protocols is examined. Interestingly, as shown in Sec. \[sec3d\], we find that for certain parameter settings, dark states can be stabilized even in the absence of any drive. Finally, in Sec. \[sec3e\], we discuss additional points, e.g., concerning the role of Majorana state overlaps or how to perform a parity readout of the stabilized states. - In Sec. \[sec4\], we turn to a setup with two coupled boxes and present our DD stabilization and manipulation protocols for quantum states that belong to a dark space manifold. The Lindblad equation for this setting is derived in Sec. \[sec4a\]. We explain how one can engineer a degenerate dark space in Sec. \[sec4b\]. This topic is the main focus of Ref. [@ourprl], and the discussion is therefore kept rather short here. Finally, in Sec. \[sec4c\], we show how to stabilize Bell states in the two-box setting. - The paper concludes with a summary and an outlook in Sec. \[secConc\]. Technical details and additional information can be found in three Appendices. Let us also remark that we often use units with $\hbar=k_B=1$. Symbol Meaning First appearance --------------------------------------------------------------- ------------------------------------------------------------------------------- ------------------ *Model parameters:* $A$ drive amplitude $\alpha$ dimensionless system-bath coupling for Ohmic bath $\beta_j$ phases of the tunnel couplings $\lambda_{j\nu}$ $E_C$ charging energy of the Majorana box $\epsilon_{j}$ level energy of the respective quantum dot $g_0$ cotunneling scale for single-box setup, $g_0=t_0^2/E_C$ $\tilde g_0^{}$ cotunneling scale for double-box setup $\lambda_{j\nu}$ tunnel coupling between QD fermion $d_j$ and Majorana operator $\gamma_{\nu}$ (‘state design parameters’) $M$ number of MBSs on Majorana box $\omega_0$ drive frequency $\omega_c$ cut-off frequency for Ohmic bath $T$ temperature $t_0$ overall scale of tunnel couplings between QDs and Majorana box $t_{LR}$ tunnel coupling connecting both Majorana boxes, see Sec. \[sec4\] *Dynamical quantities:* $D$ dark space dimension Sec. \[sec3e4\] $\Delta_{z,x,y,m}$ dissipative gap for the respective dark state e.g., see $h_{\pm}, \tilde h^{}_\pm$ Lamb shift parameters for full and reduced Lindblad eq., respectively , $J_\pm, \Gamma_{\pm}, H_{\rm L}$ jump operators, transition rates, and Hamiltonian for full Lindblad eq. , , $\tilde J^{}_\pm, \tilde\Gamma^{}_{\pm}, \tilde H^{}_{\rm L}$ jump operators, transition rates, and Hamiltonian for reduced Lindblad eq. , , $K_{j=1,\ldots,6}, \tilde \Gamma^{}_{j}$ jump operators and transition rates for two-box setup , $p$ occupation probability of high-lying QD $\rho(t)$ reduced density matrix for combined QD-Majorana system $\rho_{\rm M}(t)$ reduced density matrix for the Majorana sector $(\tau_x,\tau_y,\tau_z)$ Pauli operators for QD pair in single-occupancy regime $N_{\rm d}=1$ $\theta_{j\nu},\theta$ fluctuating electromagnetic phases , $\hat W_{jk}, \hat W_{x,y,z}$ fluctuating cotunneling operators , $W_{jk}, W_{x,y,z}$ cotunneling operators for $\theta_{j,\nu}=0$ $(X,Y,Z)$ Pauli operators of Majorana box Driven dissipative Majorana dynamics {#sec2} ==================================== We start this section by discussing the Majorana box [@Plugge2017; @Karzig2017]. Our DD model as well as the physical assumptions behind it are explained in Sec. \[sec2a\]. We then derive the Lindblad master equation governing the dynamics of the reduced density matrix of the Majorana sector. To that end, we first trace over the environmental degrees of freedom in Sec. \[sec2b\], and then over the QD fermions in Sec. \[sec2c\]. Model and low-energy theory {#sec2a} --------------------------- In this subsection, we introduce the model for the DD Majorana setup illustrated in Fig. \[fig1\]. We also outline the hardware ingredients needed for implementing our dark state stabilization and manipulation protocols. For concreteness, we refer to a possible realization using proximitized semiconductor nanowires [@Plugge2017; @Karzig2017]. In addition, we describe the effective low-energy Hamiltonian obtained after the high-energy charge states on the Majorana island are projected away. ### Majorana box Consider the setup depicted in Fig. \[fig1\], where a floating topological superconductor island harbors $M$ zero-energy MBSs. For this case we have $M=4$, but for generality, we shall allow for general (even) values of $M$. The MBSs correspond to the Majorana operators $\gamma^{}_\nu=\gamma_\nu^\dagger$, with anticommutator $\{\gamma_\nu,\gamma_{\nu'}\}=2\delta_{\nu\nu'}$ and $\nu=1,\ldots,M$. As indicated in Fig. \[fig1\], they could be realized as end states of two parallel InAs/Al nanowires [@Lutchyn2018]. We consider class-$D$ topological superconductor wires, where time reversal symmetry is broken by a magnetic field [@Alicea2012]. Both nanowires are electrically connected by a superconducting bridge such that the entire island has a common charging energy, $E_C=e^2/(2C)$, with typical values of the order $E_C\approx 1$ meV [@Lutchyn2018]. The isolated island (‘box’) has the Hamiltonian (we work in the Schrödinger picture for now) $$\label{Hbox} H_{\rm box}=E_C( \hat N-N_g)^2.$$ The operator $\hat N$ refers to the total electron number on the box, and $N_g$ is a tunable backgate parameter. In Eq.  we have neglected hybridization energies resulting from a finite overlap between different MBS pairs. These energy scales are exponentially small in the respective MBS-MBS distance. As will be discussed in Sec. \[sec3e\], a small hybridization between MBSs is often tolerable for DD-generated dark states or dark spaces. For the native Majorana qubit, such effects cause dephasing. Our theory requires several conditions to be satisfied. First, we assume that our DD protocols only involve energy scales well below both $E_C$ and the superconducting (proximity) gap $\Delta$. This assumption implies that the ambient temperature satisfies $T\ll {\rm min}\{E_C,\Delta\}$. We can then neglect the effects of above-gap continuum quasiparticles, as has tacitly been assumed in Eq. , which otherwise constitute an intrinsic source of dissipation in the Majorana sector. In practice, one also needs to ensure that accidental low-energy Andreev states are not accessible, see Ref. [@Manousakis2019] for a recent discussion. Second, we consider Coulomb valley conditions [@Nazarov; @AltlandBook], i.e., $N_g$ is tuned close to an integer value and the box is only weakly coupled to the QDs in Fig. \[fig1\]. In that case, $H_{\rm box}$ leads to charge quantization, which dictates the fermion number parity of the island. At temperatures well below the superconducting gap, only the Majorana sector of the full Hilbert space of the box has to be kept [@Fu2010]. For $M=4$, we arrive at a parity constraint in the Majorana sector, $\gamma_1\gamma_2\gamma_3\gamma_4=\pm 1$ [@Beri2012], and the lowest-energy island state is then doubly degenerate. The corresponding Pauli operators associated with the resulting Majorana qubit are represented by Majorana bilinears [@Beri2012; @Landau2016; @Plugge2016], $$\label{PauliOp} X=i\gamma_1\gamma_3,\quad Y=i\gamma_3\gamma_2,\quad Z=i\gamma_1\gamma_2.$$ The fact that Pauli operators correspond to spatially separated pairs of Majorana operators allows for unusually versatile qubit access options. ### Quantum dots We next turn to the Hamiltonian describing the two QDs, $H_{\rm d}$, in Fig. \[fig1\]. We start from a general single-dot Hamiltonian, $H_{\rm QD}=\sum_{\alpha} h_{\alpha} d_{\alpha}^{\dagger}d^{}_{\alpha}+\epsilon_C\left(\hat n-n_g\right)^2$, where $\alpha$ labels electron spin and orbital degrees of freedom, $d_{\alpha}$ are fermion operators with $\hat n=\sum_{\alpha} d_{\alpha}^{\dagger}d^{}_{\alpha}$, $h_{\alpha}$ describes a single-particle energy, and $\epsilon_C$ is the (large) dot charging energy [@Karzig2017; @Flensberg2011; @Nazarov; @AltlandBook]. On low energy scales, the dot can then effectively be described by a single spinless fermion level. Denoting the corresponding level energy by $\epsilon_j$ for QD $j=1,2$, one arrives at $$\label{HDots} H_{\rm d}=\sum_{j=1,2} \epsilon_j d_j^{\dagger} d^{}_j,$$ see Ref. [@Karzig2017] for details. The energies $\epsilon_j$ can be controlled by variation of the gate voltage parameter $n_g$. Without loss of generality, we take $\epsilon_2>\epsilon_1$ throughout, where both energies should satisfy $|\epsilon_j|\ll {\rm min}\{E_C,\Delta\}$. In addition, we employ a time-dependent electromagnetic driving field which can pump single electrons between the two QDs via a tunnel link. To that end, a suitable AC voltage can be applied to a gate electrode located near this link. The respective Hamiltonian contribution is given by [@Platero2004] $$\label{Hdriv} H_{{\rm drive}}(t) = w(t) d_1^{\dagger} d_2^{}+{\rm h.c.},\quad w(t) = t_{12}+2A \cos\left(\omega_0 t\right),$$ where $\omega_0$ denotes the drive frequency and $A$ the drive amplitude. In what follows, we assume that the static contribution vanishes, $t_{12}=0$, because a small coupling $t_{12}\ne 0$ will not affect the dissipator in the Lindblad equation, see Eq.  below, and thus does not change the physics in a qualitative manner. In this work, we consider the Coulomb valley regime where the total charge on the box is fixed by the charging term in Eq.  on time scales $\delta t>1/E_C$ [@Romito2014]. The total particle number on the QDs, $N_{\rm d}=\sum_j d_j^\dagger d_j^{}$, is therefore also conserved on such time scales. For even $N_{\rm d}\in\{0,2\}$, the inter-QD dynamics is effectively frozen out. We here mainly focus on the case $N_{\rm d}=1$, where the pair of QDs forms a spin-1/2 degree of freedom corresponding to Pauli operators $\tau_{x,y,z}$ with $\tau_\pm=(\tau_x\pm i \tau_y)/2$, $$\tau_+ = \tau_-^\dagger = d_1^{\dagger}d_2^{}, \quad \tau_z = d_1^{\dagger}d_1^{} - d_2^\dagger d_2^{} = 2\tau_+\tau_--1.\label{taudef}$$ We next turn to the tunnel couplings connecting the QDs to the island. ### Tunnel couplings and electromagnetic environment In the above parameter regime, tunneling to the box has to proceed via MBSs since no other low-energy island states are available. Such processes can be inelastic due to the coupling to a bosonic environment. We here consider the case of a dissipative electromagnetic environment, which can be modeled by including fluctuating phases $\theta_{j\nu}$ in the tunneling matrix elements [@Nazarov; @Devoret1990; @Girvin1990], $$\label{hatlambda} \hat\lambda_{j\nu}=\lambda_{j\nu}e^{i\theta_{j\nu}},$$ with dimensionless complex-valued parameters $\lambda_{j\nu}$ subject to ${\rm max}\{|\lambda_{j\nu}|\}=1$. Here $\lambda_{j\nu}$ determines the transparency of the tunnel link between the QD fermion $d_j$ and the Majorana state $\gamma_\nu$ in the absence of electromagnetic noise [@Zazunov2016]. With the overall hybridization energy $t_0$ characterizing the QD-MBS couplings, the tunneling Hamiltonian is given by [@Nazarov; @Devoret1990; @Girvin1990] $$\label{Htun} H_{{\rm tun}} =t_0 e^{-i\hat\phi}\sum_{j,\nu} \hat \lambda_{j\nu} d^{\dagger}_j \gamma_\nu + {\rm h.c.}$$ The phase operator $\hat\phi$ of the island has the commutator $[\hat N,\hat\phi]=-i$ with the number operator $\hat N$ in Eq. . The $e^{i\hat\phi}$ ($e^{-i\hat \phi}$) factor in Eq.  thus ensures that an electron charge is added to (subtracted from) the island in a tunneling process. It is well known that the electromagnetic potential fluctuations predominantly couple to the phase of the wave function [@Devoret1990; @Girvin1990]. This fact is expressed by the appearance of the fluctuating tunnel couplings $\hat\lambda_{j\nu}$, see Eq. , in the tunneling Hamiltonian . For concreteness, we assume that the electromagnetic environment can be modeled by a single bosonic bath, see also Ref. [@Munk2019]. Representing the bath by an infinite set of harmonic oscillators [@Weiss2007; @Breuer2006], the environmental Hamiltonian is $H_{\rm env}=\sum_m E_m b_m^{\dagger} b_m^{}$, with the energy $E_m>0$ of the photon mode described by the boson annihilation operator $b_m$. In practice, the relevant bath energies $E_m$ are strongly suppressed above a cutoff frequency $\omega_c$. With dimensionless real-valued couplings $g_{j\nu,m}$, the stochastic phase operators $\theta_{j\nu}$ are written as $$\label{phasedef} \theta_{j\nu}=\sum_{m} g_{j\nu,m} \left(b^{}_m+b_m^\dagger\right).$$ Clearly, they commute with each other, $[\theta_{j\nu},\theta_{j'\nu'}]=0$. ### Low-energy theory {#spsec1} We are interested in the parameter regime defined by the conditions $$\label{condit} {\rm max}\{T,A,t_0,\omega_0,\omega_c,|\epsilon_j|\}\ll {\rm min}\{E_C,\Delta\}.$$ The parameters on the left side of Eq.  affect the dissipative transition rates in the Lindblad equation below. These rates in turn set the time scale on which dark states are approached. We will adopt a concise description, whereby for engineering a stabilization protocol targeting a specific dark state, it suffices to adjust the complex-valued tunnel link parameters $\lambda_{j\nu}$, see Sec. \[sec3\]. In practice, those *state design parameters* can be changed via gate voltages. We also note that under the conditions in Eq. (\[condit\]), boson-assisted processes can neither excite above-gap quasi-particles nor higher-energy charge states on the island. The full Hamiltonian can then be projected onto the doubly degenerate ground-state space of the box, $H(t)\to H_{\rm eff}(t)$. Using a Schrieffer-Wolff transformation to implement this projection, and noting that $H_{\rm box}$ then reduces to an irrelevant constant energy shift, we arrive at the effective low-energy Hamiltonian $$\label{heff0} H_{\rm eff}(t)=H_{\rm d}+ H_{\rm env} + H_{\rm drive}(t)+H_{\rm cot},$$ with the drive term in Eq.  and the cotunneling contribution $$H_{\rm cot} = g_0\sum_{j,k=1,2}\hat W_{jk} \left( 2 d_j^\dagger d_k^{}-\delta_{jk} \right), \quad g_0\equiv \frac{t_0^2}{E_C}.\label{Heff}$$ We here use the operators $$\label{Wopdef} \hat W_{jk}= \sum_{1\le\mu<\nu\le M} \left( \hat \lambda_{j\nu} \hat \lambda_{k\mu}^\dagger - \hat \lambda_{j\mu}\hat \lambda_{k\nu}^\dagger \right)\gamma_\mu\gamma_\nu.$$ Equation describes cotunneling paths through the box, where the energy of the intermediate virtual state has been approximated by $E_C$, cf. Eq. , and photon emission and absorption processes are encoded by the $\hat \lambda$ factors in Eq. . For even QD occupation number $N_{\rm d}$, Eq.  reduces to $$\label{heffeven} H_{\rm cot}^{(N_{\rm d}=0,2)}=g_0 \ {\rm sgn}(N_{\rm d}-1) \ \sum_j \hat W_{jj}.$$ For $N_{\rm d}=1$, using the notation $$\begin{aligned} \nonumber \hat W_+&\equiv& \hat W_{12}, \quad \hat W_-=\hat W_+^\dagger,\quad \hat W_x=\hat W_++\hat W_-,\\ \label{wdef1} \hat W_y&=&i(\hat W_+ -\hat W_-),\quad \hat W_z=\hat W_{11}-\hat W_{22},\end{aligned}$$ we find that Eq.  can instead be expressed in the form $$\label{cot1} H_{\rm cot}^{(N_{\rm d}=1)} = g_0 \sum_{a=x,y,z} \hat W_a \tau_a,$$ with the QD Pauli operators $\tau_a$ in Eq. . We emphasize that like the $\hat W_{jk}$ operators in Eq. , also the $\hat W_a$ still contain the phase fluctuation operators due to the electromagnetic environment. In order to realize the most general qubit-qubit exchange coupling between the QD spin $\{\tau_a\}$ and the $M=4$ Majorana box spin $(X,Y,Z)$ in the cotunneling regime, one has to specify nine independent (tunable) real-valued coupling constants. For the $M=4$ case in Fig. \[fig1\], taking into account gauge invariance — which allows us to set one of the $\lambda_{j\nu}$ to a real value —, the five different complex-valued hopping parameters $\lambda_{j \nu}$ are sufficient. On top of that, the direct tunnel amplitude between the QDs is assumed to be real-valued after setting $t_{12} = 0$ in Eq. . To simplify the subsequent analysis, we assume that the dominant contribution to the environmental electromagnetic noise comes from the long wavelength part. In effect, such contributions will cause dephasing of the QDs, e.g., due to the presence of a backgate electrode. This assumption is also consistent with the picture of a single bath. To good accuracy, the couplings $g_{j\nu,m}$ in Eq.  then do not depend on the Majorana index $\nu$, i.e., $g_{j\nu,m}=g_{j,m}$. As a consequence, also the fluctuating phases become $\nu$-independent, $\theta_{j\nu}=\theta_j$. In that case, the diagonal entries $\hat W_{jj}$ are insensitive to electromagnetic noise and the bath completely decouples for even $N_{\rm d}$, see Eq. . From now on, we therefore focus on the case of a single electron shared by the QDs, $N_{\rm d}=1$. Defining the phase operator $$\theta\equiv \theta_1-\theta_2 = \sum_m (g_{1,m}-g_{2,m}) \left(b_m^{}+b_m^\dagger\right),$$ Eq.  then yields $$\label{cotfinal} H_{\rm cot}= 2g_0 \left(e^{i\theta} W_+\tau_+ + {\rm h.c.}\right)+ g_0 W_z \tau_z.$$ The operators $W_+$ and $W_z$ correspond to ‘undressed’ ($\theta_{j\nu}\to 0$) versions of $\hat W_+$ and $\hat W_z$, respectively. These operators act only on the Hilbert space sector describing Majorana states. Comparing to Eq. , we have $$\label{wdef2} W_{jk} = \sum_{\mu<\nu}^M \left( \lambda_{j\nu} \lambda_{k\mu}^\ast - \lambda_{j\mu} \lambda_{k\nu}^\ast \right)\gamma_\mu\gamma_\nu.$$ For the device in Fig. \[fig1\], the $W_{jk}$ operators can be expressed in terms of the Pauli operators $(X,Y,Z)$ in Eq. , see below. ### Bath correlation functions The equilibrium density matrix of the thermal environment is given by $$\label{envtrace} \rho_{\rm env}=Z^{-1}_{\rm env} e^{-H_{\rm env}/T}\quad {\rm with} \quad Z_{\rm env}={\rm tr}_{\rm env} \ e^{-H_{\rm env}/T},$$ with ‘tr$_{\rm env}$’ denoting a trace operation over the environmental bosons. Using $\braket{\hat O}_{\rm env}\equiv {\rm tr}_{\rm env} (\hat O\rho_{\rm env})$, we define the correlation function [@Weiss2007] $$\begin{aligned} \nonumber && J_{\rm env}(t)=\braket{[\theta(t)-\theta(0)] \theta(0)}_{\rm env} = \int_0^\infty \frac{d\omega}{\pi} \frac{{\cal J}(\omega)}{\omega^2}\times \\ &&\quad \times \left\{ [\cos(\omega t)-1] \coth\left(\frac{\omega}{2T}\right) - i \sin(\omega t) \right\},\label{BathCorr}\end{aligned}$$ with the spectral density $$\label{spectraldensity} {\cal J}(\omega)=\pi\sum_m (g_{1,m}-g_{2,m})^2 E_m^2 \delta(\omega-E_m).$$ Switching to the continuum limit in bath frequency space, we focus on the practically most important Ohmic case with ${\cal J}(\omega)\propto \omega$ in the low-frequency limit. In concrete calculations, we use the model spectral density [@Weiss2007] $$\label{Ohmic} {\cal J}(\omega) = \alpha \omega e^{-\omega/\omega_c},$$ where $\alpha$ is a dimensionless system-bath coupling and frequencies above $\omega_c$ are exponentially suppressed. For a related discussion in the context of Majorana qubits, see Ref. [@Munk2019]. The parameter $\alpha$ is related to the environmental impedance $Z(\omega)$ [@Devoret1990], $$\label{alphadef} \alpha= \frac{e^2}{2h} {\rm Re} Z(\omega=0).$$ We consider the case $\alpha<1$ below. For the subsequent discussion, we rewrite $H_{\rm cot}$ in normal-ordered form relative to the phase fluctuations, $$\label{normalorder1} H_{\rm cot} = H_{\rm cot}^{(0)} + V,$$ where $H_{\rm cot}^{(0)}$ is the expectation value of $H_{\rm cot}$ with respect to phase fluctuations and $V$ represents the coupling of the combined QD-MBS system to phase fluctuations. Since $\langle\theta^2\rangle_{\rm env}$ diverges in the Ohmic case, we have $\langle e^{i\theta}\rangle_{\rm env}=0$, resulting in $$\label{normalorder2} H_{\rm cot}^{(0)} \equiv \braket{H_{\rm cot}}_{\rm env}= g_0 W_z\tau_z.$$ The interaction term in Eq.  is then given by $$\label{normalorder3} V= 2g_0 \left( e^{ i\theta} W_+ \tau_+ + {\rm h.c.}\right) .$$ By construction, $\braket{V}_{\rm env}=0$. Correlation functions of exponentiated phase fluctuations are given by ($s=\pm 1$) $$\label{xicor} \braket{e^{is\theta (t)} e^{-is\theta(0)}}_{\rm env} = e^{J_{\rm env}(t)}$$ with $J_{\rm env}(t)$ in Eq. . ### Interaction picture and Rotating Wave Approximation From now on, we shall switch to the interaction picture with respect to $H_{\rm d}+H_{\rm env}$. The Hamiltonian then takes the form, see Eqs.  and , $$\begin{aligned} \label{Hfull} H_{{\rm eff},I}(t) &=& H_{0,I}(t)+ V_I(t),\\ \nonumber H_{0,I}(t) &=& H_{{\rm drive},I}(t) + H^{(0)}_{{\rm cot},I}(t).\end{aligned}$$ For simplicity, we drop the ‘$I$’ index (for interaction picture) in what follows and focus on resonant drive conditions, $$\label{omegadef} \omega_0=\epsilon_2-\epsilon_1.$$ In the regime $\omega_0\gg T$ considered below, see Eq. , we can then apply the rotating wave approximation (RWA) [@Gardiner2004]. As a consequence, $H_{\rm drive}(t)\to \tilde H_{\rm drive}$ with $$\label{Hdriv2} \tilde H_{\rm drive}=A\left( d_1^{\dagger} d^{}_2+ d_2^{\dagger} d^{}_1\right) = A \tau_x,$$ resulting in a time-independent drive Hamiltonian in the interaction picture. If the drive frequency is slightly detuned, $\omega_0=\epsilon_2-\epsilon_1+ \delta\omega_0$, a residual time dependence remains, $H_{\rm drive}(t)= e^{-i\delta\omega_0 t} A d_1^{\dagger} d^{}_2 +$ h.c., after applying the RWA. However, we find that the final Lindblad equation for the dynamics of the Majorana sector in Sec. \[sec2c\] is not affected to leading order in $\delta\omega_0$. A small mismatch in the resonance condition will therefore not obstruct our findings. We then put $\delta\omega_0=0$ from now on. Master equation {#sec2b} --------------- In this subsection, we consider the time evolution of the reduced density matrix, $\rho(t)$, describing the coupled system defined by the MBSs and the pair of QD fermions. After tracing over the environmental bosons, we arrive at a Lindblad master equation for the dynamics of $\rho(t)$. In Sec. \[sec2c\], we will subsequently trace over the QD fermions to obtain a Lindblad equation for the Majorana sector only. With $\omega_0=\epsilon_2-\epsilon_1$ and $g_0=t_0^2/E_C$, we consider the regime $$\label{basiccond} g_0\ll T\ll \omega_0, \quad A\alt g_0.$$ In particular, $T\ll \omega_0$ is needed to justify the RWA, while $g_0\ll T$ is required for the Born-Markov approximation. In addition, the regime $g_0\ll T$ allows us to neglect emission and absorption processes taking place only in the Majorana sector since the bath is then unable to resolve such transitions. Of course, we will account for boson-assisted inter-QD transitions resulting from cotunneling processes. Finally, Eq.  states that we study a weakly driven system with drive amplitude $A\alt g_0$. The opposite strongly driven case is briefly discussed in App. \[appA\] and will be studied in detail elsewhere. We note that inelastic corrections to the drive Hamiltonian due to electromagnetic phase fluctuations, see Eq. , can be neglected by the secular approximation, cf. Sec. II.B of Ref. [@Shavit2019]. ### Lindblad master equation for $\rho(t)$ The master equation governing the dynamics of the density matrix $\rho(t)$ for the combined system (QDs and Majorana sector) is obtained by following the standard derivation of Born-Markov master equations [@Weiss2007; @Breuer2006; @Gardiner2004]. We assume a factorized initial (time $t=0$) density matrix of the total system, $\rho_{\rm tot}(0)=\rho(0) \otimes \rho_{\rm env}$, with $\rho_{\rm env}$ in Eq. . Starting from the von-Neumann equation for $\rho_{\rm tot}(t)$ subject to $H_{\rm eff}(t)$ in Eq. , we trace over the environmental modes and apply the Born-Markov approximation [@Weiss2007; @Breuer2006; @Gardiner2004]. As a result, $\rho(t)$ obeys the master equation $$\begin{aligned} \label{UnwantedTerms} &&\partial_t\rho(t)=-i\left[ H_{0}(t),\rho(t)\right] \\ \nonumber && -\, {\rm tr}_{\rm env} \int_0^{\infty} d\tau\left[ V(t-\tau),\left[ V(t)+H_0(t),\rho(t)\otimes\rho_{\rm env}\right]\right],\end{aligned}$$ where we have used that, by construction, ${\rm tr}_{\rm env}\left[ V(t),\rho(0)\otimes\rho_{\rm env}\right]=0$. Similarly, the mixed term involving $V(t-\tau)$ and $H_{0}(t)$ vanishes identically. We are then left with the coherent evolution term due to $H_{0}(t)$, and the double commutator containing two $V$ terms. Unfolding the double commutator, we arrive at a master equation of Lindblad [@Lindblad1976; @Lindblad1983] type, $$\label{GeneralLindblad} \partial_t\rho(t) = -i\left[ H_{\rm L},\rho(t)\right]+ \sum_{\pm} \Gamma_\pm \mathcal{L}[J_\pm]\rho(t) .$$ The subscript ‘L’ in $H_{\rm L}$ is meant to clarify that this Hamiltonian appears in a Lindblad equation. The dissipator ${\cal L}$ acts as superoperator on $\rho$ [@Breuer2006], $$\label{Dissipator} \mathcal{L}[J]\rho=J\rho J^\dagger -\frac{1}{2}\lbrace J^{\dagger} J,\rho \rbrace.$$ The two *jump operators* in Eq.  are given by $$\label{jumpops} J_\pm = 2W_\pm \tau_\pm = J_\mp^\dagger,$$ with the corresponding dissipative transition rates, $$\label{dissrate} \Gamma_{\pm}= 2g_0^2\, {\rm Re} \Lambda_\pm .$$ Here, we define the quantities $$\label{lambdadef} \Lambda_\pm= \int_0^\infty dt \, e^{\pm i\omega_0 t} e^{J_{\rm env}(t)},$$ with the bath correlation function . Their imaginary parts give Lamb shift parameters, $$\label{shifts2} h_\pm = g_0^2 \, {\rm Im}\Lambda_\pm ,$$ which appear in the Hamiltonian governing the coherent time evolution in Eq. , $$\label{Hq} H_{\rm L} = A \tau_x + g_0 W_z\tau_z + \sum_{\pm} h_\pm J_{\pm}^{\dagger} J_\pm^{}.$$ The first two terms in $H_{\rm L}$ originate from $H_0$ in Eq. , while the third term contains the Lamb shifts . Next we observe that Eq.  implies the general relation $$\label{detbal} J_{\rm env}\left(-t-i/T\right)=J_{\rm env}(t)$$ in the complex-time plane. Using Eq.  in Eq.  then results in a detailed balance relation, $\Lambda_-=e^{-\omega_0/T} \Lambda_+$. As a consequence, for arbitrary parameters, we find $$\Gamma_-=e^{-\omega_0/T}\Gamma_+, \quad h_-=e^{-\omega_0/T} h_+.$$ In particular, for $T\ll \omega_0$, the dissipative rate $\Gamma_-$ associated with the jump operator $J_-$ will be exponentially suppressed against the rate $\Gamma_+$. The dissipative part of the Lindblad equation is therefore completely dominated by the jump operator $J_+$. It is a distinguishing feature of our DD platform that jump operators can be directly implemented by designing *unidirectional* inelastic cotunneling paths connecting pairs of QDs via the box, with the overall energy scale $g_0$. The QDs are also directly coupled by a driven tunnel link $w(t)$, see Eq. , with overall energy scale $A$. For $T\ll \omega_0$, as far as inter-dot transitions via the box are concerned, only photon emission processes are relevant. As a consequence, only transitions from the energetically high-lying QD 2 to QD 1 may take place, corresponding to the jump operator $J_+\propto \tau_+$, see Eqs.  and . Such transitions act on the Majorana state according to the operator $W_+$. As we show below, this operator can be engineered at will by adjusting the tunneling parameters $\lambda_{j\nu}$, which in turn is possible by changing suitable gate voltages. The driving field pumps the dot electron in the opposite direction, i.e., from QD $1\to 2$, and for a small pumping rate, $A\alt g_0$, we obtain a steady state circulation $1\to 2\to 1$ by alternating pumping and cotunneling processes. On the other hand, for $A>g_0$, pumping processes will dominate and the cotunneling channel is effectively suppressed, see App. \[appA\]. To facilitate analytical progress, we consider the case $\omega_0\ll \omega_c$. (Otherwise Eq.  can be solved numerically in a straightforward manner.) One then finds [@Weiss2007] $$J_{\rm env}(t) \simeq -2\alpha \ln\left( \frac{\omega_c}{\pi T} \sinh(\pi T t)\right) - i\pi\alpha \,{\rm sgn}(t),$$ and with the Gamma function $\Gamma(z)$, we arrive at $$\begin{aligned} \label{rate2} \Gamma_+ &\simeq& \Gamma(1-2\alpha)\sin(2\pi\alpha) \left(\frac{\omega_0}{\omega_c}\right)^{2\alpha} \frac{2 g_0^2 }{\omega_0} ,\\ \nonumber h_+ &\simeq& \frac12 \cot(2\pi \alpha) \Gamma_+.\end{aligned}$$ For the device in Fig. \[fig1\], using the Pauli operators , the jump operators $J_\pm^{}=J_\mp^\dagger$ follow from Eq.  in the general form $$\begin{aligned} \nonumber J_{+}&=& \tilde J_+ \tau_+ , \\ \tilde J_+ &=& 2ie^{i\beta_2}|\lambda_{23}|\left(e^{-i\beta_3}|\lambda_{11}|X-e^{-i\beta_1}|\lambda_{12}|Y\right) \nonumber \\ &-& 2i\left[e^{-i\beta_1}|\lambda_{12}\lambda_{21}|-e^{i\beta_4}|\lambda_{11}\lambda_{22}|\right]Z, \label{jumpN}\end{aligned}$$ where the phases $\beta_{1,2,3,4}$ are indicated in Fig. \[fig1\]. They are connected to the phases $\chi_{j\nu}$ in the tunneling parameters, $\lambda_{j\nu}=|\lambda_{j\nu}|e^{-i\chi_{j\nu}}$, by the relations $$\label{betadef} \beta_1 = \chi_{12}, \quad \beta_2 = \chi_{23},\quad \beta_3= \chi_{11}, \quad \beta_4=\chi_{22},$$ with the gauge choice $\chi_{21}=0$. In particular, $\beta_1-\beta_3$ ($\beta_2$) is the loop phase accumulated along the shortest closed tunneling trajectory involving only QD 1 (QD 2), cf. Eq. . These phases, as well as the absolute values $|\lambda_{j\nu}|$, can be experimentally varied, e.g., by changing the voltages on nearby gates. We emphasize that $\tilde J_+$ is fully determined by selecting the state design parameters $\lambda_{j\nu}$. The Hamiltonian $H_{\rm L}$ then follows as $$\begin{aligned} H_{\rm L}&=& A\tau_x + 2g_0 \tilde J_z \tau_z + \sum_\pm h_\pm J_\pm^\dagger J_\pm^{},\nonumber \\ \label{jzdef} \tilde J_z &=& \frac12\bar\lambda^2+\sin\beta_2|\lambda_{21}\lambda_{23}| X+\\ \nonumber &+& \sin\left(\beta_4-\beta_2\right)|\lambda_{22}\lambda_{23}|Y+\\ \nonumber &+&\left[\sin\beta_4|\lambda_{21}\lambda_{22}|-\sin\left(\beta_1-\beta_3\right)|\lambda_{11}\lambda_{12}|\right]Z ,\end{aligned}$$ where $\bar \lambda^2\equiv |\lambda_{11}|^2+|\lambda_{12}|^2+|\lambda_{21}|^2+|\lambda_{22}|^2+|\lambda_{23}|^2$. It is worth mentioning that the operators $\tilde J_\pm$ and $\tilde J_z$ act only on the Majorana subsector. To illustrate the above general expressions, let us consider a simple example. We take stabilization parameters subject to the conditions $$\begin{aligned} \label{parameterchoice1} |\lambda_{11}| &=& |\lambda_{12}|, \quad \lambda_{22}=0, \\ \nonumber \beta_1 &=&- \beta_2 =\pi/2, \quad \beta_3 = \beta_4 =0.\end{aligned}$$ Using Eq. , the dominant jump operator contributing to the Lindbladian is then given by $$J_+=2|\lambda_{11}| \left(2|\lambda_{23}|\sigma_+ + |\lambda_{21}|Z\right)\tau_+, \label{NumericJumpOperator}$$ where $\sigma_\pm=(X\pm iY)/2$. For $|\lambda_{23}|\gg |\lambda_{21}|$, the Lindbladian will then automatically drive an arbitrary Majorana state $\rho_{\rm M}$ towards $|0\rangle\langle 0|$, with the $Z$-eigenstate $|0\rangle$ to eigenvalue $+1$, i.e., $Z|0\rangle=|0\rangle$. Here, the reduced density matrix $\rho_{\rm M}(t)$ describes the Majorana sector only, see Sec. \[sec2c\]. However, the operator $\tilde J_z$ appearing in the Hamiltonian $H_{\rm L}$ still contains a small $X$ component, see Eq. , which could potentially disrupt the action of the dissipator. Nonetheless, we find below that for small $|\lambda_{21}|$, the desired state $|0\rangle$ is approached with high fidelity, regardless of the initial system state $\rho(0)$. An optimized parameter choice for stabilizing $|0\rangle$ will be discussed in Sec. \[sec3\]. ### Numerical results {#specialsec} ![ Driven dissipative dynamics for the setup in Fig. \[fig1\], illustrating the time-dependent expectation values of the Pauli operators $\tau_{x,y,z}$ describing the QDs, see Eq. . We also show the purity, $P_s(t)$, of the system state, see Eq. . All results were obtained by numerical integration of the Lindblad equation for the density matrix $\rho$ describing the QDs and the Majorana sector, with $H_{\rm L}$ in Eq. . We used the parameters in Eq. , with $T/g_0=4$, $\omega_0/g_0=40$, $\omega_c/g_0=200$, $A/g_0=0.1$, $\alpha=1/4$, $|\lambda_{11}|=|\lambda_{12}|=|\lambda_{23}|= 1$, and $|\lambda_{21}|=0.1$. Fast transient oscillations in $\langle\tau_a(t)\rangle$ are not resolved on the shown time scale, corresponding to shaded regions. The respective dynamics in the Majorana sector is depicted in Fig. \[fig3\]. []{data-label="fig2"}](f2){width="\columnwidth"} ![Time evolution of the Bloch vector, $(\langle X\rangle,\langle Y\rangle,\langle Z\rangle)(t)$, describing the Majorana state $\rho_{\rm M}(t)$ for the same parameters as in Fig. \[fig2\]. The expectation value is computed by numerically integrating the Lindblad equation. Starting from the initial $X$-eigenstate $|+\rangle$, the DD protocol stabilizes the dark state $|0\rangle$ at long times, corresponding to the north pole of the Bloch sphere. The intermediate states (with alternating colors) were obtained at times $g_0t\in\lbrace 5\times 10^{3}, 10\times 10^3,\ldots,15\times10^4\rbrace$. \[fig3\]](f3){width="0.7\columnwidth"} We next turn to a numerical integration of Eq.  using the approach of Refs. [@Johansson2012; @Johansson2013]. Numerical results for the above parameters are shown in Figs. \[fig2\] and \[fig3\]. While the goal of the DD protocol is to stabilize a selected state in the Majorana sector, it is useful to also study the dynamics in the QD sector, see Fig. \[fig2\]. We start from a pure initial state, $\rho(0)=|\Psi(0)\rangle\langle\Psi(0)|$, with $|\Psi(0)\rangle= |+\rangle \otimes |0\rangle_{\rm d}$, where the $\tau_z=+1$ QD eigenstate, $|0\rangle_{\rm d}$, describes an electron located in the energetically lower QD 1, with QD 2 left empty, see Eq. . The initial Majorana state has been chosen as the $X$-eigenstate $|+\rangle$ with eigenvalue $+1$. However, we have checked that the same long-time limit of $\rho(t)$ is reached for other initial states. We define the purity of the system state as $$\label{puritydef} P_s(t)= {\rm tr} \rho^2(t).$$ The upper left panel of Fig. \[fig2\] shows that the purity approaches a value close to the largest possible value ($P_s=1$) at long times. Moreover, as observed from Fig. \[fig3\], the DD protocol steers the Majorana state towards the pure state $|0\rangle$, i.e., towards the north pole of the corresponding Bloch sphere. For the shown example, the QD state $\rho_{\rm d}$ has most probability weight in the energetically lower QD 1. Indeed, Fig. \[fig2\] shows that at long times, the electron shared by the two QDs will predominantly relax to QD 1, corresponding to the state $|0\rangle_{\rm d}$. Nonetheless, it is of crucial importance that the occupation probability $p$ for encountering the electron in the energetically higher QD 2 (corresponding to the state $|1\rangle_{\rm d}$) remains finite at long times. We find $p\approx 0.001$ for the parameters in Fig. \[fig2\]. We conclude that the system state factorizes at long times, $\rho(t)\simeq \rho_{\rm M}\otimes \rho_{\rm d}$ with $\rho_{\rm M}=|0\rangle \langle 0|$. The approach of the Majorana state towards $|0\rangle$ takes place on a time scale given by the inverse of the dissipative gap of the reduced Lindbladian describing the Majorana sector only, see Sec. \[sec3\] below. The relaxation time scales for the QD subsystem can be longer, cp. Figs. \[fig2\] and \[fig3\]. Finally, we remark that for the special case $\lambda_{21}=0$, the electron shared by the two QDs will *not* predominantly relax to the energetically lower QD $1$. One here has only two cotunneling paths between both QDs, namely the constituents forming the operator $4|\lambda_{11}\lambda_{23}|\sigma_+$ in Eq. . Both paths interfere destructively once the Majorana island is stabilized in the state $\ket{0}$. An arbitrarily weak drive can then overcome all dissipative effects in the long-time limit. In contrast to what happens for $\lambda_{21}\ne 0$, the QDs will thus realize an equal-weight mixture of $|0\rangle_{\rm d}$ and $|1\rangle_{\rm d}$. Nonetheless, the reduced Lindblad equation below still applies, with $p\to 1/2$ and $p_\perp\to 0$ in Eq. . We note that those parameters are also appropriate in the strongly driven case, cf. App. \[appA\]. Lindblad equation for the Majorana sector {#sec2c} ----------------------------------------- The above observations allow us to derive a reduced Lindblad equation, which directly describes the dynamics of $\rho_{\rm M}(t)$ in the Majorana sector alone. To that end, we now trace also over the QD subspace. At long times, our numerical simulations generically show that $\rho(t)$ factorizes into a Majorana part, $\rho_{\rm M}(t)$, and a QD contribution, $\rho_{\rm d}(t)$, $$\label{factorize} \rho(t \to \infty) \simeq \rho_{\rm M}(t) \otimes \rho_{\rm d}(t).$$ For tracing over the QD part, we can effectively use a time-independent *Ansatz*, $$\label{ssform} \rho_{\rm d}=\left( \begin{array}{cc} 1-p & p_\perp \\ p^\ast_\perp & p \end{array}\right),$$ written in the basis $\{ |0\rangle_{\rm d},|1\rangle_{\rm d}\}$ selected by the coupling to the QDs. Here, $p\ne 0$ refers to the occupation probability of the energetically higher QD 2. This probability can be determined by numerically solving Eq. , cf. Sec. \[sec2b\], or it may be treated as phenomenological parameter. A simple estimate predicts $p\approx {\rm max}(A,g_0)/\omega_0$. Noting that a small but finite expectation value $\langle\tau_x\rangle\ne 0$ is observed in Fig. \[fig2\] at long times, we have also included an off-diagonal term $(p_\perp)$ in Eq. . Inserting Eq.  into Eq.  and tracing over the QD subsystem, we arrive at a Lindblad equation for the $2\times 2$ density matrix $\rho_{\rm M}(t)$ only, $$\label{LindbladMBQ} \partial_t\rho_{\rm M}(t) = -i[\tilde H_{\rm L},\rho_{\rm M} ]+ \sum_{s=\pm} \tilde\Gamma_s \mathcal{L}[\tilde J_s]\rho_{\rm M}(t),$$ where the jump operators $\tilde J_\pm$ have been defined in Eq. . The dissipative transition rates $\tilde \Gamma_\pm$ in Eq.  are given by $$\label{mod1} \tilde \Gamma_+ = p \Gamma_+,\quad \tilde \Gamma_- = (1-p)\Gamma_-,$$ cf. Eqs.  and . The coherent time evolution in Eq.  is governed by the Hamiltonian $$\label{mod22} \tilde H_{\rm L} = 2(1-2p)g_0 \tilde J_z + \sum_\pm \tilde h_\pm \tilde J_\pm^\dagger \tilde J_\pm^{},$$ where $\tilde J_z$ has been specified in Eq.  and the Lamb shifts $\tilde h_\pm$ are given by $$\label{mod2} \tilde h_+ = p h_+ ,\quad \tilde h_- = (1-p) h_-.$$ The drive amplitude $A$ then appears only implicitly through the dependence $p=p(A)$. We note that within the RWA, no contributions $\propto p_\perp$ appear in Eq. . Indeed, the RWA allows one to neglect terms $\propto \tau_+\rho\tau_+$ which stem from $p_\perp\ne 0$. Importantly, apart from the initial transient behavior, all of our numerical results for the Majorana dynamics obtained from the full Lindblad equation for the combined QD-MBS system, Eq. , are quantitatively reproduced by using the simpler Lindblad equation . This statement is valid for arbitrary model parameters subject to Eqs.  and . We emphasize that the integration over the QD degrees of freedom as carried out above relies on the facts that (i) the convergence towards the target state is dictated by the Majorana sector, and that (ii) the QD and MBS degrees of freedom always decouple in the long-time limit, see Eq. . The latter feature has been established by extensive numerical simulations of Eq. . The reduced Lindblad equation is applicable as long as transient behaviors are not of interest. In particular, when studying, e.g., the dynamics of $\rho_{\rm M}(t)$ in the presence of time-dependent QD level energies $\epsilon_j(t)$, Eq.  should only be used for very slow (adiabatic) time dependences. For rapidly varying QD level energies, one has go back to the full Lindblad equation for the combined QD-MBS system in Eq. . Dark state stabilization {#sec3} ========================= Using the Lindblad master equation and the Choi isomorphism [@Albert2014] summarized in App. \[appB\], we now turn to a detailed analysis of our stabilization protocols for the single-box device in Fig. \[fig1\]. The parameter values for stabilizing a specific dark state can be determined by solving the zero-eigenvalue condition of the Lindbladian, cf. App. \[appB\]. We recall that the key state design parameters of our DD protocol are given by the complex-valued tunneling amplitude parameters $\lambda_{j\nu}$, which also define the phases $\beta_j$ in Fig. \[fig1\]. In Sec. \[sec3a\], we show how to stabilize Pauli operator eigenstates. In Sec. \[sec3b\], we discuss magic state stabilization protocols, followed by a study of temperature effects in Sec. \[sec3c\]. We show in Sec. \[sec3d\] that in certain cases, a dark state can be stabilized even in the absence of any driving field. Finally, we conclude in Sec. \[sec3e\] with several remarks. Pauli operator eigenstates {#sec3a} -------------------------- ![Dark-state stabilization protocols for Pauli operator eigenstates. Left side panels (blue curves): Stabilization of $|0\rangle$. Right side panels (red curves): Stabilization of $|+\rangle$, where $X|+\rangle=|+\rangle$. In both cases, the Majorana island has initially been prepared in the $Y$-eigenstate with eigenvalue $+1$. We use the parameters in Eq.  with $p=1/2$, all other parameters are as in Fig. \[fig2\]. With $E_C=1$ meV and $g_0/E_C=2.5\times 10^{-3}$, the time units follow as shown. As explained in the main text, for the chosen parameter set, Rabi oscillations are absent. []{data-label="fig4"}](f4){width="\columnwidth"} We start by discussing DD protocols targeting Pauli operator eigenstates. Typical numerical results obtained by solving Eq.  are illustrated in Fig. \[fig4\]. Following the method in App. \[appB\], the $Z=\pm 1$ eigenstates can be realized by choosing $$\label{sigmazcond} |\lambda_{11}|=|\lambda_{12}|,\quad \lambda_{21}=\lambda_{22}=0, \quad \beta_1-\beta_3=\pm\pi/2,$$ with arbitrary $\lambda_{23}$ and $\beta_{2,4}$, see Eq. . (We note that for $\lambda_{23}=0$, the phases $\beta_{2,4}$ are not defined.) At this point, we use the concept of a *dissipative map* $\hat E$ [@Breuer2006], which is defined in terms of a jump operator mapping the system onto a specific state when acting inside the Lindblad dissipator. For example, the dissipative maps targeting the $Z=\pm 1$ eigenstates are $$\hat E_{\pm}=\sigma_\pm=(X\pm iY)/2.$$ For the stabilization parameters in Eq. , the jump operator $\tilde J_+\propto \hat E_\pm$, with the $\pm$ sign determined by Eq. , completely dominates the Lindbladian part of Eq.  at low temperatures, $T\ll\omega_0$. The dissipative dynamics then maps every input state to $|0\rangle$ (for the $+$ sign) or $|1\rangle$ (for the $-$ sign). At the same time, the Hamiltonian evolution in Eq.  comes from $\tilde H_{\rm L}\propto Z$, see Eq. . Evidently, this Hamiltonian commutes with the targeted state $\rho_{\rm M}(\infty)$, and therefore does not affect the dynamics towards the steady state generated by the dissipative map $\hat E_\pm$. The Majorana state $\rho_{\rm M}(t)$ is thus automatically steered towards the corresponding $Z$-eigenstate by the Lindbladian, with no obstruction from the Hamiltonian dynamics. For the above protocol, the *dissipative gap* is given by, cf. App. \[appB\], $$\label{deltaz} \Delta_z = |4\lambda_{11}\lambda_{23}|^2\sum_{s=\pm}\tilde\Gamma_{s}.$$ In general terms, the dissipative gap is defined as the real part of the smallest non-vanishing eigenvalue of the Lindbladian (the dark state itself has eigenvalue zero) [@Breuer2006]. The time scale on which the dark state will be approached is therefore given by $\Delta_z^{-1}$. Moreover, the approach of the Bloch vector towards the dark state $|0\rangle$ is in general accompanied by damped oscillations in the $(X,Y)$ components, where $\Delta_z$ is the damping rate and the Rabi frequency follows from Eq.  as $$\label{rabiz} \Omega_z \simeq \left|2g_0(1-2p)|\lambda_{11}|^2-8|\lambda_{11}\lambda_{23}|^2\tilde h_+\right|.$$ For the special case $\lambda_{21}=0$ with $p=1/2$, cf. Sec. \[sec2b\], and noting that $\tilde h_+=0$ for $\alpha=1/4$, cf. Eq. , we obtain $\Omega_z=0$ in Eq. . The left panels in Fig. \[fig4\] therefore exhibit only damping in the $(X,Y)$ components, without Rabi oscillations. Next, $X=\pm 1$ eigenstates are realized by choosing $$|\lambda_{21}|=|\lambda_{23}|,\quad \lambda_{11}=\lambda_{22}=0,\quad \beta_2=\mp\pi/2,$$ with the dissipative gap $\Delta_x=|4\lambda_{12}\lambda_{21}|^2\sum_s\tilde\Gamma_s.$ As shown in the right panels of Fig. \[fig4\], $X$-eigenstates, e.g., the state $|+\rangle$ for eigenvalue $+1$, can be stabilized using the setup in Fig. \[fig1\]. As for the $Z$-stabilization shown in the left panels, there are no Rabi oscillations for this parameter set. Finally, for stabilizing the $Y$-eigenstates with eigenvalue $\pm 1$, one requires $$|\lambda_{22}|=|\lambda_{23}|, \quad \lambda_{12}= \lambda_{21}=0,\quad \beta_2-\beta_3-\beta_4=\pm\pi/2,$$ with the dissipative gap $\Delta_y = |4\lambda_{11}\lambda_{22}|^2\sum_s\tilde\Gamma_s$. In all these examples, the target axis (say, $\hat e_z$ for $Z$-eigenstates) is controlled by selecting appropriate tunneling amplitude parameters $\lambda_{j\nu}$. Two links are switched off, and two are matched in amplitude such that the desired jump operator $\tilde J_+$ is implemented. For $T\ll \omega_0$, dissipative transitions are fully governed by this jump operator which is due to inelastic cotunneling transitions from QD $2\to 1$. Under these conditions, we find that $\tilde H_{\rm L}$ commutes with the Pauli operator $\hat\sigma$ corresponding to the target axis (e.g., $\hat\sigma=Z$ for $Z$-states). Finally, by adjusting the phases $\beta_j$, one can select the stabilized state, say, $|0\rangle$ or $|1\rangle$. It is a remarkable feature of our Majorana-based DD setup that the Hamiltonian $\tilde H_{\rm L}$ can be engineered to only generate $\hat\sigma$. As a consequence, the Lindbladian dissipator already drives the system to the desired dark state. Magic states {#sec3b} ------------ ![ Fidelity for a stabilization protocol targeting the magic state $|m\rangle$. Here the Majorana state follows by numerical integration of Eq.  using the parameters in Eq.  with $|\lambda_{23}|=1$. Other parameters are $E_C = 1$ meV, $g_0/E_C=2.5\times 10^{-3}, T/g_0=4$, $\omega_0/g_0=40$, $\omega_c/g_0 =200$, $\alpha=1/4$, and $p = 0.01$. Main panel: Time dependence of the fidelity for ideal parameters \[Eq. \] (red curve), with a mismatch of order $10\%$ in all state design parameters \[$|\lambda_{11}|=-0.1+1/\sqrt2,|\lambda_{21}|=+0.1+1/\sqrt{2},|\lambda_{12}|=|\lambda_{22}|=0.9,\beta_3=-\beta_2=11 \pi/20$\] (blue), and a mismatch of order $20\%$ in the same parameters (orange). Inset: Steady-state fidelity vs deviation $\Delta\beta_2$ with otherwise ideal parameters, where $\beta_2=-\frac{\pi}{2}(1+\Delta\beta_2)$. \[fig5\]](f5){width="\columnwidth"} In order to highlight the power of our DD stabilization protocols, we next consider the magic state [@Nielsen] $$\label{magicstate} |m\rangle = e^{-i\frac{\pi}{8}Y} |0\rangle.$$ The practical importance of this state comes from the fact that a large number of ancilla qubits approximately prepared in the state $|m\rangle$ are needed for the magic state distillation protocol. The latter is an essential ingredient for implementing the $T$-gate required for universal surface code quantum computation [@Fowler2012; @Vijay2015; @Landau2016; @Plugge2016; @Nielsen]. Targeting $|m\rangle$, the stabilization conditions now involve all tunnel links in Fig. \[fig1\] and are given by $$\begin{aligned} \label{magiccond} |\lambda_{12}|&=&|\lambda_{23}|, \quad |\lambda_{21}|=|\lambda_{11}|=|\lambda_{23}|/\sqrt2,\\ \nonumber \lambda_{22}&=&0,\quad \beta_3=\beta_1+\beta_2, \quad \beta_2 = -\pi/2.\end{aligned}$$ We here define the *fidelity* of the state $\rho_{\rm M}(t)$ with respect to a specific pure state, $\rho_{\rm M}^{(0)}=|\psi\rangle\langle \psi|$, as $$\label{fidelity} F(t)={\rm tr}\left[ |\psi\rangle\langle \psi|\rho_{\rm M}(t)\right].$$ We show numerical results for the magic state fidelity with $|\psi\rangle=|m\rangle$ in Fig. \[fig5\], using the parameters in Eq. . We find $F=1$ at long times for the ideal parameter choice in Eq. . Figure \[fig5\] also illustrates the long-time fidelity when allowing for small deviations from Eq.  which are inevitable in practical implementations. Remarkably, even for sizeable deviations from the ideal parameter set, the fidelity remains close to unity. By determining the spectrum of the Lindbladian, we obtain the dissipative gap as $$\label{dissgapm} \Delta_m = |4\lambda_{11}\lambda_{23}|^2\sum_s\tilde \Gamma_s.$$ Using the parameters in Fig. \[fig5\], we find $\Delta_m^{-1}\simeq 80$ ns. Even though our magic state stabilization protocol requires more parameter fine tuning than the stabilization of $|0\rangle$, the dark state $|m\rangle$ is reached on essentially the same time scale. Effect of temperature {#sec3c} --------------------- ![ Steady-state fidelity, $F(\infty)$, and purity, $P(\infty)$, vs temperature (in Kelvin) for the state $|0\rangle$ and for the magic state $|m\rangle$. We use ideal state design parameters, see Eqs.  and , with all other parameters as in Figs. \[fig4\] and \[fig5\], respectively. The numerical results for both states cannot be distinguished for these parameter choices on the shown scales. The frequency $\omega_0$ corresponds to a temperature of $\approx 2.5$ K. []{data-label="fig6"}](f6){width="0.95\columnwidth"} We next address the effect of raising temperature within the conditions set by Eq. , in particular $T\ll \omega_0$. Figure \[fig6\] shows numerical results for the $T$-dependent steady state fidelity $F(\infty)$ with respect to the states $|0\rangle$ and $|m\rangle$, choosing ideal parameters as in Eqs.  and , respectively. At very low temperatures, the fidelity stays very close to the ideal value ($F=1$) since here only the rate $\tilde \Gamma_+$, see Eqs.  and , is significant. In this limit, corrections to $F=1$ are exponentially small and appear to be governed by the dissipative gap, $1-F\propto \exp(-\Delta_{z/m}/T)$. The same scaling behavior also applies to the purity. As temperature increases, the thermal excitation rate $\tilde\Gamma_-=e^{-\omega_0/T}\tilde \Gamma_+$ cannot be neglected anymore. Focusing on the stabilization of the state $|0\rangle$, we have $\tilde J_-\propto \sigma_-$. The Lindblad dissipator $\tilde \Gamma_- {\cal L}[\tilde J_-]$ will then target the ‘wrong’ $Z$-eigenstate $|1\rangle$. The competition between ${\cal L}[\tilde J_+]$ and ${\cal L}[\tilde J_-]$ implies that the fidelity will deteriorate as temperature increases. This expectation is confirmed by our numerical results. For the parameters in Fig. \[fig6\], the fidelity noticeably drops once $T$ exceeds the crossover temperature $T_c\approx 250$ mK. Figure \[fig6\] also shows the temperature dependent purity of the steady state, $P(\infty) ={\rm tr}\rho_{\rm M}^2(t\to \infty)$. For $T\ll T_c$, we find $P(\infty) \simeq 1$. As $T$ increases, however, the maximally mixed state $\rho_{\rm M}(\infty)=\frac12 \mathbb{1}$ with $F(\infty)= P(\infty)=1/2$ is approached, and consequently the purity also becomes smaller. Stabilization without driving field {#sec3d} ----------------------------------- In certain cases, it is possible to stabilize dark states even without drive Hamiltonian, $H_{\rm drive}=0$. In this subsection, we demonstrate the feasibility of this idea for special choices of the state design parameters. We are not aware of other DD systems allowing for dark states in the absence of driving. In our setup, we will see that the dissipative dynamics can also generate terms that mimic the effects of a weak driving field. To be specific, we apply the Lindblad equation to setups where $\lambda_{j\nu}\ne 0$ only for $(j \nu) \in \{ 11,12,23 \}$. In particular, since $\lambda_{21}=0$, this case corresponds to the special parameter regime discussed in Sec. \[specialsec\]. For simplicity, below we drop the exponentially small contribution to the dissipator due to $\tilde J_-$. From Eq. , the only relevant jump operator is then given by $$\label{jplus1} \tilde J_{+} = 2i \lambda_{23}^\ast \left( \lambda_{11} X - \lambda_{12} Y \right).$$ In addition, we keep Lamb shift effects implicit. In particular, they can be taken into account by renormalizing $B_z$ in Eq.  below. The operator $\tilde J_z$ entering $\tilde H_{\rm L}$, see Eqs.  and , has the form $$\label{jz1} \tilde J_z =-\sin\beta_1\left| \lambda_{11} \lambda_{12} \right| \, Z .$$ We now study the undriven ($A=0$) scenario for two parameter sets allowing for analytical progress. The stabilization of pure dark states may then be possible because the Hamiltonian $\tilde H_L$ can effectively take over the role of the drive. The frequency $\omega_0$ now simply represents the (positive) energy difference $\epsilon_2-\epsilon_1$, see Eq. , instead of a drive frequency. Moreover, we assume $p_\perp = 0$ while the probability $p$ in Eq.  is estimated by $p\approx g_0/\omega_0$. We note in passing a finite static contribution to the inter-QD tunnel coupling, $t_{12}\ne 0$ in Eq. , can be taken into account here. This coupling will modify $p$ according to $p \approx{\rm max}(t_{12}, g_0)/\omega_0$. We also recall that for $A\ne 0$, one instead finds $p=1/2$ since we have $\lambda_{21}=0$, cf. Sec. \[specialsec\]. ### Case 1: $\lambda_{11}=\pm i \lambda_{12}$ {#case-1-lambda_11pm-i-lambda_12 .unnumbered} The first case is defined by $\lambda_{11}=is \lambda_{12}$, with $s=\pm 1$. We observe that the dot fermion operator $d_1$ corresponding to QD 1 is then tunnel-coupled to a nonlocal fermion formed from the Majorana operators, $c=(\gamma_1-is \gamma_2)/2$. With $\sigma_\pm=(X\pm iY)/2$, Eqs.  and simplify to $$\label{case1} \tilde J_+=4i\lambda_{23}^\ast \lambda_{11}\sigma_{-s}, \quad\tilde J_z=-s |\lambda_{11}|^2 \, Z.$$ The Lindblad equation is then given by $$\label{nodrive:LBMcase1} \partial_t \rho_{\rm M}(t) = - i [ \tilde H_{\rm L}, \rho_{\rm M}(t) ] + \Gamma_1 {\cal L} \left[ \sigma_{-s} \right] \rho_{\rm M}(t),$$ where the Hamiltonian follows from Eq.  as $$\label{HM1} \tilde H_{\rm L} = -2s (1-2p) g_0 |\lambda_{11}|^2 Z=s B_z Z.$$ We note that the Lamb shift $\tilde h_+$ can be taken into account by redefining $B_z$. Furthermore, the rate $\Gamma_1$ in Eq.  is proportional to $\tilde \Gamma_+$ in Eq. . The only zero eigenstate of the Lindbladian is the $Z$-eigenstate $|0\rangle$ (for $s=-1$) \[or $|1\rangle$ (for $s=+1$)\], e.g., ${\cal L} \left[ \sigma_+ \right] |0 \rangle \langle 0 | = 0$. The same $Z$-eigenstate is also the lowest energy eigenstate of $\tilde H_{\rm L}$ in Eq. . Using the $Z$-eigenstate basis $\{ |0\rangle , |1\rangle \}$ for $s=-1$ \[and $\{|1\rangle,|0\rangle \}$ for $s=+1$\], we can parametrize the time-dependent density matrix $\rho_{\rm M}(t)$ solving Eq.  with real-valued $x(t)$ subject to $0\le x\le 1$ and complex-valued $y(t)$ as $$\rho_{\rm M}(t) = \left( \begin{array}{cc} 1 - x(t) & y(t) \\ y^\ast(t) &x(t)\end{array}\right).$$ The quantities $x(t)$ and $y(t)$ represent the diagonal and off-diagonal density matrix deviations, respectively, from the steady-state density matrix corresponding to the stabilized $Z$-eigenstate. Using Eq. , these deviations obey the equations of motion $$\label{relax1} \partial_t x = - \Gamma_1 x,\quad \partial_t y = - 2 i B_z y - \frac{\Gamma_1}{2} y,$$ which explicitly shows the relaxation and decoherence dynamics of $\rho_{\rm M}(t)$ towards the stabilized pure state. The above example demonstrates that the dissipative stabilization of a dark state can be achieved even in the absence of a driving field in our Majorana box setup. ### Case 2: $\beta_1=0$ {#case-2-beta_10 .unnumbered} Putting the phase $\beta_1$ to zero, $d_1$ is effectively coupled to a single Majorana operator, $\gamma_{\rm eff}=\gamma_1 \cos \delta + \gamma_2\sin \delta$, with $\delta= \tan^{-1} \left| \lambda_{12} / \lambda_{11} \right|$. One then obtains $\tilde J_z=0$. The jump operator $\tilde J_+$ is now given by $$\label{case2} \tilde J_+= B_{\perp}\sigma_+ e^{i\delta}+{\rm h.c.},\quad B_\perp= 2i \lambda_{23}^\ast \lambda_{11}/|\cos\delta|.$$ Noting that the Lamb shifts in $\tilde H_{\rm L}$ only give an irrelevant constant, we arrive at the Lindblad equation $$\label{nodrive:LBMcase2} \partial_t \rho_{\rm M}(t) = \frac{\Gamma_2}{4} {\cal L} \left[ \sigma_{\bf n} \right] \rho_{\rm M}(t),$$ where we define $$\sigma_{\bf n} = {\bf n \,\cdot} {\bm \sigma} = \sigma_+ e^{i \delta} + \sigma_-e^{-i\delta},$$ with the unit vector ${\bf n} = (\cos \delta, - \sin \delta, 0)$. Again, the rate $\Gamma_2$ is proportional to the respective rate $\tilde \Gamma_+$ in Eq. . For the case in Eq. , the Lindbladian has two zero eigenstates, ${\cal L} \left[ \sigma_{\bf n} \right] | s \rangle \langle s | = {\cal L} \left[ \sigma_{\bf n} \right] | a \rangle \langle a | = 0$, corresponding to the eigenstates of ${\bf \sigma}_{\bf n}$, i.e., $\sigma_{\bf n}| s \rangle = | s \rangle$ and $\sigma_{\bf n}| a \rangle =- | a \rangle$. Using the $X$-eigenstates $|\pm\rangle$, one finds $$| s / a \rangle =\frac{1}{ \sqrt{2}} \left( e^{i \delta} |+\rangle \pm e^{-i \delta} |-\rangle \right).$$ In the $\{ |s\rangle,|a\rangle\}$ basis, $\rho_{\rm M}(t)$ can be parametrized as $$\rho_{\rm M}(t) = \left( \begin{array}{cc} \frac12 + x(t) & y(t) \\ y^\ast(t) & \frac12-x(t) \end{array} \right),$$ where the real-valued parameter $x(t)$ has to satisfy $|x|\le 1/2$. Equation  then yields $$\partial_t x = 0,\quad \partial_t y = - \frac{\Gamma_2}{2} y.$$ Clearly, there is no relaxation in the basis selected by the environment via the QDs, i.e., $x(t)$ remains constant. Only the off-diagonal elements of the density matrix are subject to decay with the rate $\Gamma_2/2.$ One can therefore prepare an arbitrary mixed state as steady state. Discussion {#sec3e} ---------- We conclude this section with several additional points. ### Mixed states As pointed out in Sec. \[sec3d\], one can also use our protocols for stabilizing mixed states, see also Ref. [@Kumar2020]. To give another example, now for $A\ne 0$, we consider changing the above phase conditions such that a mixture of Pauli eigenstates can be prepared as dark state. For instance, by choosing the state design parameters as in Eq.  but keeping $\bar\beta=\beta_1-\beta_3$ arbitrary, one obtains the dark state $$\rho_{\rm M}(\infty)=\frac{1+\sin\bar\beta}{2}|0\rangle\langle 0|+\frac{1-\sin\bar\beta}{2}|1\rangle\langle 1|.$$ The relative weight of the two components can then be altered by adjusting the phase difference $\bar\beta$. ### Majorana overlaps So far we have assumed that the overlap between different MBSs is negligibly small. What are the effects of a finite (but small) hybridization between different MBS pairs on the above stabilization protocols? Such terms could arise, e.g., due to the finite nanowire length [@Alicea2012]. They are described by a Hamiltonian term $ H'=\sum_{\nu<\nu'}i\epsilon_{\nu\nu'}\gamma_\nu\gamma_{\nu'}$, with hybridization energies $\epsilon_{\nu\nu'}$. By construction, such a term survives the RWA and the Schrieffer-Wolff projection in Sec. \[sec2\] and thus contributes to the Hamiltonian $\tilde H_{\rm L}$ in the Lindblad equation without affecting the Lindbladian dissipator. In the Pauli operator language, such terms act like a weak magnetic Zeeman field. If the corresponding field is parallel to the target axis of the dark state, it does not cause any dephasing. For instance, for the stabilization of the $Z$-eigenstate $|0\rangle$, the hybridization parameters $\epsilon_{12}$ and $\epsilon_{34}$ can be tolerated as they only couple to the Pauli operator $Z$ in Eq. . Clearly, such couplings have no detrimental effects on our stabilization protocols. ### Readout dynamics For reading out a stabilized dark state, it is possible to use the same techniques suggested previously for the native Majorana qubit [@Plugge2017; @Karzig2017; @Munk2019]. In particular, one can perform capacitance spectroscopy using additional single-level QDs that are tunnel-coupled to MBS pairs. These QDs are used for measurements only, where the spectroscopic signal contains an interference term which depends on the respective Pauli matrix in Eq. . This projective readout yields the Pauli eigenvalue $\pm 1$ with a state-dependent probability [@Karzig2017]. Of course, this method can also be used to prepare the Majorana island in a Pauli eigenstate before the DD protocol is started. In order for the readout not to interfere with the DD stabilization protocol, one has to make sure that the characteristic projective measurement time scale (see Refs. [@Plugge2017; @Karzig2017] for detailed expressions) is much longer than the typical inelastic cotunneling time $\tilde\Gamma_+^{-1}$. Similarly, single-electron pumping protocols via a pair of QDs attached to different MBSs allow one to apply a Pauli operator to the tetron state in a topologically protected manner [@Plugge2017]. ### Beyond the horizon of a dark state {#sec3e4} So far we have discussed DD stabilization protocols targeting a desired dark state. The dark space dimension for those protocols is $D=1$, see App. \[appC\]. Since there is a unique dark state for a given choice of the state design parameters, one could utilize a DD single-box device as a self-correcting quantum memory. By means of adiabatic changes of the state design parameters, one can in principle steer the Majorana state on its Bloch sphere. However, for general state manipulation protocols, it is advantageous to have access to a dark space manifold with $D>1$, which may be engineered in systems with more than four MBSs. We address this case in the next section. Dark space engineering {#sec4} ======================= We continue with DD protocols targeting quantum states within a dark space manifold. A degenerate manifold of dark states may be engineered by employing a device with at least two Majorana boxes as depicted in Fig. \[fig7\]. After introducing our model and the corresponding Lindblad equation in Sec. \[sec4a\], we show in Sec. \[sec4b\] how a dark space can be created and classified. In Sec. \[sec4c\], we then describe how to stabilize Bell states. In Ref. [@ourprl], we describe external perturbations for moving the dark state to another state within the protected dark space manifold, and we show how to create a dark space manifold realizing a ‘dark Majorana qubit’. In such a system, topological and DD mechanisms reinforce each other and thereby can provide exceptionally high levels of fault tolerance. Moreover, we remark that the stabilization of Bell states can also be implemented in a hexon device (i.e., a Majorana box with six MBSs [@Karzig2017]), see Ref. [@GauThesis]. ![Schematic two-box layout for DD dark space stabilization and manipulation protocols, cp. Fig. \[fig1\] for the single-box case. The left (right) box harbors four MBSs described by $\gamma_{\nu}^L$ ($\gamma^R_\nu$). The tunneling bridge with amplitude $t_{LR}$ connects $\gamma_4^L$ and $\gamma_2^R$. QD 3 has independently driven tunneling bridges to QD 1 and to QD 2 (solid lines). The three QDs are operated in the single-electron regime, $N_{\rm d}=1$. The electromagnetic environment affects the phases of the tunnel links betweens QDs and MBSs (dashed lines). The phases $\beta_{j}$ for this geometry are also indicated.[]{data-label="fig7"}](f7){width="\columnwidth"} Lindblad equation for two coupled boxes {#sec4a} --------------------------------------- ### Model Following the discussion in Sec. \[sec2a\], we describe the two islands in Fig. \[fig7\] by $H_{\rm box}=H_{{\rm box},L}+H_{{\rm box},R}$, with $H_{{\rm box},L/R}$ as in Eq. . Here, the four MBSs on the left (right) box correspond to Majorana operators $\gamma_\nu^L$ ($\gamma_\nu^R$). Both islands are separately operated under Coulomb valley conditions. For notational simplicity, we assume that they have the same charging energy, $E_{C,L}=E_{C,R}=E_C$. Focusing on the long-wavelength components of the electromagnetic environment, we again work with a single bosonic bath, $H_{\rm env}=\sum_m E_m b_m^\dagger b_m^{}$, where photons couple to the QDs and MBSs via fluctuating phases, $\theta_j$, in the tunneling Hamiltonian, see Sec. \[sec2a\]. The setup in Fig. \[fig7\] requires up to three single-level QDs, $H_{\rm d}=\sum_{j=1}^3 \epsilon_j d_j^\dagger d_j^{}$, where QD 3 couples to both other QDs via independently driven tunnel links. We consider the regime $N_{\rm d}=1$, where on time scales $\delta t>1/E_C$, the three QDs share a single electron. Using the interaction picture with respect to the dot Hamiltonian $H_{\rm d}$, the full Hamiltonian is then given by $$\label{Hfull2MBQ} H(t) = H_{\rm box} + H_{\rm env}+ H_{LR} + H_{\rm drive}(t) + H_{\rm tun}(t),$$ where a phase-coherent tunnel link couples the boxes. Without loss of generality, we assume a real-valued tunneling amplitude $t_{LR}>0$, $$\label{LRCoup} H_{LR} = i t_{LR} \gamma_4^L \gamma_2^R.$$ The drive Hamiltonian now has the form $$\label{Hdrive2} H_{\rm drive}(t)=\sum_{j=1,2} 2A_j\cos\left(\omega_jt\right)e^{i\left(\epsilon_j-\epsilon_3\right)}d_j^{\dagger}d^{}_3 + {\rm h.c.},$$ where the two driving fields have the respective amplitude $A_{1,2}$ and frequency $\omega_{1,2}$. In analogy to Eq. , the QD-MBS tunnel links are described by $$\label{Htun2} H_{\rm tun}(t)=t_0 \sum_{j\nu, \kappa=L/R}\lambda_{j,\nu\kappa}^{} e^{-i\phi_\kappa} e^{i\theta_j} e^{i\epsilon_jt} d^\dagger_j \gamma_{\nu}^\kappa +{\rm h.c.},$$ with the phase operators $\phi_{L/R}$ for the left/right Majorana island. Using the same approximations as in Sec. \[spsec1\], the electromagnetic environment enters Eq.  through the fluctuating phases $\theta_j$. With the overall energy scale $t_0$, the complex-valued parameters $\lambda_{j,\nu\kappa}$ parametrize the transparency of the tunnel contact between $d_j$ and $\gamma_\nu^{\kappa=L/R}$. Similar to Eq. , the phases $\beta_j$ in Fig. \[fig7\] follow from the phases of these parameters. Since $\beta_4$ can be absorbed by a renormalization of $\beta_3$ for the purposes below, we put $\beta_4=0$. To simplify the presentation, we next assume that QDs 1 and 2 have the same energy level, $\epsilon_1=\epsilon_2$. Moreover, we consider the case of equal drive frequencies, $\omega_1=\omega_2\equiv \omega_0$, and identical drive amplitudes, $A_1=A_2\equiv A$, and again impose a resonance condition, $\omega_0=\epsilon_3-\epsilon_1$. However, in analogy to our discussion in Sec. \[sec2\], we expect that overly precise fine tuning with respect to those conditions is not necessary. We now proceed in analogy to Sec. \[sec2a\] with the construction of an effective low-energy model by means of a Schrieffer-Wolff transformation to the lowest-energy charge state in each box. We can then define Pauli operators $(X_\kappa,Y_\kappa,Z_\kappa)$ with $\kappa=L,R$ referring to the left and right box, respectively, using the Majorana representation in Eq. . In the present case, it is crucial to keep all terms up to third order in the expansion parameters when accounting for cotunneling trajectories connecting pairs of QDs, cf. Fig. \[fig7\]. (For the single-box case in Sec. \[sec2a\], it is sufficient to go to second order only.) The electromagnetic environment then enters the low-energy theory via the three phase differences $\theta_j-\theta_k$ with $j<k$. This fact implies that, in general, we have six different spectral densities ${\cal J}_{jk;j'k'}(\omega)$. We model these spectral densities by the Ohmic form in Eq. (\[Ohmic\]), with system-bath couplings $\alpha_{jk;j'k'}$. For simplicity, we employ an average value $\alpha$ for these couplings below. The bath is then described by a single spectral density ${\cal J}(\omega)$ again. Importantly, the physics is not changed in an essential manner by this approximation. In particular, no additional jump operators appear when allowing for different $\alpha_{jk;j'k'}$. ### Lindblad equation We consider again the weak driving regime with $T\ll \omega_0$. Under these conditions, proceeding along similar steps as in Sec. \[sec2b\], one obtains a Lindblad master equation for the density matrix, $\rho(t)$, describing both the Majorana sector and the QD degrees of freedom. In order to arrive at a Lindblad equation for the reduced density matrix, $\rho_{\rm M}(t)$, which refers only to the Majorana sector of both boxes, we next trace over the QD subsector, see Sec. \[sec2c\]. For the QD steady-state density matrix, $\rho_{\rm d}$, we use the *Ansatz* $$\label{ansatz2} \rho_{\rm d}= {\rm diag}\left(\frac{1-p}{2}, \frac{1-p}{2},p\right),$$ expressed in the basis $\{ |100\rangle,|010\rangle,|001\rangle \}$ with QD occupation states $|n_1,n_2,n_3\rangle$ for $N_{\rm d}=1$. Note that since we assumed $\epsilon_1=\epsilon_2$, the occupation probabilities of QDs 1 and 2 are equal. The occupation probability $0<p\ll 1$ refers to the energetically highest QD 3. Equation is consistent with our numerical analysis of the Lindblad equation for $\rho(t)$, where we again find a factorized density matrix at long times, $\rho(t)\simeq \rho_{\rm M}(t)\otimes \rho_d$. We note that the dark space turns out to be independent of the concrete value of $p$. Going through the corresponding steps in Sec. \[sec2c\], we arrive at a Lindblad equation for $\rho_{\rm M}(t)$, $$\label{LindbladTwoMBQ} \partial_t\rho_{\rm M}(t)=-i[\tilde H_{\rm L},\rho_{\rm M}(t)]+ \sum_{a=1}^6 \tilde \Gamma_a \mathcal{L}[K_a] \rho_{\rm M}(t).$$ The six jump operators are denoted by $K_a$, with the respective dissipative transition rates $\tilde\Gamma_a$. With $\lambda_{LR}\equiv t_{LR}/E_C \ll 1$, we obtain $$\begin{aligned} \nonumber K_1^{} &=& K_4^\dagger = ie^{i(\beta_3-\beta_1)}\frac{|\lambda_{1,1R}\lambda_{3,3R}|}{\lambda_{LR}} X_R \\ &&\qquad - \,e^{i\beta_3} |\lambda_{1,3L}\lambda_{3,3R}| Z_L Y_R,\nonumber\\ \label{jumpK} K_2^{} &=& K_5^\dagger =-ie^{i(\beta_3-\beta_2)}\frac{|\lambda_{2,4R}\lambda_{3,3R}|}{\lambda_{LR}}Z_R\\ &&\qquad + \, e^{i\beta_3}|\lambda_{2,2L}\lambda_{3,3R}| X_LY_R,\nonumber\\ K_3^{} &=& K_6^\dagger = i\frac{|\lambda_{1,3L}\lambda_{2,2L}|}{\lambda_{LR}}Y_L- ie^{i(\beta_2-\beta_1)} \frac{|\lambda_{1,1R}\lambda_{2,4R}|}{\lambda_{LR}} Y_R\nonumber\\ &&\qquad+ \,e^{-i\beta_1} |\lambda_{1,1R}\lambda_{2,2L}| \,X_LZ_R \nonumber \\ \nonumber && \qquad -\,e^{i\beta_2}|\lambda_{1,3L}\lambda_{2,4R}| \,Z_LX_R. \nonumber\end{aligned}$$ The coherent evolution in Eq.  is governed by the Hamiltonian $$\label{h2qdef} \tilde H_{\rm L} = 2p\tilde g_0 K_z + \sum_{a=1}^6 \tilde h_a K_a^\dagger K_a,$$ with the operator $$\label{kz} K_z^{} =\sin\beta_1|\lambda_{1,1R}\lambda_{1,3L}| Z_LZ_R+ \sin\beta_2|\lambda_{2,2L}\lambda_{2,4R}| X_L X_R.$$ We here used the energy scale $$\label{tildeg0} \tilde g_0= \lambda_{LR} g_0 = \frac{t_0^2 t_{LR}}{E_C^2},$$ which characterizes the relevant inelastic cotunneling processes in the double-box setup. The transition rates $\tilde\Gamma_a$ follow in the form $$\begin{aligned} \tilde\Gamma_1 &=& \tilde \Gamma_2 = 2p\tilde g_0^2 \,{\rm Re} \int_0^\infty dt e^{i\omega_0 t} e^{J_{\rm env}(t)},\nonumber\\ \tilde\Gamma_3 &=& \tilde \Gamma_6 =(1-p)\tilde g_0^2\, {\rm Re} \int_0^\infty dt e^{J_{\rm env}(t)} ,\label{trans22}\\ \tilde\Gamma_4 &=&\tilde\Gamma_5= \frac{(1-p)}{2p} e^{-\omega_0/T}\tilde \Gamma_1,\nonumber\end{aligned}$$ and the Lamb shifts $\tilde h_a$ are given by $$\begin{aligned} \tilde h_1 &=& \tilde h_2 = p\tilde g_0^2 \,{\rm Im} \int_0^\infty dt e^{i\omega_0 t} e^{J_{\rm env}(t)},\nonumber\\ \tilde h_3 &=& \tilde h_6 = \frac12 (1-p)\tilde g_0^2 \,{\rm Im} \int_0^\infty dt e^{J_{\rm env}(t)} ,\label{lamb22}\\ \tilde h_4 &=&\tilde h_5= \frac{(1-p)}{2p} e^{-\omega_0/T}\tilde h_1.\nonumber\end{aligned}$$ For $\omega_0\ll \omega_c$, we can then make further analytical progress. Explicit expressions for $\tilde \Gamma_{1,2}$ and $\tilde h_{1,2}$ follow by comparison with Eq. . In addition, we find $$\begin{aligned} \nonumber \tilde \Gamma_{3,6} &\simeq& (1-p) \frac{\cos(\pi \alpha)\Gamma(\alpha)\Gamma(1-2\alpha)}{2^{1-2\alpha} \Gamma(1-\alpha)} \left( \frac{\pi T}{\omega_c}\right)^{2\alpha-1} \frac{2g_0^2}{\omega_c} ,\\ \tilde h_{3,6}&=& -\frac{1}{2}\tan(\pi \alpha) \tilde \Gamma_{3,6}.\end{aligned}$$ By following the derivation of the reduced master equation , we observe that the operator $K_1$ ($K_2$) comes from unidirectional transitions transferring an electron from the energetically high-lying QD 3 to QD 1 (QD 2) via the double-box setup, collecting all possible cotunneling trajectories allowed by third-order perturbation theory. Likewise, the jump operator $K_4$ ($K_5$) describes the reversed process, with a cotunneling transition from QD 1 (QD 2) to QD 3. For $T\ll \omega_0$, the transition rates $\tilde \Gamma_{4,5}$ and Lamb shifts $\tilde h_{4,5}$ are exponentially suppressed, $\propto e^{-\omega_0/T}$, against the respective contributions from $K_{1,2}$. Moreover, the jump operators $K_3$ and $K_6$ in Eq.  describe cotunneling transitions between QDs 1 and 2. Since these QDs are not directly connected by a driven tunnel link and have the same energy, $\epsilon_1=\epsilon_2$, the corresponding rates and Lamb shifts coincide, $\tilde\Gamma_3=\tilde\Gamma_6$ and $\tilde h_3=\tilde h_6$. Importantly, for $1/2< \alpha < 1$, these quantities are reduced by a factor $(T/\omega_0)^{2\alpha-1}\ll 1$ against $\tilde\Gamma_{1,2}$ and $\tilde h_{1,2}$, respectively. In the remainder of this section, we shall study this parameter regime where the most important jump operators in Eq.  are given by $K_1$ and $K_2$. Nonetheless, we retain the other jump operators in our numerical analysis as well. Finally, we note that all terms without the factor $\lambda^{-1}_{LR}\gg 1$ in Eqs.  and stem from third-order processes. While one *a priori* expects that the corresponding dissipative terms in Eq.  are suppressed against second-order contributions, by careful tuning of the link transparencies $\lambda_{j,\nu\kappa}$, they can become of comparable magnitude. As a consequence, all relevant cotunneling paths will then have amplitudes corresponding to third-order processes. This means that for the present two-box setup, the energy scale $g_0=t_0^2/E_C$ appearing in Eq.  has to be replaced by $\tilde g_0$ in Eq. . The Lindblad equation describing the weak driving limit is therefore valid under the conditions $$\label{basiccond2} \tilde g_0\ll T\ll \omega_0,\quad A \alt \tilde g_0.$$ ### Dissipative maps Before entering our discussion of stabilization protocols for the layout in Fig. \[fig7\], it is convenient to introduce the dissipative maps [@Barreiro2011] $$\label{bellmap} \hat E_{1,\pm} = (\mathbb{1}\pm Z_L Z_R) X_R,\quad \hat E_{2,\pm} = (\mathbb{1}\pm X_L X_R) Z_R.$$ These maps can be used to target the four Bell states, $$\label{bellstates} |\psi_{\pm} \rangle =\frac{1}{\sqrt2}(|00\rangle\pm|11\rangle), \quad|\phi_{\pm}\rangle=\frac{1}{\sqrt2}(|01\rangle\pm|10\rangle),$$ which are eigenstates of both $Z_LZ_R=\pm 1$ and $X_LX_R=\pm 1$. We observe that $\hat E_{1,-}$ maps even-parity onto the respective odd-parity states, $\hat E_{1,-}|\psi_{\pm}\rangle=|\phi_{\pm}\rangle$, while odd-parity states do not evolve in time, $\hat E_{1,-}|\phi_{\pm}\rangle=0$. Under this dissipative map, the system will thus be driven into the degenerate odd-parity subsector spanned by the $|\phi_\pm\rangle$ states. Similarly, $\hat E_{2,-}$ can drive the system into the antisymmetric subsector spanned by $|\phi_-\rangle$ and $|\psi_-\rangle$. The key idea in our DD protocols below is to identify state design parameters such that the jump operators effectively realize the needed dissipative map(s) in Eq. . Recalling that a dissipative map breaks a number of conserved quantities (and therefore symmetries) in our system, see Refs. [@Albert2014; @Albert2016] and App. \[appC\], we here employ this insight to either stabilize a dark space, see Sec. \[sec4b\] and Ref. [@ourprl], or to target protected and maximally entangled two-qubit dark states, see Sec. \[sec4c\]. Stabilization of a dark space {#sec4b} ----------------------------- In this subsection, we briefly outline how one can stabilize a dark space in the setup of Fig. \[fig7\], see also Ref. [@ourprl]. For convenience, we decouple QD 2 from the system by using the parameter choice $$\label{decouple2} \lambda_{2,2L}=\lambda_{2,4R}=0,\quad \beta_2=0.$$ We note that this is not the only possible parameter set for constructing a dark space. As a consequence of Eq. , many of the jump operators in Eq.  vanish identically, $K_2=K_3=K_5=K_6=0$. The jump operator $K^{}_1=K_4^\dagger$ then yields the dissipative map $\hat E_{1,-}$ in Eq.  upon choosing $$\beta_1=-\pi, \quad \beta_3 = -\pi/2, \quad |\lambda_{1,1R}|=\lambda_{LR} |\lambda_{1,3L}|. \label{spacecond}$$ Noting that $\hat E_{1,-}= X_R-i Z_L Y_R$, see Eq. , we indeed arrive at $K_1\propto \hat E_{1,-}$ from Eq. . In addition, Eq.  shows that under the above conditions, $\tilde H_{\rm L}$ only generates terms $\propto Z_L Z_R$ which do not obstruct the dissipative dynamics. For $T\ll \omega_0$, we next observe that to exponential accuracy, $K_1$ is the only jump operator contributing to the Lindbladian in Eq.  for the parameters in Eqs.  and . The DD protocol therefore will stabilize the system in the odd-parity ($Z_L Z_R=-1$) Bell state manifold spanned by $\{ |\phi_+\rangle, |\phi_-\rangle \}$. We show in Ref. [@ourprl] that this degenerate manifold has the dark space dimension $D=4$, see also App. \[appC\], which is equivalent to a degenerate qubit space [@Albert2014]. It is possible to manipulate dark states within a dark space by following different strategies [@ourprl]. For instance, one can adiabatically switch on a perturbation that breaks at least one conservation law. An alternative possibility is to employ single-electron pumping protocols, in analogy to previous proposals for native Majorana qubits [@Plugge2017; @Karzig2017]. Stabilizing Bell states {#sec4c} ----------------------- We next turn to the stabilization of Bell states in the setup of Fig. \[fig7\], where the couplings between QD 2 and the Majorana islands are now assumed finite again. In that case, the jump operator $K_2$ in Eq.  does not vanish anymore. In the low temperature regime, the corresponding Lindbladian term in Eq.  contributes with the same transition rate, $\tilde\Gamma_2=\tilde\Gamma_1$, as for $K_1$, see Eq. . Importantly, $K_2$ breaks additional conservation laws and thereby allows one to engineer stabilization protocols targeting maximally entangled two-qubit states. We again study the regime $1/2< \alpha<1$, where the jump operators $K_{3,6}$ give only subleading contributions. Let us start with the Bell singlet state $|\phi_-\rangle$ in Eq. , where $Z_LZ_R=-1$ and $X_LX_R=-1$. By choosing the state design parameters as $$\begin{aligned} \label{bellcond} \beta_1 &=& -\pi,\quad \beta_2 = 0,\quad \beta_3 = -\pi/2,\\ |\lambda_{1,1R}|&=&\lambda_{LR}|\lambda_{1,3L}|,\quad |\lambda_{2,4R}|=\lambda_{LR}|\lambda_{2,2L}|, \nonumber\end{aligned}$$ we observe from Eq.  that $K_1\propto \hat E_{1,-}$ and $K_2\propto \hat E_{2,-}$ are directly expressed in terms of the corresponding dissipative maps, see Eq. . The Lindbladian will therefore drive the system to the dark state $|\phi_-\rangle$. The dark space dimension is thus given by $D=1$. ![Fidelity for stabilizing the Bell singlet state $|\phi_-\rangle$ in the setup of Fig. \[fig7\]. We show numerical results obtained from Eq.  with the parameters in Eq.  and $|\lambda_{1,3L}|=|\lambda_{2,2L}|=|\lambda_{3,3R}|=1$, using the initial state $\rho_{\rm M}(0)=|00\rangle\langle00|$. Other parameters are $E_C=1$ meV, $\tilde g_0/E_C=10^{-5}$, $T/\tilde g_0=2, \omega_0/\tilde g_0=2\times 10^3, \omega_c/\tilde g_0=10^4, \alpha=0.99$, and $p=0.01$. Main panel: Time dependence of $F(t)$ for ideal parameters \[Eq. \] (red curve), and for a mismatch of order $10\%$ in all state design parameters \[$|\lambda_{1,1R}|=1.1\lambda_{LR}|\lambda_{1,3L}|, \, |\lambda_{2,4R}|=0.9\lambda_{LR}|\lambda_{2,2L}|, \, \beta_1=-1.1\pi, \, \beta_3=-9\pi/20$\] (blue). Inset: Steady-state fidelity vs deviation $\Delta\beta_1$ from the ideal value, i.e., $\beta_1=-\pi(1+\Delta\beta_1)$, with otherwise ideal parameters.[]{data-label="fig8"}](f8){width="\columnwidth"} As is shown in Fig. \[fig8\], the numerical solution of Eq.  confirms this expectation. For the stabilization parameters in Eq. , the Bell singlet state is reached with nearly perfect fidelity when taking ideal parameter values. One can rationalize the almost perfect fidelity by noting that the coherent evolution due to $\tilde H_{\rm L}$, see Eq. , involves only the operators $Z_L Z_R$ and $X_L X_R$. As a consequence, the dynamics induced by the dissipative maps $K_{1,2}\propto \hat E_{1/2,-}$ will not be disturbed. Note that the parameters in Fig. \[fig8\] were chosen such that $\tilde\Gamma_1\gg\tilde\Gamma_3$ while staying in the regime specified in Eq. . Indeed, the observed small deviations from the ideal value $F=1$, see Fig. \[fig8\], can be traced back to the jump operators $K_3$ and $K_6$, which give nominally subleading but practically important contributions to the Lindblad equation. Figure \[fig8\] shows that the stabilization protocol is rather robust against deviations of state design parameters from their ideal values in Eq. , see Sec. \[sec3\]. Following the approach in App. \[appB\], we find that the dissipative gap for stabilizing $|\phi_-\rangle$ is given by $$\label{bellgap} \Delta_{\rm Bell}=|2\lambda_{3,3R}|^2\left(|\lambda_{1,3L}|^2+|\lambda_{2,2L}|^2\right)\sum_{a=1,2,4,5}\tilde\Gamma_a.$$ Due to the importance of third-order inelastic cotunneling processes, this dissipative gap is several orders of magnitude below the corresponding gaps in the single-box case, cf. Sec. \[sec3\]. For the parameters in Fig. \[fig8\], we obtain the time scale $\Delta_{\rm Bell}^{-1}\approx 0.3$ ms. The other Bell states in Eq.  can be targeted by changing the phases $\beta_j$ in Eq. . The jump operators $K_1$ and $K_2$ will then directly implement the desired dissipative maps, with the dissipative gap still given by Eq. . For stabilization of the Bell state $|\psi_+\rangle$ ($|\psi_-\rangle$), one has to put $\beta_1=0, \, \beta_2=\pi \, (\beta_2=0)$, and $\beta_3=\pi/2$. Similarly, $|\phi_+\rangle$ is stabilized for $\beta_1=-\pi, \beta_2 = \pi,$ and $\beta_3= -\pi/2$. We thus always have $\beta_3-\beta_1=\pi/2$, and the remaining two phases select the targeted Bell state. In particular, $\beta_1$ selects the parity of the target state while $\beta_2$ determines the symmetric vs antisymmetric state. Summary and prospects {#secConc} ===================== In this paper, we have described DD protocols in Majorana-based layouts for stabilizing as well as manipulating dark states and dark spaces. For devices with one or two Majorana boxes coupled to driven QDs and subject to electromagnetic noise, we have shown that in a wide parameter regime the dynamics in the Majorana sector is accurately described by Lindblad master equations. The underlying topological nature of the Majorana states significantly boosts the power of DD schemes in several directions. First, the role of uncontrolled environmental noise sources should be suppressed compared to topologically trivial realizations, which is a key advantage for high-dimensional dark space constructions. Second, the fact that Pauli operators describing native Majorana qubits correspond to products of Majorana operators (pertaining to spatially separated MBSs), see Eq. , allows for unique addressability options. Only through this feature, which is rooted in topology, it is possible to design the special unidirectional cotunneling paths which directly implement the jump operators appearing in the Lindblad equation. In the simplest single-box case, see Fig. \[fig1\], the basic pumping-cotunneling cycle involves (i) pumping the dot electron from QD 1 to the high-lying QD 2 by means of a weak driving field, and (ii) the back transfer of the electron from QD 2 to QD 1 by cotunneling through the box. In general, competing transfer mechanisms may also contribute to both steps, and the parameter regime has to be carefully adjusted to minimize their impact. Taking step (ii) as example, the drive Hamiltonian in Eq. , possibly together with photon emission processes, may provide such a competing rate. By choosing both a sufficiently small drive amplitude, $A<g_0$, and a very small direct tunnel coupling $t_{12}$ between both QDs, these competing rates can be systematically suppressed against the cotunneling rates through the box. We also note that in most cases of interest, the Lindbladian dissipator alone is responsible for driving the system into the desired dark state or dark space, i.e., the Hamiltonian appearing in coherent part of the Lindblad equation does not obstruct the dissipative dynamics. For a single-box architecture, we have shown how to stabilize arbitary pure dark states, i.e., states that are fault tolerant and stable on arbitrary time scales. For multiple-box devices, one can also stabilize dark spaces, i.e., manifolds of degenerate dark states, as well as protected two-qubit Bell states. In our accompanying short paper [@ourprl], we show that a two-box device allows one to implement a dark Majorana qubit, which in turn could serve as basic ingredient for dark space quantum computation schemes. Our stabilization and manipulation protocols can be implemented with available hardware elements once a working Majorana platform becomes available. The above concepts and ideas raise many interesting perspectives for future research. First, we expect that one can devise robust Majorana braiding protocols [@Alicea2012; @Leijnse2012; @Beenakker2013] that are stabilized by working within a dark space manifold. Second, for chains of many boxes, our DD stabilization protocols may allow for interesting quantum simulation applications, e.g., a realization of the topologically nontrivial ground state of spin ladders [@Ebisu2019] or of the Affleck-Kennedy-Lieb-Tasaki (AKLT) spin chain [@Kraus2008; @Affleck1987]. For clarifying the feasibility of such ideas, one needs to analyze the spectrum of the Lindbladian for DD multiple-box networks. We leave this endeavor to future work. We thank A. Altland, S. Diehl, and K. Snizhko for discussions. This project has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Grant No.  277101999, TRR 183 (project C01), under Germany’s Excellence Strategy - Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1 - 390534769, and under Grant No. EG 96/13-1. In addition, we acknowledge funding by the Israel Science Foundation. On the strong driving limit {#appA} =========================== We here briefly discuss the strong driving limit for the single-box device in Fig. \[fig1\], with total QD occupancy $N_{\rm d}=1$ and under resonant driving conditions, $\omega_0=\epsilon_2-\epsilon_1$. We consider the regime $$g_0 \ll T<A\ll \omega_0,$$ with otherwise identical conditions as in Sec. \[sec2\]. After imposing the RWA, the steady-state density matrix of the QDs is given by Eq.  with $p=1/2$ and $p_\perp=0$. Starting from the effective Hamiltonian $H_{\rm eff}(t)$ in Eq. , we then arrive at a Lindblad equation for the density matrix $\rho(t)$ describing the combined system of MBSs and QDs, $$\label{AAA} \partial_t \rho(t) = -i [ H_{\rm L}, \rho(t) ] + 2 g_0^2 \sum_{a = 1}^3 \sum_{s = \pm} {\rm Re}\,\Lambda_{a,s} \, {\cal L} [ J_{a,s} ] \rho(t),$$ with the effective Hamiltonian $$H_{\rm L} = A\tau_x + g_0^2 \sum_{a = 1}^3 \sum_{s = \pm} {\rm Im}\,\Lambda_{a,s} \, J_{a,s}^\dagger J_{a,s}^{}.$$ We here encounter *six* jump operators ($s=\pm$), $$\label{rwa:Jaq} J_{1,s} = \tilde J_s \tau_x,\quad J_{2,s} = J_{3,-s}^\dagger = \tilde J_s (\tau_z + i \tau_y )/2,$$ with the operators $\tilde J_\pm$ in Eq. . The dissipative transition rates as well as the Lamb shifts follow from $\Lambda_{1,\pm}\equiv \Lambda_\pm$, see Eq. , and $$\label{rwa:Lambdaq} \Lambda_{2/3,s} = \int_0^\infty dt \, e^{i s \omega_0 t \pm i A t} e^{J_{\rm env}(t)},$$ with the bath correlation function . Comparing to the weakly driven case in Sec. \[sec2b\], the strong driving field $A$ splits the two jump operators $J_s$ in Sec. \[sec2b\] into the six jump operators in Eq. . Tracing over the QD degrees of freedom, we arrive at a Lindblad equation for the density matrix $\rho_{\rm M}(t)$, cf. Sec. \[sec2c\], $$\label{rwa:LBM} \partial_t \rho_{\rm M}(t) = - i [ \tilde H_{\rm L}, \rho_{\rm M}(t) ] + \sum_{s = \pm} \tilde\Gamma_s \, {\cal L }[ \tilde J_s ] \rho_{\rm M}(t),$$ with $\tilde H_{\rm L} = {\rm Tr}_{\rm d} \left\{ \rho_{\rm d} H_{\rm L} \right\}$. In this expression, $\rho_{\rm d}$ follows from Eq.  with $p\to 1/2$ and $p_\perp \to 0$. Only the two jump operators $\tilde J_\pm$ appear in the reduced Lindblad equation , with the dissipative transition rates $$\label{rwa:Gammaq} \tilde\Gamma_s =2 g_0^2 \, {\rm Re}\left[\Lambda_{1,s} +\frac12 \left( \Lambda_{2,s} + \Lambda_{3,s} \right) \right].$$ Finally, we note that for $T>A\gg g_0$, the Lindblad equation holds with $p \to 1/2$. On the dissipative gap {#appB} ====================== An elegant way to study the spectrum of a general Lindbladian uses the so-called Choi isomorphism in order to map the $N\times N$ system density matrix, $\rho(t)$, to an $N^2\times 1$ vector, $|\rho(t)\rangle$, and the Liouvillian, $\mathcal{\hat L}$, to an $N^2\times N^2$ superoperator ${\bm L}$ [@Albert2014]. We here include the Hamiltonian part in $\mathcal{\hat L}$. Let us consider a general Lindblad master equation, cf. Eq. , $$\label{genB1} \partial_t\rho(t)=\mathcal{\hat L}\rho(t)=-i[ H,\rho(t)]+\sum_{a}\Gamma_{a}{\cal L} [ J_{a}] \rho(t),$$ with jump operators $J_a$ and the corresponding transition rates $\Gamma_a$. Using the isomorphism, we have the correspondence $J\rho J^\dagger \leftrightarrow ( J\otimes J^*)|\rho\rangle$, and Eq.  takes the equivalent form $\partial_t|\rho(t)\rangle={\bm L}|\rho(t)\rangle$ with $$\begin{aligned} \label{genB2} {\bm L}&=&-i\left(H\otimes \mathbb{1}- \mathbb{1}\otimes H^\ast\right)+ \sum_{a}\frac{\Gamma_{a}}{2} \times \\ \nonumber &\times& \left(2 J_{a}^{}\otimes J^{\ast}_{a}-\mathbb{1}\otimes \bigl(J_a^{\dagger} J_{a}^{}\bigr)^\ast -J_{a}^{\dagger} J_{a}\otimes\mathbb{1}\right).\end{aligned}$$ In this language, the steady state, $\rho_{\rm ss}$, follows as right eigenvector of ${\bm L}$ with eigenvalue zero, $${\bm L}\left|\rho_{\rm ss}\right\rangle=0. \label{Criterion}$$ Equation allows one to systematically search for stabilization conditions targeting a desired dark state. Moreover, the spectrum of the Lindbladian coincides with the eigenvalues of the superoperator $\bm L$. In particular, the number of zero eigenvalues defines the dark space dimension, $D$, and the dissipative gap equals the real part of the smallest non-zero eigenvalue [@Albert2014]. On conserved quantities {#appC} ======================= For an open quantum system described by a Lindbladian as in Eq. , where we assume that $\mathcal{\hat L}$ has no purely imaginary eigenvalues, it is known that all conserved quantities are linked to the basis states spanning the dark space [@Albert2014]. For a Lindbladian with $D$ conserved quantities $C_{\mu=1,\ldots,D}$, we have the commutation relations $$[H, C_\mu]=[J_a, C_\mu] = 0.$$ Using an orthonormal basis, $\lbrace M_\mu\rbrace_{\mu=1}^D$, to span the resulting $D$-dimensional dark space, the steady state can be written as $$\rho_{\rm ss} = \underset{t\to \infty}{\rm lim} e^{\mathcal{\hat L}t} \rho(0)=\sum_{\mu=1}^D c_\mu M_{\mu}, \label{generaldarkspace}$$ where $\rho(0)$ is the initial density matrix and the $c_\mu={\rm tr}[C_\mu^{\dagger} \rho(0)]$ are weights determining in which of the degenerate steady states the system ends up. As first illustration, let us consider the stabilization of the dark state $|0\rangle$ for a single-box device, cf. Sec. \[sec3a\] and Eq. . The jump operators are then given by $\tilde J_\pm\propto \sigma_\pm$. The only operator commuting with both $\tilde J_+$ and $\tilde J_-$ is the identity, $C_\mu = \mathbb{1}$, and thus the dark space dimension is $D=1$. For this example, we also have $H=\tilde H_{\rm L}\propto Z$, see Eq. . We conclude that $M_1=|0\rangle\langle 0|$ spans the corresponding space. As second example, we discuss the dark space stabilization for a two-box device in Sec. \[sec4b\]. Using the Lindblad equation and assuming that QD $2$ remains decoupled from the system, see Eq. , the four conserved quantities $C_{\mu}$ listed in Ref. [@ourprl] are readily identified. Given these quantities, a basis spanning the dark space can be constructed from Eq. . One may view the basis elements, $M_{\mu}$, as linearly independent ‘vectors’ with the orthogonality relation ${\rm tr}( M_{\mu}^{\dagger}M_{\nu}^{})=\delta_{\mu\nu}$. The existence of four conserved operators $C_{\mu}$ now implies that we have four basis vectors spanning the dark space, see Ref. [@ourprl] for explicit expressions. 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--- author: - 'E. Daddi' - 'F. Valentino' - 'R. M. Rich' - 'J. D. Neill' - 'M. Gronke' - 'D. O’Sullivan' - 'D. Elbaz' - 'F. Bournaud' - 'A. Finoguenov' - 'A. Marchal' - 'I. Delvecchio' - 'S. Jin' - 'D. Liu' - 'A. Calabro' - 'R. Coogan' - 'C. D’Eugenio' - 'R. Gobat' - 'B. S. Kalita' - 'P. Laursen' - 'D.C. Martin' - 'A. Puglisi' - 'E. Schinnerer' - 'V. Strazzullo' - 'T. Wang' date: 'Received / Accepted ' title: 'Three Lyman-$\alpha$ emitting filaments converging to a massive galaxy group at z=2.91: a case for cold gas infall' --- Introduction ============ A fundamental phenomenon required to explain the evolution of massive galaxies at high redshifts is the efficient accretion of cold gas streaming along filaments, surviving the shocks at the virial radii of their massive halos and delivering the required fuel to galaxies (Dekel et al. 2009; Kere[š]{} et al. 2005). This scenario is intimately connected to our current understanding of the star formation and growth of galaxies at [*cosmic noon*]{} $1<z<3$ (and earlier), whose key observational features might be summarized with two basic tenets: the existence of tight correlations between the stellar mass and star formation rates (SFRs) in galaxies (the so-called Main Sequence of star formation; Noeske et al 2007; Elbaz et al 2007; Daddi et al 2007; and many others works) and the systematic increase of gas fractions along with specific SFRs as a function of redshift for typical Main Sequence galaxies (Daddi et al 2008; 2010; Tacconi et al 2010; Magdis et al 2012; Genzel et al. 2015; plus many others). The finding that star forming galaxies at these redshifts are much more common than quiescent systems (e.g., Ilbert et al 2010), coupled to the tight Main Sequence correlations imply that star formation in galaxies occurs and is persistent over timescales much longer than their typical stellar doubling times and gas consumption timescales, which requires constant replenishment of their gas reservoirs (e.g., Lilly et al 2013). Cold accretion frameworks quite satisfactorily account for this observational evidence, as they predict that cold material, nearly ready to form stars, accretes at rates proportional to the hosting halo mass (Neinstein & Dekel 2008; Dekel et al. 2013), thus naturally resulting in Main-Sequence like behaviour (as recognised by theory even before observational confirmation, see e.g. Finlator et al. 2006). Also, accretion rates at fixed mass are predicted to evolve rapidly with redshift, with trends (scaling as $(1+z)^{\alpha}$ with $\alpha\approx2$–3) that correspond well to the evolving behaviour of the Main Sequence normalization (e.g., Sargent et al 2012) and gas fractions (Magdis et al 2012; Genzel et al 2015). Not everything is fully reconciled, as for example a tension between predicted versus observed star formation rates in typical galaxies at [*cosmic noon*]{} has persisted for over a decade (e.g., Daddi et al. 2007) but it is generally understood as due to limitations in the modelling of feedback and the subsequent implications for gas consumption and the baryon cycle (Somerville & Dave 2015). Now, despite more than a decade of effort, direct, convincing observational confirmation of the existence such cold accreting gas are still lacking, so that the theory is being necessarily questioned. On the observational side it appears that outflows are actually widespread in the circumgalactic gas around galaxies, with hardly any sign of inflows (Steidel et al. 2010). From the theoretical side, the latest generation, high resolution simulations now call into question whether streams can survive the interaction with the hot baryons in halos and remain stable (Nelson et al 2015; Mandelker et al 2019). Also, numerical simulations of cold streams have been questioned for not having the required resolution to capture the small scale gas physics (Cornuault et al. 2018), making it unclear whether predictions can be taken quantitatively. This uncertainty on the feeding of galaxy activity also limits our understanding of feedback processes (e.g., Gabor & Bournaud 2014; Dekel & Mandelker 2014) It is widely recognised that the most promising avenue to reveal these cold gas streams is through their collisionally excited  emission (Dijkstra & Loeb 2009; Goerdt et al. 2010; Rosdhal & Blaizot 2012), possibly enhanced by hydrodynamical instabilities (e.g., Mandelker et al 2020a). Much more difficult is to ascertain whether any observed extended  emission is due to collisions or rather to recombinations following photo-ionization from star formation and/or AGN activity. Even more fundamental is the difficulty to properly distinguish  emission as coming from outflowing or infalling gas, given that broadly they would give rise to similar instability-driven phenomenology (e.g., Cornuault et al. 2018). Giant  nebulae are now routinely discovered around QSOs at redshifts $2<z<4$ (e.g., Borisova et al 2016; Arrigoni Battaia et al 2019; Cai et al 2019; O’Sullivan et al 2020) with detections as high as $z\sim6.6$ (Farina et al. 2019) and could potentially provide large samples to statistically search for the role of infall. Filamentary structures, sometimes found in QSO  nebulae (Cantalupo et al. 2014; Hennawi et al. 2015), might be consistent with gas infall (e.g., Martin et al. 2015a; 2019). However, it is not easy to rule out alternative interpretations: outflows (Fiore et al. 2017; Guo et al 2020; Veilleux et al 2020) overshadow expected infall in luminous QSO hosting halos by orders of magnitudes for both energy and gas flows (see quantitative discussion later in this work). Also, the  emission is there certainly photoionized by the QSO hard UV emerging photons, making it prohibitive not only to gauge if any gravitational driven  is at all present in QSO nebulae but also if any infall is actually taking place. Filaments shining in  have been recently found also in the SSA22a-LAB1 protocluster environment (Umehata et al. 2020), but the relatively giant  nebula does not appear to be consistent with arising from infall (Herenz et al. 2020) and the filaments are situated at locations that are currently impossible to directly connect to individual dark matter (DM) halos. Both QSO filaments and the SSA22a-LAB1 filaments are remarkably extended over Mpc scales, much more than any putative hosting halo virial radius. These features led these studies to assert connections to the cosmic web. However, theory predicts that cold streams should have a detectable Lya surface brightness only within the virial radius of massive halos, where the hot gas can efficiently confine them and enhance their density (Dekel et al. 2009, Dijkstra & Loeb 2009). We conclude that the nature and origin of reported filaments detected up to date are therefore in doubt. A critical test for models would then be the search for cold accreting gas in distant and massive halos and in environments where the contrast with competing mechanism for gas flows and for powering the detectable Lya emission is maximal. The first requirement follows from the fact that the dark and baryonic matter accretion rates increase with both the halo mass and redshift (Neinstein & Dekel 2008, Dekel et al. 2009), and is not trivial to address when considering that massive halos become more rare in the distant Universe because of their hierarchical assembly. Moreover, the necessity to exclude alternative mechanisms suggests a move away from extreme sources such as QSOs, focusing on structures where the black hole and star formation activities proceed at a standard pace. Both lines of argument point to high redshift clusters or groups as ideal testbeds for comparing theory to observations and searching for the evidence of cold accreting gas, as already seminally suggested in Valentino et al. (2015; see their Fig. 17 and related discussion) and Overzier et al. (2016; see their Fig. 11 and related discussion). This is because such clusters/group would provide the opportunity to search for non-photoionized  in an environment where the role of outflows could be minimal and where filaments could be studied in connection to the halo they are streaming into, thus enabling quantitative comparison to cold accretion theory. This work presents one such plausible candidate. Following the serendipitous discovery (Valentino et al. 2016) of a giant  halo centered on the X-ray detected cluster CL 1449 at $z=1.99$, we have pursued this avenue and started systematic observations of several structures at $2<z<3.5$ with the Keck Cosmic Web Imager (KCWI), searching for redshifted . This is reversing the standard strategy of discovering  nebulae from blind (e.g., narrow-band) surveys and following them up to find that they are typically hosted in moderately dense environments, by starting with a systematic investigation of the prevalence of  emission inside massive halos at high redshifts. Among the great advantages of integral field spectroscopy, as now provided routinely by MUSE and KCWI, and as opposed to earlier narrow-band imaging attempts, is the potential to unveil the kinematics and spectral properties of the  emission that, keeping in mind the uncertainties linked to resonant scattering effects, can provide valuable diagnostics on the presence of accretion (see, e.g., Ao et al. 2020). As part of these efforts, we used KCWI to search for redshifted  in RO-1001, a massive group of galaxies at $z=2.91$ that currently is our best studied target with the deepest and widest observations, that we present in this work. Results for our full KCWI survey of distant structures will be presented elsewhere (E. Daddi., et al., in preparation). This paper is organized as follows: we present in Section 2 the observational characterization of the RO-1001 structure, starting from the observations of the giant  halo which motivates a detailed look into the overall properties of galaxies hosted thereby. In Sect. 3 we present the spectral properties of the  emission including moments (velocity and dispersion fields), interpret them with the aid of simplified resonant scattering modeling and multi-gaussian decomposition, and compare to cold accretion theory predictions informed by the estimated DM halo. In Sect. 4 we discuss the overall energetics and gas flows that characterise the system, particularly in comparison to  nebulae observed around QSO. Conclusions and summary are provided in Section 5. We adopt a standard cosmology and a Chabrier IMF. [{width="21cm"}]{} [{width="\textwidth"}]{} Observational characterization of RO-1001 ========================================== RO-1001 was selected in the COSMOS 2-square-degree field as a 12$\sigma$ overdensity of optically faint radio sources centered at RA 10:01:23.064 and DEC 2:20:04.86, following a recently proposed technique (Daddi et al. 2017): it was found to contain 3 VLA detections with $S_{\rm 3\,GHz}>30\,\mu$Jy and $z_{\rm phot}>2.5$ within a radius of 10$''$ (80 kpc; proper scales are used throughout the paper), i.e. the size of a massive halo core (Strazzullo et al. 2016; Wang et al. 2016). These three VLA galaxies are also bright in ALMA sub-mm imaging ($200\,\mu$m rest-frame), i.e. they are highly star-forming galaxies (not AGNs), as discussed in the following. NOEMA CO\[3-2\] line observations confirm their $z_{\rm spec}\sim2.91$. In this section we first present the observations and reduction of KCWI data, with the discovery of a  nebula in this structure. We then discuss the stellar mass of its member galaxies, and constrain the hosting halo mass. Finally, we present the exploration of their star formation and AGN content and the derivation of redshifts from CO observations. KCWI observations and analysis. ------------------------------- We observed RO-1001 with KCWI on January 16th 2018 for 1 h using the BM grism ($R=2000$ with the adopted large field of view), and on February 3rd and 4th 2019 for 3.5 h and 4 h, respectively, using the lower resolution BL grism ($R=900$), giving a total 8.5 h on-source when combining all observations. Conditions were excellent with dark sky and seeing typically in the range of 0.4–0.7$''$. We used integration times of 900 s in 2018 and of 1800 s in 2019, with dithering and large offsets to eventually cover a region corresponding to 2$\times$2 KCWI fields of view of $40''\times60''$ (i.e. $300\times500$ kpc$^2$). In fact, already with the first 1 h integration obtained in 2018 it was clear that the nebula extends well beyond the usable KCWI field of view of about $18''\times31''$ for the adopted configuration with large slices ($1.35''$). The low-resolution BL data covers the full 3500–5500Å range (corresponding to 900–1400 Å rest-frame at the redshift of RO-1001), while the BM observations cover a shorter range across the  emission. We used the standard KCWI pipeline [*Kderp*]{} (Morrissey et al. 2018; Neill et al. 2018) for the data reduction, including twilight flats to obtain accurate illumination corrections. We further used [*CubEx*]{} tools (Cantalupo et al. 2019) to refine the flat-fielding slice by slice, thus allowing us to improve the sky subtraction by removing a median sky value at each wavelength layer, after masking sources detected in the stacked (continuum) cubes. This step was first performed over the whole frame and subsequently iterated by masking regions where  emission had been detected, to avoid self-subtraction of the  signal. Further reduction and analysis steps were performed with the [*CWItools*]{} scripts (O’Sullivan et al. 2020; Martin et al. 2019). We estimate variance cubes from the original, non-resampled cubes and propagate the uncertainties through the combination to obtain a final variance cube. We combined the dithered and offset observations based on the astrometry of each frame that was derived by cross-correlating to B-band Suprime CAM imaging of the area publicly available from the COSMOS survey, and resampling into a final pixel scale of 0.29$''$, corresponding to the finer grid in the original slices. We then subtracted any continuum emission from objects in the combined cubes by fitting a 7th order polynomial as a function of wavelength at each spatial pixel, avoiding the wavelength range where  emission is present. From compact objects in the final cube we estimate an average image quality of 0.6$''$ (FWHM). The  nebula is very clearly detected (Fig. 1 and 2). We produced a low-resolution cube containing all 8.5 h observations, used for most of the analysis in this paper, and a higher spectral resolution cube using only 2018 BM data, used to obtain higher quality  spectra in the core (Sect. 3). We used adaptive smoothing (O’Sullivan et al. 2020; Martin et al. 2019) to recover the full extent of the nebula in the low-resolution cube, thresholding at the 3$\sigma$ level. We started by smoothing with a spatial kernel equal to the seeing and averaging three (1Å wide) spectral layers (roughly corresponding to the spectral resolution). This already selects 93% of the pixels eventually detected in the  nebula when allowing for larger spatial and spectral smoothing scales. The observations revealed faint low-surface brightness, large area filamentary structures that converge onto a bright  nebula at the center of the potential well of the group (Figs.1, 2), with a total $L_{Lya}=1.3\pm0.2\times10^{44}$ erg s$^{-1}$ (the uncertainty includes the effect of SNR thresholding on flux detection and accounting for possible absorption[^1] from $z=2.91$). Three filaments can be readily recognized from the surface brightness profile of the nebula (Figs.1, 2). The two most prominent, extending South-East and West of the nebula, respectively, appear to be traceable over a projected distance of 200 kpc from the core, at the current 3$\sigma$ surface brightness limit of $10^{-19}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$, reconstructed from adaptive smoothing. A third structure that we can identify with a filament extends towards the North-West (Figs. 1,2), likely affected by projection effects. [{width="9cm"}]{} [ {width="99.00000%"} ]{} Stellar masses and hosting halo mass. ------------------------------------- There are four massive ($M>10^{11}\,M_\odot$) galaxies within $13''$ ($\sim$100 kpc proper at $z=2.91$) of RO-1001, with an estimated (Muzzin et al. 2013; Laigle et al. 2016) photometric redshift in the range $2.5<z<3.5$, see Table 1. One source (D in Fig. 2) is blended with a close neighbor. We obtained a revised stellar mass estimate, empirically calibrated on sources at similar redshift from the COSMOS2015 catalog (Laigle et al. 2016), and based on J, H, K photometry from the DR4 UltraVISTA imaging and 3.6, 4.5$\mu$m Spitzer/IRAC imaging. We estimate a stellar mass completeness limit of log(M/M$_{\odot}$)=10.8. Three of the massive galaxies are shown to be at $z=2.91$ from CO\[3-2\] spectroscopy, as discussed later in this section, but source D remains unidentified as it is likely passive (Sect. 2.3). Given the similarity in the optical rest-frame colors and SED to other group members, and the negligible probability of such a red and massive galaxy to be there by chance (recall galaxy D was not detected in the radio), we will assume in the following that it also resides in the RO-1001 group. There are no further $z\sim3$ massive $\sim10^{11}M_\odot$ galaxies out to a distance of $1'$ from RO-1001. The total stellar mass of $\log(M/M_\odot)>10.8$ galaxies in the structure thus adds up to $5.4^{+2}_{-0.5} \times 10^{11}\,M_\odot$, where we assumed that the uncertainty on individual stellar mass estimates is at least a factor $\sim$50%, see e.g. Muzzin et al. 2013. We extrapolate a total stellar mass down to $10^{7}\,M_\odot$ of 1.0$^{+0.7}_{-0.2} \times10^{12}$M$_{\odot}$ assuming the stellar mass function of field galaxies, again from Muzzin et al. 2013, at $2.5<z<3$. Adopting the scaling between total stellar and halo mass derived from $z\sim1$ clusters with masses in the range 0.6–16 $\times 10^{14}\,M_\odot$ (van der Burg et al. 2014), would yield a halo mass of $M_{200} \sim 6 \times 10^{13}\,M_\odot$. We note that if scaling by the difference between the cluster and field galaxy stellar mass function at $z\sim 1$ as in van der Burg et al. 2013, we would obtain a lower estimate for the total stellar mass of $7.6^{+5}_{-1.5} \times10^{11}$M$_{\odot}$, and thus a lower estimate for the total halo mass of $M_{200}\sim4\times10^{13}\,M_\odot$. However, the environmental dependence of stellar mass functions at $z\sim1$ and $z\sim3$ may be significantly different, so a range of $M_{200}\sim4$–$6\times10^{13}\,M_\odot$ brackets plausible estimates. We note that the stellar mass-to-DM mass scaling that we obtain is similar to what is estimated for Cl-1001 and for Cl-1449, which are supported by X-ray detections (Gobat et al. 2011; Wang et al. 2016) and SZ for Cl-1449 (Gobat et al. 2019). ID A B C D -------------------------- ----------------------- --------------- ---------------- ---------------- -------------- RA 10:01:23.174 10:01:22.964 10:01:22.369 10:01:23.438 DEC 02:20:05.57 02:20:05.87 02:20:02.63 02:20:01.10 $z_{\rm spec}$ 2.9214 2.9156 2.9064 2.9 (1) $\log M_{\star}$ $M_\odot$ 11.13 11.13 11.23 11.00 SFR (2) $M_\odot$ yr$^{-1}$ 306 706 266 $<30$ $S_\nu$(870 $\mu$m) mJy $4.0\pm0.1$ $9.1\pm0.1$ $3.39\pm0.15$ $<0.3$ $S_\nu$(1.25 mm) mJy $1.2\pm0.1$ $3.4\pm0.1$ $1.3\pm0.2$ $<0.3$ $S_\nu$(3.4 mm) $\mu$Jy $40\pm5$ $88\pm6$ $39\pm5$ $<15$ $S_\nu$(10 cm) $\mu$Jy $38\pm3$ $34\pm3$ $69\pm6$ I$_{\rm CO[3-2]}$ Jy$\times$km s$^{-1}$ $0.10\pm0.03$ $0.69\pm0.05$ $0.63\pm0.05$ $<0.1$ (3) FWZV$_{\rm CO[3-2]}$ (4) km s$^{-1}$ 381 1114 1098 $v_{\rm CO[3-2]}$ (5) km s$^{-1}$ 460 13 -690 $r_{\rm 1/2}$ (6) $''$ $0.07\pm0.01$ $0.10\pm0.003$ $0.11\pm0.007$ \ [Notes: (1) photometric redshifts; (2) derived from the measurement of individual galaxies assuming the same SED shape as for their coaddition (Fig. 5). (3) assuming a linewidth of 500 km s$^{-1}$. (4) Full Width at Zero Velocity corresponding to the full extraction range of the emission line in velocity. (5) systemic velocities of the galaxies are relative to the average, flux-weighted redshift of the  emission ($z=2.9154$). (6) reported sizes are half-light radii from a circular Gaussian fit. Errors are much smaller than the beam size given the high SNR detections. The average size of $0.1''$ corresponds to 800 pc at $z=3$. ]{} ALMA dust continuum observations -------------------------------- We recovered publicly available ALMA band 7 data covering RO-1001, which consist of three pointings from projects 2015.1.00137.S (PI: N. Scoville) and 2016.1.00478.S (PI: O. Miettinen). They are imaged with a common restoring beam of 0.15 arcsec with natural weighting (given the maximum baseline of 1107m), then corrected for primary beam attenuation and combined. The continuum rms reaches about 50$\mu$Jybeam$^{-1}$ in the central region at the restored resolution (and about $\times$2 higher if tapered to a beam of 0.6$''$). Three galaxies are very clearly detected (Fig. 3; Table 1). No other significant detection is present in the ALMA imaging. In order to measure the size of the dust emission, we modeled the ALMA observations in the [*uv*]{} space, combining all datasets. We fitted circular Gaussian sources for simplicity: sizes for individual objects are reported in Table 1. NOEMA CO observations. ---------------------- We observed the RO-1001 field with the IRAM NOEMA interferometer covering the CO\[3-2\] line emission redshifted to 88.3 GHz for $z=2.91$, with the main aims of confirming the redshift of cluster galaxies and measuring accurately their systemic velocities. The field was observed during November 2018 to March 2019. A total of seven tracks were obtained. The data were calibrated in a standard way using GILDAS [*clic*]{} software packages, and analyzed with [*mapping*]{}. The image has an rms sensitivity of $5\,\mu$Jy/beam in the continuum and of 24 mJy km s$^{-1}$ over 300 km s$^{-1}$ for emission lines. The primary beam is about 1 arcmin, thus covering a large area around the nebula. The resulting synthesized beam at 88.3 GHz is rather elongated at $4.0\times1.8''$, with a position angle of 15$^\circ$. None of the ALMA detected sources are resolved at this resolution. We thus extracted their spectra by fitting PSFs in the [*uv*]{} space at the known spatial positions from ALMA. We simultaneously fit all galaxies in the field, i.e. the three ALMA galaxies and a bright interloper falling by chance in the large NOEMA field, in order to avoid being affected by sidelobes. The continuum is strongly detected in all three ALMA sources in the RO-1001 structure. We searched for emission lines in the spectra by identifying channel ranges with excess positive emission and identified the strongest line in each galaxy spectrum as the one with the lowest chance significance (Jin et al. 2019; Coogan et al. 2018). Sources B and C have very strong CO\[3-2\] detections with $\mathrm{SNR}\sim14$ at $z=2.91$ and very broad emission lines with full width zero intensity of FWZI$\sim1000$ km s$^{-1}$. For source A the strongest feature in its spectrum is a fairly weak, 3.5$\sigma$ emission with 380 km s$^{-1}$ of FWZI. While in itself its reality could be questionable, it turns out that this feature is offset by only 460 km s$^{-1}$ from the average  velocity from the whole system ($z=2.9154$), when we cover CO\[3-2\] over about 50,000 km s$^{-1}$ in total. The probability of a line with such SNR to be found by chance so close to the structure redshift is about 1%. If we also take into account that the weak line falls almost exactly on top of the  velocity at the position of the galaxy (see also Fig. 10a), we conclude that the identification of this weak feature as CO\[3-2\] is quite certain, and thus the redshift of the galaxy from the simultaneous detection of CO\[3-2\] and . A summary of the properties of the three detections by ALMA and NOEMA is given in Table 1. We note that there are large variations in the CO\[3-2\] flux to underlying continuum ratio (Table 1). This can be due to several reasons, including variations in the dust temperature and/or CO excitation ratio. They might also be connected to rapid SFR variations given that the dust continuum timescale is 50–100 Myr while the CO\[3-2\] line is sensitive to the instantaneous dense gas content. However, we note that galaxy A has the lowest CO\[3-2\] to dust continuum ratio, and is also the most compact: a factor of two smaller in radius then the other two (which are also extremely compact already). We speculate that the lack of CO\[3-2\] might be due to high optical depths as recently claimed for high-$z$, dusty galaxies (Jin et al. 2019; Cortzen et al. 2020). The fourth-most massive galaxy in the system (object D) remains undetected in ALMA and NOEMA continuum and has no CO\[3-2\] detection. Assuming the average SED temperature as seen in the group to convert the ALMA upper limits into SFR, we place an upper limit of $\mathrm{SFR}<30\,M_\odot$ yr$^{-1}$ (sSFR$<0.3$ Gyr$^{-1}$), which locates this galaxy one dex below the main sequence. It is thus a candidate quiescent system in the group. [ {width="49.00000%"}]{} [ {width="49.00000%"}]{} [{width="9cm"}]{} Integrated star formation activity. ----------------------------------- The total IR luminosity of the group is derived by fitting (Jin et al. 2018) Herschel, SCUBA2, and ALMA and NOEMA continuum fluxes. The Herschel/SPIRE images are fitted using two PSF components on the images on the position of sources C and the average position of A and B, respectively. Due to the higher spatial resolution, we fit at the position of all three ALMA sources in the SCUBA2 image (Figs. 5; 6). We fit the SED of the group, including the summed photometry from Spitzer, Herschel, SCUBA2, ALMA and NOEMA, and obtain a best fitting SFR $=1200\ {\rm M}_\odot$ yr$^{-1}$ for a SED with an average intensity of the radiation field $\langle U\rangle = 45$ (Fig. 5). Both the dust temperature and average specific star formation rate (sSFR; 3 Gyr$^{-1}$ on average over the three ALMA galaxies) are in agreement with those of typical main sequence galaxies at $z\sim3$ (Bethermin et al. 2015; Schreiber et al 2018). Assuming that individual SFRs of the ALMA-detected galaxies scale like the ALMA/NOEMA continuum fluxes, we conclude that also the sSFRs of the individual galaxies place them within the main sequence. This does not imply that they are normal galaxies, as the very compact ALMA sizes betray some ongoing/past starbursting activity (Puglisi et al 2019). The ALMA 870 $\mu$m emission from the three detected galaxies is consistent within the uncertainties with the SCUBA2 signal (Fig. 5), implying that the bulk of the IR emission and SFR in the RO-1001 group comes from the three ALMA detections. We identified additional star-forming galaxies in the structure at $z=2.91$ via their  emission, or in two cases via possible UV absorption lines (Fig. 7). They are not detected in the near-IR to current depths, impliying they are lower-mass, star-forming galaxies. AGN limits. ----------- A cross-match between RO 1001 and the deepest Chandra COSMOS+Legacy images (Civano et al. 2016) yields no X-ray point-source detection. We stacked the observed soft (0.5–2 keV) and hard (2–10 keV) bands at the position of the three ALMA sources using CSTACK[^2]. We estimate an average $L_{\rm X}<3.5\times10^{43}$ erg s$^{-1}$ (3$\sigma$ upper limit) in the rest-frame 2–10 keV directly from the soft X-ray fluxes (0.5–2 keV observed). The observed hard X-ray emission maps directly into 8–40 keV rest-frame and is much less affected by obscuration. It provides a limit of $L_{\rm X}<5.4\times10^{43}$ erg s$^{-1}$ in the rest-frame 2–10 keV, K-corrected assuming a power-law X-ray spectrum with photon index $\Gamma=1.4$ (Gilli, Comastri & Hasinger 2007). When spread over the three sources, this gives an integrated limit of $L_{\rm X}<1.5\times10^{44}$ erg s$^{-1}$ in the rest-frame 2–10 keV. We further estimated 3$\sigma$ $L_{\rm X}$ upper limits of $L_{\rm X}<7\times10^{42}$ erg s$^{-1}$ (rest-frame 0.5–2 keV) corresponding to an AGN bolometric luminosity of $L_{\rm AGN}<2\times10^{45}$ erg s$^{-1}$, fairly independent of obscuration. This is corroborated by analysis of the individual broad-band SEDs (Jin et al. 2018), in which the mid-IR AGN component is always negligible relative to the host galaxy, providing similar limits on any possible AGN bolometric luminosity. We further calculated the average L$_{\rm X}$ expected from the integrated SFR of the group/cluster, assuming empirical $M_{\star}$-dependent $L_{\rm X}$/SFR relations (Mullaney et al. 2012; Rodighiero et al. 2015; Delvecchio et al. 2020) for star-forming galaxies. On average, we would expect from this structure an AGN activity at the level of L$_{\rm AGN}\sim$2$\times$10$^{45}$ erg s$^{-1}$. We use this bolometric luminosity for our energetics estimates. Using modeling that reproduces the evolution of the X-ray luminosity function through cosmic time on the basis of the mass function and SFR distributions statistically observed in galaxies (Delvecchio et al. 2020), we infer that, given the massive galaxies in the RO-1001 structure and their SFRs, the probability to observe one of them with QSO luminosities as high as those in Borisova et al. 2016 is of the order of $10^{-4}$. This shows that the time during which our RO-1001 group could be selected as part of the QSO nebulae surveys is very small. The RO-1001 structure was selected due to the presence of three VLA-detected sources at 3 GHz. All have moderate radio power $L_{\rm 3\,GHz} \sim10^{24.2-24.6}$ W Hz$^{-1}$. Given the integrated SFR of the group/cluster ($\sim1200\,M_\odot$ yr$^{-1}$) and a redshift-dependent infrared-radio correlation, radio emission can be broadly explained by consistent levels of star formation within less than a factor of 2 (Fig. 5). Source C contains a weak radio feature elongated to the W, visible at 3 GHz, which might trace (past or relatively weak) AGN activity in this source. This is consistent with the weak  (Fig. 1, 2) and HST i-band continuum (Fig. 7) detections of this galaxy. [ {width="50.00000%"} ]{} X-ray constraint and halo masses -------------------------------- In order to further constrain the hosting halo mass, we searched for extended X-ray emission from the hot gas at the position of RO-1001 (lacking any detectable point source X-ray emission, see previous section). We used an X-ray image in the 0.5–2 keV range, produced by combining the Chandra and XMM-Newton images after background and point source removal (Fig.8). We used a 24$''$ radius aperture to place the flux estimates on the source, obtaining a value of $5.8\times10^{-16}$ erg s$^{-1}$ cm$^{-2}$, which is a 3.1$\sigma$ excess over the background. The source has a 230 ks Chandra exposure, with a corresponding upper limit on the point source contamination of $1\times10^{-16}$ erg s$^{-1}$ cm$^{-2}$ (using the same count-rate to flux-conversion-rate as for the source). At a redshift of $z=2.9$ this corresponds (Leauthaud et al. 2010) to rest-frame 0.1–2.4 keV $L_\mathrm{X}$ of $1.1\times10^{44}$ erg s$^{-1}$ and a mass of $M_{\rm DM}\sim4\times10^{13}M_\odot$. The temperature of the ICM implied by the same correlations is 2.1 keV. Results: general and spectral properties of the  emission nebula in RO-1001 =========================================================================== The RO-1001 structure appears to be a group hosted in a single, fairly massive dark matter halo. This mass estimate is important to quantitatively compare our observations and overall properties of its  nebula (Sect. 2.1) to predictions from cold accretion theory. The halo mass of RO-1001 is estimated using three methods returning consistent results: 1. The integrated stellar mass of 5.4$\times10^{11}M_\odot$ of its four most massive galaxies (Fig. 2; Section 2.2) above a mass completeness limit of ${\rm log(M}/M_\odot)>10.8$ and scaled based on stellar to DM ratios corresponds to $M_{\rm DM}\sim$4–6$\times10^{13}M_\odot$. 2. Herschel+ALMA reveal a total infrared luminosity $L_{\rm IR}=1.2\times10^{13}L_\odot$ from star formation (Sect.2.5): scaling from the $z=2.5$ cluster CL-1001 (Wang et al. 2016) this returns $M_{\rm DM}\sim4\times10^{13}M_\odot$ (accounting for the expected cosmic increase of the SFR over $z=2.5$–2.9). 3. A blind (no free parameters) X-ray measurement centered at the barycenter of the 4 massive galaxies returns a 3.1$\sigma$ excess over the background corresponding to $M_{\rm DM}\sim4\times10^{13}M_\odot$ (Sect.2.7; contamination by X-rays from star formation is negligible, while no individual point-like X-ray sources are present). RO-1001 thus appears roughly half as massive to Cl-1001 at $z=2.51$ (Wang et al. 2016), the previously most distant X-ray structure known, but more distant, at $z=2.91$. This is 400 Myr earlier (15% of Hubble time), which is substantial because at these epochs the halo mass function is rapidly evolving (e.g., Mo & White 2002). RO-1001 falls close to the $z$-$M_{\rm DM}$ regime where cosmological cold flows might penetrate the hot halo, as inferred from simulations and models (Dekel et al. 2009; Behroozi et al. 2018; cold gas would not penetrate at lower redshifts). Substantial accretion of cold gas in this structure would not be surprising, given the ongoing $\mathrm{SFR}=1200\,M_\odot\,\mathrm{yr}^{-1}$ of the group galaxies. Theory predicts gas accretion from the IGM scaling as $M_{\rm DM}^{1.15}\times(1+z)^{2.25-2.5}$ (Dekel et al. 2013; Neinstein & Dekel 2008), or about $10,000\,M_\odot$ yr$^{-1}$ in RO-1001. This would be sufficient to feed the ongoing SFR, provided that a non-negligible fraction of the likely multi-phase inflow (Cornuault et al. 2018) remains cold. Additionally, local cooling due to the interactions and shocks between the infalling gas and the hot cluster gas might also provide the required cold gas fuel for star formation (Mandelker et al. 2019; Zinger et al. 2018). The  emission does not appear to be generally centered on individual galaxies: the luminous core peaks in an empty region located at the center of the halo potential well (defined as the barycenter of the stellar mass distribution; Fig. 9). The 150–200 kpc radius traced by filaments corresponds to 50–70% (and up to 100% accounting for possible projection effects) of the virial radius of its hosting DM halo ($R_{\rm V}\sim280$kpc): their disappearance beyond the virial radius at our current sensitivities is explained by the lack of hot gas compressing the cold material (Dekel et al. 2009; Dijkstra & Loeb 2009; Rosdhal & Blaizot 2012). These filaments have substantial transverse diameters (50–70 kpc; 10$''$; $\sim20$% of the virial radius), as expected for flows into the most massive halos, broadened due to their initially higher pressure and instabilities (Cornuault et al. 2018; Rosdhal & Blaizot 2012). The average surface brightness in the filaments is of order $1\times10^{-18}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$, with a total area above this surface brightness limit of 210 arcsec$^2$ ($1.3\times10^4$ kpc$^2$). Again, all of this is quite comparable to theoretical expectations given the hosting halo mass. The circularly averaged surface brightness profile follows $r^{-2.1}$ at large distances, consistent with predictions (e.g., Rosdhal & Blaizot 2012).  moment maps ------------ We can search for evidence of the ongoing physical processes from the  spectral properties. The moment 1 (velocity) map of the  emission in RO-1001 is shown in Fig. 10a. The velocity relative to the core increases in absolute value towards the outer region of the filaments reaching 400–500 km s$^{-1}$ (and possibly up to 600–700 km s$^{-1}$ when statistically correcting for the unknown inclinations), similar to the virial velocity and as predicted by theory (Rosdhal & Blaizot 2012; Cornuault et al. 2018). Any attempt to reproduce this feature with outflows would need to be excessively fine-tuned, given that material ejected from outflows would not accelerate while moving outwards. In the case of infall along the filaments our observations suggest that the initial (virial) velocity is progressively reduced as the flows proceed into the hot medium with which they interact, or betray the changing direction of the gas filaments while inspiraling towards the center of the potential well. Indeed, inflow models (Danovich et al. 2015; Mandelker et al. 2020a) do not predict a reduction in the absolute speed of the filaments while moving from the outskirts to the core but do predict projection effects where the gradient arises from the bending of the filaments inside the halo. This seems more consistent also with the kinematics of the SE filament that shows evidence for a local velocity gradient from the velocity map (Fig. 10a). If due to rotation, this might also represents a lower-mass dark matter halo in the process of merging into the larger system. Intriguingly, we have currently no obvious individual galaxy associated with this putative sub-halo. The moment-2 (velocity dispersion; Fig. 10b) map shows a typical velocity dispersion at the level of 200–300 km s$^{-1}$ (de-convolved by the instrumental resolution), about half of the virial velocity (classifying RO-1001 as a dynamically [*cold*]{} nebula) but higher than the expected thermal broadening of the cold $\sim10^4$ K gas (few tens km s$^{-1}$), as expected for multiphased, cloudy accretion flows in which streams do not remain highly collimated (McCourt et al. 2018; Cornuault et al. 2018). The observed velocity gradient between the edge of the filaments and the core together with the velocity dispersion increasing towards the core allows us to obtain a rough estimate of the cold gas mass flow, assuming that the initial kinetic energy of the cold gas ($\dot{M}v_{\rm virial}^2$) is partly transformed into turbulence ($\dot{M}\sigma_{\rm turbulence}^2$) and partly radiated away (mainly via ). Developing such calculations suggest that 10–30% of the initial energy is converted into turbulence (Cornuault et al. 2018), with a cold mass flow rate of 1000–2000 $M_\odot$ yr$^{-1}$, thus a penetration efficiency (as defined by Dekel et al. 2013) of order of 10–20%. This infall rate approximates the ongoing SFR, but only a fraction of this would reach the galaxies, consistent with the requirement that the system must be heading to a downfall of the activity in a few dynamical timescales (a few Gyr), and eventually quench (e.g., by $z<2$). [ {width="99.00000%"} ]{} [ {width="89.00000%"} ]{}  radiative transfer modeling. ----------------------------- The typical spectra of the  emission in the core of the nebula is reported in Fig. 11d, showing a double-peaked shape, with the blue component stronger than the red one. This is reversed from what is observed in most  emitting galaxies at both low- (Kunth et al. 1998; Henry et al. 2015; Yang et al. 2016; Rivera-Thorsen et al. 2015) and high-redshift (Erb et al. 2014; Orlitov[á]{} et al. 2016; Matthee et al. 2017; Herenz et al. 2017) where the observation of a stronger red peak is associated with outflows (Kulas et al. 2012), hence here suggesting infall (see Ao et al. 2020 for a  blob showing similar spectral properties). We modeled the spectra accounting for radiative transfer effects, assuming the geometry of a shell. The ‘shell model’ consists of a moving concentric shell of neutral hydrogen and dust which surrounds a central - (and continuum-) emitting source. This setup introduces at least five parameters: the neutral hydrogen column density $N_{\mathrm{HI}}$, the dust optical depth $\tau_{\mathrm{d}}$, the velocity of the shell $v_{\mathrm{exp}}$ (defined to be positive when outflowing), the (effective) temperature of the shell $T$ (incorporating any small-scale turbulence), and the width of the intrinsic Gaussian line $\sigma_{\mathrm{i}}$. This is admittedly simplistic but providing qualitatively similar results to spherical clouds, with either an illuminating central point source or uniformly distributed in the volume (Verhamme et al. 2006), and even a multiphase medium with high clumping factor as discussed above (Gronke et al. 2017). Specifically, we fitted the observed (continuum-subtracted) spectrum using an improved pipeline originally described in Gronke et al. (2015). This pipeline consists of $12,960$ models covering the $(N_{\mathrm{HI}},\,v_{\mathrm{exp}},\,T)$ parameter space, and the other parameters are modeled in post-processing. To the five parameters above, we also added the systemic redshift which we allowed to vary within $[-700,\,400]\,\mathrm{km}\,\mathrm{s}^{-1}$ of $z=2.9154$ (corresponding to the velocity range spanned by individual CO\[3-2\] detections in the group). Furthermore, we modelled the effect of the spectral resolution by smoothing the synthetic spectra with a Gaussian with FWHM of $150\,\mathrm{km}\,\mathrm{s}^{-1}$, equal to the resolution of the BM dataset. In order to sample the posterior distribution sufficiently, we used the tempered affine invariant Monte Carlo sampler of the `Python` package `emcee` (Goodman & Weare 2010) We chose to employ $10$ temperatures, $200$ walkers, and $2000$ steps which we found to be sufficient to sample the posterior. For all fitted spectra (panels b, c, d in Fig. 11) the fitting returns moderately low column densities of neutral hydrogen, at the level of $10^{17-18.5}$ cm$^{-2}$, suggesting fairly reduced radiative transfer effects implying that the inferred systemic velocities and dispersions previously measured from the   moments are reliable, as often observed in  blobs (Herenz et al. 2020). The blueshift-dominated spectral shape requires an overall average infalling velocity of $v_{\rm infall}\sim150$ km s$^{-1}$. The systemic velocity is close to our zero-velocity scale, which is defined as the  flux weighted velocity of the full nebula, and is consistent with the average CO\[3-2\] redshifts of the three ALMA galaxies. This implies that such emission originates from gas at rest with respect to the center of the potential well, on average. Observations of non-resonant lines would be decisive to confirm this feature and all results from shell modeling, in general. Several hotter regions with 400–500 km s$^{-1}$ dispersion are evident (red spots in Fig. 10b): it is tempting to interpret them as possible shock fronts from the incoming material surrounding the core regions where gas density is highest. Modeling the spectra supports this idea, recovering higher HI column densities at one of the most conspicuous locations (Fig. 11b). For a region around the galaxy ‘A’ the spectral shape is reversed (Fig. 11c), with the red component being stronger. This red-dominated spectrum is fitted as originating at a systemic velocity of 326 km s$^{-1}$; shifted in the direction of the CO\[3-2\] velocity inferred for the underlying galaxy ‘A’ (460 km s$^{-1}$). It is difficult to distinguish such a velocity offset as due to relative motion with respect to the center of the system, or due to a Hubble flow effect that would require a physical separation of about 1 Mpc along the line-of-sight. The prevalence of the red peak suggests in any case, an overall outflowing velocity of $v_{\rm outflow}\sim200$ km s$^{-1}$. The red-core emission is thus best understood as coming from outflows likely originating from galaxy ‘A’. If this is the case, the modest velocity offsets and flow-velocities suggest star formation-driven outflows (consistent with our estimates in Table 2), given that for AGN driven ones some 1000–2000 km s$^{-1}$ velocity offsets would be expected.  double Gaussian decomposition. ------------------------------- It would be very instructive to be able to carry out a detailed spectral analysis also along the filaments, evaluating whether their spectral shape is also mostly blueshifted. However, they are detected at high SNRs only in the lower spectral resolution data. We thus decompose pixel-by-pixel with double Gaussians the lower-resolution  spectra over the whole nebula, using `ROHSA` (Marchal et al. 2019), a multi-Gaussian decomposition algorithm based on a regularized nonlinear least-square criterion that takes into account the spatial coherence of the emission. Here we choose to decompose the signal into a sum of $N=2$ Gaussians, in order to capture the basic effects of resonant scattering, producing blueshifted and redshifted components (Fig. 11). The model of the emission is then $$\tilde I\big(\nu, \thetab({\bm{r}})\big) = \sum_{n=1}^{N=2} G\big(\nu, \thetab_n({\bm{r}})\big) \label{eq::model_gauss} ,$$ with $\thetab({\bm{r}}) = \big(\thetab_1({\bm{r}}), \dots, \thetab_n({\bm{r}})\big)$ and where $$G\big(\nu, \thetab_n({\bm{r}})\big) = {\bm{a}}_n({\bm{r}}) \exp \left( - \frac{\big(\nu - {\bm{\mu}}_n({\bm{r}})\big)^2}{2 {\bm{\sigma}}_n({\bm{r}})^2} \right)$$ is parametrized by $\theta_n = \big({\bm{a}}_n, {\bm{\mu}}_n, {\bm{\sigma}}_n\big)$ with ${\bm{a}}_{n} \geq \bm{0}$ being the amplitude, ${\bm{\mu}}_{n}$ the position, and ${\bm{\sigma}}_{n}$ the standard deviation 2D maps of the $n$-th Gaussian profile. The estimated parameters $\hat{\theta}$ are obtained by minimizing the cost function as described in Marchal et al. 2019. The latter includes a Laplacian filtering to penalize the small-scale fluctuation of each 2D map of parameters. Note that in order to perform this minimization, the whole emission cube is fitted at once. The strength of this filtering is controlled by three hyper-parameters $\lambda_{{\bm{a}}}$=10, $\lambda_{{\bm{\mu}}}$=4000, and $\lambda_{{\bm{\sigma}}}$=4000. These parameters have been empirically chosen to obtain a spatially coherent solution of $\hat{\theta}$ with the smallest residual $\tilde I\big(\nu, \theta({\bm{r}})\big)$ - $I\big(\nu)$. We find a median absolute residual of 2.3%, showing that $N$=2 provides a good spatially coherent description of the signal. We obtain spatially resolved maps of the ratio of redshifted to blueshifted components (Fig. 11a), that can be used to gauge where infall vs. outflows might prevail. The result of this analysis is that infall appears to dominate also along the two main filaments, except for a few regions that appear to be instead dominated by outflows, similarly to the core (including possibly the whole Northern filament). [c|ccc|ccc]{} & &\ Energy source & Constrain & Total & Effective & Constrain & Total & Effective\ AGN photo. & $L_{\rm AGN}\lesssim2\times10^{45}$ ergs$^{-1}$ & $\lesssim0.6$ & $<0.2$ & $L_{\rm AGN}\sim10^{47}$ergs$^{-1}$ & [**40**]{} & $\approx1$\ SF photo. & 1200 $M_\odot$ yr$^{-1}$ & 10 & $<0.03$ & 120 $M_\odot$ yr$^{-1}$ & 1 & $<1$\ AGN outflows & $\lesssim200\,M_\odot$ yr$^{-1}$ & 0.3 & $\ll$0.1 & [**8000**]{} $M_\odot$ yr$^{-1}$ & [**20**]{} & $\approx1$?\ SF outflows & 1200 $M_\odot$ yr$^{-1}$ & 1 & $<0.1$ & 120 $M_\odot$ yr$^{-1}$ & negl. & negl.\ Gravity & -------------------------------------- $M_{\rm DM}=4\times10^{13}\,M_\odot$ [**10000**]{} $M_\odot$ yr$^{-1}$ -------------------------------------- & [**160**]{} & $\approx1$ ? & -------------------------------------- $M_{\rm DM}=3\times10^{12}\,M_\odot$ $500\,M_\odot$ yr$^{-1}$ -------------------------------------- & $\approx1$ & $<0.01$ ?\ \ [Total and Effective energy rates for  ionization are given in units of $1.5\times10^{44}$ergs$^{-1}$ for the case of outflows and gravity, i.e. the typical  luminosity of both the RO-1001 and also of the QSO nebulae (Borisova et al. 2016), and relative to the required numbers of ionizing photons for the case of AGN/SF photoionization. The typical SFR of bright QSO fields is from Schulze et al. 2019. ]{} Discussion ========== [As typically the case with giant  nebulae, there are two fundamental questions that arise concerning their nature: 1) what is the origin of the cold -emitting gas? and 2) what is the energy source of the emission? In order to gather physical insights into these questions for the RO-1001 nebula, it is relevant to compare with nebulae found in QSO/radio-galaxy fields]{} (Borisova et al. 2016), which can reach similar luminosities and maximum spatial extent. However, in RO-1001 there is no evidence of ongoing (obscured or not) AGN activity from ultra-deep Chandra and mid-IR constraints (see Sect. 2.6). A summary of relevant physical quantities and energetics comparing the RO-1001 nebula and QSO nebulae is reported in Table 2. Origin of the cold gas: inflow and outflow rates. ------------------------------------------------- Regarding question 1) on the origin of the cold, diffuse, intra-group gas shining in , there are two main primary channels that need to be evaluated and, ideally, distinguished: cosmological inflows, mainly expected to be a function of the DM halo mass and redshift, and outflows from AGNs/galaxies, which depend on the AGN bolometric luminosity and galaxy SFRs. Secondary channels are also important to evaluate, namely the returning gas from outflows in the form of galactic fountains, and the cold gas that results from cooling due to instabilities in the intra-group medium. However, they are directly related to the primary channels, given that e.g. the amount of cooling that will results from outflows largely depends on the outflow rates and the same for inflows. Similarly, we would ascribe the galactic fountains as mainly related to outflows (even if the gas is infalling). Hence we focus the discussion on the primary channels. For the case of the QSO nebulae, outflow rates driven by AGNs (derived using the scaling relations in Fiore et al. 2017 based on the average bolometric luminosity of $10^{47}$ erg s$^{-1}$) are expected to exceed inflow rates by at least two orders of magnitude (Table 2), making the search for infalling gas in those systems very difficult. The QSO-fields inflow rates are conservatively (over-)estimated using a maximally large hosting DM halo for QSOs of $3\times10^{12}M_\odot$, where the best estimate from clustering is instead $1\times10^{12}M_\odot$ (see Pezzulli et al. 2019). The situation is reversed for the case of RO-1001: given the AGN luminosity upper limits and ongoing SFRs, infalling gas rates are expected to exceed outflow rates by one order of magnitude. Most of the outflowing gas mass is expected to be SFR driven rather than AGN driven, see Table 2. For SF-driven outflows we assume a loading factor of 1, with outflow rates from star-formation equal to SFRs, plausible given the large stellar masses (Hopkins et al. 2012; Newman et al. 2012; Gabor & Bournaud 2014, Hayward et al. 2015). Hence, in relative terms, the RO-1001 group, respect to QSO fields, has a 3 orders of magnitude (factor of 1000) higher ratio between gas mass inflow and outflow rates. On the other hand, the larger hosting DM halo of RO-1001 will result in a higher proportion of the outflowing gas being retained in the system deeper potential well and being recycled onto the galaxies via galactic fountains, thus likely reducing somewhat the expected order of magnitude contrast between nearly pristine cosmological inflowing gas and material that came out of member galaxies via outflows (see, e.g., Valentino et al. 2016)[^3]. Similarly, not all the predicted cosmological inflows might remain cold while penetrating the halo, as discussed in the previous section, although instabilities from the flowing gas will generate secondary cold gas within the halo. Finally, when considering that the gas migth be accumulating over longer timescales, the impact of AGNs outflows in QSO fields might be somewhat reduced, having in mind that the survival time of the cold gas (Klein et al. 1994; Valentino et al. 2016; Schneider & Robertson 2017) is not necessarily much longer than QSO variation timescales (both of order of 10 Myr), although this point is still debated and longer gas survival time might be possible (Gronke & Oh 2018; Mandelker et al. 2019). Still, the advantage over QSO fields clearly remains large. Energetics ---------- Regarding question 2) on the powering source, relevant channels are: ionization from AGN or star formation, dissipation of kinetic energy carried out either by outflows from the same AGN/SFR or from inflows from the cosmic web, ultimately due to gravitational energy. ### AGN photoionization For the case of photoionization, the observed  luminosity in RO-1001 corresponds to $\sim1.5\times10^{55}$ hydrogen ionizing photons. Scaling from CL 1449 calculations (Valentino et al. 2016), this ionizing photon rate requires an AGN with bolometric luminosity of L$_{\rm AGN}\sim$3.5$\times$10$^{45}$ erg s$^{-1}$. In the RO-1001 nebula our limit on AGN activity (or, equivalently, assuming ongoing AGN activity at the cosmic average given the stellar mass and SFR present; Delvecchio et al. 2020), implies that $<60$% of the required photons are produced. When considering that the typical Lyman continuum escape for moderate AGNs is $\sim30$%, and that not all photons will likely power the  nebula due to geometry constraints, covering factors, etc., we conclude that AGN ionization cannot produce the observed . As discussed in Sect. 2.6, the possibility that a bright QSO just switched off seems very unlikely: the probability of finding a luminous QSO associated with our massive galaxies is $<10^{-4}$. Finally, a skeleton with three filaments onto a giant  nebula is hardly compatible with a spherical or conical geometry generally associated with QSO illumination. For the QSO nebulae,  emission is generally assumed to be powered by QSO ionization and subsequent recombination, which indeed is the most energetic source (Table 2) given the average QSO luminosity of $10^{47}$ erg s$^{-1}$, providing 40 times more ionizing photons than required to account for the typical  nebulae in these environments. ### Energy injected from AGN and SF outflows We compute luminosity rates that can be induced by outflows as their mass flows times their typical velocities squared. Calculation of gas mass outflow rates for QSOs and for the RO-1001 nebula have been reported in Sect. 4.1 and Table 2. We assume typical velocities of 500 km s$^{-1}$ (1000 km s$^{-1}$) for SF (AGN) outflows. For the RO-1001 group the SF-driven outflows (with a smaller, possible contribution from AGN outflows) would carry out of the galaxies just as much energy as that observed in . However, most of it will be dissipated via thermal instabilities (Valentino et al. 2016) and only a (small) fraction of this energy would emerge reprocessed by . Also, their modest velocities might prevent them from reaching large distances from the ALMA galaxies (which contain most SFR in the RO-1001 structure). Hence we expect that outflows could contribute locally, but likely not dominate. For QSOs, outflows carry 20 times the energy required to power the typical observed  luminosities and with much higher velocities than star formation driven outflows, thus possibly far reaching. They might thus contribute a sizeable amount of the observed  energy when dissipating in the circumgalactic medium (see also Ji et al. 2019; Gronke & Oh 2020a; Fielding et al. 2020). Even accounting for substantial inefficiencies in  production (as discussed previously for RO-1001), their role in contributing to the powering of QSO nebulae might have been underestimated so far (Table 2). ### Ionization from star forming galaxies While this channel is negligible for QSOs (when compared to the QSO ionization), for RO-1001 the output from the ongoing SFR from the three ALMA galaxies is potentially capable of producing 10 times more ionizing photons than required to ionize the  nebula. This channel has therefore to be carefully investigated for the RO-1001 group. The ALMA galaxies are highly obscured. Even assuming typical attenuation properties of normal main sequence galaxies of the same stellar mass (Pannella et al. 2015) would imply that only a small fraction of ionizing photons and  photons can actually escape such galaxies, on average. This number could be even smaller if these sources are obscured like typical SMGs (Simpson et al. 2017; Jin et al. 2019; Calabro et al. 2018; 2019), as IR-luminous massive galaxies at high $z$ tend to be (Elbaz et al. 2018; Puglisi et al. 2019). We emphasize that, unlike AGN obscurations, dust attenuation in star-forming galaxies is not expected to be highly anisotropic in the UV (it is not driven by a torus), as demonstrated, e.g., by the tight relation between dust extinction and stellar mass (Garn & Best 2010; Kashino et al. 2013). Hence we estimate what fraction of the ionizing photons (or  photons) can escape the SF galaxies in RO-1001 by, first, evaluating the UV rest-frame output of these sources towards our direction. The brightest object in the UV is galaxy ’C’ (Fig. 7), that also has some weak  enhancement at its position (Fig. 2), with a UV SFR of $2.5M_\odot$ yr$^{-1}$ at the level of 1% of the IR SFR for this galaxy (that contains 20% of the total IR from the group). Galaxies A and B have much less than 1% of the intrinsic UV radiation (as inferred from the IR) being emitted in the UV after escaping dust extinction. So in total the emerging, unattenuated UV SFR corresponds to $<3\times10^{-3}$ of the SFR$=1200M_\odot$ yr$^{-1}$ seen in the IR, corresponding already to a negligible $<3$% fraction of the photons required to power the observed  nebula. The overall effective output from SF galaxies would hardly change when considering additional UV-selected galaxies in the RO-1001 overdensity (Fig. 7; notice that additional UV-bright sources aligned with the nebula as visible in Figs 2 and 7 are in the foreground). Still, further reductions have to be considered. For example, to predict the emerging  flux from the ongoing SFR one has to include additional differential attenuation between the UV rest frame and . It has been shown locally that in ULIRGs the relative escape fraction of  respect to far-UV unattenuated regions is typically at the level of 0.1%, altough reaching 10% in one peculiar case (Martin et al. 2015b). Hence we can expect  directly produced by the SFR in RO-1001 to be entirely negligible, including  photons directly produced at the sites of the ALMA galaxies and any contribution from scattering out of these sources. In terms of computing the Lyman continuum escape from the SFR in RO-1001, required to ionize hydrogen and produce  by recombinations, one has to further account for additional extinction between  and wavelengths below the Lyman break, including absorption by the HI gas within the galaxies. Finally, one can wonder if a large, additional population of faint low-mass star forming galaxies, unseen to current limits and potentially attenuation-free, could produce the observed . Using deep Subaru Suprime-CAM imaging in VRI bands places a $5\sigma$ lower limit of $EW>500$ Å (rest-frame) on the diffuse  equivalent width, using regions where no detectable continuum is present in the current UV rest frame imaging. This is high enough to exclude diffuse in-situ star formation (e.g.,  originating in large numbers of low-mass, star-forming galaxies). ### Gravitational energy. For the dark matter halo estimated for RO-1001, the energy associated (Faucher-Giguère et al. 2010) with cosmological gas accretion is 160$\times$ what is required to power the nebula. This exceeds all other source of energy by about two orders of magnitudes (Table 2). In the case of QSOs the same calculations show that gravitational energy connected with gas accretion is barely comparable to the energy required to power the nebulae, and 100 times smaller than QSO photoionization efficiency, hence it plays quite clearly a negligible role given the overall poor efficiency of converting such energy into  photons. In a RO-1001-like halo, the emerging  radiation powered by gravitational energy has been predicted by existing models (Dijkstra et al. 2009; Goerdt et al. 2010; Rosdhal & Blaizot 2012; Laursen et al. 2019) to be of the order of $10^{44}$erg s$^{-1}$, fully consistent with what is observed here, and at the level of few percent of the total available gravitational energy. On the other hand, predictions for the much smaller QSO halos from the same works halos return average  luminosities much smaller than what is observed on average. ### Cooling from the X-ray gas Localized runaway cooling might occur in the densest regions of the hot X-ray halo due to the onset of thermal instabilities (Gaspari et al. 2012, Sharma et al. 2012). The thermal cascade would then result in the emission of Lya photons which might contribute on scales smaller than our resolution. This mechanism is at the base of the self-regulated feedback successfully explaining several observed features of the ionized filaments in local cool-core clusters (Voit et al. 2017). However, this mechanism is unlikely to explain the extreme /X-ray luminosity ratio of the giant nebula in RO1001 and similar objects, considering both a classical stationary cooling flow (Geach et al. 2009) or empirically comparing to the observed values /X-ray ratios of cool-core clusters that are orders of magnitude lower than what we observe in RO-1001 (see also discussions about this point in Valentino et al. 2016, for Cl1449 at $z=1.99$). ### Conclusions on the energetics In conclusion, energetic arguments strongly favor cooling via radiation-dissipated gravitational energy as the most plausible channel for the (collisional or shock) excitation of  emission in RO-1001 nebula. This channel is providing two orders of magnitude more energy than any other plausible channel for RO-1001. This is opposed to QSO nebulae, where the combined effect of QSO photoionization and outflows provides two orders of magnitude more energy than gravitation. Hence this results in a contrast of 4 orders of magnitude (factor of $10^4$) in relative terms between the RO-1001 and QSO fields in terms of likelyhood of revealing  powered ultimately by gravitational energy release. The origin of  in RO-1001 ------------------------- In light of these results it is relevant to re-evaluate which process might be responsible for the  emission. The classic expectation would be collisionally excited  from the cold gas dissipating kinetic energy acquired via the gravitational energy. In order to be viable this would require a non negligible neutral fraction in the gas. The formal HI column density inferred from  modeling (Sect.3.1; Fig.11) is pretty low, and not obvious to be sufficient for the purpose. For example, assuming the rough calculation of flowing gas mass reported in Sect. 3.1 and using the virial velocity and virial radius to compute the timescale corresponds to a fraction of neutral hydrogen mass over total gas mass of order $10^{-3}$. This is still compatible with collisional excitation without the need to advocate photo-ionization from currently unknown sources if the cold gas temperature is a few $10^4$ K (e.g., see Fig.1 in Cantalupo et al 2008). All the more, it should be emphasized that the HI column density and infall/outflow velocities inferred from the shell modelling reported in Sect.3.1 and Fig. 11 must be considered as strict lower limits. This is because the emerging spectrum is strongly weighted to lines of sights with the lowest column densities, given that  escapes through the paths of least resistance (e.g., Eide et al. 2018), generally orthogonal to the stream velocities in case of infall (Gronke et al. 2017). We notice though that an alternative scenario suggested by recent modeling is that radiation is emitted through the cooling of mixed gas occurring at the boundary between the phases (Mandelker et al. 2019; Cornuault et al. 2018). In that case  radiation would originate from the combined dissipation of kinetic energy of the stream and of thermal energy from the mixed gas that cools down (Gronke & Oh 2020a). This scenario, and the recalled calculations of gas mass flows in Sect. 3.1 also imply a total cold gas mass present at each moment in the diffuse streams of order of $10^{11}M_\odot$ (to be compared to lower limit of $\approx10^{8}M_\odot$ of neutral hydrogen). Is that a reasonable gas mass to be diffused in the intra-group gas? On one hand this is comparable or smaller than the ISM mass of all group galaxies combined, and less than 10% of all baryons expected to be in the group given its DM estimate and assuming a universal baryon fraction. Assuming pressure equilibrium the flowing gas would need to be confined in clouds with quite small volume filling factor, of order of $10^{-3}$ to $10^{-5}$, qualitatively similar to what discussed in theory work (e.g., Cornuault et al 2018; McCourt et al. 2018; Gronke & Oh 2020b). This is required to reach a high enough density to compensate for the much larger temperature of the hot medium. Such dense gas clouds would produce copious recombination  photons following photo-ionization, if a source emitting enough ionizing photons towards this medium were to be present, which does not seem to be the case for RO-1001, at least given present evidence. Summary and conclusions ======================= The main findings of this work can be summarized as follows: 1. RO-1001 is a group at $z=2.91$ defined by 4 galaxies with stellar masses above $10^{11}M_\odot$, originally selected as an overdensity of faint radio sources (12$\sigma$ excess) over a small, 10$''$ region. 2. It contains an ongoing SFR of 1200 M$_\odot$ yr$^{-1}$ mostly spread among 3 ALMA detected, IR-luminous sources characterised by very compact IR sizes, for which we reported CO\[3-2\] detections with NOEMA. 3. RO-1001 does not contain any detectable evidence of ongoing AGN activity, down to limits that are consistent with a cosmic average co-existence of AGN and SFR at typical levels. 4. The hosting dark matter halo mass is estimated to be typical of a group, with $\sim4\times10^{13}M_\odot$ consistently derived using three methods, including a blind X-ray measurement with significance of 3.1$\sigma$. 5. RO-1001 hosts a giant  halo with a luminosity of $1.3\times10^{44}$ erg s$^{-1}$ as revealed by KCWI observations. Three  filaments are observed extended over an overall 300 kpc area (comparable to the virial diameter of RO-1001) and converging into the bright  core whose luminosity peak is well aligned with the center of mass of the group. 6. The  emission is fairly ’cold’, with a velocity dispersion of about 250 km s$^{-1}$. The absolute velocity at the edge of the filaments is of order of 400-500 km s$^{-1}$ higher than in the core, comparable to the virial velocity. The increase of velocity in absolute value with respect to the core is difficult to reconcile with the idea of filaments representing streams of outflowing gas. 7. The  spectral profile in most of the core appears to be dominated by blushifted components, as well as in the two main filaments as derived by multi-gaussian decompositions. Evidence for regions dominated by redshifted components in  exists as well. 8. Shell modeling of the blushifted emission in the core suggests the presence of inflowing gas with moderate velocity (150 km s$^{-1}$) and column densities of neutral gas (a few $10^{17}$ cm$^{-2}$) arising from a rest-frame velocity consistent with the average one as traced by CO in the ALMA galaxies. This supports the  moments analysis as being relatively unaffected by resonant scattering. 9. From the point of view of expected rates of gas flows given the hosting DM halo masses and ongoing SF and AGN activity, cold gas inflows from the cosmic web are expected to dominate over outflows by up to an order of magnitude in RO-1001. This corresponds to a relative contrast of 3 orders of magnitude with respect to  nebulae hosted by luminous QSO (where inflows from the cosmic web are expected to be two orders of magnitude smaller than outflows). 10. From the point of view of  powering and energetics, the gravitational energy associated with the gas infall can provide two orders of magnitude more energy than required to power the observed  nebula in RO-1001, and over two orders of magnitude more energy than any other plausible source. Again, this corresponds to a relative contrast of 4 orders of magnitude respect to QSO fields (where the  is powered by photo-ionisation and sub-sequent recombinations, with possibly a contribution from outflows). In conclusion, RO-1001 at $z=2.91$ is currently the first plausible case of direct observations of gas accretion towards a massive potential well (Fig.9), with its filaments possibly identifiable with the long sought-after cold accreation streams, but where the effect of phase mixing, dissipation and local cooling seems also important. Knowledge of the mass and position of the center of mass of the RO-1001 group is crucial information that was not available for other known filamentary nebulae and sets a clear new precedent for future research along this line. Of course, several uncertainties remain. We do see evidence that even in RO-1001 outflowing gas is still playing some non-negligible role, and it is extremely difficult to definitely rule out photo-ionization as the dominant mechanism for  emission. More insights could be obtained if we were to be able to obtain measurements of non resonant lines that, as discussed in Sect.3.2, would allow a more solid modeling of the  emission in terms of unveiling the supposed prevalence of inflows while at the same time providing more robust kinematics and velocity dispersion fields. Observations of H$\alpha$ with JWST could be illuminating, also keeping in mind that fairly weak H$\alpha$ emission would be expected if  is predominantly collisionally excited. Similarly, UV metal line observations would provide constraints on ionization and enrichment, potentially clarifying if we are seeing fairly pristine gas being accreted, e.g. at least at the edge of the filaments. All of this will have to wait for future follow up of RO-1001 and other structures in coming years. It is not obvious that trying to observe  around other (non-QSO) structures hosted in lower mass halos would provide an advantage in terms of finding even more direct probes of infalling gas then what we could gather so far in RO-1001. In relative terms, the higher fraction of the inflowing gas that could remain cold after entering lower mass halos would be counterbalanced by the lower contrast between infall and outflows owing to the different scaling of these terms with mass and to the increased loading factor of outflows from lower mass galaxies. And in absolute terms the  luminosities of infalling gas would be suppressed, roughly proportionally to hosting halo mass, according to model predictions. Of course, it is also not obvious that model predictions are, even roughly, correct. We have emphasized how predictions of  emission from cold streams for a RO-1001-like halo are very close to the observed  luminosity. On the other hand, models might be simplistic. Faucher-Giguère et al. (2010) suggested that when accounting for self-shielding and properly treating sub-resolution effects could easily lead to reduced forecasts of the  emission by 1 or more orders of magnitude. Rosdhal & Blaizot (2012) predict substantially higher neutral fraction than what inferred by our shell modeling of the emerging spectrum. If this were to be true (but beware that column density from shell modeling are strict lower limits to real average column densities, see comments in Sect. 4.3), than their calculations of emerging  emission might also be overestimated, given that collisional excitation luminosity scales with neutral gas density. Needless to say, if models are optimistic by large factors then it might become prohibitive to ever detect any signature of cold accretion from . Ultimately, future generations of models capturing physical effects that currently remain sub-resolution will be crucial for interpreting this and future observations in terms of cold gas accretion. We are indebted to Sebastiano Cantalupo for use of his CubeEx software and for enlightening discussions. We also thank Matt Lehnert, Avishai Dekel, Nir Mandelker and Anne Verhamme for discussions. This work includes observations carried out with the IRAM NOEMA Interferometer. F.V. acknowledges support from the Carlsberg Foundation Research Grant CF18-0388 “Galaxies: Rise And Death” and the Cosmic Dawn Center of Excellence funded by the Danish National Research Foundation under then Grant No. 140. MG was supported by NASA through the NASA Hubble Fellowship grant HST-HF2-51409 and acknowledges support from HST grants HST-GO-15643.017-A, HST-AR-15039.003-A, and XSEDE grant TG-AST180036. AP acknowledges financial support from STFC through grants ST/T000244/1 and ST/P000541/1. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain). This paper makes use of data from ALMA: a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This work was supported by the Programme National Cosmology et Galaxies (PNCG) of CNRS/INSU with INP and IN2P3, co-funded by CEA and CNES. Ao Y., et al., 2020, NatAst, in press (arXiv:2003.06099) Arrigoni Battaia F., Hennawi J. F., Prochaska J. X., O[ñ]{}orbe J., Farina E. P., Cantalupo S., Lusso E., 2019, MNRAS, 482, 3162 Behroozi, P. & Silk, J. 2018, MNRAS 477, 5382-5387. 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Yang H., et al., 2016, ApJ, 820, 130 Zinger, E., Dekel, A., Birnboim, Y., Nagai, D., Lau, E., Kravtsov, A. V. 2018, MNRAS 476, 56. [^1]: See Laursen et al. 2011; Dijkstra & Loeb 2009. [^2]: CSTACK is publicly available at: <http://lambic.astrosen.unam.mx/cstack/> [^3]: This is equivalent to saying that the outflowing gas would be seen when outflowing, and when part of it will be recycled back to the galaxies, implying that the outflowing contributions listed in Table 2 would have to be counted with a factor slightly above 1 (but well less than 2).

[**Comment on “Domain Structure in a Superconducting Ferromagnet”**]{}\ According to Faurè and Buzdin [@FB] in a superconducting ferromagnet a domain structure with a period small compared with the London penetration depth $\lambda$ can arise. They claim that this contradicts the conclusion of Ref. that ferromagnetic domain structure in the Meissner state of a superconducting ferromagnet is absent at equilibrium. Actually, there is no contradiction: The results of Ref. have only been misunderstood. First of all it is necessary to properly define what is a ferromagnetic domain structure. A distinctive feature of a ferromagnetic state is a nonzero average spontaneous magnetization $\vec M$ in a [*macroscopic*]{} volume. This takes place even in a ferromagnet with domains, since in ferromagnets the domain size $l$ is macroscopic. It depends on the size and shape of the sample and on the orientation of $\vec M$ with respect to the sample surface. For example, in a ferromagnetic slab of thickness $L$, but infinite in other directions, there are no domains if $\vec M$ is parallel to the slab surface. But if $\vec M$ is normal to the surface the stripe domains of the macroscopic size $l \propto \sqrt{L}$ appear at equilibrium [@LL]. On the other hand, from the very beginning of studying the coexistence of the ferromagnetism and superconductivity it was known that competition between ferromagnetism and superconductivity may lead to structures with periodic variation of the $\vec M$ direction in space. The period of these structures is determined by the intrinsic parameters of the material, is normally smaller than $\lambda$, and does not depend on the size and shape of the sample. Appearance of this structure means that ferromagnetism has lost competition with superconductivity and the “superconducting ferromagnet” is not a ferromagnet in a strict sense: this is an antiferromagnetic structure with a large but finite period. Various types of such structures were known: cryptoferromagnet alignment of Anderson and Suhl [@AS], spiral structure of Blount and Varma [@BV], or domain structure of Krey [@Krey]. One can find these and other references in the review [@BB] cited in Ref. . The second paragraph in Ref. clearly emphasized the difference between the ferromagnetic macroscopic domains and these structures (let us call them intrinsic domain structures) and specifically warned that the paper addressed the case when the material is stable with respect to formation of intrinsic domains. Faurè and Buzdin [@FB] considered the intrinsic domain structure, which was analyzed by Krey [@Krey] more than 30 years ago. They rederived the structure parameters obtained by him. The domain size given by Faurè and Buzdin in Eq. (7), $l \sim \tilde w^{1/3} \lambda^{2/3}$, coincides with that given by Krey in his Eq. (30) (apart from notations). Here $\tilde w \sim (K/2\pi M^2) \delta$, $K$ is the energy of the easy-axis anisotropy, and $\delta$ is the domain-wall thickness. The condition for formation of this structure obtained by Krey also coincides with that of Faurè and Buzdin: $\lambda > \tilde w$. Thus in the limit $L\to \infty$ they obtained the intrinsic domain structure in the state which is globally antiferromagntic. The structure can appear in any sample whatever its demagnetization factors are, in particular, in the slab of thickness $L$ independently of whether $\vec M$ is normal or parallel to the slab plane. Certainly the results of Ref. cannot be relevant for this state as clearly warned there. Faurè and Buzdin claimed that their results for thin slabs (small $L$) disagree with Ref. , though Ref. did not consider finite-$L$ corrections at all addressing (like Refs. ) only the macroscopic limit, when $L$ exceeds any intrinsic scales (including $\lambda$) or any combination of them. Only then the difference between intrinsic domains and [*macroscopic*]{} domains has a clear meaning. Though time and again Faurè and Buzdin stressed contradiction to Ref. , in reality they confirmed its conclusion: If the superconducting ferromagnet is stable with respect to formation of intrinsic domains, macroscopic domains also do not appear. They claim that the area of stability, for which the analysis of Ref. is relevant, corresponds to “the nonrealistic limit of vanishing $\lambda$”. In reality Krey’s stability condition $ \lambda < \tilde w \sim (K/2\pi M^2)\delta$ (not $ \lambda \ll \tilde w $ !) is not so severe and allows the values of $\lambda$ essentially larger than the domain-wall width $\delta$. Indeed, the ratio $K/2\pi M^2$, which is called the quality factor of the magnetic material, can be rather high. This is required for various applications of magnetic materials [@MS]. The quality factor is especially high for weak ferromagnetism, which is the most probable case for the coexistence of ferromagnetism and superconductivity. E.B. Sonin\ Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel\ PACS numbers: 75.60.Ch, 74.25.Ha, 74.90.+n [99]{} M. Faurè and A.I. Buzdin, Phys. Rev. Lett [**94**]{}, 187202 (2005). E.B. Sonin, Phys. Rev. B [**66**]{}, 100504(R) (2002). L.D. Landau and E.M. Lifshitz, [*Electrodynamics of Continuous Media*]{} (Pergamon Press, Oxford, 1984). P.W. Anderson and H. Suhl, Phys. Rev. [**116**]{}, 898 (1959). E.L. Blount and C.M. Varma, Phys. Rev. Lett [**42**]{}, 1079 (1979). U. Krey, Intern. J. Magnetism, [**3**]{}, 65 (1972). L. N. Bulaevskii, A. I. Buzdin, M. L. Kulić, and S. V.  Panjukov, Adv. Phys. [**34**]{}, 176 (1985). A.P. Malozemoff and J.C. Slonczewski, [*Magnetic Domain Walls in Bubble Materials*]{} (Academic Press, N.Y., 1979).

--- abstract: | Let $U$ be an operator in a Hilbert space $\mathcal{H}_{0}$, and let $\mathcal{K}\subset\mathcal{H}_{0}$ be a closed and invariant subspace. Suppose there is a period-$2$ unitary operator $J$ in $\mathcal{H}_{0}$ such that $JUJ=U^{\ast}$, and $PJP\geq0$, where $P$ denotes the projection of $\mathcal{H}_{0}$ onto $\mathcal{K}$. We show that there is then a Hilbert space $\mathcal{H}\left( \mathcal{K}\right) $, a contractive operator $W\colon\mathcal{K}\rightarrow\mathcal{H}\left( \mathcal{K}\right) $, and a selfadjoint operator $S=S\left( U\right) $ in $\mathcal{H}\left( \mathcal{K}\right) $ such that $W^{\ast}W=PJP$, $W$ has dense range, and $SW=WUP$. Moreover, given $\left( \mathcal{K},J\right) $ with the stated properties, the system $\left( \mathcal{H}\left( \mathcal{K}\right) ,W,S\right) $ is unique up to unitary equivalence, and subject to the three conditions in the conclusion. We also provide an operator-theoretic model of this structure where $U|_{\mathcal{K}}$ is a pure shift of infinite multiplicity, and where we show that $\ker\left( W\right) =0$. For that case, we describe the spectrum of the selfadjoint operator $S\left( U\right) $ in terms of structural properties of $U$. In the model, $U$ will be realized as a unitary scaling operator of the form$$f\left( x\right) \longmapsto f\left( cx\right) ,\qquad c>1,$$ and the spectrum of $S\left( U_{c}\right) $ is then computed in terms of the given number $c$. address: | Department of Mathematics\ The University of Iowa\ Iowa City, IA 52242-1419\ U.S.A. author: - 'Palle E. T.  Jorgensen' bibliography: - 'jorgen.bib' title: Diagonalizing operators with reflection symmetry --- [^1] \[Int\]Introduction =================== The paper is motivated by two problems one from mathematical physics, and the other from the interface of integral transforms and interpolation theory. The first problem is that of changing the spectrum of an operator, or a one-parameter group of operators, with a view to getting a new spectrum with physical desiderata (see, e.g., [@Seg98]), for example creating a mass gap, and still preserving quasi-equivalence of the two underlying operator systems. In the other problem we study how Hilbert space functional completions change under the variation of a parameter in the integral kernel of the transform in question. The motivating example here is derived from a certain version of the Segal–Bargmann transform. For more detail on the background and the applications alluded to in the Introduction, we refer to the two previous joint papers [@JoOl98] and [@JoOl99], as well as [@Nee94] and [@Hal98]. Let $U$ be an operator in a Hilbert space $\mathcal{H}_{0}$, and let $J$ be a period-$2$ unitary operator in $\mathcal{H}_{0}$ such that$$JUJ=U^{\ast}. \label{eqInt.1}$$ We think of (\[eqInt.1\]) as a reflection symmetry for the given operator $U$. In this case, $U$ and its adjoint $U^{\ast}$ have the same spectrum, but, of course, $U$ need not be selfadjoint. Nonetheless, we shall think of (\[eqInt.1\]) as a notion which generalizes selfadjointness. As an example, let the Hilbert space $\mathcal{H}_{0}=L^{2}\left( \mathbb{T}\right) $, $$\left( Uf\right) \left( z\right) =zf\left( z\right) ,\qquad f\in L^{2}\left( \mathbb{T}\right) ,\;z\in\mathbb{T}, \label{eqInt.2ins}$$ and$$Jf\left( z\right) =f\left( \bar{z}\right) . \label{eqInt.3ins}$$ The space $L^{2}\left( \mathbb{T}\right) $ is from Haar measure on the circle group $\mathbb{T}=\left\{ z\in\mathbb{C}\mathrel{;}\left| z\right| =1\right\} $. It clear that (\[eqInt.1\]) then holds. If $\mathcal{K}=H^{2}\left( \mathbb{T}\right) $ is the Hardy space of functions, $f\left( z\right) =\sum_{n=0}^{\infty}c_{n}z^{n}$, with $\left\| f\right\| ^{2}=\sum_{n=0}^{\infty}\left| c_{n}\right| ^{2}<\infty$, then we also have$$PJP\geq0 \label{eqInt.2bis}$$ where $P$ denotes the projection onto $H^{2}\left( \mathbb{T}\right) $. In fact$$\left\langle f,Jf\right\rangle =\left| c_{0}\right| ^{2}, \label{eqInt.3bis}$$ where $\left\langle \,\cdot\,,\,\cdot\,\right\rangle $ denotes the inner product in $L^{2}\left( \mathbb{T}\right) $. While our result applies to the multiplicity-one shift, this is a degenerate situation, and the nontrivial applications are for the case of infinite multiplicity. There is in fact an infinite-multiplicity version of the above which we proceed to describe. Let $0<s<1$ be given, and let $\mathcal{H}_{s}$ be the Hilbert space whose norm $\left\| f\right\| _{s}$ is given by$$\left\| f\right\| _{s}^{2}=\int_{\mathbb{R}}\int_{\mathbb{R}}\overline {f\left( x\right) }\,\left| x-y\right| ^{s-1}f\left( y\right) \,dx\,dy. \label{eqInt.4}$$ Let $a\in\mathbb{R}_{+}$ be given, and set$$\left( U\left( a\right) f\right) \left( x\right) =a^{s+1}f\left( a^{2}x\right) . \label{eqInt.5}$$ It is clear that then $a\mapsto U\left( a\right) $ is a unitary representation of the multiplicative group $\mathbb{R}_{+}$ acting on the Hilbert space $\mathcal{H}_{s}$. It can be checked that $\left\| f\right\| _{s}$ in (\[eqInt.4\]) is finite for all $f\in C_{c}\left( \mathbb{R}\right) $ ($=$ the space of compactly supported functions on the line). Now let $\mathcal{K}$ ($=\mathcal{K}_{s}$) be the closure of $C_{c}\left( -1,1\right) $ in $\mathcal{H}_{s}$ relative to the norm $\left\| \,\cdot\,\right\| _{s}$ of (\[eqInt.4\]). It is then immediate that $U\left( a\right) $, for $a>1$, leaves $\mathcal{K}_{s}$ invariant, i.e., it restricts to a semigroup of isometries $\left\{ U\left( a\right) \mathrel {;}a>1\right\} $ acting on $\mathcal{K}_{s}$. Setting$$\left( Jf\right) \left( x\right) =\left| x\right| ^{-s-1}f\left( \frac{1}{x}\right) ,\qquad x\in\mathbb{R}\setminus\left\{ 0\right\} , \label{eqInt.6}$$ we check that $J$ is then a period-$2$ unitary in $\mathcal{H}_{s}$, and that $$JU\left( a\right) J=U\left( a\right) ^{\ast}=U\left( a^{-1}\right) \label{eqInt.7}$$ and$$\left\langle f,Jf\right\rangle _{\mathcal{H}_{s}}\geq0,\qquad\forall \,f\in\mathcal{K}_{s}, \label{eqInt.8}$$ where $\left\langle \,\cdot\,,\,\cdot\,\right\rangle _{\mathcal{H}_{s}}$ is the inner product $$\left\langle f_{1},f_{2}\right\rangle _{\mathcal{H}_{s}}:=\int_{\mathbb{R}}\int_{\mathbb{R}}\overline{f_{1}\left( x\right) }\,\left| x-y\right| ^{s-1}f_{2}\left( y\right) \,dx\,dy. \label{eqInt.9}$$ In fact, if $f\in C_{c}\left( -1,1\right) $, the expression in (\[eqInt.8\]) works out as the following reproducing kernel integral:$$\int_{-1}^{1}\int_{-1}^{1}\overline{f\left( x\right) }\left( 1-xy\right) ^{s-1}f\left( y\right) \,dx\,dy, \label{eqInt.10}$$ and we refer to [@JoOl98; @JoOl99] for more details on this example. As an application of our result, we will show that, if $a>1$, then $U\left( a\right) |_{\mathcal{K}_{s}}$ induces a selfadjoint operator $S\left( a\right) $ in a Hilbert space $\mathcal{H}\left( \mathcal{K}_{s}\right) $, and there is a contraction $W\colon\mathcal{K}_{s}\rightarrow\mathcal{H}\left( \mathcal{K}_{s}\right) $, with$$\ker\left( W\right) =0, \label{eqInt.11}$$ such that$$W^{\ast}W=PJP, \label{eqInt.12}$$$$S\left( a\right) W=WU\left( a\right) P, \label{eqInt.13}$$ and$$\operatorname*{spectrum}\left( S\left( a\right) \right) =\left\{ a^{s-1-2n}\mathrel{;}n=0,1,2,\dots\right\} . \label{eqInt.14}$$ What is important in this application is the property (\[eqInt.11\]). So the properties in this case for $W$ are $\left\| W\right\| \leq1$, $\ker\left( W^{\ast}\right) =\ker\left( W\right) =0$. While of course $U\left( a\right) |_{\mathcal{K}_{s}}$ and $S\left( a\right) $ cannot be unitarily equivalent, then $W$ nonetheless defines a strong notion of equivalence (quasi-equivalence) for the two semigroups $U\left( a\right) |_{\mathcal{K}_{s}}$ and $S\left( a\right) $, $a>1$, specified by the intertwining property$$S\left( a\right) W=WU\left( a\right) P. \label{eqInt.15}$$ In particular, since both $W$ and $W^{\ast}$ have dense range in the respective Hilbert spaces $\mathcal{K}$ and $\mathcal{H}\left( \mathcal{K}\right) $, it follows that the partial isometry part $L$ in the polar decomposition $W=L\left( W^{\ast}W\right) ^{1/2}=L\left( PJP\right) ^{1/2}$, is in fact a *unitary* isomorphism of $\mathcal{K}$ onto $\mathcal{H}\left( \mathcal{K}\right) $. The intertwining property for $W^{\ast}W$ of the polar decomposition is$$\left( W^{\ast}W\right) UP=PU^{\ast}\left( W^{\ast}W\right) . \label{eqInt.16}$$ But this cannot be iterated, so there is *not* an analogous relation for the factors $\left( W^{\ast}W\right) ^{1/2}$ and $L$. The properties of $W$ and $S$ in this example imply that $UP$ is in fact a pure shift (i.e., the unitary part of the isometry $U|_{\mathcal{K}_{s}}$ of the Wold decomposition is trivial, and moreover the backwards shift $PU^{\ast}$ has a cyclic vector. The second conclusion is unique to this example, and follows from the fact that $S=S\left( a\right) $ has simple spectrum. \[ProInt.1\]The isometry $UP$ is a pure shift. The result may be read off from the following estimate:$$\left\| PU^{\ast\,k}W^{\ast}\psi\right\| =\left\| W^{\ast}S\left( a^{k}\right) \psi\right\| \leq\left\| S\left( a^{k}\right) \psi\right\| \leq a^{k\left( s-1\right) }\left\| \psi\right\| \underset{k\rightarrow \infty}{\longrightarrow}0, \label{eqInt.17}$$ the estimate being valid for all $\psi\in\mathcal{H}\left( \mathcal{K}\right) $. Since $\ker\left( W\right) =0$, $W^{\ast}\mathcal{H}\left( \mathcal{K}\right) $ is dense in $\mathcal{K}$, so we have $\lim _{k\rightarrow\infty}\left\| PU^{\ast\,k}\varphi\right\| =0$ for all $\varphi\in\mathcal{K}$, and this last property is equivalent to $U|_{\mathcal{K}_{s}}$ being a pure shift on $\mathcal{K}_{s}$. The restriction on $s$ remains $0<s<1$. It follows in fact from [@JoOl98; @JoOl99] that the multiplicity of this shift is $\infty$, i.e., that if $a>1$, the dimension of $\mathcal{K}_{s}\ominus U\left( a\right) \mathcal{K}_{s}$ is infinite. The simplest case of a system $\left( \mathcal{H}_{0},J\right) $ with $J$ as a reflection is that of $\mathcal{H}_{0}=\mathcal{H}\oplus\mathcal{H}$ and $J=I\oplus\left( -I\right) $, i.e., $$J\left( h_{1}\oplus h_{2}\right) =h_{1}\oplus\left( -h_{2}\right) ,\qquad h_{1},h_{2}\in\mathcal{H}. \label{eqInt.a}$$ In many applications of this, it will further be given that $\mathcal{H}$ is a reproducing kernel Hilbert space in the sense of [@Aro50]. Suppose this is the case, and that $Q\left( \,\cdot\,,\,\cdot\,\right) $ is the corresponding reproducing kernel. We then have $\mathcal{H}$ realized as a Hilbert space of $\mathbb{C}$-valued functions $h\left( \,\cdot\,\right) $ defined on some set $\Omega$, and $Q$ is a function on $\Omega\times\Omega$ such that $Q\left( z,\,\cdot\,\right) \in\mathcal{H}$ for all $z\in\Omega$, and $$\left\langle Q\left( z,\,\cdot\,\right) ,h\right\rangle =h\left( z\right) \text{\qquad for all }h\in\mathcal{H}. \label{eqInt.b}$$ In this case, we will use $Q$ in identifying a class of subspaces $\mathcal{K}\subset\mathcal{H}\oplus\mathcal{H}$ such that$$\left\langle k,Jk\right\rangle \geq0\text{\qquad for all }k\in\mathcal{K}. \label{eqInt.c}$$ We now describe such a class of spaces $\mathcal{K}$. Let $D:=\left\{ z\in\mathbb{C}\mathrel{;}\left| z\right| <1\right\} $. It will be stated in an abstract setting, and the applications to interpolation theory will be given in Section \[Han\] below. \[ProInt.3\]Let $\mathcal{H}$ be a reproducing kernel Hilbert space corresponding to a kernel function$$Q\colon\Omega\times\Omega\longrightarrow\mathbb{C}, \label{eqInt.d}$$ and let $\Omega_{0}\subset\Omega$ be a subset. Let a function $$\varphi\colon\Omega_{0}\longrightarrow\bar{D} \label{eqInt.e}$$ be given, and let $\mathcal{K}_{\varphi}\subset\mathcal{H}\oplus\mathcal{H}$ be defined as the closed span of$$\left\{ \begin{pmatrix} Q\left( z,\,\cdot\,\right) \\ \varphi\left( z\right) Q\left( z,\,\cdot\,\right) \end{pmatrix} \mathrel{;}z\in\Omega_{0}\right\} \subset\begin{pmatrix} \mathcal{H}\\ \mathcal{H}\end{pmatrix} ^{\oplus}. \label{eqInt.f}$$ 1. \[ProInt.3(1)\]Then holds for $J=\left( \begin{smallmatrix} I & 0\\ 0 & -I \end{smallmatrix} \right) $ if and only if $$\left( z_{1},z_{2}\right) \longmapsto\left( 1-\overline{\varphi\left( z_{1}\right) }\varphi\left( z_{2}\right) \right) Q\left( z_{1},z_{2}\right)$$ is positive definite on $\Omega_{0}$. 2. \[ProInt.3(2)\]If instead $\varphi\colon\Omega_{0}\rightarrow \mathbb{C}$, and $J=\left( \begin{smallmatrix} 0 & I\\ I & 0 \end{smallmatrix} \right) $, then holds if and only if $$\left( z_{1},z_{2}\right) \longmapsto\left( \overline{\varphi\left( z_{1}\right) }+\varphi\left( z_{2}\right) \right) Q\left( z_{1},z_{2}\right)$$ is positive definite on $\Omega_{0}$. The result follows from a substitution of the vectors in (\[eqInt.f\]) into the positivity requirement (\[eqInt.c\]), and computing out the answer for the two cases of reflection $J$, i.e., $J=\left( \begin{smallmatrix} I & 0\\ 0 & -I \end{smallmatrix} \right) $ and $J=\left( \begin{smallmatrix} 0 & I\\ I & 0 \end{smallmatrix} \right) $. We refer to Section \[Han\] for more details, and additional comments on applications to interpolation theory. \[Pur\]Pure isometries ====================== It is well known that pure isometries (alias shifts) of infinite multiplicity play a role in the harmonic analysis of wavelets, see [@BrJo97b], and in the Lax–Phillips version of scattering theory for the wave equation [@LaPh89]. Let $V$ be a shift in a Hilbert space $\mathcal{K}$, and let$$\mathcal{L}:=\mathcal{K}\ominus V\mathcal{K}; \label{eqPur.1}$$ then$$\mathcal{K}=\sideset{}{^{\smash{\oplus}}}{\sum}\limits_{n=0}^{\infty}V^{n}\mathcal{L} \label{eqPur.2}$$ as a direct sum. But for every nonzero $l\in\mathcal{L}$, and $z\in D:=\left\{ z\in\mathbb{C}\mathrel{;}\left| z\right| <1\right\} $, the vector$$f=f\left( l,z\right) :=l\oplus zVl\oplus z^{2}V^{2}l\oplus\cdots \label{eqPur.3}$$ is an eigenvector of $V^{\ast}$, i.e.,$$V^{\ast}f=zf, \label{eqPur.4}$$ and $\left\| f\right\| ^{2}=\left( 1-\left| z\right| ^{2}\right) ^{-1}\left\| l\right\| ^{2}$. In fact, as $l$ varies over $\mathcal{L}\setminus\left\{ 0\right\} $, the vectors $$\left\{ f\left( l,z^{n}\right) \mathrel{;}n=1,2,\dots\right\} \label{eqPur.5}$$ span a dense subspace in $\mathcal{K}$. This is true for every $z\in D$ fixed; so it is clear from this that there is a variety of ways of creating selfadjoint, and normal, realizations of a given $V$, i.e., solutions to the problem$$WV=NW. \label{eqPur.6}$$ Specifically, there is a Hilbert space $\mathcal{H}\left( \mathcal{K}\right) $, a bounded operator $W\colon\mathcal{K}\rightarrow\mathcal{H}\left( \mathcal{K}\right) $, and a normal operator $N$ in $\mathcal{H}\left( \mathcal{K}\right) $ such that (\[eqPur.6\]) holds. This problem has been studied recently by Feldman [@Fel99], and Agler et al. [@AgMc98], but it is a different focus from ours. The reflection $J$ plays a crucial role in our approach. It also makes our setting considerably more restrictive and it allows us to get solutions to the diagonalization problem which are *unique up to unitary equivalence.* More importantly, it gives an answer to a reflection problem from mathematical physics which we proceed to describe. The approach (\[eqPur.4\]) for $V^{\ast}$ works for a wider class of operators than the backwards shift, namely the operators in the Cowen–Douglas classes, see [@CoDo78], but we have not yet checked which of the Cowen–Douglas operators admit reflection symmetry. Our next result will be stated for general bounded operators $U$ which have reflection symmetry, and the symmetry is given in terms of a period-$2$ unitary $J$ and a subspace $\mathcal{K}$ which is invariant under $U$. From this we will then arrive at a selfadjoint realization $S$ of $U$, and when $\left( \mathcal{K},J\right) $ is given, we will show that $S$ is determined uniquely up to unitary equivalence. The result is interesting even if $U$ is given at the outset to be unitary. In fact in an application from quantum field theory, $U$ will be rather a unitary one-parameter group $\left\{ U\left( t\right) \right\} _{t\in\mathbb{R}}$ of operators acting on a Hilbert space $\mathcal{H}_{0}$, and $\mathcal{K}$ will be a subspace in $\mathcal{H}_{0}$ which is invariant under $U\left( t\right) $ for $t\geq0$. By Stone’s theorem [@Var85], there is a selfadjoint Hamiltonian operator $H$ (generally unbounded) in $\mathcal{H}_{0}$ such that$$U\left( t\right) =e^{-itH},\qquad t\in\mathbb{R}. \label{eqPur.7}$$ In this application, we will have$$JU\left( t\right) J=U\left( -t\right) ,\qquad t\in\mathbb{R}, \label{eqPur.8}$$ and $J$ is referred to as “time-reversal” or “time-reflection”. The initial Hamiltonian might not have the right “physical” spectrum; for example, the spectrum of $H$ might be all of $\mathbb{R}$, and what is desired would be a spectrum which is contained in $\mathbb{R}_{+}$ with a positive gap between $0$ and the bottom of the “physical” spectrum. We will show that this can be achieved; in fact we will describe a selfadjoint realization $S=S\left( U\right) $ in the form of a semigroup$$S\left( t\right) =e^{-t\hat{H}} \label{eqPur.9}$$ where $\hat{H}$ is a selfadjoint operator in the new Hilbert space $\mathcal{H}\left( \mathcal{K}\right) $, and the spectrum of $\hat{H}$ will be “physical” in that it will be positive and there will be a “mass gap”, i.e., a positive gap between $0$ and the lower bound for $\operatorname*{spectrum}\left( \hat{H}\right) $. But the key to passing from $H$ to $\hat{H}$ will be the given $\left( \mathcal{K},J\right) $ when $\mathcal{K}\subset\mathcal{H}_{0}$ is assumed invariant under $U\left( t\right) $, $t\geq0$, and $J$ is a time-reflection, i.e., $J$ and $\left\{ U\left( t\right) \right\} $ will satisfy (\[eqPur.8\]). As we noted, the construction $H\rightsquigarrow\hat{H}$ with $\hat{H}$ having a mass-gap will show, after the fact, that the initial semigroup of isometries $U\left( t\right) |_{\mathcal{K}}$, $t\geq0$, will necessarily be a pure shift (and of infinite multiplicity). By this we mean that there is a unitary isomorphism between $\mathcal{H}_{0}$ and $L^{2}\left( \mathbb{R},\mathcal{M}\right) $ for some infinite-dimensional Hilbert space $\mathcal{M}$ which intertwines $\left\{ U\left( t\right) \right\} _{t\in\mathbb{R}}$ with translation on $L^{2}\left( \mathbb{R},\mathcal{M}\right) $. Specifically, there is a unitary isomorphism$$Y\colon\mathcal{H}_{0}\longrightarrow L^{2}\left( \mathbb{R},\mathcal{M}\right) \text{, onto,} \label{eqPur.10}$$ such that$$YU\left( t\right) Y^{-1}f\left( x\right) =f\left( x-t\right) ,\qquad f\in L^{2}\left( \mathbb{R},\mathcal{M}\right) ,\;t\in\mathbb{R}, \label{eqPur.11}$$ with the further property that$$Y\left( \mathcal{K}\right) =L^{2}\left( \mathbb{R}_{+},\mathcal{M}\right) , \label{eqPur.12}$$ i.e., the functions in $L^{2}\left( \mathbb{R},\mathcal{M}\right) $ which are supported in the positive half line. \[Ref\]Reflection symmetry ========================== The following result provides the axiomatic setup for reflection symmetry in the form described above. With the given symmetry axioms, it provides the step $U\mapsto S\left( U\right) $ from a general operator $U$ with symmetry to its selfadjoint version $S\left( U\right) $, and we show that $S\left( U\right) $ is unique up to unitary equivalence. The data that emerges is $\left( \mathcal{H}\left( \mathcal{K}\right) ,W,S\right) $, where$$SW=WUP. \label{eqRef.1}$$ Here $P$ denotes the projection onto the subspace $\mathcal{K}$ which both is invariant for $U$ and satisfies reflection positivity relative to the period-$2$ unitary $J$ (i.e., the reflection). But in the general setting, the axioms allow $W\colon\mathcal{K}\rightarrow\mathcal{H}\left( \mathcal{K}\right) $ to have nonzero kernel, and this represents some degree of non-uniqueness: for example, $W$ may be a “small” (rank-one, say) projection, and $S$ might be zero. Hence we shall focus on the setting when $\ker\left( W\right) =0$, and we will say then that the two operators $U|_{\mathcal{K}}$ and $S$ are *quasi-equivalent.* While the intertwining operator $W$ is $1$–$1$ with dense range, its inverse $W^{-1}$ will be unbounded. \[ThmRef.1\] Let $U$ be a bounded operator in a Hilbert space $\mathcal{H}_{0}$. Let $\mathcal{K}\subset\mathcal{H}_{0}$ be an invariant subspace, and let $P$ denote the projection of $\mathcal{H}_{0}$ onto $\mathcal{K}$. Let $J$ be a period-$2$ unitary operator in $\mathcal{H}_{0}$ which satisfies$$JUJ=U^{\ast} \label{eqThmRef.1(1)}$$ and$$PJP\geq0. \label{eqThmRef.1(2)}$$ 1. \[ThmRef.1(1)\]Then there is a Hilbert space $\mathcal{H}\left( \mathcal{K}\right) $ and a contractive operator$$W\colon\mathcal{K}\longrightarrow\mathcal{H}\left( \mathcal{K}\right)$$ with dense range, and a bounded selfadjoint operator $S=S\left( U\right) $ in $\mathcal{H}\left( \mathcal{K}\right) $ such that$$SW=WUP, \label{eqThmRef.1(3)}$$$$W^{\ast}W=PJP, \label{eqThmRef.1(4)}$$ and$$\left\| S\left( U\right) \right\| \leq\left( \operatorname*{sp}\left( U^{2}\right) \right) ^{\frac{1}{2}}, \label{eqThmRef.1(5)}$$ where $\operatorname*{sp}\left( U^{2}\right) $ denotes the spectral radius of $U^{2}$. 2. \[ThmRef.1(2)\]Given the data $\left( \mathcal{H}\left( \mathcal{K}\right) ,W,S\right) $ is unique up to unitary equivalence subject to the axioms Specifically, suppose $\left( \mathcal{H}_{i}\left( \mathcal{K}\right) ,W_{i},S_{i}\right) $, $i=1,2$, are two systems which both solve the extension problem, i.e., are extensions satisfying Then there is a unitary isomorphism $T\colon \mathcal{H}_{1}\left( \mathcal{K}\right) \rightarrow\mathcal{H}_{2}\left( \mathcal{K}\right) $ of $\mathcal{H}_{1}\left( \mathcal{K}\right) $ onto $\mathcal{H}_{2}\left( \mathcal{K}\right) $ which satisfies $$TW_{1}=W_{2} \label{eqThmRef.1(6)}$$ and$$TS_{1}=S_{2}T. \label{eqThmRef.1(7)}$$ 3. \[ThmRef.1(3)\]There are operators $U$, with reflection symmetry, such that $W$ from $\left( \mathcal{H}\left( \mathcal{K}\right) ,W,S\right) $ has$$\ker\left( W\right) =0. \label{eqThmRef.1(8)}$$ The proof is rather long and will be broken up into its three parts (\[ThmRef.1(1)\]), (\[ThmRef.1(2)\]), and (\[ThmRef.1(3)\]). Part (\[ThmRef.1(1)\]) asserts the existence of a selfadjoint realization of the given operator $U$, while part (\[ThmRef.1(2)\]) is uniqueness up to unitary equivalence. Part (\[ThmRef.1(3)\]) is an explicit construction which takes place in a certain reproducing kernel Hilbert space. The following observation gives a more concrete understanding of axiom (\[eqThmRef.1(2)\]) in part (\[ThmRef.1(1)\]) of Theorem \[ThmRef.1\]. Let $J$ be a period-$2$ unitary operator in a Hilbert space $\mathcal{H}_{0}$, and let $\mathcal{H}_{\pm}$ be the respective eigenspaces corresponding to eigenvalues $\pm1$ of $J$. If $P_{+}$ is the projection onto $\mathcal{H}_{+}$, then $J=2P_{+}-I$. \[LemRef.2\]A closed subspace $\mathcal{K}\subset\mathcal{H}_{0}$ satisfies if and only if $\mathcal{K}$ is the graph of a contractive operator $\Lambda$ from $\mathcal{H}_{+}$ to $\mathcal{H}_{-}$. By this we mean that $\Lambda$ is defined on a closed subspace $\mathcal{P}\subset\mathcal{H}_{+}$ and $\Lambda$ maps $\mathcal{P}$ contractively into $\mathcal{H}_{-}$. Hence $\mathcal{K}\simeq\left\{ \left( p,\Lambda p\right) \mathrel{;}p\in\mathcal{P}\right\} $, or we will write simply $\mathcal{K}=G\left( \Lambda\right) $ and $\mathcal{P}=D\left( \Lambda\right) $ where $G$ and $D$ are used for graph and domain, respectively. The main idea in the proof is in [@Phil], but we include a sketch. This will also give us a chance for introducing some terminology which will be needed later anyway. Suppose $\mathcal{K}\subset\mathcal{H}_{0}$ is a closed subspace which satisfies (\[eqThmRef.1(2)\]). For $k\in\mathcal{K}$ we have $k=P_{+}k+P_{-}k$, where $P_{-}:=I-P_{+}$ and $J=P_{+}-P_{-}$. But $\left\langle k,Jk\right\rangle =\left\| P_{+}k\right\| ^{2}-\left\| P_{-}k\right\| ^{2}$ for all $k\in\mathcal{K}$ by (\[eqThmRef.1(2)\]), and if we define $\Lambda P_{+}k:=P_{-}k$, then $\Lambda$ is well-defined and contractive from $\mathcal{P}=P_{+}\mathcal{K}$ to $P_{-}\mathcal{K}$. The reasoning shows that the converse argument is also valid, so the lemma follows except for the assertion that $\mathcal{P}:=P_{+}\mathcal{K}$ must be automatically closed. Let $k_{n}$ be a sequence of vectors in $\mathcal{K}$ such that $P_{+}k_{n}\rightarrow h_{+}\in\mathcal{H}_{+}$. Then by (\[eqThmRef.1(2)\]),$$\left\| P_{-}\left( k_{n}-k_{m}\right) \right\| \leq\left\| P_{+}\left( k_{n}-k_{m}\right) \right\| \longrightarrow0\text{\qquad as }n,m\longrightarrow\infty.$$ So the limit $\lim_{n\rightarrow\infty}P_{-}k_{n}=h_{-}$ exists in $\mathcal{H}_{-}$, and $$k_{n}=P_{+}k_{n}+P_{-}k_{n}\longrightarrow h_{+}+h_{-}.$$ Since $\mathcal{K}$ is assumed closed in $\mathcal{H}_{0}$, we get $h_{+}+h_{-}\in\mathcal{K}$, and $h_{+}=P_{+}\left( h_{+}+h_{-}\right) =\lim_{n\rightarrow\infty}P_{+}k_{n}$. This shows that $P_{+}\mathcal{K}$ is closed, and the proof is completed. \[Proof of Theorem continued\](\[ThmRef.1(1)\]) Let the operator $U$ be given as in the statement of the theorem. Let $\mathcal{K}\subset\mathcal{H}_{0}$ be the invariant subspace with projection $P$, and let $J$ be the reflection. It is assumed to satisfy (\[eqThmRef.1(1)\])–(\[eqThmRef.1(2)\]). In view of (\[eqThmRef.1(2)\]), we have $$\left\langle k,Jk\right\rangle \geq0\text{\qquad for all }k\in\mathcal{K}, \label{eqRef.2}$$ where $\left\langle \,\cdot\,,\,\cdot\,\right\rangle $ denotes the given inner product from $\mathcal{H}_{0}$. (Note that $\mathcal{K}$ is not invariant under $J$, so the vector $Jk$ is typically not in $\mathcal{K}$ if $k$ is.) Applying the Cauchy–Schwarz inequality, we get$$\left| \left\langle k_{1},Jk_{2}\right\rangle \right| ^{2}\leq\left\langle k_{1},Jk_{1}\right\rangle \left\langle k_{2},Jk_{2}\right\rangle \text{\qquad for all }k_{1},k_{2}\in\mathcal{K}. \label{eqRef.3}$$ The idea is to get a new Hilbert space $\mathcal{H}\left( \mathcal{K}\right) $ from the form $\left\langle k_{1},Jk_{2}\right\rangle $, i.e., that this form should be the new inner product. So we must form the quotient space $\mathcal{K}/\mathcal{N}$ where$$\mathcal{N}=\left\{ k\in\mathcal{K}\mathrel{;}\left\langle k,Jk\right\rangle =0\right\} . \label{eqRef.4}$$ In view of (\[eqRef.3\]), we get$$\mathcal{N}=\left\{ k_{0}\in\mathcal{K}\mathrel{;}\left\langle k_{0},Jk\right\rangle =0\text{ for all }k\in\mathcal{K}\right\} . \label{eqRef.5}$$ Since$$\left\langle k_{1},JUk_{2}\right\rangle =\left\langle k_{1},U^{\ast}Jk_{2}\right\rangle =\left\langle Uk_{1},Jk_{2}\right\rangle \text{\qquad for all }k_{1},k_{2}\in\mathcal{K}, \label{eqRef.6}$$ we conclude that $U$ passes to the quotient $\mathcal{K}/\mathcal{N}$ and defines there a symmetric operator. When $\mathcal{K}/\mathcal{N}$ is completed in the new norm $\left\| \,\cdot\,\right\| _{J}$,$$\left\| k\right\| _{J}^{2}:=\left\langle k,Jk\right\rangle , \label{eqRef.7}$$ the induced operator becomes selfadjoint in this Hilbert space$$\mathcal{H}\left( \mathcal{K}\right) :=\left( \mathcal{K}/\mathcal{N}\right) \sptilde. \label{eqRef.8}$$ The induced operator will be denoted $S=S\left( U\right) $, and we will now show that it satisfies conditions (\[eqThmRef.1(3)\])–(\[eqThmRef.1(5)\]), starting with (\[eqThmRef.1(5)\]), i.e., showing first that $S\left( U\right) $ is a bounded operator in the Hilbert space $\mathcal{H}\left( \mathcal{K}\right) $. The argument for boundedness is essentially in [@JoOl98], but we include it here for the convenience of the reader. Let $k\in\mathcal{K}$, and use recursion on (\[eqRef.3\]) as follows:$$\begin{aligned} \left\| Uk\right\| _{J}^{2} & =\left\langle Uk,JUk\right\rangle =\left\langle Uk,U^{\ast}Jk\right\rangle =\left\langle U^{2}k,Jk\right\rangle \\ & \leq\left\langle U^{2}k,JU^{2}k\right\rangle ^{\frac{1}{2}}\left\langle k,Jk\right\rangle ^{\frac{1}{2}}\\ & \leq\left\langle U^{4}k,JU^{4}k\right\rangle ^{\frac{1}{4}}\left\langle k,Jk\right\rangle ^{\frac{1}{2}+\frac{1}{4}}\\ & \leq\vphantom{\left\langle U^{4}k,JU^{4}k\right\rangle^{\frac{1}{4}}\left\langle k,Jk\right\rangle^{\frac{1}{2}+\frac{1}{4}}}\cdots\\ & \leq\left\langle U^{2^{n}}k,JU^{2^{n}}k\right\rangle ^{\frac{1}{2^{n\mathstrut}}}\cdot\left\langle k,Jk\right\rangle ^{\frac{1}{2^{\mathstrut}}+\frac{1}{4^{\mathstrut}}+\dots+\frac{1}{2^{n\mathstrut}}}\\ & \leq\left\langle U^{2^{n+1}}k,Jk\right\rangle ^{\frac{1}{2^{n\mathstrut}}}\cdot\left\| k\right\| _{J}^{2}\\ & \leq\left\| U^{2^{n+1}}\right\| ^{\frac{1}{2^{n\mathstrut}}}\cdot\left\| k\right\| ^{\frac{1}{2^{n-1\mathstrut}}}\cdot\left\| k\right\| _{J}^{2}.\end{aligned}$$ We have $\lim\limits_{n\rightarrow\infty}\left\| U^{2^{n+1}}\right\| ^{\frac{1}{2^{n\mathstrut}}}=\operatorname*{sp}\left( U^{2}\right) =$ the spectral radius, and $\lim\limits_{n\rightarrow\infty}\left\| k\right\| ^{\frac{1}{2^{n-1\mathstrut}}}=1$ if $k\neq0$. We have therefore proved the estimate$$\left\| Uk\right\| _{J}\leq\left( \operatorname*{sp}\left( U^{2}\right) \right) ^{\frac{1}{2}}\left\| k\right\| _{J}$$ for $k\in\mathcal{K}$, and it follows that the induced operator $S=S\left( U\right) $ on $\mathcal{H}\left( \mathcal{K}\right) =\left( \mathcal{K}/\mathcal{N}\right) \sptilde$ satisfies (\[eqThmRef.1(5)\]), as claimed. Since we already showed that $S$ is selfadjoint, we conclude that $S$ has bounded spectrum inside the interval$$\left[ -\left( \operatorname*{sp}\left( U^{2}\right) \right) ^{\frac {1}{2}},\left( \operatorname*{sp}\left( U^{2}\right) \right) ^{\frac{1}{2}}\right] \subset\mathbb{R}. \label{eqRef.9}$$ If $U$ on $\mathcal{H}_{0}$ is unitary, this is the interval $\left[ -1,1\right] $. If $U=U\left( t\right) $, $t\in\mathbb{R}$, is a group of operators, then $S=S\left( t\right) $, $t\geq0$, is a semigroup of selfadjoint operators, and so$$S\left( t\right) =S\left( \frac{t}{2}\right) ^{2}\geq0 \label{eqRef.10}$$ for all $t\geq0$, and the spectrum of $S\left( t\right) $ is therefore positive in that case, and we get the representation$$S\left( t\right) =e^{-t\hat{H}},\qquad t\geq0, \label{eqRef.11}$$ for some (generally unbounded) selfadjoint operator $\hat{H}$ in $\mathcal{H}\left( \mathcal{K}\right) $. \[Proof of part \]For a given operator $U$ which has a pair $\left( \mathcal{K},J\right) $ defining a reflection symmetry, we showed in (\[ThmRef.1(1)\]) that there is a system $\left( \mathcal{H}\left( \mathcal{K}\right) ,W,S\right) $ with a selfadjoint operator $S$ in $\mathcal{H}\left( \mathcal{K}\right) $, and an intertwining operator $W$, which satisfy (\[eqThmRef.1(3)\])–(\[eqThmRef.1(5)\]) in the statement of the theorem. We now prove that this system is unique up to unitary equivalence. So suppose there are two systems $\left( \mathcal{H}_{i}\left( \mathcal{K}\right) ,W_{i},S_{i}\right) $, $i=1,2$, both satisfying (\[eqThmRef.1(3)\])–(\[eqThmRef.1(4)\]) and with the two “extension” operators $S_{1}$ and $S_{2}$ both selfadjoint and bounded. We will now show that there is then a unitary isomorphism $T\colon\mathcal{H}_{1}\left( \mathcal{K}\right) \rightarrow\mathcal{H}_{2}\left( \mathcal{K}\right) $ which defines the equivalence, i.e., which satisfies (\[eqThmRef.1(6)\]) and (\[eqThmRef.1(7)\]) in the theorem. We will make (\[eqThmRef.1(6)\]) into a definition, setting$$TW_{1}k=W_{2}k, \label{eqRef.12}$$ for $k\in\mathcal{K}$. Since both $W_{1}$ and $W_{2}$ satisfy (\[eqThmRef.1(4)\]), we conclude that$$\left\| W_{1}k\right\| _{J}=0\iff k\in\mathcal{N}\iff\left\| W_{2}k\right\| _{J}=0,$$ or, stated equivalently,$$\ker\left( W_{i}\right) =\mathcal{N}\text{\qquad for }i=1,2,$$ where $\mathcal{N}$ is defined in (\[eqRef.4\]). Hence, formula (\[eqRef.12\]) makes a good definition of a linear operator $T$ mapping a dense subspace in $\mathcal{H}_{1}\left( \mathcal{K}\right) $ into one in $\mathcal{H}_{2}\left( \mathcal{K}\right) $. But property (\[eqThmRef.1(4)\]) for $W_{1}$ and $W_{2}$ implies that $T$ is also isometric, indeed$$\left\| TW_{1}k\right\| _{J}^{2}=\left\| W_{2}k\right\| _{J}^{2}=\left\langle k,Jk\right\rangle =\left\| W_{1}k\right\| _{J}^{2}.$$ Hence $T$ is a unitary isomorphism of $\mathcal{H}_{1}\left( \mathcal{K}\right) $ onto $\mathcal{H}_{2}\left( \mathcal{K}\right) $. Using now (\[eqThmRef.1(3)\]) for the two systems, we get$$\left( TS_{1}\right) W_{1}k=TW_{1}Uk=W_{2}Uk=S_{2}W_{2}k=\left( S_{2}T\right) W_{1}k\text{\qquad for all }k\in\mathcal{K}.$$ Since $W_{1}$ has dense range, we get the desired intertwining property (\[eqThmRef.1(7)\]) as claimed in the theorem. \[Proof of part \]The assertion in part (\[ThmRef.1(3)\]) is that there are examples where the induction $U\rightsquigarrow S\left( U\right) $ has intertwining operator $W$ with zero kernel, or equivalently, $\mathcal{N}=\left\{ 0\right\} $. We already mentioned this in (\[eqInt.4\])–(\[eqInt.6\]) of Section \[Int\], and in fact this is a one-parameter semigroup of isometries $U\left( a\right) P_{\mathcal{K}_{s}}$, $a>1$. In fact, it arises as the restriction to an invariant subspace of a unitary one-parameter group. It is a representation $U\left( a\right) $, $a\in\mathbb{R}_{+}$, of the multiplicative group $\mathbb{R}_{+}$, or equivalently, via $a=e^{t}$, a representation of the additive group $\mathbb{R}$. We get as a corollary of (\[ThmRef.1(3)\]) that $\left\{ U_{s}\left( e^{t}\right) \right\} _{t\in\mathbb{R}}$ is equivalent to the group of translations on $L^{2}\left( \mathbb{R},\mathcal{M}\right) $ for some infinite-dimensional Hilbert space $\mathcal{M}$ as described in (\[eqPur.10\])–(\[eqPur.12\]) in the conclusion of Section \[Pur\] above. Now recall the Hilbert space $\mathcal{H}_{s}$ and its subspace $\mathcal{K}_{s}$ from Section \[Int\]. When $0<s<1$, $\mathcal{H}_{s}$ is defined by the norm $\left\| \,\cdot\,\right\| _{s}$ from (\[eqInt.4\]) and the subspace $\mathcal{K}_{s}$ is the completion of $C_{c}\left( -1,1\right) $ in the $\left\| \,\cdot\,\right\| _{s}$-norm. We may pick some $a>1$, and consider the isometry $U_{s}\left( a\right) |_{\mathcal{K}_{s}}$ of $\mathcal{K}_{s}$. From (\[eqInt.6\]) we see that $J$ also depends on $s$. The new inner product is $$\left\langle k_{1},k_{2}\right\rangle _{J}:=\left\langle k_{1},Jk_{2}\right\rangle _{\mathcal{H}_{s}}\label{eqRef.13}$$ (defined for $k_{1},k_{2}\in\mathcal{K}_{s}$), and depends on $s$ as well. It is worked out explicitly in (\[eqInt.10\]). It follows from (\[eqInt.9\]) that $\left\langle \,\cdot\,,\,\cdot\,\right\rangle _{\mathcal{H}_{s}}$ is defined from the integral kernel $\left| x-y\right| ^{s-1}$. The corresponding operator $A_{s}$ is a special case of the Knapp–Stein intertwining operator, see [@KnSt80]. (See also [@Sal62] and [@Rad98].) This operator $A_{s}\left( n\right) $ is defined more generally and also in $\mathbb{R}^{n}$. Then the integral kernel is $\left| x-y\right| ^{s-n}$, and $0<s<n$. If $\Delta$ is the positive Laplace operator in $\mathbb{R}^{n}$, i.e., $\Delta=\sum_{j=1}^{n}\left( \frac{1}{i}\frac{\partial\,}{\partial x_{j}}\right) ^{2}$, then it is shown in [@Ste70 Lemma 2, p. 117] that $A_{s}=\Delta^{-\frac{s}{2}}$, and the Fourier transform of $\left| x\right| ^{s-n}$ is $$\left( \pi^{-\frac{s}{2}}\Gamma\left( \frac{s}{2}\right) \biggm/\Gamma\left( \frac{n-s}{2}\right) \right) \cdot\left| \xi\right| ^{-s}.$$ Hence up to a constant, the norm $\left\| \,\cdot\,\right\| _{s}$ of (\[eqInt.9\]) may be rewritten as$$\int_{\mathbb{R}}\left| \xi\right| ^{-s}\left| \hat{f}\left( \xi\right) \right| ^{2}\,d\xi,\label{eqRef.14}$$ and the inner product $\left\langle \,\cdot\,,\,\cdot\,\right\rangle _{s}$ as $$\int_{\mathbb{R}}\left| \xi\right| ^{-s}\overline{\hat{f}_{1}\left( \xi\right) }\hat{f}_{2}\left( \xi\right) \,d\xi,\label{eqRef.15}$$ where $$\hat{f}\left( \xi\right) =\int_{\mathbb{R}}e^{-i\xi x}f\left( x\right) \,dx\label{eqRef.16}$$ is the usual Fourier transform suitably extended to $\mathcal{H}_{s}$, using Stein’s singular integrals. Intuitively, $\mathcal{H}_{s}$ consists of functions on $\mathbb{R}$ which arise as $\left( \frac{d\,}{dx}\right) ^{s}f_{s}$ for some $f_{s}$ in $L^{2}\left( \mathbb{R}\right) $. This also introduces a degree of “non-locality” into the theory, and the functions in $\mathcal{H}_{s}$ cannot be viewed as locally integrable, although $\mathcal{H}_{s}$ for each $s$, $0<s<1$, contains $C_{c}\left( \mathbb{R}\right) $ as a dense subspace. In fact, formula (\[eqRef.14\]), for the norm in $\mathcal{H}_{s}$, makes precise in which sense elements of $\mathcal{H}_{s}$ are “fractional” derivatives of locally integrable functions on $\mathbb{R}$, and that there are elements of $\mathcal{H}_{s}$ (and of $\mathcal{K}_{s}$) which are not locally integrable. On the other hand, vectors in $\mathcal{H}_{s}$ are not too singular: for example the Dirac function $\delta$ is not in $\mathcal{H}_{s}$. To see this, pick some approximate identity $\varphi_{\varepsilon}\underset{\varepsilon\rightarrow 0}{\longrightarrow}\delta$, say $\varphi\in C_{c}\left( -1,1\right) $, $\varphi>0$, $\int\varphi\left( x\right) \,dx=1$, and set $\varphi _{\varepsilon}\left( x\right) =\frac{1}{\varepsilon}\varphi\left( \frac {x}{\varepsilon}\right) $; then a calculation shows that$$\left\| \varphi_{\varepsilon}\right\| _{\mathcal{H}_{s}}^{2}=C_{s}\varepsilon^{s-1}\label{eqRef.17}$$ for some positive constant $C_{s}$. Hence $\delta$ is not in $\mathcal{H}_{s}$, and then of course also not in the subspace $\mathcal{K}_{s}$. Nonetheless, if we pass to the new norm $\left\| f\right\| _{J}^{2}=\left\| f\right\| _{\mathcal{H}\left( \mathcal{K}_{s}\right) }^{2}=\left\langle f,Jf\right\rangle _{s}$ of (\[eqRef.13\]), then from (\[eqInt.10\]) we get$$\left\| \varphi_{\varepsilon}\right\| _{J}^{2}=\mathcal{O}\left( \varepsilon^{2}\right) . \label{eqRef.18}$$ Hence the limit $\varphi_{\varepsilon}\rightarrow\delta$ defines a bounded linear functional on $\mathcal{H}\left( \mathcal{K}_{s}\right) $, relative to the norm $\left\| \,\cdot\,\right\| _{J}$ on that Hilbert space. From the Riesz lemma, and the definition of $\mathcal{H}\left( \mathcal{K}_{s}\right) $, we conclude that $\delta$ is in $\mathcal{H}\left( \mathcal{K}_{s}\right) $. The same argument shows that the distributions $\delta^{\left( n\right) }:=\left( \frac{d\,}{dx}\right) ^{n}\delta$ given by $$\delta^{\left( n\right) }\left( \phi\right) =\left( -1\right) ^{n}\frac{d^{n}\phi}{dx^{n}}\left( 0\right) \label{eqRef.19}$$ for $\phi\in C_{c}^{\infty}\left( -1,1\right) $, are also in $\mathcal{H}\left( \mathcal{K}_{s}\right) $. In fact, the norm computes out as$$\left\| \delta^{\left( n\right) }\right\| _{J}^{2}=n!\left( 1-s\right) \left( 2-s\right) \cdots\left( n-s\right) \text{\qquad for }n=0,1,2,\dots. \label{eqRef.20}$$ In the next lemma we provide the detailed proof of the fact that the iterated derivatives $\left( \frac{d\,}{dx}\right) ^{n}\delta=:\delta^{\left( n\right) }$ of the Dirac delta function are all in the completion of $C_{c}^{\infty}\left( -1,1\right) $ relative to the “new” norm of the Hilbert space $\mathcal{H}\left( \mathcal{K}_{s}\right) $. But recall that $\delta$, or its derivatives, are not in $\mathcal{K}_{s}$. \[LemDiracsinHKs\]For the Dirac mass and its derivatives, we have $\delta^{\left( n\right) }\in\mathcal{H}\left( \mathcal{K}_{s}\right) $, $n=0,1,2,\dots$. The restriction on $s$ is, as before, $0<s<1$. First note that if $\phi\in C_{c}^{\infty}\left( -1,1\right) $, then $$\int_{-1}^{1}\phi\left( x\right) \left( 1-xy\right) ^{s-1}\,dx \label{eqDiracsinHKs.1}$$ restricts to a $C^{\infty}$-function on $\left[ -1,1\right] $. By this we mean that there is a $C^{\infty}$-function $\varphi_{s}$ on $\mathbb{R}$ such that$$\varphi_{s}\left( y\right) =\int_{-1}^{1}\phi\left( x\right) \left( 1-xy\right) ^{s-1}\,dx \label{eqDiracsinHKs.2}$$ holds for all $y$ in $\left[ -1,1\right] $. Hence, if $F$ is a distribution with compact support in $\left[ -1,1\right] $, then$$\left\langle \varphi_{s},F\right\rangle =F\left( \varphi_{s}\right) \label{eqDiracsinHKs.3}$$ is well-defined. The same argument shows that $\left\langle \left( 1-\,\cdot\;y\right) ^{s-1},F\right\rangle $ is well-defined, and that$$y\longmapsto\left\langle \left( 1-\,\cdot\;y\right) ^{s-1},F\right\rangle$$ is also $C^{\infty}$ up to the endpoints in the closed interval $I=\left[ -1,1\right] $. Hence, the distribution $F$ may be applied again, and we get the expression$$\left\| F\right\| _{\mathcal{H}\left( \mathcal{K}_{s}\right) }^{2}:=\int_{I}\int_{I}\overline{F\left( x\right) }\left( 1-xy\right) ^{s-1}F\left( y\right) \,dx\,dy. \label{eqDiracsinHKs.4}$$ Moreover, if $\phi\in C_{c}^{\infty}\left( -1,1\right) $, then $$\left\langle W\phi,F\right\rangle _{\mathcal{H}\left( \mathcal{K}_{s}\right) }=\int_{I}\int_{I}\overline{\phi\left( x\right) }\left( 1-xy\right) ^{s-1}F\left( y\right) \,dx\,dy$$ is well-defined in the distribution sense, and$$\left| \left\langle W\phi,F\right\rangle _{\mathcal{H}\left( \mathcal{K}_{s}\right) }\right| \leq\left\| W\phi\right\| _{\mathcal{H}\left( \mathcal{K}_{s}\right) }\left\| F\right\| _{\mathcal{H}\left( \mathcal{K}_{s}\right) },$$ where $\left\| F\right\| _{\mathcal{H}\left( \mathcal{K}_{s}\right) }$ is the expression (\[eqDiracsinHKs.4\]). Hence for each $n=0,1,2,\dots$, we must show the following implication:$$\left\langle W\phi,F\right\rangle _{\mathcal{H}\left( \mathcal{K}_{s}\right) }=0\text{ for all }\phi\in C_{c}^{\infty}\left( -1,1\right) \Longrightarrow \left\langle \delta^{\left( n\right) },F\right\rangle _{\mathcal{H}\left( \mathcal{K}_{s}\right) }=0. \label{eqDiracsinHKs.5}$$ The interpretation of the brackets $\left\langle \,\cdot\,,\,\cdot \,\right\rangle _{\mathcal{H}\left( \mathcal{K}_{s}\right) }$ is in the sense of distributions as noted. In particular, $$\left\langle \delta^{\left( n\right) },F\right\rangle _{\mathcal{H}\left( \mathcal{K}_{s}\right) }=\left( s-1\right) \cdots\left( s-n\right) \int_{I}y^{n}F\left( y\right) \,dy, \label{eqDiracsinHKs.6}$$ where $\int_{I}y^{n}F\left( y\right) \,dy$ is really the compactly supported distribution $F$ evaluated at the monomial $y^{n}$. Recall, it is assumed that the distribution $F$ is supported in $I$. Now pick $\phi\in C_{c}^{\infty }\left( -1,1\right) $ such that $\phi>0$, and $\int_{I}\phi\left( x\right) \,dx=1$, and let $\phi_{\varepsilon}\left( x\right) =\frac{1}{\varepsilon }\phi\left( \frac{x}{\varepsilon}\right) $, for $0<\varepsilon<1$. We prove next that $$\lim_{\varepsilon\rightarrow0}\left\langle W\phi_{\varepsilon}^{\left( n\right) },F\right\rangle _{\mathcal{H}\left( \mathcal{K}_{s}\right) }=\left\langle \delta^{\left( n\right) },F\right\rangle _{\mathcal{H}\left( \mathcal{K}_{s}\right) }, \label{eqDiracsinHKs.7}$$ where both sides are understood in the sense of distributions. But we also have $\left\langle W\phi_{\varepsilon}^{\left( n\right) },F\right\rangle =0$ for all $\varepsilon>0$, by the assumption in (\[eqDiracsinHKs.5\]). To complete the proof we will then only need to check that$$\sup_{0<\varepsilon<1}\left\| W\phi_{\varepsilon}^{\left( n\right) }\right\| _{\mathcal{H}\left( \mathcal{K}_{s}\right) }<\infty. \label{eqDiracsinHKs.8}$$ Explicitly,$$\left\| W\phi_{\varepsilon}^{\left( n\right) }\right\| _{\mathcal{H}\left( \mathcal{K}_{s}\right) }^{2}=\int_{I}\int_{I}\phi_{\varepsilon }^{\left( n\right) }\left( x\right) \left( 1-xy\right) ^{s-1}\phi_{\varepsilon}^{\left( n\right) }\left( y\right) \,dx\,dy, \label{eqDiracsinHKs.9}$$ and this last expression can be estimated directly: If $n\in\left\{ 0,1,2,\dots\right\} $, there is a constant $C_{n}$ ($<\infty$) such that the $\int_{I}\int_{I}\cdots\,dx\,dy$ term in (\[eqDiracsinHKs.9\]) is estimated by $C_{n}$. In particular, we have the desired estimate (\[eqDiracsinHKs.8\]). The left-hand side of (\[eqDiracsinHKs.7\]) may therefore be estimated by $\left\| F\right\| _{\mathcal{H}\left( \mathcal{K}_{s}\right) }\cdot C_{n}$. Since $\left\langle W\phi_{\varepsilon}^{\left( n\right) },F\right\rangle _{\mathcal{H}\left( \mathcal{K}_{s}\right) }=0$ for all $n$ and all $\varepsilon$, by assumption, see (\[eqDiracsinHKs.5\]), we will then have $\left\langle \delta^{\left( n\right) },F\right\rangle _{\mathcal{H}\left( \mathcal{K}_{s}\right) }=0$, which is the claim. It remains to check that the limit (as $\varepsilon\rightarrow0$) in (\[eqDiracsinHKs.7\]) is as stated. The argument is much as the previous one, so we will merely sketch the details for the case of $n=0$: Since $F$ is an distribution with support in $I=\left[ -1,1\right] $, we need to check that $$\lim_{\varepsilon\rightarrow0}\frac{1}{\varepsilon}\int_{I}\phi\left( \frac{x}{\varepsilon}\right) \left( 1-xy\right) ^{s-1}\,dx=1 \label{eqDiracsinHKs.10}$$ and$$\lim_{\varepsilon\rightarrow0}\frac{1}{\varepsilon}\left( \frac{d\,}{dy}\right) ^{m}\int_{I}\phi\left( \frac{x}{\varepsilon}\right) \left( 1-xy\right) ^{s-1}\,dx=0 \label{eqDiracsinHKs.11}$$ for all $m\in\mathbb{N}$; and both of these limits can be verified by calculus. Indeed the left-hand side in (\[eqDiracsinHKs.10\]) is of the order$$L_{\varepsilon}^{\left( s\right) }\left( y\right) :=\begin{cases} \displaystyle\frac{\left( 1+\varepsilon y\right) ^{s}-\left( 1-\varepsilon y\right) ^{s}}{2\varepsilon sy}&\text{if }y\neq0,\\1&\text{if }y=0, \end{cases} $$ which is differentiable in $y$, for every $\varepsilon\in\mathbb{R}_{+}$. The corresponding expression in (\[eqDiracsinHKs.11\]) is $\mathcal{O}\left( \varepsilon^{m}\right) $, $m=1,2,\dots$. Since the distribution is of compact support (in $I$) we also have, for some $m\in\mathbb{N}$, the estimate $$\left| F\left( \psi\right) \right| \leq{\operatorname*{Const.}}\cdot\max_{0\leq k\leq m}\max_{x\in I}\left| \psi^{\left( k\right) }\left( x\right) \right|$$ for all $\psi\in C^{\infty}\left( \mathbb{R}\right) $. Applying this to the functions $\psi$ ($=L_{\varepsilon}$) in the left-hand side of (\[eqDiracsinHKs.10\]), we finally arrive at the desired conclusion (\[eqDiracsinHKs.7\]). This completes the proof of the lemma. Hence if $f\in\mathcal{K}_{s}$, $Wf\in\mathcal{H}\left( \mathcal{K}_{s}\right) $, we get the inner product $\left\langle \delta^{\left( n\right) },Wf\right\rangle _{J}$ is well-defined. A calculation yields$$\left\langle \delta^{\left( n\right) },Wf\right\rangle _{J}=\left( s-1\right) \left( s-2\right) \cdots\left( s-n\right) \int_{-1}^{1}x^{n}f\left( x\right) \,dx. \label{eqRef.21}$$ However, if $f$ is not locally integrable, then the right-hand side in (\[eqRef.21\]) must be understood as a singular integral, see, e.g., [@Ste70 Chapters V.1–2]. Recall that $\mathcal{K}_{s}$ is obtained as the completion of $C_{c}\left( -1,1\right) $ relative to the norm $\left\| \,\cdot\,\right\| _{s}$ of (\[eqInt.9\]). If $f$ is in $C_{c}\left( -1,1\right) $, then the Fourier transform$$\hat{f}\left( \xi\right) =\int_{-1}^{1}f\left( x\right) e^{-ix\xi}\,dx \label{eqRef.22}$$ of (\[eqRef.16\]) clearly has an entire analytic extension, i.e., it extends to complex values of $\xi$ as an entire analytic function with exponential growth factor $e^{\left| \operatorname{Im}\xi\right| }$, $\xi\in\mathbb{C}$. We wish to show that this also holds for $f\in\mathcal{N}\subset \mathcal{K}_{s}$. Note if $f\in\mathcal{N}$, it has finite $\left\| \,\cdot\,\right\| _{s}$-norm, and$$\int_{-1}^{1}\int_{-1}^{1}\overline{f\left( x\right) }\left( 1-xy\right) ^{s-1}f\left( y\right) \,dx\,dy=0, \label{eqRef.23}$$ or rather $\left\| f\right\| _{J}=0$. Since $f$ can be rather singular, the claim requires a proof. We have $Wf=0$, and the Dirac measures $\delta_{x}$, for $x\in\mathbb{R}$, $\left| x\right| <1$, are in $\mathcal{H}\left( \mathcal{K}_{s}\right) $. Hence $\left\langle \delta_{x},Wf\right\rangle _{J}=0$. But a calculation yields, for $x\in\left( -1,1\right) =:I$,$$\left\langle \delta_{x},Wf\right\rangle _{J}=\int_{-1}^{1}\left( 1-xy\right) ^{s-1}f\left( y\right) \,dy. \label{eqRef.24}$$ Let $x\in I\setminus\left\{ 0\right\} $, and multiply by $\left| x\right| ^{1-s}$, to get$$\int_{-1}^{1}\left| \frac{1}{x}-y\right| ^{s-1}f\left( y\right) \,dy=0,$$ and so $\left( A_{s}f\right) \left( \frac{1}{x}\right) =0$. We conclude that $Af$ is supported in the interval if $f$ is in $\mathcal{N}$. This localizes the computation of $$\left\| f\right\| _{s}^{2}=\int_{\mathbb{R}}\overline{f\left( x\right) }A_{s}f\left( x\right) \,dx, \label{eqRef.25}$$ but still interpreted as a singular integral. Since $\left\| f\right\| _{s}<\infty$, and $f\in\mathcal{K}_{s}$, there is a sequence $\varphi_{n}\in C_{c}^{\infty}\left( -1,1\right) $ such that $\lim_{n\rightarrow\infty}\left\| f-\varphi_{n}\right\| _{s}=0$. Then of course also$$\lim_{n\rightarrow\infty}\left\| \varphi_{n}\right\| _{s}=\left\| f\right\| _{s}<\infty. \label{eqRef.26}$$ But$$\left\| \varphi_{n}\right\| _{s}^{2}=C_{s}\int_{\mathbb{R}}\left| \xi\right| ^{-s}\left| \hat{\varphi}_{n}\left( \xi\right) \right| ^{2}\,d\xi\label{eqRef.27}$$ by (\[eqRef.14\]). It follows that there is a subsequence $\varphi_{n_{i}}$ such that $\hat{\varphi}_{n_{i}}\left( \,\cdot\,\right) $ converges pointwise almost everywhere on $\mathbb{R}$. We wish to use Montel’s theorem [@Hil62 v. II, Theorem 15.3.1] to conclude that the Fourier transform $\hat{f}$ of $f$ also has an entire analytic extension. To do this we need only check that $\hat{\varphi}_{n_{i}}\left( \zeta\right) $, $\zeta \in\mathbb{C}$, is an equicontinuous family. Now pick $\zeta_{1},\zeta_{2}\in\mathbb{C}$, and consider$$\hat{\varphi}_{n_{i}}\left( \zeta_{1}\right) -\hat{\varphi}_{n_{i}}\left( \zeta_{2}\right) =\int_{-1}^{1}\varphi_{n_{i}}\left( x\right) \left\{ e^{-ix\zeta_{1}}-e^{-ix\zeta_{2}}\right\} \,dx.$$ Let $E\left( x\right) :=e^{-ix\zeta_{1}}-e^{-ix\zeta_{2}}$, and pick $\psi\in C_{c}^{\infty}\left( \mathbb{R}\right) $ such that $\psi\equiv1$ on $\bar{I}=\left[ -1,1\right] $. Continuing the calculation, we get$$\begin{aligned} \int_{-1}^{1}\varphi_{n_{i}}\left( x\right) E\left( x\right) \,dx & =\int_{\mathbb{R}}\varphi_{n_{i}}\left( x\right) \psi\left( x\right) E\left( x\right) \,dx\\ & =\int_{\mathbb{R}}\left( \Delta^{-\frac{s}{2}}\varphi_{n_{i}}\left( x\right) \right) \left( \Delta^{\frac{s}{2}}\psi E\left( x\right) \right) \,dx\end{aligned}$$ and$$\begin{aligned} \left| \int_{-1}^{1}\varphi_{n_{i}}\left( x\right) E\left( x\right) \,dx\right| & \leq\left\| \varphi_{n_{i}}\right\| _{s}\cdot\left\| \Delta^{\frac{s}{2}}\psi E\right\| _{L^{2}\left( \mathbb{R}\right) }\\ & \leq\left\| \varphi_{n_{i}}\right\| _{s}\cdot\left\{ \int_{\mathbb{R}}\left| \frac{d\,}{dx}\left( \psi E\right) \left( x\right) \right| ^{2}\,dx\right\} ^{\frac{1}{2}}.\end{aligned}$$ But we have from (\[eqRef.26\]) that $\sup_{i}\left\| \varphi_{n_{i}}\right\| _{s}<\infty$, and the second term is independent of $n_{i}$, and it can be estimated in terms of $\left| \zeta_{1}-\zeta_{2}\right| $ by calculus. This shows that the entire functions $\left\{ \hat{\varphi}_{n_{i}}\left( \zeta\right) \right\} $ do form an equicontinuous family. Since $\hat{\varphi}_{n_{i}}\left( \xi\right) $ is convergent $\mathrm{a.e.}\;\xi\in\mathbb{R}$ as noted, we conclude that the entire functions $\hat{\varphi}_{n_{i}}\left( \zeta\right) $ converge uniformly for $\zeta$ in compact subsets of $\mathbb{C}$, and that the limit function is also entire analytic. But by the argument above, this limit is an extension of $\hat {f}\left( \xi\right) $, for $\xi\in\mathbb{R}$. From (\[eqRef.21\]), we have$$\left\langle \delta^{\left( n\right) },Wf\right\rangle _{J}=\left( s-1\right) \left( s-2\right) \cdots\left( s-n\right) i^{n}\left( \frac{d\,}{d\zeta}\right) ^{n}\hat{f}\left( \zeta\right) |_{\zeta=0}.$$ Since $f\in\mathcal{N}$, $Wf=0$, and the left-hand side vanishes for all $n=0,1,2,\dots$. Hence all the derivatives $\left( \frac{d\,}{d\zeta}\right) ^{n}\hat{f}\left( \zeta\right) $ vanish at $\zeta=0$. Since $\hat{f}$ is analytic, it must vanish identically. Finally use (\[eqRef.14\]) to conclude that $f=0$ as an element of $\mathcal{K}_{s}$. This completes the proof of (\[ThmRef.1(3)\]), and therefore the proof of the theorem. In Section \[Han\], we will consider more systematically the structure of systems $\left( \mathcal{H}_{0},\mathcal{K},J,U\right) $ for which $W\colon\mathcal{K}\rightarrow\mathcal{H}\left( \mathcal{K}\right) $ is $1$–$1$. The present construction (i.e., Theorem \[ThmRef.1\](\[ThmRef.1(3)\])) has the initial operator $U$ unitary in $\mathcal{H}_{0}$, and in fact part of a unitary one-parameter group. If the unitarity restriction on $U$ is relaxed, then there is a richer variety of examples with $\ker\left( W\right) =\left\{ 0\right\} $. For example, let $A$ denote the *unilateral shift* in $H^{2}=H^{2}\left( \mathbb{T}\right) $, and set $$U=\begin{pmatrix} A & 0\\ 0 & A^{\ast}\end{pmatrix} ,\qquad J=\begin{pmatrix} 0 & I\\ I & 0 \end{pmatrix}$$ on $\mathcal{H}_{0}=H^{2}\oplus H^{2}$. Then we show in Section \[Han\] that the subspaces $\mathcal{K}$ described axiomatically in Theorem \[ThmRef.1\] above, and which are further assumed maximal, are in $1$–$1$ correspondence with finite positive Borel measures on $\left[ -1,1\right] $, such that $n\mapsto\int x^{n}\,d\mu\left( x\right) $ is in $\ell^{2}$. For those examples, the condition $\ker\left( W_{\mu}\right) =\left\{ 0\right\} $ holds if and only if $\operatorname*{supp}\left( \mu\right) $ has accumulation points in $\left( -1,1\right) $. It holds, for example, if $\mu$ is the restriction to $\left[ -1,1\right] $ of Lebesgue measure. \[Rep\]Reproducing kernels ========================== In the proof of part (\[ThmRef.1(3)\]) of Theorem \[ThmRef.1\], we used the reflection $J$ to arrive at a new Hilbert space $\mathcal{H}\left( \mathcal{K}_{s}\right) $. Recall that $\mathcal{K}_{s}$ is the closure of $C_{c}\left( -1,1\right) $ in the norm $\left\| \,\cdot\,\right\| _{s}$ defined as in (\[eqInt.9\]) from the Knapp–Stein operator $A_{s}$. But in part (\[ThmRef.1(2)\]) of Theorem \[ThmRef.1\], we showed that the system $\left( \mathcal{H}\left( \mathcal{K}_{s}\right) ,W,S\right) $ is determined uniquely from $\left( \mathcal{K}_{s},J\right) $ up to unitary equivalence. In proving part (\[ThmRef.1(3)\]), we selected a particular version of $\mathcal{H}\left( \mathcal{K}_{s}\right) $ which turned out to contain distributions, specifically, we showed that $\left\{ \delta^{\left( n\right) }=\left( \frac{d\,}{dx}\right) ^{n}\delta\mathrel{;}n=0,1,\dots\right\} $ forms an orthogonal basis in $\mathcal{H}\left( \mathcal{K}_{s}\right) $. Our interpretation of this is that we make the Taylor expansion around $x=0$ into an orthogonal expansion relative to the inner product in $\mathcal{H}\left( \mathcal{K}_{s}\right) $. But there is an alternative construction of $\mathcal{H}\left( \mathcal{K}_{s}\right) $ consisting of analytic functions in $$D:=\left\{ z\in\mathbb{C}\mathrel{;}\left| z\right| <1\right\} . \label{eqRep.1}$$ This is a Hilbert space $\mathcal{H}_{\mathrm{rep}}\left( s\right) $ constructed as a reproducing kernel Hilbert space from the kernel$$Q_{s}\left( z,w\right) =\left( 1-z\bar{w}\right) ^{s-1},\qquad\left( z,w\right) \in D\times D. \label{eqRep.2}$$ It is known that there is a unique Hilbert space $\mathcal{H}_{\mathrm{rep}}\left( s\right) $ consisting of analytic functions on $D$ such that$$f\left( w\right) =\left\langle Q_{s}\left( \,\cdot\,,w\right) ,f\right\rangle _{\mathcal{H}_{\mathrm{rep}}\left( s\right) }, \label{eqRep.3}$$ where $\left\langle \,\cdot\,,\,\cdot\,\right\rangle _{\mathcal{H}_{\mathrm{rep}}\left( s\right) }$ is the inner product of this Hilbert space. It has the monomials $\left\{ z^{n}\mathrel{;}n=0,1,2,\dots\right\} $ as an orthogonal basis, and we refer to [@ShSh62] and [@Aro50] for more details on these Hilbert spaces. It will be convenient for us to denote the kernel functions in $\mathcal{H}_{\mathrm{rep}}\left( s\right) $, $$q_{w}\left( z\right) :=\left( 1-\bar{w}z\right) ^{s-1}. \label{eqRep.4}$$ An application of (\[eqRep.3\]) then yields$$Q_{s}\left( w_{1},w_{2}\right) =\left\langle q_{w_{1}},q_{w_{2}}\right\rangle _{\mathcal{H}_{\mathrm{rep}}\left( s\right) }. \label{eqRep.5}$$ \[CorRep.1\]The two Hilbert spaces $\mathcal{H}\left( \mathcal{K}_{s}\right) $ and $\mathcal{H}_{\mathrm{rep}}\left( s\right) $, $0<s<1$, are naturally isomorphic with a unitary isomorphism$$T\colon\mathcal{H}\left( \mathcal{K}_{s}\right) \longrightarrow \mathcal{H}_{\mathrm{rep}}\left( s\right) \label{eqRep.6}$$ which intertwines the respective selfadjoint scaling operators$$\left( S_{a}f\right) \left( x\right) =a^{s+1}f\left( a^{2}x\right) \label{eqRep.7}$$ and$$\left( S_{a}^{\mathbb{C}}F\right) \left( z\right) =a^{s-1}F\left( a^{-2}z\right) , \label{eqRep.8}$$ for $f\in\mathcal{H}\left( \mathcal{K}_{s}\right) $, $x\in\mathbb{R}$, $F\in\mathcal{H}_{\mathrm{rep}}\left( s\right) $, $z\in D$, $a>1$. Specifically, we have $$TS_{a}^{{}}=S_{a}^{\mathbb{C}}T. \label{eqRep.9}$$ While it is possible to give a direct proof along the lines of the last two pages in section 9 of [@JoOl99], we will derive the result here as a direct corollary to Theorem \[ThmRef.1\](\[ThmRef.1(2)\]), i.e., the uniqueness up to unitary equivalence. Given $a>1$, we already established the system $\left( \mathcal{H}\left( \mathcal{K}_{s}\right) ,W,S_{a}\right) $ in part (\[ThmRef.1(3)\]) of Theorem \[ThmRef.1\]. We wish to show that there is a second system $$\left( \mathcal{H}_{\mathrm{rep}}^{{}}\left( s\right) ,W_{{}}^{\mathbb{C}},S_{a}^{\mathbb{C}}\right) ,\qquad W_{{}}^{\mathbb{C}}=W_{s}^{\mathbb{C}}, \label{eqRep.10}$$ which also satisfies axioms (\[eqThmRef.1(3)\])–(\[eqThmRef.1(4)\]) in part (\[ThmRef.1(2)\]). The $s$-dependence of $W=W_{s}$ will be suppressed in the proof for simplicity. For $S_{a}^{\mathbb{C}}$ we take the transformation defined in (\[eqRep.8\]) above, and we get $W^{\mathbb{C}}\colon\mathcal{K}_{s}\rightarrow\mathcal{H}_{\mathrm{rep}}\left( s\right) $ by the following formula:$$\left( W^{\mathbb{C}}k\right) \left( z\right) =\int_{-1}^{1}k\left( x\right) \left( 1-xz\right) ^{s-1}\,dx \label{eqRep.11}$$ for $k\in\mathcal{K}_{s}$, and $z\in D$. To see that $S_{a}^{\mathbb{C}}$ in (\[eqRep.8\]) is selfadjoint in $\mathcal{H}_{\mathrm{rep}}\left( s\right) $, we compute the inner products as follows:$$\begin{aligned} \left\langle S_{a}^{\mathbb{C}}q_{w_{1}}^{{}},q_{w_{2}}^{{}}\right\rangle _{\mathrm{rep}} & =a^{s-1}\left\langle q_{w_{1}}\left( a^{-2}\,\cdot\,\right) ,q_{w_{2}}\right\rangle _{\mathrm{rep}}\\ & =a^{s-1}\left\langle q_{a^{-2}w_{1}}\left( \,\cdot\,\right) ,q_{w_{2}}\right\rangle _{\mathrm{rep}}\\ & =a^{s-1}Q_{s}\left( a^{-2}w_{1},w_{2}\right) \\ & =a^{s-1}\left( 1-a^{-2}w_{1}\bar{w}_{2}\right) ^{s-1}\\ & =a^{s-1}Q_{s}\left( w_{1},a^{-2}w_{2}\right) \\ & =\left\langle q_{w_{1}}^{{}},S_{a}^{\mathbb{C}}q_{w_{2}}^{{}}\right\rangle _{\mathrm{rep}}\text{\qquad for all }w_{1},w_{2}\in D.\end{aligned}$$ Since the kernel functions $\left\{ q_{w}^{\left( s\right) }\mathrel{;}w\in D\right\} $ are dense in $\mathcal{H}_{\mathrm{rep}}\left( s\right) $ by construction, we conclude that $S_{a}^{\mathbb{C}}$ is indeed selfadjoint in $\mathcal{H}_{\mathrm{rep}}\left( s\right) $ when $a>1$ and $0<s<1$. We now show that $W^{\mathbb{C}}\colon\mathcal{K}_{s}\rightarrow \mathcal{H}_{\mathrm{rep}}\left( s\right) $ in (\[eqRep.11\]) is contractive. For $k\in\mathcal{K}_{s}$, we have $$\begin{aligned} \left\| W^{\mathbb{C}}k\right\| _{\mathrm{rep}}^{2} & =\int_{-1}^{1}\int_{-1}^{1}\overline{k\left( x\right) }\left\langle q_{x},q_{y}\right\rangle _{\mathrm{rep}}k\left( y\right) \,dx\,dy\\ & =\int_{-1}^{1}\int_{-1}^{1}\overline{k\left( x\right) }\left( 1-xy\right) ^{s-1}k\left( y\right) \,dx\,dy\\ & =\int_{\mathbb{R}}\overline{k\left( x\right) }A_{s}Jk\left( x\right) \,dx\\ & =\left\langle k,Jk\right\rangle _{\mathcal{H}_{s}}\leq\left\| k\right\| _{s}^{2},\end{aligned}$$ which shows that $W_{s}^{\mathbb{C}}$ is contractive as claimed. But we also proved that$$\left\langle W^{\mathbb{C}}k_{1},W^{\mathbb{C}}k_{2}\right\rangle _{\mathrm{rep}}=\left\langle k_{1},Jk_{2}\right\rangle _{\mathcal{H}_{s}}$$ for all $k_{1},k_{2}\in\mathcal{K}_{s}\subset\mathcal{H}_{s}$. Hence$$\left( W^{\mathbb{C}}\right) ^{\ast}W^{\mathbb{C}}=P_{s}JP_{s}, \label{eqRep.12}$$ where $P_{s}$ denotes the projection of $\mathcal{H}_{s}$ onto $\mathcal{K}_{s}$. Hence axiom (\[eqThmRef.1(4)\]) in the statement of Theorem \[ThmRef.1\](\[ThmRef.1(2)\]) is also satisfied. We leave the verification of$$S_{a}^{\mathbb{C}}W_{{}}^{\mathbb{C}}=W_{{}}^{\mathbb{C}}UP_{s}^{{}} \label{eqRep.13}$$ from (\[ThmRef.1(2)\])(\[eqThmRef.1(3)\]) to the reader. The conclusion of Corollary \[CorRep.1\] is now immediate from Theorem \[ThmRef.1\](\[ThmRef.1(2)\]). Let $T\colon\mathcal{H}\left( \mathcal{K}_{s}\right) \rightarrow \mathcal{H}_{\mathrm{rep}}\left( s\right) $ be the unitary isomorphism from (\[eqRep.9\]) in the statement of Corollary \[CorRep.1\]. We saw in Theorem \[ThmRef.1\](\[ThmRef.1(2)\]) that$$TW_{s}^{{}}=W_{s}^{\mathbb{C}}T.$$ Recall that $\delta^{\left( n\right) }=\left( \frac{d\,}{dx}\right) ^{n}\delta$ is in $\mathcal{H}\left( \mathcal{K}_{s}\right) $, and we conclude that $$T\left( \delta^{\left( n\right) }\right) \left( z\right) =\left( s-1\right) \left( s-2\right) \cdots\left( s-n\right) z^{n}.$$ Since $T$ is isometric, and $$\left\| \delta^{\left( n\right) }\right\| _{\mathcal{H}\left( \mathcal{K}_{s}\right) }^{2}=\left( 1-s\right) \cdots\left( n-s\right) n!\,,$$ we conclude that$$\left\| z^{n}\right\| _{\mathcal{H}_{\mathrm{rep}}\left( s\right) }^{2}=\frac{n!}{\left( 1-s\right) \left( 2-s\right) \cdots\left( n-s\right) }.$$ We have proved the following \[CorRep.2\]Elements of $\mathcal{H}_{\mathrm{rep}}\left( s\right) $ may be characterized by the orthogonal expansion$$\begin{aligned} f\left( z\right) & =\sum_{n=0}^{\infty}c_{n}z^{n},\\ \left\| f\right\| _{\mathcal{H}_{\mathrm{rep}}\left( s\right) }^{2} & =\sum_{n=0}^{\infty}\left| c_{n}\right| ^{2}\frac{n!}{\left( 1-s\right) \left( 2-s\right) \cdots\left( n-s\right) }.\end{aligned}$$ \[Har\]The Hardy space $H^{2}\left( \mathbb{T}\right) $ ========================================================= In this section, we return to the space $L^{2}\left( \mathbb{T}\right) $ and its subspace $H^{2}\left( \mathbb{T}\right) $ introduced in Section \[Int\]. Relative to the reflection $Jf\left( z\right) =f\left( \bar {z}\right) $, $f\in L^{2}\left( \mathbb{T}\right) $, we describe a family of positive subspaces defined from $H^{2}\left( \mathbb{T}\right) $. The individual subspaces $\mathcal{K}\left( b\right) $ are positive relative to $J$ and indexed by some function, $b$, say, in $H^{\infty}\left( \mathbb{T}\right) $. However, unless $b\equiv1$, the subspace $\mathcal{K}\left( b\right) $ is not shift invariant. We first return to the axiomatic setup from Section \[Int\], and we derive a formula for the contractive operator$$W\colon\mathcal{K}\longrightarrow\mathcal{H}\left( \mathcal{K}\right) \label{eqHar.1}$$ constructed from a given positive subspace $\mathcal{K}\subset\mathcal{H}_{0}$. Let $\mathcal{H}_{0}$ be a Hilbert space, and let $J$ be a period-$2$ unitary operator in $\mathcal{H}_{0}$. Let $\mathcal{H}_{\pm}$ be the $J$-eigenspaces corresponding to eigenvalues $\pm1$, and let $P_{\pm}$ be the respective projections onto $\mathcal{H}_{\pm}$, specifically$$P_{\pm}=\frac{1}{2}\left( I\pm J\right) . \label{eqHar.1prime}$$ We say that a closed subspace $\mathcal{K}\subset\mathcal{H}_{0}$ is *positive* if$$\left\langle k,Jk\right\rangle \geq0\text{\qquad for all }k\in\mathcal{K}. \label{eqHar.2}$$ In Section \[Int\], we proved the following: \[LemHar.1\] 1. \[LemHar.1(1)\]There is a $1$–$1$ correspondence between the following data and 1. \[LemHar.1(1)(1)\]closed positive subspaces $\mathcal{K}$, and 1. \[LemHar.1(1)(2)\]closed subspaces $\mathcal{K}_{+}\subset \mathcal{H}_{+}$, and contractive linear operators$$\Lambda\colon\mathcal{K}_{+}\longrightarrow\mathcal{H}_{-}. \label{eqHar.3}$$ 2. \[LemHar.1(2)\]Given set$$\mathcal{K}_{+}:=P_{+}\mathcal{K}, \label{eqHar.4}$$ and$$\Lambda\left( P_{+}k\right) :=P_{-}k\text{\qquad for }k\in\mathcal{K}. \label{eqHar.5}$$ 3. \[LemHar.1(3)\]Given set $\mathcal{K}:=G\left( \Lambda\right) =$ the graph of the contraction $\Lambda$ in i.e.,$$\mathcal{K=}\left\{ k_{+}\oplus\Lambda k_{+}\mathrel{;}k_{+}\in \mathcal{K}_{+}\right\} . \label{eqHar.6}$$ While the details are essentially in Section \[Int\], we sketch (\[LemHar.1(1)(1)\]) $\leftrightarrow$ (\[LemHar.1(1)(2)\]). (\[LemHar.1(2)\]) Given (\[LemHar.1(1)(1)\]), and defining $\mathcal{K}_{+}$ and $\Lambda$ by (\[eqHar.4\])–(\[eqHar.5\]), we saw that $\mathcal{K}_{+}$ is *closed,* and that, by (\[eqHar.2\]), $\Lambda$ is well-defined and contractive. (\[LemHar.1(3)\]) Given (\[LemHar.1(1)(2)\]), the subspace $\mathcal{K}$ in $\mathcal{H}_{0}$, defined in (\[eqHar.6\]), is *positive.* Indeed, if $k=k_{+}+\Lambda k_{+}$, $k_{+}\in\mathcal{K}_{+}$, then$$\left\langle k,Jk\right\rangle =\left\| k_{+}\right\| ^{2}-\left\| \Lambda k_{+}\right\| ^{2}\geq0, \label{eqHar.7}$$ since $\Lambda$ is assumed contractive. We also easily check that $\mathcal{K}$ in (\[eqHar.6\]) is *closed* when (\[LemHar.1(1)(2)\]) holds, i.e., $\mathcal{K}_{+}$ is closed, and the operator $\Lambda$ in (\[eqHar.3\]) is contractive. \[CorHar.2\]Let $\mathcal{K}\subset\mathcal{H}_{0}$ be a closed positive subspace as defined in Lemma from a given $J$. Let $\Lambda\colon\mathcal{K}_{+}\rightarrow\mathcal{H}_{-}$ be the corresponding contraction with closed domain $\mathcal{K}_{+}\subset\mathcal{H}_{+}$, and set$$\mathcal{N}_{+}=\left\{ k_{+}\in\mathcal{K}_{+}\mathrel{;}\Lambda^{\ast }\Lambda k_{+}=k_{+}\right\} . \label{eqHar.8}$$ Let$$\mathcal{H}_{+}\left( \Lambda\right) =\left( \mathcal{K}_{+}/\mathcal{N}_{+}\right) \sptilde\label{eqHar.9}$$ be the Hilbert space obtained by completing the quotient space $\mathcal{K}_{+}/\mathcal{N}_{+}$ relative to the Hilbert norm$$k_{+}\longmapsto\left\| \left( I-\Lambda^{\ast}\Lambda\right) ^{\frac{1}{2}}k_{+}\right\| , \label{eqHar.10}$$ and let$$W_{+}\colon\mathcal{K}_{+}\longrightarrow\mathcal{K}_{+}/\mathcal{N}_{+}\longrightarrow\mathcal{H}_{+}\left( \Lambda\right) \label{eqHar.11}$$ be the natural contractive mapping. Then$$W_{+}=P_{\mathcal{K}}P_{+}\left( I-\Lambda^{\ast}\Lambda\right) ^{\frac {1}{2}}P_{+}P_{\mathcal{K}}, \label{eqHar.12}$$ where $P_{\mathcal{K}}$ denotes the projection of $\mathcal{H}_{0}$ onto $\mathcal{K}$, and $P_{\pm}$ are given by Finally there is a unitary isomorphism$$T\colon\mathcal{H}_{+}\left( \Lambda\right) \longrightarrow\mathcal{H}\left( \mathcal{K}\right)$$ which is determined by the formula$$W=TW_{+}P_{+}P_{\mathcal{K}}. \label{eqHar.13}$$ Let $\mathcal{K}$ be a positive subspace, and let $\Lambda$ be the corresponding contraction with closed domain $\mathcal{K}_{+}$, see Lemma \[LemHar.1\]. We saw that then $\mathcal{K}=G\left( \Lambda\right) $; and, if $$k=k_{+}+\Lambda k_{+},\qquad k_{+}\in\mathcal{K}_{+}, \label{eqHar.14}$$ then$$\left\langle k,Jk\right\rangle =\left\| k_{+}\right\| ^{2}-\left\| \Lambda k_{+}\right\| ^{2}=\left\langle k_{+},k_{+}-\Lambda^{\ast}\Lambda k_{+}\right\rangle =\left\| \left( I-\Lambda^{\ast}\Lambda\right) ^{\frac{1}{2}}k_{+}\right\| ^{2}. \label{eqHar.15}$$ It follows that the assignment $k_{+}\mapsto k$ then passes to respective quotients$$\mathcal{K}_{+}/\mathcal{N}_{+}\longrightarrow\mathcal{K}/\mathcal{N},$$ where $\mathcal{N}_{+}$ is defined in (\[eqHar.8\]). If $T_{0}$ is the corresponding operator $\mathcal{K}_{+}/\mathcal{N}_{+}\rightarrow \mathcal{K}/\mathcal{N}$ induced by $k_{+}\mapsto k_{+}+\Lambda k_{+}$, then $T_{0}$ is isometric relative to the two new norms, and it passes to the respective completions$$T=\tilde{T}_{0}\colon\underset{\begin{array} [c]{c}\shortparallel\\ \mathcal{H}_{+}\left( \Lambda\right) \end{array} }{\left( \mathcal{K}_{+}/\mathcal{N}_{+}\right) \sptilde}\longrightarrow \underset{\begin{array} [c]{c}\shortparallel\\ \mathcal{H}\left( \mathcal{K}\right) \end{array} }{\left( \mathcal{K}/\mathcal{N}\right) \sptilde}.$$ From (\[eqHar.14\])–(\[eqHar.15\]), we read off formula (\[eqHar.12\]) for the contraction $W_{+}\colon\mathcal{K}_{+}\rightarrow\mathcal{H}_{+}\left( \Lambda\right) $. Using again (\[eqHar.15\]), we conclude that $T$ satisfies (\[eqHar.13\]). Conversely, if $W$ and $W_{+}$ are constructed from $\mathcal{K}$ and $\Lambda$, respectively, then, if we set $TW_{+}k_{+}=Wk$, $k\in\mathcal{K}$, then $T$ is isometric, and extends naturally to a unitary isomorphism of $\mathcal{H}_{+}\left( \Lambda\right) $ onto $\mathcal{H}\left( \mathcal{K}\right) $. \[RemHar.pound\]Recent work of Arveson [@Arv98] suggests a multivariable version of the construction in Section above, i.e., reproducing kernels in several variables, as a candidate for a model in multivariable operator theory. With this in view, one should generalize Corollary above to the case of a system of commuting operators $\Lambda_{i}\colon\mathcal{K}_{+}\rightarrow\mathcal{H}_{-}$, $i=1,\dots,d$, such that$$\left\| \sum_{i=1}^{d}\Lambda_{i}k_{i}\right\| ^{2}\leq\sum_{k=1}^{d}\left\| k_{i}\right\| ^{2}$$ for all $k_{1},\dots,k_{d}$, $k_{i}\in\mathcal{K}_{+}$. To make the connection to the setup in the present Corollary note that the condition of Arveson is equivalent to the operator estimate$$\Lambda_{1}^{{}}\Lambda_{1}^{\ast}+\dots+\Lambda_{d}^{{}}\Lambda_{d}^{\ast }\leq I,$$ and the analogue of our operator from is then $$\left( I-\sum_{i=1}^{d}\Lambda_{i}^{{}}\Lambda_{i}^{\ast}\right) ^{\frac {1}{2}}.$$ The following observations make connections between the reflection-symmetric operator $U$ and the subspace $\mathcal{K}$. Let $\mathcal{H}_{+}$ and $\mathcal{H}_{-}$ be Hilbert spaces, set $$\mathcal{H}_{0}=\mathcal{H}_{+}\oplus\mathcal{H}_{-},\qquad J=\begin{pmatrix} I & 0\\ 0 & -I \end{pmatrix} , \label{eqHar118.1}$$ and let $a\colon\mathcal{H}_{+}\rightarrow\mathcal{H}_{-}$ be an arbitrary operator. Then set $$U=U\left( a\right) =\begin{pmatrix} a^{\ast}a & a^{\ast}\\ -a & aa^{\ast}\end{pmatrix} . \label{eqHar118.2}$$ It follows that$$JU\left( a\right) J=U\left( a\right) ^{\ast}=U\left( -a\right) , \label{eqHar118.3}$$ i.e., $U\left( a\right) $ is reflection-symmetric. Moreover, $U=U\left( a\right) $ satisfies$$U^{\ast}U=\begin{pmatrix} a^{\ast}a + \left( a^{\ast}a \right) ^{2} & 0\\ 0 & aa^{\ast}+\left( aa^{\ast}\right) ^{2}\end{pmatrix} .$$ Conversely, every operator $U\colon\mathcal{H}_{0}\rightarrow\mathcal{H}_{0}$ which satisfies$$JUJ=U^{\ast}, \label{eqHar118.4}$$ and$$U^{\ast}U=\left( \begin{tabular} [c]{c|c}$\operatorname*{operator}_{1}$ & $0$\\\hline $0$ & $\operatorname*{operator}_{2}$\end{tabular} \right) \label{eqHar118.5}$$ relative to the decomposition (\[eqHar118.1\]) is of the form $$U=\begin{pmatrix} s_{1} & a^{\ast}\\ -a & s_{2}\end{pmatrix} \label{eqHar118.6}$$ for some operator $a\colon\mathcal{H}_{+}\rightarrow\mathcal{H}_{-}$, and for two selfadjoint operators $s_{1}$ and $s_{2}$ in the respective Hilbert spaces $\mathcal{H}_{+}$ and $\mathcal{H}_{-}$, and satisfying the intertwining relation:$$as_{1}=s_{2}a. \label{eqHar118.7}$$ Returning to the classical example from Section \[Int\] above, let $\mathcal{H}_{0}:=L^{2}\left( \mathbb{T}\right) $, and set$$Jf\left( z\right) :=f\left( \bar{z}\right) ,\qquad f\in L^{2}\left( \mathbb{T}\right) ,\;z\in\mathbb{T}. \label{eqHar.16}$$ \[ProHar.query\]Let $H^{2}=H^{2}\left( \mathbb{T}\right) $, and $H^{\infty}=H^{\infty}\left( \mathbb{T}\right) $ be the usual Hardy spaces of harmonic analysis. Let $b\in H^{\infty}$ be given, and suppose that $\left\| b\right\| _{\infty}\leq1$. Define the subspace $\mathcal{K}\left( b\right) \subset\mathcal{H}_{0}$ $=L^{2}\left( \mathbb{T}\right) $ as follows:$$\mathcal{K}\left( b\right) =\left\{ \left( 1-b\left( \bar{z}\right) \right) k\left( \bar{z}\right) +\left( 1+b\left( z\right) \right) k\left( z\right) \mathrel{;}k\in H^{2}\right\} \label{eqHar.17}$$ Then $\mathcal{K}\left( b\right) $ is a maximal positive subspace of $\mathcal{H}_{0}$ relative to the given reflection operator $J$ from Moreover, the space $\mathcal{K}\left( b\right) $ is invariant under the shift$$Uf\left( z\right) =zf\left( z\right) ,\qquad f\in L^{2}\left( \mathbb{T}\right) ,\;z\in\mathbb{T},\label{eqHar.18}$$ if and only if $b\equiv1$. In that case, $\mathcal{H}\left( \mathcal{K}\right) $ is one-dimensional, and $S\left( U\right) =0$. The proof is based on Corollary \[CorHar.2\] above. Since $J$ is given by (\[eqHar.16\]) at the outset, the two subspaces $\mathcal{H}_{\pm}\subset L^{2}\left( \mathbb{T}\right) $ are then determined from (\[eqHar.1prime\]), applied to $J$. Let $\mathcal{K}=H^{2}\left( \mathbb{T}\right) $, and set $\mathcal{K}_{\pm}:=P_{\pm}\mathcal{K}$. Then $\mathcal{K}_{\pm}=\mathcal{H}_{\pm}$, where $$\mathcal{K}_{\pm}=\left\{ k\left( z\right) \pm k\left( \bar{z}\right) \mathrel{;}k\in H^{2}\right\} . \label{eqHar.19}$$ Let $b\in H^{\infty}$, $\left\| b\right\| _{\infty}\leq1$, be given, and define $\Lambda=\Lambda_{b}$ by$$\Lambda\left( P_{+}k\right) :=P_{-}\left( bk\right) \text{,\qquad for all }k\in H^{2}. \label{eqHar.20}$$ Then it follows from $\mathcal{K}_{+}=\mathcal{H}_{+}$ that $\Lambda$ is a contractive operator with domain $\mathcal{H}_{+}$ and mapping into $\mathcal{H}_{-}$. The corresponding positive subspace, see Lemma \[LemHar.1\], is that which is given by (\[eqHar.17\]). The space $\mathcal{K}\left( b\right) $ is maximally positive. A positive subspace $\mathcal{K}^{\prime}$ satisfying $\mathcal{K}\left( b\right) \subset \mathcal{K}^{\prime}$ would correspond to a contractive operator $\Lambda^{\prime}$ mapping $\mathcal{H}_{+}$ into $\mathcal{H}_{-}$ and extending $\Lambda$, in the sense that the graph of $\Lambda^{\prime}$ contains that of $\Lambda$. But then $\Lambda=\Lambda^{\prime}$ and therefore $\mathcal{K}\left( b\right) =\mathcal{K}^{\prime}$ by the uniqueness part in Lemma \[LemHar.1\]. This proves that $\mathcal{K}\left( b\right) $ is maximally positive in $L^{2}\left( \mathbb{T}\right) $. The contractive property for the operator $\Lambda=\Lambda_{b}$ in (\[eqHar.20\]) follows from the two assumptions on $b$, i.e., $b\in H^{\infty}$, and $\left\| b\right\| _{\infty}\leq1$. Indeed, if $k\in H^{2}$, then$$\begin{gathered} \left\| P_{-}\left( bk\right) \right\| _{2}^{2}=\left\| \frac{1}{2}\left( -b\left( \bar{z}\right) k\left( \bar{z}\right) +b\left( z\right) k\left( z\right) \right) \right\| _{2}^{2}\\ =\frac{1}{2}\left( \left\| bk\right\| _{2}^{2}-\left| b\left( 0\right) k\left( 0\right) \right| ^{2}\right) \\ \leq\frac{1}{2}\left\| bk\right\| _{2}^{2}\leq\frac{1}{2}\left\| b\right\| _{\infty}^{2}\left\| k\right\| _{2}^{2}\leq\frac{1}{2}\left\| k\right\| _{2}^{2}=\left\| P_{+}k\right\| _{2}^{2}.\end{gathered}$$ This proves that the operator $\Lambda=\Lambda_{b}$ in (\[eqHar.20\]) is indeed well-defined and contractive. We then conclude from Lemma \[LemHar.1\](\[LemHar.1(3)\]) that the corresponding positive subspace $\mathcal{K}\left( b\right) $ is the graph of $\Lambda_{b}$. An application of (\[eqHar.6\]) from Lemma \[LemHar.1\] then finally yields (\[eqHar.17\]) as claimed. If it were the case that $\mathcal{K}\left( b\right) $ ($=G\left( \Lambda_{b}\right) $) were invariant under the shift $U$ of (\[eqHar.18\]), then from Beurling’s theorem, there would be a unitary function $u\in L^{\infty}\left( \mathbb{T}\right) $ such that$$\mathcal{K}\left( b\right) =uH^{2}. \label{eqHar.21}$$ (Recall $u\in L^{\infty}$ is said to be unitary if the corresponding multiplication operator $M_{u}$ on $L^{2}$ is unitary.) But identity in (\[eqHar.21\]) for some unitary $u\in L^{\infty}$ is possible only if the factor $\left( 1-b\left( \bar{z}\right) \right) $ in (\[eqHar.17\]) vanishes identically on $\mathbb{T}$, and it follows therefore that $\mathcal{K}\left( b\right) $ can only be shift-invariant if $b\equiv1$. In this case, $\mathcal{K}\left( b\right) =\mathcal{K=}H^{2}$ reduces to the special case which we studied in Section \[Int\]. In that case, the contraction $\Lambda$ from (\[eqHar.20\]) reduces to $\Lambda\left( P_{+}k\right) =P_{-}k$, and$$\left\langle k,Jk\right\rangle =\left\| P_{+}k\right\| ^{2}-\left\| P_{-}k\right\| ^{2}=\left| c_{0}\right| ^{2}\text{\quad if\quad}k\left( z\right) =\sum_{n=0}^{\infty}c_{n}z^{n}\in H^{2}.$$ Hence $\mathcal{H}\left( \mathcal{K}\right) $ is one-dimensional. Since$$Uk\left( z\right) =zk\left( z\right) =c_{0}z+c_{1}z^{2}+\cdots$$ has zero constant term, the selfadjoint operator $S\left( U\right) $ on $\mathcal{H}\left( \mathcal{K}\right) $, induced from $U$, is zero, and the proof is completed. Elaborating on the abstract setup in Proposition \[ProInt.3\], we conclude with a family of finite-dimensional positive subspaces in $H^{2}\oplus H^{2}$. The simplest situation when a triple $\left( \mathcal{H}_{0},\mathcal{K},J\right) $ arises in an application is the case of the Pick–Nevanlinna interpolation problem. In that case, let$$\mathcal{H}_{0}=\ell_{+}^{2}\oplus\ell_{+}^{2},\qquad J=\begin{pmatrix} I & 0\\ 0 & -I \end{pmatrix} ,$$ $N\in\mathbb{N}$, distinct points $z_{1},\dots,z_{N}\in D=\left\{ z\in\mathbb{C}\mathrel{;}\left| z\right| <1\right\} $, and $w_{1}\dots,w_{N}\in\mathbb{C}$, be given. The Pick–Nevanlinna theorem states that there exists a function $\varphi\in H^{\infty}\left( D\right) $ such that $\varphi\left( z_{i}\right) =w_{i}$ for each $i$, and $\left\| \varphi\right\| _{\infty}\leq$ $1$ if and only if the corresponding $N\times N$ matrix $\left( \frac{1-\bar{w}_{i}w_{j}}{1-\bar{z}_{i}z_{j}}\right) $ is positive semidefinite. We will now assume the latter, and relate it to the $\mathcal{K}$-problem. Then set$$\mathcal{K}:=\left\{ \begin{pmatrix} \left( \sum_{i}c_{i}^{{}}z_{i}^{n}\right) _{n=0_{\mathstrut}}^{\infty}\\ \left( \sum_{i}c_{i}^{{}}w_{i}^{{}}z_{i}^{n}\right) _{n=0}^{\infty ^{\mathstrut}}\end{pmatrix} \mathrel{;}c_{1},c_{2},\dots,c_{N}\in\mathbb{C}\right\} \subset\begin{pmatrix} \ell_{+}^{2}\\ \ell_{+}^{2}\end{pmatrix} ^{\oplus}.$$ It is an $N$-dimensional subspace, and so closed. For general vectors $k=k\left( c\right) $, $c=\left( c_{1},\dots,c_{N}\right) $ in $\mathcal{K}$, the term $\left\langle k,Jk\right\rangle =\left\| P_{+}k\right\| ^{2}-\left\| P_{-}k\right\| ^{2}$ computes out as$$\sum_{n}\left| \sum_{i}c_{i}^{{}}z_{i}^{n}\right| ^{2}-\sum_{n}\left| \sum_{i}c_{i}^{{}}w_{i}^{{}}z_{i}^{n}\right| ^{2}=\sum_{i}\sum_{j}\frac{1-\bar{w}_{i}w_{j}}{1-\bar{z}_{i}z_{j}}\bar{c}_{i}c_{j}\geq0,$$ assuming the Pick–Nevanlinna condition. Since we also work with the $H^{2}$-version of $\ell_{+}^{2}$, we note that the above positive subspace $\mathcal{K}$ has an equivalent form in $\mathcal{H}_{0}=H^{2}\oplus H^{2}$. There we have the reproducing kernel $q_{z}\left( \zeta\right) =\left( 1-\bar{z}\zeta\right) ^{-1}$, and $\mathcal{K}$ then takes the form of column vectors as follows:$$\mathcal{K}=\left\{ \begin{pmatrix} \sum_{i}c_{i}q_{z_{i}}\\ \sum_{i}c_{i}w_{i}q_{z_{i}}\end{pmatrix} \mathrel{;}c_{1},c_{2},\dots,c_{N}\in\mathbb{C}\right\} .$$ The Pick–Nevanlinna problem was stated in terms of the pair $\mathcal{K}$, $J=\left( I\oplus\left( -I\right) \right) $, but if we use instead $J=\left( \begin{smallmatrix} 0 & I\\ I & 0 \end{smallmatrix} \right) $, then it is easy to check that the corresponding condition, $\left\langle k,Jk\right\rangle \geq0$ for $k\in\mathcal{K}$, is now equivalent to the matrix order relation, $\left( \frac{\bar{w}_{i}+w_{j}}{1-\bar{z}_{i}z_{j}}\right) \geq0$, i.e., equivalent to$$\sum_{i=1}^{N}\sum_{j=1}^{N}\bar{c}_{i}\left( \frac{\bar{w}_{i}+w_{j}}{1-\bar{z}_{i}z_{j}}\right) c_{j}\geq0\text{\qquad for all }c_{1},\dots ,c_{N}\in\mathbb{C}.$$ This alternative is in turn equivalent to a solution to the interpolation problem $\varphi\left( z_{i}\right) =w_{i}$ for each $i$, and $\operatorname{Re}\varphi\geq0$ in $D$ for some interpolating analytic function $\varphi$. Hence both of the classical interpolation problems correspond to positivity for a pair $\left( \mathcal{K},J\right) $ where $\mathcal{K}\subset H^{2}\oplus H^{2}$ is as stated, but where $J$ changes from one problem to the other. A nice solution to both problems is presented in the classic paper [@Sar67]. (See also [@FaKo94].) \[Han\]Hankel operators ======================= In this section, we consider the direct sum of the unilateral shift $A$ and its adjoint $A^{\ast}$, i.e., $U=A\oplus A^{\ast}$. If $J=\left( \begin{smallmatrix} 0 & I\\ I & 0 \end{smallmatrix} \right) $, then $JUJ=U^{\ast}$, and we solve the problem of finding the subspaces $\mathcal{K}\subset\ell_{+}^{2}\oplus\ell_{+}^{2}$ which satisfy the positivity (\[eqThmRef.1(2)\]) of Theorem \[ThmRef.1\], and are invariant under $U$. This is analogous to (and yet very different from) the classical solution of Beurling [@Hel95 chapter 6] which gives the invariant subspaces for $A$. Recall the invariant subspaces for $A$ are in $1$–$1$ correspondence with the *inner functions,* i.e., functions $\xi\in H^{\infty}$ such that $\left| \xi\left( e^{i\theta}\right) \right| =1$ $\mathrm{a.e.}$ $\theta\in\left[ -\pi,\pi\right) $. For our present problem with $A\oplus A^{\ast}$, we will first reduce the analysis to considering closed invariant subspaces $\mathcal{K}\subset\ell_{+}^{2}\oplus\ell_{+}^{2}$ which are maximally positive. This reduction follows in fact from an application of Beurling’s theorem. We then show that those invariant subspaces $\mathcal{K}$ are in $1$–$1$ correspondence with positive and finite Borel measures $\mu$ on $\left[ -1,1\right] $ in such a way that the corresponding induced selfadjoint operator $S_{\mu}\left( A\oplus A^{\ast}\right) $, acting on $\mathcal{H}\left( \mathcal{K}\right) $, is unitarily equivalent to multiplication by the real variable $x$ on $L_{\mu}^{2}\left( \left[ -1,1\right] \right) $, i.e., $f\left( x\right) \mapsto xf\left( x\right) $, on the $L^{2}$ space given by $\int_{-1}^{1}\left| f\left( x\right) \right| ^{2}\,d\mu\left( x\right) <\infty$, and defined from a finite positive measure $\mu$ on $\left[ -1,1\right] $. We also make explicit how a subspace $\mathcal{K}=\mathcal{K}_{\mu}$ with the desired properties may be reconstructed from some given measure $\mu$ as specified. We first give some Hilbert-space background: Let $\mathcal{H}$ be a Hilbert space, and let $A$ be a bounded operator in $\mathcal{H}$. Then$$U:=\begin{pmatrix} A & 0\\ 0 & A^{\ast}\end{pmatrix} \text{\qquad on }\mathcal{H}_{0}:=\mathcal{H}\oplus\mathcal{H} \label{eqHan.1}$$ satisfies$$JUJ=U^{\ast} \label{eqHan.2}$$ relative to$$J=\begin{pmatrix} 0 & I\\ I & 0 \end{pmatrix} , \label{eqHan.3}$$ i.e., the operator $J$ on $\mathcal{H}_{0}$ is given by $J\left( h\oplus k\right) =k\oplus h$. This observation also shows that the identity (\[eqHan.2\]) typically does not imply any special property for the operators making up $U$. On the other hand, the example in Section \[Ref\] had $U$ unitary relative to the original Hilbert space $\mathcal{H}_{0}$. We wish to compute the correspondence $U\mapsto S\left( U\right) $ of Theorem \[ThmRef.1\] in the case of (\[eqHan.1\]) and (\[eqHan.3\]). Given a subspace $\mathcal{K}\subset\mathcal{H}_{0}$ such that$$U\left( \mathcal{K}\right) \subset\mathcal{K}, \label{eqHan.3prime}$$ we will pass to the new Hilbert space $$\mathcal{H}\left( \mathcal{K}\right) =\left( \mathcal{K}/\mathcal{N}\right) \sptilde, \label{eqHan.3primeprime}$$ where $\mathcal{N}=\left\{ k\in\mathcal{K}\mathrel{;}\left\langle k,Jk\right\rangle =0\right\} $. We say that $\mathcal{K}$ is the *graph* of some operator from a domain $D\left( \Gamma\right) \subset\mathcal{H}$ into $\mathcal{H}$, if$$\begin{pmatrix} 0\\ h \end{pmatrix} \in\mathcal{K}\Longrightarrow h=0. \label{eqHan.3primeprimeprime}$$ But in view of (\[eqHan.3\]), vectors of the form $\left( \begin{smallmatrix} 0\\ h \end{smallmatrix} \right) $ are automatically in $\mathcal{N}$, and so do not contribute to $\mathcal{H}\left( \mathcal{K}\right) $ of (\[eqHan.3primeprime\]). We will suppose, therefore, that the spaces $\mathcal{K}$ of (\[eqHan.3prime\]) have the form $\mathcal{K}=G\left( \Gamma\right) $. Note that the operator $\Gamma$ of which $\mathcal{K}$ is the graph need not have dense domain. The subspace $\mathcal{K}$ is said to be *positive* if $\left\langle k,Jk\right\rangle \geq0$ for $k\in\mathcal{K}$, and *maximally positive* if it is maximal (relative to inclusion) with respect to this property. It follows from (\[eqHan.3\]) that the maximally positive subspaces $\mathcal{K}$ of the form $\mathcal{K}=G\left( \Gamma\right) $ correspond to operators $\Gamma$ which are *dissipative, closed,* and have *dense domain* in $\mathcal{H}$. The corresponding Cayley transform$$\Lambda:=\left( I-\Gamma\right) \left( I+\Gamma\right) ^{-1} \label{eqHan.3iv}$$ is then contractive and everywhere defined on $\mathcal{H}$, and it corresponds to the contraction also denoted $\Lambda$ from Lemma \[LemRef.2\]. This contraction derives from the general contractive transformation$$P_{+}k\longmapsto P_{-}k,\qquad k\in\mathcal{K}, \label{eqHan.3v}$$ where $P_{\pm}=\frac{1}{2}\left( I\pm J\right) $. Using (\[eqHan.3\]) we get$$P_{\pm}\begin{pmatrix} h\\ \Gamma h \end{pmatrix} =\frac{1}{2}\begin{pmatrix} h\pm\Gamma h\\ h\pm\Gamma h \end{pmatrix} \text{\qquad for }h\in D\left( \Gamma\right) ,$$ and so$$\left\| P_{\pm}\begin{pmatrix} h\\ \Gamma h \end{pmatrix} \right\| =\frac{1}{\sqrt{2}}\left\| h\pm\Gamma h\right\| .$$ Since (\[eqHan.3v\]) is contractive, it follows that $\Lambda$ in (\[eqHan.3iv\]) is well-defined and also contractive. Let $A$ in (\[eqHan.1\]) be the unilateral shift. Then of course $U$ will not even be normal. Nonetheless, the possibilities for reflection symmetry yield a richer family, and we will show here that the possibilities can even be classified, i.e., if $A$ in (\[eqHan.1\]) is the unilateral shift. Let $\mathcal{H}=H^{2}$. We will use both of the representations $f\left( z\right) =\sum_{n=0}^{\infty}c_{n}z^{n}$, and $\left( c_{0,}c_{1},c_{2},\dots\right) $ for elements in $H^{2}$, i.e., the function vs. its Fourier series. Hence $A$ takes alternately the form$$\left( Af\right) \left( z\right) =zf\left( z\right) ,\qquad f\in H^{2},\;z\in\mathbb{T}, \label{eqHan.4}$$ or$$A\left( c_{0},c_{1},c_{2},\dots\right) =\left( 0,c_{0},c_{1},c_{2},\dots\right) ,\qquad\left( c_{n}\right) _{n=0}^{\infty}\in\ell^{2}, \label{eqHan.5}$$ and $A^{\ast}$ given by $A^{\ast}\left( c_{0},c_{1},c_{2},\dots\right) =\left( c_{1},c_{2},c_{3},\dots\right) $. It is immediate that, if $\Gamma$ is an operator in $\mathcal{H}=H^{2}$, with domain $D\left( \Gamma\right) $, and graph $G\left( \Gamma\right) =\left\{ \left( \begin{smallmatrix} h\\ \Gamma h \end{smallmatrix} \right) \mathrel{;}h\in D\left( \Gamma\right) \right\} $, then $\mathcal{K}:=G\left( \Gamma\right) $ satisfies the positivity$$\left\langle k,Jk\right\rangle \geq0\text{\qquad for all }k\in\mathcal{K} \label{eqHan.6}$$ if and only if $\Gamma$ is *dissipative,* meaning $$\operatorname{Re}\left\langle h,\Gamma h\right\rangle \geq0\text{\qquad for all }h\in D\left( \Gamma\right) . \label{eqHan.7}$$ It is easy to show, see, e.g., [@Phil], that if $\Gamma$ is dissipative, then the closure of $G\left( \Gamma\right) $, i.e., $\overline{G\left( \Gamma\right) }$, is also the graph of a dissipative operator, denoted $\bar{\Gamma}$. (An operator is said to be *closed* if its graph is closed.) We will consider subspaces $\mathcal{K}$ which are invariant under $U=\left( \begin{smallmatrix} A & 0\\ 0 & A^{\ast}\end{smallmatrix} \right) $. But if $\mathcal{K}$ is invariant, then so is $\overline {\mathcal{K}}$, and we will restrict attention to closed subspaces, and corresponding closed operators. \[LemHan.1\]Let $U=\left( \begin{smallmatrix} A & 0\\ 0 & A^{\ast}\end{smallmatrix} \right) $ be built from the shift $A$, see and let $\Gamma$ be an operator with domain $D\left( \Gamma\right) $ in $H^{2}$, and graph $G\left( \Gamma\right) $ in $H^{2}\oplus H^{2}$. Then$$U\left( G\left( \Gamma\right) \right) \subset G\left( \Gamma\right) \label{eqHan.8}$$ if and only if $D\left( \Gamma\right) $ is $A$-invariant and $$\Gamma A=A^{\ast}\Gamma\text{\qquad on }D\left( \Gamma\right) . \label{eqHan.9}$$ Since$$U\begin{pmatrix} h\\ \Gamma h \end{pmatrix} =\begin{pmatrix} Ah\\ A^{\ast}\Gamma h \end{pmatrix} \text{\qquad for }h\in D\left( \Gamma\right) ,$$ we see that (\[eqHan.8\]) holds if and only if $\Gamma Ah=A^{\ast}\Gamma h$, which is the conclusion. However, the operators $\Gamma$ satisfying (\[eqHan.9\]) are the Hankel operators. Relative to the standard basis in $H^{2}$, such a $\Gamma$ has the form$$\left( \Gamma x\right) _{n}=\sum_{m=0}^{\infty}\gamma_{n+m}x_{m} \label{eqHan.10}$$ for $n=0,1,\dots$, where $\gamma$ is some sequence, $\gamma\in\ell^{2}$. While the bounded Hankel operators are known, the interesting ones, for reflection positivity, will be unbounded ones. (Recall $\Gamma=\Gamma_{\gamma}$ is bounded in $H^{2}\ $if and only if there is some $\varphi\in L^{\infty}\left( \mathbb{T}\right) $ such that $\gamma_{n}=\hat{\varphi}\left( -n\right) $, $n=0,1,\dots$, see, e.g., [@Pow82].) While we can reduce to the case when $\mathcal{K}=G\left( \Gamma\right) $ is closed in $H^{2}\oplus H^{2}$, the domain $D\left( \Gamma\right) $ is not closed in $H^{2}$, but only dense. \[LemHan.2\]Let $\Gamma=\Gamma_{\gamma}$ be the closed operator defined in when it is assumed that$$\operatorname{Re}\gamma_{n}\geq0\text{\qquad for all }n=0,1,2,\dots. \label{eqHan.12}$$ Then$$\left( I+\Gamma\right) D\left( \Gamma\right) =H^{2}. \label{eqHan.13}$$ It follows from (\[eqHan.10\]) that the condition (\[eqHan.12\]) on the sequence $\left( \gamma_{n}\right) _{n=0}^{\infty}$ is equivalent to $\Gamma_{\gamma}$ being dissipative. Hence, since $\left( \gamma_{n}\right) \in\ell^{2}$, the operator $\Gamma$ has a dense domain $D\left( \Gamma\right) $, and the closure of $\Gamma$ is well-defined. We will work with the closure, and refer to $\Gamma$ as the closed operator. Notice that if $\Gamma$ is defined from a sequence $\left( \gamma_{n}\right) $, then the adjoint operator $\Gamma^{\ast}$ is defined from the sequence $\left( \bar{\gamma}_{n}\right) $; and so, by (\[eqHan.12\]), both are dissipative. In particular,$$\operatorname{Re}\left\langle h,\Gamma^{\ast}h\right\rangle \geq0 \label{eqHan.14}$$ for all $h\in D\left( \Gamma^{\ast}\right) $. To prove (\[eqHan.13\]), suppose $h\perp\left( I+\Gamma\right) D\left( \Gamma\right) $. Then $h\in D\left( \Gamma^{\ast}\right) $, and $\Gamma^{\ast}h=-h$. Since then $\operatorname{Re}\left\langle h,\Gamma^{\ast}h\right\rangle =-\left\| h\right\| ^{2}$, this contradicts (\[eqHan.14\]), unless $h=0$. Hence $\left( I+\Gamma\right) D\left( \Gamma\right) $ is dense in $H^{2}$. But it is also closed since $\Gamma$ is closed and dissipative. \[ThmHan.3\]The maximally positive subspaces $\mathcal{K}\subset H^{2}\oplus H^{2}$ which are invariant under $U=\left( \begin{smallmatrix} A & 0\\ 0 & A^{\ast}\end{smallmatrix} \right) $, $A$ the unilateral shift, have the form$$\mathcal{K}=G\left( \Gamma_{\gamma}\right) \mod{\mathcal{N}}, \label{eqHan.15}$$ where the sequence $\gamma\in\ell^{2}$ satisfies $$2\operatorname{Re}\gamma_{n}=\int_{\mathbb{R}}x^{n}\,d\mu\left( x\right) \label{eqHan.16}$$ for some positive and finite Borel measure $\mu$ on the interval $I=\left[ -1,1\right] \subset\mathbb{R}$. If $\mathcal{K}$ comes from such a measure $\mu$, then $\mu$ is unique, and the pair $\left( \mathcal{H}\left( \mathcal{K}\right) ,S\left( U\right) \right) $ may be taken to be $L^{2}\left( I,d\mu\right) $ for the Hilbert space $\mathcal{H}\left( \mathcal{K}_{\mu}\right) $, and multiplication by $x$ on $L^{2}\left( I,d\mu\right) $ for the induced selfadjoint operator $S_{\mu}\left( U\right) $. We begin with a lemma. \[LemHan154\]Let $\gamma_{n}\in\mathbb{C}$, $n=0,1,2,\dots$, be a sequence such that all the sums$$S_{\gamma}\left( \zeta\right) :=\sum_{n}\sum_{m}\bar{\zeta}_{n}\gamma _{n+m}\zeta_{m}$$ satisfy $S_{\gamma}\left( \zeta\right) \geq0$ for sequences $\left( \zeta_{n}\right) $ which are eventually zero. Let $\mu$ be a positive Borel measure on $I:=\left[ -1,1\right] $ with finite moments$$\gamma_{n}=\int_{-1}^{1}x^{n}\,d\mu\left( x\right) ,\qquad n=0,1,2,\dots.$$ Let $\Gamma$ be the possibly unbounded Hankel operator with symbol sequence $\left( \gamma_{n}\right) $. 1. \[LemHan154(1)\]Then the following are equivalent: 1. \[LemHan154(1)(1)\]$\openone\in D\left( \Gamma\right) $, 2. \[LemHan154(1)(2)\]$e_{n}\left( z\right) :=z^{n}\in D\left( \Gamma\right) $ for *some* $n\in\left\{ 0,1,2,\dots\right\} $, 3. \[LemHan154(1)(3)\]$e_{n}\left( z\right) :=z^{n}\in D\left( \Gamma\right) $ for *all* $n\in\left\{ 0,1,2,\dots\right\} $, and 4. \[LemHan154(1)(4)\]$\left( \gamma_{n}\right) _{n=0}^{\infty}\in\ell^{2}$. 2. \[LemHan154(2)\]If one, and therefore all, the conditions hold, then$$\lim_{n\rightarrow\infty}\left\| \Gamma\left( e_{n}\right) \right\| =0.$$ 3. \[LemHan154(3)\]The conditions are satisfied if$$\int_{-1}^{1}\left( 1-x^{2}\right) ^{-\frac{1}{2}}\,d\mu\left( x\right) <\infty. \label{eqHan154.1}$$ But is more restrictive than in We view $\Gamma=\Gamma_{\gamma}$ as an operator on $H^{2}$, and note that, if $z^{n}\in D\left( \Gamma\right) $, then$$\Gamma\left( z^{n}\right) =\sum_{m=0}^{\infty}\gamma_{n+m}z^{m}.$$ Equivalently, setting $e_{n}\left( z\right) :=z^{n}$, $$\Gamma\left( e_{n}\right) \left( z\right) =\sum_{m}\gamma_{n+m}z^{m}.$$ The equivalence of conditions (\[LemHan154(1)(1)\])–(\[LemHan154(1)(4)\]) of (\[LemHan154(1)\]) is immediate from this. Indeed, if $e_{n}\in D\left( \Gamma\right) $, then $\left\| \Gamma\left( e_{n}\right) \right\| ^{2}=\sum_{k=n}^{\infty}\left| \gamma_{k}\right| ^{2}$. So this decides (\[LemHan154(1)(4)\]); and (\[LemHan154(2)\]) also follows. Hence for (\[LemHan154(3)\]), it is enough to show that (\[LemHan154(1)(1)\]) follows from (\[eqHan154.1\]). Let $\left( c_{0},c_{1},\dots\right) $ be a sequence which is eventually zero. Then $$\begin{gathered} \left| \sum_{n=0}^{\infty}\gamma_{n}c_{n}\right| =\left| \sum_{n=0}^{\infty}\int_{I}x^{n}c_{n}\,d\mu\left( x\right) \right| \leq\int_{I}\sum_{n=0}^{\infty}\left| x^{n}c_{n}\right| \,d\mu\left( x\right) \\ \leq\int_{I}\left( \sum_{n=0}^{\infty}x^{2n}\right) ^{\frac{1}{2}}\left( \sum_{n=0}^{\infty}\left| c_{n}\right| ^{2}\right) ^{\frac{1}{2}}\,d\mu\left( x\right) =\left\| \left( c_{n}\right) \right\| _{\ell^{2}}\cdot\int_{I}\left( 1-x^{2}\right) ^{-\frac{1}{2}}\,d\mu\left( x\right) ,\end{gathered}$$ and the integral on the right is finite by assumption (\[eqHan154.1\]). It follows that the sequence $\left( \gamma_{n}\right) $ defines a bounded linear functional on $H^{2}\simeq\ell_{+}^{2}$, and so it is in $\ell_{+}^{2}$ by Riesz’s theorem. Equivalently, $\Gamma\left( e_{0}\right) \left( z\right) =\sum_{n=0}^{\infty}\gamma_{n}z^{n}$ defines an element of $H^{2}$, and so (\[LemHan154(1)(1)\]) holds, and in fact $\Gamma_{\gamma}$ is densely defined as an operator on $H^{2}$. We now continue with the proof of Theorem \[ThmHan.3\]. Let $\mathcal{K}$ be given, and assume it has the properties stated in the theorem. Then from Theorem \[ThmRef.1\], we know that there is a selfadjoint version $S\left( U\right) $ in a Hilbert space $\mathcal{H}\left( \mathcal{K}\right) $. With the data from Theorem \[ThmRef.1\], we also know that the pair $\left( \mathcal{H}\left( \mathcal{K}\right) ,S\left( U\right) \right) $ is unique up to unitary equivalence. Since the spectral radius of $U$ in the present theorem is clearly one, we get, from Theorem \[ThmRef.1\](\[eqThmRef.1(5)\]), that $\left\| S\left( U\right) \right\| \leq1$. Suppose for the moment that $S\left( U\right) $ is realized as multiplication by $x$ on $L^{2}\left( \mathbb{R},\mu\right) $. Then the spectrum of $S_{\mu}\left( U\right) $ must be contained in $I=\left[ -1,1\right] $, and so the support of $\mu$ must be contained in $I$. We saw in Lemmas \[LemHan.1\] and \[LemHan.2\] that $\mathcal{K}$ must have the desired form (\[eqHan.15\]) for some dissipative operator $\Gamma$ with dense domain $D\left( \Gamma\right) $ in $\mathcal{H}$. Since $G\left( \Gamma\right) $ is mapped into itself by $\left( \begin{smallmatrix} A & 0\\ 0 & A^{\ast}\end{smallmatrix} \right) $, we get the commutation identity (\[eqHan.9\]). Writing out the positivity (\[eqHan.6\]) for $k=\left( \begin{smallmatrix} h\\ \Gamma h \end{smallmatrix} \right) $, $h\in D\left( \Gamma\right) $, $h\left( z\right) =\sum _{n=0}^{\infty}c_{n}z^{n}$, we get$$\begin{gathered} \left\langle k,Jk\right\rangle =2\operatorname{Re}\left\langle h,\Gamma h\right\rangle =2\operatorname{Re}\left( \sum_{n=0}^{\infty}\sum _{m=0}^{\infty}\bar{c}_{n}\gamma_{n+m}c_{m}\right) \label{eqHan.17}\\ =2\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\bar{c}_{n}\operatorname{Re}\left( \gamma_{n+m}\right) c_{m}\geq0.\end{gathered}$$ But this means that the Hamburger moment problem is solvable for the sequence $\left( \operatorname{Re}\left( \gamma_{n}\right) \right) _{n=0}^{\infty}$. If the solution is represented as in (\[eqHan.16\]), then it follows that $S_{\mu}\left( U\right) $ is represented as multiplication by $x$ on $L^{2}\left( \mathbb{R},\mu\right) $, and we saw (using Theorem \[ThmRef.1\](\[eqThmRef.1(5)\])) that this forces $\mu$ to be supported in the interval $I=\left[ -1,1\right] $. Since $\gamma\in\ell^{2}$, it is known from the theory of moments that $\mu$ is unique from $\Gamma_{\gamma}$. We include the argument for why $S_{\mu}\left( U\right) $ is indeed multiplication by $x$ on $L^{2}\left( \mathbb{R},\mu\right) $. Returning to (\[eqHan.17\]), we note that $S\left( U\right) $ is determined from the identity$$\left\langle k,JUk\right\rangle =\left\langle k,S\left( U\right) k\right\rangle _{J}$$ for $k=\left( \begin{smallmatrix} h\\ \Gamma h \end{smallmatrix} \right) $, $h\in D\left( \Gamma\right) $; and we have:$$JUk=\begin{pmatrix} 0 & I\\ I & 0 \end{pmatrix}\begin{pmatrix} A & 0\\ 0 & A^{\ast}\end{pmatrix}\begin{pmatrix} h\\ \Gamma h \end{pmatrix} =\begin{pmatrix} A^{\ast}\Gamma h\\ Ah \end{pmatrix} =\begin{pmatrix} \Gamma Ah\\ Ah \end{pmatrix} .$$ Consider finite sums $h_{1}\left( z\right) =\sum_{n}a_{n}z^{n}$ and $h_{2}\left( z\right) =\sum_{n}b_{n}z^{n}$, and the corresponding restrictions to $z=x\in\mathbb{R}$. Using $k_{1}=\left( \begin{smallmatrix} h_{1}\\ \Gamma h_{1}\end{smallmatrix} \right) $ and $k_{2}=\left( \begin{smallmatrix} h_{2}\\ \Gamma h_{2}\end{smallmatrix} \right) $, we get$$\begin{gathered} \left\langle k_{1},S\left( U\right) k_{2}\right\rangle _{J}=\left\langle h_{1},\Gamma Ah_{2}\right\rangle +\left\langle \Gamma h_{1},Ah_{2}\right\rangle =2\sum_{n}\sum_{m}\bar{a}_{n}\operatorname{Re}\left( \gamma_{n+m}\right) b_{m-1}\\ =\sum_{n}\sum_{m}\bar{a}_{n}\int_{\mathbb{R}}x^{n+m}\,d\mu\left( x\right) \,b_{m-1}=\int_{\mathbb{R}}\overline{h_{1}\left( x\right) }xh_{2}\left( x\right) \,d\mu\left( x\right) .\end{gathered}$$ This concludes the proof of existence. \[Proof of uniqueness in Theorem \]Let $\mu$ be a finite positive Borel measure on $\mathbb{R}$ which is supported in $\left[ -1,1\right] $, and assume that $n\mapsto\int_{-1}^{1}x^{n}\,d\mu\left( x\right) $ is in $\ell^{2}$. We wish to reconstruct $\mathcal{K}=G\left( \Gamma\right) $ such that $\Gamma$ is a closed dissipative operator with dense domain in $H^{2}$. Note that if $\Gamma$ has been found, then $$\left\| \begin{pmatrix} h\\ \Gamma h \end{pmatrix} \right\| _{J}^{2}=\left\langle h,\Gamma h\right\rangle +\left\langle \Gamma h,h\right\rangle =\left\langle h,\left( \Gamma+\Gamma^{\ast}\right) h\right\rangle . \label{eqHan.a}$$ It follows that if $\Gamma\sim\left( \gamma\right) $ for some $\gamma\in \ell^{2}$, then $\left\| \left( \begin{smallmatrix} h\\ \Gamma h \end{smallmatrix} \right) \right\| _{J}$ and therefore the corresponding norm-completion $\mathcal{H}_{J}\left( G\left( \Gamma\right) \right) $ only depends on the sequence $\left( \operatorname{Re}\gamma_{n}\right) $, i.e., from (\[eqHan.a\]), $\Gamma+\Gamma^{\ast}\sim\left( 2\operatorname{Re}\gamma _{n}\right) $. Equivalently, we may assume without loss of generality that the sequence $\left( \gamma_{n}\right) $ is real-valued. Now set $$\gamma_{n}:=\frac{1}{2}\int_{-1}^{1}x^{n}\,d\mu\left( x\right) , \label{eqHan.b}$$ and let $\Gamma$ be the corresponding positive Hankel operator. For domain $D\left( \Gamma\right) $, take the functions $h\in H^{2}$ which derive from corresponding $\phi\in L_{\mu}^{2}\left( \left[ -1,1\right] \right) $ as$$h\left( z\right) =\int_{-1}^{1}\left( 1-xz\right) ^{-1}\phi\left( x\right) \,d\mu\left( x\right) . \label{eqHan.c}$$ Recall $A$ is the unilateral shift, and therefore$$A^{\ast\,n}\gamma=\left( \gamma_{n},\gamma_{n+1},\dots\right) ,$$ or, in function form,$$\left( A^{\ast\,n}\gamma\right) \left( z\right) =\gamma_{n}+\gamma _{n+1}z+\gamma_{n+2}z^{2}+\cdots. \label{eqHan.d}$$ We then set $$\left( \Gamma h\right) \left( z\right) =\sum_{n=0}^{\infty}\left( A^{\ast\,n}\gamma\right) \left( z\right) \int_{-1}^{1}x^{n}\phi\left( x\right) \,d\mu\left( x\right) \label{eqHan.e}$$ and note that $\Gamma$ is a Hankel operator, which is closed with dense domain $D\left( \Gamma\right) \subset H^{2}$ and given by (\[eqHan.c\]). Moreover, $\mathcal{K}=G\left( \Gamma\right) $ has the desired properties, with$$W_{\mu}\begin{pmatrix} h\\ \Gamma h \end{pmatrix} \left( x\right) =h\left( x\right) \text{\qquad for }h\in D\left( \Gamma\right) \subset H^{2}, \label{eqHan.f}$$ and restricting $h$ to $\left( -1,1\right) \subset D$. Moreover, for $\phi\in L_{\mu}^{2}\left( \left[ -1,1\right] \right) $, $$\left( W_{\mu}^{\ast}\phi\right) \left( z\right) =\int_{-1}^{1}\left( 1-xz\right) ^{-1}\phi\left( x\right) \,d\mu\left( x\right) \label{eqHan.g}$$ is the function $h\left( z\right) $ given in (\[eqHan.c\]) above. \[RemHan154boundedness\] The conditions of Lemma are satisfied if $\gamma_{n}=\mathcal{O}\left( \frac{1}{n}\right) $, but, of course, for many examples which are not $\mathcal{O}\left( \frac{1}{n}\right) $ as well. It is known in fact that the Hankel operator $\Gamma_{\gamma}$ is bounded if and only if $\gamma _{n}=\mathcal{O}\left( \frac{1}{n}\right) $. A theorem of Widom [@Wid66] shows further that boundedness of the Hankel operator $\Gamma_{\gamma}$ from $\gamma_{n}=\int_{-1}^{1}x^{n}\,d\mu\left( x\right) $ with $\mu$ a positive Borel measure holds if and only if $\mu$ is a Carleson measure. A positive Borel measure $\mu$ on $I=\left[ -1,1\right] $ is said to be a Carleson measure [@Car62] if and only if $\mu\left( I\setminus\left( -x,x\right) \right) =\mathcal{O}\left( 1-x\right) $ for $0<x<1$. It follows in particular that condition in the Lemma is satisfied whenever $\Gamma_{\gamma}$ is assumed bounded; and further that is more restrictive than requiring that $\left( \gamma_{n}\right) \in\ell^{2}$ where $\left( \gamma_{n}\right) _{n=0}^{\infty}$ denotes the moment sequence of $\mu$. \[RemHan\]The moment problem *with* the finite support constraint seems to have been first studied in Devinatz [@Dev53 Lemma 1, p. 64]. \[CorHan.5\]Let $\mathcal{K}=G\left( \Gamma_{\gamma}\right) $ be a subspace of $H^{2}\oplus H^{2}$ satisfying the conditions in Theorem Let $\mu$ be the measure on $\left[ -1,1\right] $ given by$$2\operatorname{Re}\gamma_{n}=\int_{-1}^{1}x^{n}\,d\mu\left( x\right) \qquad\left( \in\ell^{2}\right) ,$$ and let$$W_{\mu}\colon\mathcal{K}\longrightarrow L^{2}\left( \left[ -1,1\right] ,\mu\right)$$ be the contractive operator which intertwines $A\oplus A^{\ast}$ with multiplication by $x$ on $L^{2}\left( \left[ -1,1\right] ,\mu\right) $, see Theorem Then$$\ker\left( W_{\mu}\right) =\left\{ 0\right\}$$ if and only if $\operatorname*{supp}\left( \mu\right) $ has points of accumulation in $\left( -1,1\right) $. So in particular, we can have $\ker\left( W_{\mu}\right) =\left\{ 0\right\} $ both for measures $\mu$ which are absolutely continuous relative to Lebesgue measure on $\left[ -1,1\right] $, as well as for singular measures. It follows from Theorem \[ThmHan.3\] that $$\left\| W_{\mu}\begin{pmatrix} h\\ \Gamma h \end{pmatrix} \right\| ^{2}=\int_{-1}^{1}\left| h\left( x\right) \right| ^{2}\,d\mu\left( x\right) \text{,\qquad for }h\in H^{2}. \label{eqHan.18}$$ So for some $\left( c_{n}\right) \in\ell_{+}^{2}$, $h\left( z\right) =\sum_{n=0}^{\infty}c_{n}z^{n}$, and we may view $h\left( x\right) $ as the restriction to $\left( -1,1\right) $ of the corresponding function $h\left( z\right) $ defined and analytic in $D=\left\{ z\in\mathbb{C}\mathrel {;}\left| z\right| <1\right\} $. If $\operatorname*{supp}\left( \mu\right) $ has accumulation points in $\left( -1,1\right) $, and $W_{\mu }h=0$, then by (\[eqHan.18\]), $h$ vanishes on a subset of $\operatorname*{supp}\left( \mu\right) $ of full measure. This subset must also have accumulation points, and since $h$ is analytic in $D$, it must vanish identically. To prove the converse, suppose $\operatorname*{supp}\left( \mu\right) $ contains only isolated points. Then $\mu$ must have the form $$\mu=\sum_{n}p_{n}\delta_{x_{n}},$$ where $p_{n}>0$, and $\sum p_{n}<\infty$ and $\sum p_{n}\left( 1-x_{n}^{2}\right) ^{-\frac{1}{2}}<\infty$. Recall $\mu$ is finite, and supported in $\left[ -1,1\right] $. Then pick $h\in H^{2}$, $\left\| h\right\| _{H^{2}}\neq0$, such that $h\left( x_{n}\right) =0$, for example $h\left( z\right) =\left( \prod_{n}\frac{x_{n}-z}{1-x_{n}z}\right) z\left( 1-z^{2}\right) $. Then $h\in\ker\left( W_{\mu}\right) $. The problems addressed in the present paper grew out of earlier joint work with G. Ólafsson [@JoOl98; @JoOl99] as well as earlier work by the present author. We are very grateful to G. Ólafsson for the benefit of ongoing discussions. We are also very grateful to Brian Treadway for excellent manuscript production. [^1]: Work supported in part by the National Science Foundation.

--- abstract: 'We discuss the properties of the distributions of energies of minima obtained by gradient descent in complex energy landscapes. We find strikingly similar phenomenology across several prototypical models. We particularly focus on the distribution of energies of minima in the analytically well-understood $p$-spin-interaction spin glass model. We numerically find non-Gaussian distributions that resemble the Tracy-Widom distributions often found in problems of random correlated variables, and non-trivial finite-size scaling. Based on this, we propose a picture of gradient descent dynamics that highlights the importance of a first-passage process in the eigenvalues of the Hessian. This picture provides a concrete link to problems in which the Tracy-Widom distribution is established. Aspects of this first-passage view of gradient-descent dynamics are generic for non-convex complex landscapes, rationalizing the commonality that we find across models.' author: - 'Horst-Holger Boltz' - Jorge Kurchan - 'Andrea J. Liu' title: Fluctuation Distributions of Energy Minima in Complex Landscapes --- Introduction ============ The notion of an underlying complex energy landscape in glassy, disordered systems is useful  [@goldstein; @stillinger; @wales; @onuchic; @heuer1997; @*heuer2008; @krzakala2007; @berthier2011; @charbonneau2014] to the extent that the landscape can be reduced to relatively few properties that are relevant to observed phenomena. The complexity, which counts stationary points in the landscape (minima, saddles, maxima) is an example of such a property. An energy landscape is complex if the number of stationary points depends exponentially on the system size. An intuitive approach to probing complexity is to do a naive search for minima using gradient descent. [@numrec] One follows an initial configuration along the (negative) gradient flow of the energy until a stationary point (vanishing gradient) is found. Because a numerical descent almost certainly ends in a minimum, gradient descent does not only constitute the simplest form of physical dynamics in a complex landscape, a quench to zero temperature, but also the most intuitive and simplest form of optimization. If one starts with flatly sampled random initial positions (corresponding to infinite-temperature $T=\infty$ configurations), gradient descent has the added advantage of sampling local minima with a probability that can be calculated because it is proportional to the volumes of their basins of attraction [@xu2011; @frenkel2017]. Finally, in addition to being a local optimization strategy, gradient descent is also the archetypal greedy algorithm, particularly if one considers a discretized version as one does with any numerical implementation: in every time step the displacement with the largest expected loss in energy is chosen. Within the field of glassy systems, gradient descent is used to obtain “inherent structures” [@stillinger1984; @*stillinger1985; @sciortino1999; @sastry2001; @debenedetti2001], *i.e.* the minima at the bottom of the local basin of attraction around which the system thermally fluctuates, while in machine learning, gradient descent is the original go-to learning strategy [@lecun2015]. Gradient descent is also used to obtain jammed packings of repulsive soft spheres, which are the least stable packings that are mechanically rigid [@ohern2003; @goodrich2014]. Here we look at the shape of the distribution of minima obtained by gradient descent for several different models, with particular focus on the spherical $p$-spin-interaction spin glass. Such distributions, for example for jamming, have been assumed to be Gaussian [@ohern2003]. Our central finding is that for all of these models, the distributions are non-Gaussian with non-trivial tail exponents on one side that are consistent with the Tracy-Widom distribution. We rationalize this finding with a novel perspective that might be the starting point for an eventual analytical approach. In Sec. II we introduce the models studied. We then present our numerical results in Sec. III and use established results for the $p$-spin model in Sec. IV to formulate a toy process that allows us to understand these numerical results. We close in Sec. V with some final remarks on the applicability of these ideas to other contexts. Models & Complexity =================== We study various models with complex landscapes. A unifying perspective is provided by all of them being random constraint-satisfaction problems, i.e. assemblies of equations or inequalities. Generically, the question of interest is whether a specific realization allows for an assignment of the variables that satisfies all constraints or whether there are frustrations (which are easily introduced in randomized problems) that prevent satisfaction of all of the constraints. Generically there is a (SAT/UNSAT) transition between a phase where a satisfying assignment is possible (SAT) and a phase where this is not possible (UNSAT) upon tweaking the hardness of the satisfaction problem, e.g. by changing a control parameter such as the ratio of (in-)equalities and variables. Versions with discrete (particularly Boolean) variables are of fundamental importance to computer science[@cook1971], whereas SAT/UNSAT transitions in continuous constraint-satisfaction problems are conjectured to form an important universality class [@franz2017] in statistical physics. The focus of our attention is the spherical $p$-spin model which we therefore introduce first, before the $k$-SAT, perceptron and jamming models. #### The p-spin model. Specifically we consider the spherical $p$-spin model[@crisanti1992; @kurchan1996]: *i.e.*, we have $N$ spins $S_i$ whose combined length is constrained to $\sum S_i^2=N$ (leaving effectively $N-1$ degrees of freedom) with an energy functional $$\begin{aligned} H &= \sum_{i_1<i_2<\ldots<i_p} J_{i_1,i_2,\ldots,i_p} S_{i_1} S_{i_2} \ldots S_{i_p}\end{aligned}$$ containing random Gaussian couplings $J$ with mean zero and variance $\langle J^2\rangle_c = N/\#J$. Here, $\#J \sim N^p$ is the number of terms appearing in the energy functional (while adhering to the constraint of ascending indices). We use this convention to account for finite-size effects from lower-order terms, but ultimately only the scaling with $N$ is important. Note that particularly in the older physics literature a different convention is used that introduces an additional factor of two here. This energy is an extensive quantity scaling with system size and we therefore also introduce the corresponding intensive quantity $\varepsilon = E/N$. As the qualitative nature of the energy landscape defined by this functional is independent of $p$ for $p>2$ ($p=2$ corresponds to a convex eigenvalue problem and therefore only has a single, trivially global minimum), we choose to limit ourselves to the numerically most accessible case of $p=3$. Still, the cost of a simple evaluation of the energy inevitably scales as $N^p$. The energy scale $\varepsilon_{th}=-2\sqrt{(p-1)/p}=-\sqrt{8/3}$ is called the threshold energy as it constitutes the upper energy boundary below which an exponentially large number of stationary points exist. This is quantified by looking at the (cumulative) complexities. If we define $\mathcal{N}_k(\varepsilon)$ to be the number of stationary points of index $k$ with an (intensive) energy not larger than $\varepsilon$, the corresponding complexity $\Sigma(\varepsilon)$ is given by $$\begin{aligned} \Sigma(\varepsilon) &= \frac{1}{N} \log \mathcal{N}_k(\varepsilon) \text{.} \label{eq:complex}\end{aligned}$$ The complexity was studied earlier within the TAP approach [@crisanti] and has been the subject of rigorous mathematical analysis in the limit of large $N$ [@auffinger]. Remarkably, a qualitatively similar structure has been found for rather small system sizes by numerical enumeration of the critical points [@mehta]. In this paper, we focus on the shape of the distribution of energies of minima, as obtained by gradient descent for finite systems. This corresponds to the shape of the normalized distribution corresponding to $\mathcal{N}_{k \equiv 0}(\varepsilon)$. The distribution of final energies found as a result of gradient descent for the $p$-spin model is shown in fig. \[fig:combo\_pdf\] (a). For a suitable choice of couplings, the $p$-spin model provides a natural energy landscape for the optimization problem corresponding to a $k$-SAT decision problem [@mezard2002]. The model also provides insight into structural glasses [@cugliandolo1993]. It is also a valuable model in its own right. The overall gestalt of the energy landscape, as captured by the complexities, is the relevant property that drives interest in the $p$-spin model as a prototypical complex energy landscape. Physical systems usually have a well-defined notion of a ground-state energy which sets a lower bound to an extensive number of minima. Additionally, the existence of an upper bound reflects that “over-frustration” of a complex system–it is exponentially hard to construct a state with an energy less favorable than some native scale. #### The $k$-SAT model. The prototypical satisfiability problem is that of Boolean (or propositional) satisfaction, see for example ref.  for an introduction. Given a number $N$ of literals (Boolean variables $s_i$ with $s_i \in \{ \text{\tt TRUE}, \text{\tt FALSE}\}$) and a number $M$ of clauses (combinations of the literals and the fundamental logical operators [OR]{} ($\lor$), [AND]{} ($\land$) and [NOT]{} ($\neg$)) which can always be brought into conjunctive normal form, which means that we consider conjunctions (AND-connected sub-clauses) of disjunctions (OR-connected (possibly negated) literals), the goal is to find a choice of the literals that satisfies the clauses (evaluates to a true statement). Particularly, we focus on the $k$-SAT version of this problem consisting only of random clauses that are disjunctions of exactly $k$ (possibly negated) literals. Interestingly, it turns out that there is a sharp change in the difficulty of the problem with $k$. For $k\leq 2$, the solution (or the existence of a solution) can be found easily in a time that depends polynomially on the problem size (see, for example, [@krom]), whereas the problem is NP-hard[@karp] for larger values of $k$ (for efficiency reasons we limit ourselves to $k=3$), meaning that the question of the existence of such an algorithm with polynomial runtime is an important outstanding problem[@fortnow]. Here, we are not interested in designing a particularly good algorithm. Instead, in analogy to gradient descent, we use a local greedy optimization strategy: pick an unsatisfied clause and an undetermined literal and set the literal to the value satisfying the clause. If at some point there are no undetermined literals left to satisfy an unsatisfied clause, the system is considered unsatisfied; if every clause is satisfied, the system is considered satisfied. The variable controlling the fraction of unsatisfied systems is $M/N$, the ratio of the number of clauses to the number of literals (the solution is obviously trivial if every literal appears in at most one clause). Because the literals are Boolean the distribution of results is not continuous, but the relevant combination $M/N$ becomes continuous in the thermodynamical limit and we will, thus, treat the data as if they were binned continuous data. We perform runs of the greedy algorithm for $k=3$ (ensuring that every literal is used in any clause at most once) with $N=128$ literals for $M=1\ldots 10^3$ and measure the fraction of unsatisfied systems. This data is shown in fig. \[fig:combo\_pdf\] (a). Reduction from $k$-SAT is usually used to prove that other Boolean satisfaction or decision problems are NP-hard; for example, there is a direct connection between the 3-Sat and the 3-coloring of a graph (by means of factor graphs). The optimization problem associated with the $k$-SAT problem is the $p$-spin model discussed above. #### The perceptron model. Generalizing from Boolean to continuous variables, there are two types of constraints: equality constraints ($f(\vec S)=0$) and inequality constraints ($f(\vec S)\geq 0$). Every independent equality constraint reduces the dimension of the solution space by one, meaning that the set solutions to problems with equality constraints is always one of zero measure. This is not only peculiar, but it is also inappropriate for many, if not most actual systems one might want to model: a very simplified descriptions of neurons, for example, is that they give an output if the input is exceeding some threshold (see jamming below for another example). This model[@mcculloch] of neurons is the origin of the perceptron model [@gardner1988]. Following the notation of Ref. , we consider continuous variables $S_i$ ($i=1...N$) subject to linear inequality constraints $h_\mu$ ($\mu=1...M$) such that $h_\mu = \vec \xi_\mu \cdot \vec S - \sigma_\mu \stackrel{!}{>} 0$. We limit the variable space to a sphere, $\vec S^2 = N$, use normally distributed $\vec \xi$ and set $\sigma_\mu \equiv \sigma$. For a given ratio $\alpha=M/N$ there is a critical value $\sigma_c(\alpha)<0 $ that marks the satisfiability transition (lower $\sigma$ corresponding to the phase in which all constraints are satisfied). Tuning $\sigma$ allows for further control of the topology (convex/non-convex) of the energy landscape that is constructed by considering $E(\vec S) = \sum_\mu h_\mu^2 \Theta(-h_\mu)$, transforming the decision into an optimization problem. At low values of $\sigma$ the system is convex and at sufficiently high values (but for $\sigma>\sigma_c$) the system is non-convex. We perform gradient descents on that energy landscape to obtain the distributions of the final energies in both the non-convex (Fig. \[fig:combo\_pdf\](c)) and convex regimes (Fig. \[fig:combo\_pdf\](d)). #### jamming We consider the packing[@ohern2003; @liu2010] of spheres with harmonic repulsions in low ($d=2,3$) dimensions. Starting from randomly placed spheres, the energy is lowered by reducing the overlap of the spheres; the constraints to be satisfied are, thus, of the form $\lvert \vec{r}_{ij} \rvert \geq \sigma_{ij}$ where $\vec{r}_{ij}$ denote the pairwise displacement vectors between particles and $\sigma_{ij}$ are the added particle radii. As a result of the inequality structure of these constraints, the perceptron model provides the appropriate mean-field framework [@franz2015; @franz2016]. The relevant control parameter that sets whether or not an unjammed configuration (which we choose to have $E=0$) is found is the packing fraction, i.e. the ratio of combined volume of the spheres to volume available in the simulation box. For finite dimensions and particle sizes, there is not a sharp satisfiability threshold, but as one increases the packing fraction the fraction of systems for which the descent ends in a jammed configuration increases. The derivative of the satisfaction curve can be interpreted as the distribution of jamming thresholds $\phi_c$ and is shown (as inferred via numerical derivation from the data of ref. ) in Fig. \[fig:combo\_pdf\] (f). Typically, these distributions are used to infer finite-size scaling properties[@ohern2003; @vagberg], such as the scaling of the width of the distribution and of the jamming threshold value in the thermodynamic (large system size) limit. In contrast, we focus on the shape of these satisfiability distributions. Additionally, we consider the distribution of energies of packings (using the same model as e.g. ref.  with $\alpha=2$) prepared[@ridout] at a fixed pressure above the jamming transition, where not all of the constraints are satisfied. These curves are shown in Fig. \[fig:combo\_pdf\](e). Numerical Results ================= {width="1.0\linewidth"} We present the results of gradient descent simulations for all the models studied in Fig. \[fig:combo\_pdf\]. Because shifting and global rescaling of the energy landscape do not qualitatively affect gradient descent, we only present histograms of normalized variables (mean zero, unit variance). The bulk of these simulations was done employing the FIRE algorithm[@bitzek] instead of a naive, direct integration of the equation of motion. This algorithm converges significantly faster, allowing for better statistics and analysis of large deviations. The additional inertial degree of freedom within the FIRE scheme can in some individual cases change the basin of attraction such that the relaxation from a specific initial condition with it leads to a different final minimum than would application of a direct gradient descent (this is also true for gradient descents with different time steps). However in smaller runs, we find no indication that this changes the statistics significantly. This is in line with previous applications of FIRE in similar quenches to zero temperature in jamming. As a visual aid and for comparison, we show the respective numerical probability functions alongside two distributions: (1) the Gaussian (normal) distribution, which is the least biased estimator for the distribution having fixed the mean and variance, and (2) the (normalized) Tracy-Widom distribution, which is characterized by tails that decays more slowly than a Gaussian on one side ($x \ll 0$) and more rapidly than a Gaussian on the other side ($x \gg 0$): $$\begin{aligned} \log W (x) \sim \begin{cases} x^{3/2} & x \ll 0 \\ -x^{3} & x\gg 0 \end{cases}. \label{eq:tw}\end{aligned}$$ There is a striking qualitative similarity across models and system sizes. The distribution functions are trivially similar to the Gaussian around zero, but the large deviations are asymmetric with a soft tail (in our presentation for $x<0$) that decays more slowly than the Gaussian and a hard tail (for $x>0$) that decay more rapidly than the Gaussian. The soft tail seems to be well-described by the Tracy-Widom distribution. The strong commonality across systems and the Tracy-Widom form of the soft tail constitute the main results of this paper. Although there is some additional $N$-dependence not eliminated by normalizing (which partially is to be expected for small systems due to corrections to scaling [@goodrich2016]), the soft tail appears robust in the thermodynamic limit, as we will elaborate below. Interestingly, the notable exception to this is the perceptron in the convex regime that has a trivial basin of attraction. This indicates that the important similarity between the analogous systems is indeed the quench from a flat measure in a complex energy landscape and the exploration of at least partially concave (some eigenvalues of the energy landscape are negative) regions. ![a) Mean and variance of the final energies as found via steepest descent in the $p$-spin as a function of system size $N$ (double logarithmic plot). We find that the energy clearly descents towards the threshold energy from above with a very clear power-law dependence (see inset). b) Collapsed distributions for $N=32,64,128$. Here, we perform collapsing using a tentative non-normalized scaling form that is inspired by the Tracy-Widom distribution. The labels I,II and III indicate the three regions used for the spectral densities in Fig. \[fig:ev\]. Inset: Large deviations, power-laws are found at the far tails of the energy distribution. []{data-label="fig:fs_distro"}](sd_distro.pdf){width="\linewidth"} We now look carefully at finite size effects to see whether the soft tail survives in the limit $N \rightarrow \infty$. We present results for the spherical $p$-spin model, for which we have the best statistics and which is also expected to have very small corrections to scaling due to its structure. This is highlighted by the extremely clear power-laws found for the first two moments of the final energies (see Fig. \[fig:fs\_distro\](A)). Using overlines to denote averages over gradient descent samples (so as not be confused with unbiased averaging over disorder for which we use angular brackets), we find that the finite-size deviations to the energy can be characterized via $$\begin{aligned} \overline{\varepsilon}-\varepsilon_\text{th} &\sim N^{-2/3}\\ \overline{\varepsilon^2}_c & \sim N^{-4/3} \text{.}\end{aligned}$$ \[eq:fs-law\] Studying the large deviation tails, we find that the “soft” tail (corresponding to low energies) has the same asymptotics as in the Tracy-Widom law, but the “hard” tail decays more like a Gaussian and therefore decays considerably more slowly than the Tracy-Widom law, $$\begin{aligned} \log P_\text{empirical} (x) \sim \begin{cases} x^{\approx 3/2} & x \ll 0 \\ -x^{\approx 2} & x\gg 0 \end{cases}\end{aligned}$$ with again the rescaled variable $x=(\varepsilon-\mu_\varepsilon(N))/\sigma_\varepsilon(N)$. This is shown in Fig. \[fig:fs\_distro\](B). From Fig. \[fig:combo\_pdf\] (a) it is hard to tell whether this holds for large $N$ as the shape of the distribution appears to cross over from something close to the Tracy-Widom distribution towards the Gaussian distribution. To address this question, we construct an estimate for the converged shape for very large $N$ using the following procedure. We sample the inverse function $Q_N(c)=C_N^{-1}(c)$ to the empirical cumulative distribution functions (CDFs) $C_N(x)$ for every $N$ (using the $N$-specific normalization) at a number of selected values $c=10^{-5},\ldots,1$. At any given value of $c$, this gives a set of pairs $(1/N,x_c=Q_N(c))$ which we use to extrapolate to $x_c(0)$. We find that the finite size effects in the shape are well described by $x(N)-x_c(0) \propto N^{-1/3}$ (corresponding to $1/N$ corrections in non-normalized variables). As the extrapolation is done at each value independently, this method does not constrain the moments of the final distribution and we therefore normalize it as a final step. The result of this procedure is shown in Fig. \[fig:extrap\], which suggests that although the final distribution is extremely close to a Gaussian, the soft tail does prevail for large $N$. {width="\linewidth"} Rationalization of Results ========================== To gain insight into the finding that the distribution of minima energies is non-Gaussian, we focus on the $p$-spin model. In this model, gradient descent is given by integration of[^1] $$\begin{aligned} \dot S_i &= -(J_{ijk}+J_{jik}+J_{jki}) S_j S_k - z S_i\end{aligned}$$ where $z$ is a Lagrange parameter ensuring the the spherical constraint, $S_i S_i = N$, and is fixed to be $$\begin{aligned} z=-3 \varepsilon\end{aligned}$$ by demanding $S_i \dot S_i = 0$ or, equivalently, $S_i S_i = \text{const}$. The descent terminates once $\dot S_i=0$ and this will be not only a stationary point, but a stable minimum. ![Left panel: Schematic sketch of the Wigner semi-circle expected in a unbiased sampling from the Hessian. The location of the mean is set by the state’s energy, the width corresponds to the threshold energy. The deviations from this average picture at finite system sizes are for the extremal eigenvalues given by a Tracy-Widom distribution with a characteristic finite-size scaling, $\sigma_{\lambda_\text{min}}\sim N^{-2/3}$. Right: eigenvalue distributions in different regimes (atypically low (I), typical (II), atypically high energy (II)), cp. Fig. \[fig:fs\_distro\]. We show the marginal semi-circle law (thin red line) as a visual guide.[]{data-label="fig:ev"}](ev.pdf){width="\linewidth"} To understand the implications of this, it is helpful to also consider the dynamical matrix ${\mathsf M}$ associated with the descent, which is given by the Hessian of the energy function (technically the Lagrange function, but we refrain from reflecting the special nature of the spherical constraint in our wording in the following) $$\begin{aligned} M_{ij} &= (J_{ijk} + \ldots) S_k + z \delta_{ij} \label{eq:hessian}\end{aligned}$$ where the omitted terms correspond to all index permutations of $i,j,k$. We can identify two contributions to the matrix ${\mathsf M}$: a Gaussian random part ($J$ and $S$ are practically independent) and a deterministic shift that only depends on the energy. From this observation, it is straightforward to infer that the spectral density of eigenvalues in the limit of large system sizes is given by a shifted Wigner semicircle law [@wigner] $$\begin{aligned} \rho(\lambda) \,\mathrm{d}\!\lambda &= \frac{1}{2\pi\sigma^2} \sqrt{(2\sigma)^2 - (\lambda-\mu)^2}\,\mathrm{d}\!\lambda\end{aligned}$$ with $\sigma=-3\varepsilon_{th}$ and $\mu = -6\varepsilon$, see left panel of Fig. \[fig:ev\]. If we ignore the shift (the second term of the Hessian in Eq. \[eq:hessian\]) for now, then we know that finite system size causes the edges of the Wigner semicircle to develop fluctuations. It is rather intuitive that these fluctuations have to be asymmetric: finding a lowest eigenvalue that is smaller than the lower boundary of the support of the semi-circle should (for sufficiently large system sizes) be entropically less costly then finding a fluctuation where the lowest eigenvalue is located somewhere within the bulk of the semi-circle; this implies that an extensive number of eigenvalues must lie at atypically large values. This intuition has been made rigorous by establishing that the edge fluctuations are described by the acclaimed Tracy-Widom distribution introduced earlier in Eq. \[eq:tw\] [@tracy1993; @*tracy1994; @deanmajumdar2006; @nadal; @*majumdar2014]. This distribution has gathered considerable interest in recent years as it has been found to appear in many systems of correlated variables that are beyond its original scope within random matrix theory. Interestingly, the charge-like repulsion between the eigenvalues that is that the core of the Tracy-Widom distribution is of purely topological origin and can be understood solely from imposing a non-crossing bias onto random walkers [@grabiner]. As of now, there is no simple closed form representation of the Tracy-Widom distribution, but the distribution of the lowest eigenvalue has a characteristic system-size dependence[^2], $$\begin{aligned} P(\lambda_\text{min}=\lambda) \sim W(-(\lambda+2\sigma-\mu)\sigma^{-1} N^{2/3}) \label{eq:t-w-law}\end{aligned}$$ with $W(x)$ being the Tracy-Widom distribution with tails described by Eq. \[eq:tw\][^3]. Note that this eigenvalue spectrum has two tails with different $N$-scalings: in the soft tail the argument of the exponential scales like $N$ whereas it scales like $N^2$ in the hard tail. This corresponds to the fact that deviations of the lowest eigenvalue to smaller values are entropically suppressed by the definition of the matrix ensemble, but fluctuations to higher values require a displacement of extensively many eigenvalues. The termination of the descent is subject to the gradient and, thus, cannot be understood by the dynamics of the eigenvalues alone, but we can identify a necessary contribution that will get us close to understanding the full dynamics. Once the lowest eigenvalue crosses zero to become positive, the descent is in its final valley and the energy will only change slightly. Neglecting this final part, gradient descent becomes a first-passage problem in the lowest eigenvalue. As time progresses and the system lowers its energy, the eigenvalues will move (with fluctuations) towards higher values while never crossing each other, until the lowest eigenvalue crosses zero so that all eigenvalues are positive. A direct empirical corroboration of the importance of eigenvalue fluctuations from the data, and a connection to problems usually connected to Tracy-Widom-laws, is that the power-laws seen in the finite size effects, see eq. , are consistent with the scaling seen in the Tracy-Widom law, cp. eq. . To start examining the descent from this spectral perspective, we calculate the spectrum of eigenvalues of the Hessian in three different ranges of energies of the minima in Fig. \[fig:fs\_distro\](B). In each of these ranges, the distribution is close to the semicircle expected at the threshold energy in unbiased sampling independent of the energy, with a shift that increases with energy. This is somewhat consistent with the finding that all the states found are close to threshold. However, they are always above the large $N$ threshold, $\varepsilon_\text{th}=-\sqrt{8/3}$, which means that the naively expected value for the lowest eigenvalue is negative and a large deviation is needed to constitute a mechanically stable state. Intuitively, the entropically least expensive way to do this is to aggregate all these eigenvalues closely above zero. This intuition has been made rigorous by an analysis by Dean and Majumdar (DM) [@deanmajumdar2008]. An important physical consequence of this aggregation around zero (forming an integrable singularity in the spectrum) would be an excess of very soft modes which is not only unphysical, but also completely contrary to the empirical findings in physical realizations of disordered systems in general or our data for the $p$-spin in particular. An immediate explanation for why such an aggregation of soft modes is not observed can be given by the sampling bias due to the gradient descent. The measure with which the minima are sampled is the relative size $\Omega$ of their basins of attraction. Both numerically (employing the Einstein method explained in [@xu2011]) and analytically (from a naive reading of the Kac-Rice formula, see for example [@auffinger]), we find that $\Omega \sim \det {\mathsf M}$, i.e. fictitious minima with an abundance of very soft modes would very likely have very small basins of attraction and, thus, not contribute significantly to the empirical distributions. This observation highlights an important and well-known aspect of the gradient descent: it is an inherently out-of-equilibrium process that should be looked at dynamically. Thus, we are not to consider the DM-ensemble with a permanently non-negative spectrum, but a transition from the initial equilibrium spectrum to a non-negative spectrum under the descent dynamics. Given that the average of the spectrum is set by the Lagrange multiplier $z$, i.e. by the energy, the constantly decreasing energy corresponds to an overall drift in the eigenvalues, shifting them towards higher values. We quantify this by expanding $S_i = S_i + \mathrm{d}\!S_i$ and $z = z +\mathrm{d}z$ in the Hessian to first order, which results in $$\begin{aligned} \mathrm{d} M_{ij} &= (J_{ijk} + \ldots ) \,\mathrm{d}S + \mathrm{d}z \delta_{ij} \text{.} \label{eq:mat}\end{aligned}$$ Our strategy to make gainful progress from this spectral perspective is to simplify the matrix dynamics of eq.  by only considering two important factors that must be there: a source of noise and an entropic confinement establishing a well-defined ensemble. The conceptual background of this approach is the seminal insight by Dyson [@dyson] that the equilibrium sampling in random matrix theory [@mehta2004] can be done by deriving the associated Langevin equation $$\begin{aligned} \mathrm{d} M^\text{eq}_{ij} &= \mathrm{d}W - \sigma_M^{-2} M^\text{eq}_{ij} \mathrm{d}t. \label{eq:mat2}\end{aligned}$$ Here, the first term is a Gaussian noise term (with $\mathrm{d}W \sim \sqrt{\beta^{-1}\mathrm{d}t}$ being a Wiener process) and the second term is an entropic spring that ensures that the matrix stays in the correct ensemble. This overall structure corresponds to the effective form of that the actual $p$-spin eigenvalue dynamics must have, aside from the energy-dependent drift ($z$-term) of eq. . The rationale for this is to note that the dynamics of the Hessian without the energy-shift are effectively uncorrelated with the change in energy (omitting index permutations) $\mathrm{d}E=J_{ijk} (S_i S_j \mathrm{d} S_{k} + \ldots)$. Thus, we can reduce the first term in eq.  to a centered Gaussian increment. The second term in eq. \[eq:mat2\] reflects the unavoidable correlations in these updates: there is a well-defined entropic ensemble (the spectrum is a semi-circle at all energies in distribution) and, thus, a restoring drift to this ensemble. Using these increments of the matrix elements, it is straightforward to use matrix perturbation theory to determine the resulting dynamics of the eigenvalues $$\begin{aligned} \mathrm{d}\lambda_n &= \mathrm{d}W + \mathrm{d}t \left[ -\frac{\lambda_n}{\sigma^2_J} + \sum_{n\neq m} \frac{1}{\lambda_n - \lambda_m} \right].\end{aligned}$$ This is Dyson Brownian motion. Here, $\sigma^2_J$ denotes the $\mathcal{O}(N^{-2})$ variance of the couplings. The repulsive third term with its divergence is the technical equivalent of the statement that two eigenvalues do not cross. Thus, the eigenvalues stay in one representation of the permutation group for all times. Comparing the two expressions of eqs  and , it is tempting on first glance to regard them as qualitatively identical, since the descent dynamics will also lead to some random noise (which ultimately will be Gaussian due to the Gaussian distribution of $J$), but with a Hessian that remains in the same well-defined ensemble. The influence of the additional drift term $z \delta_{ij}$ on the eigenvalues might at first seem trivial. However, there is an important difference related to time-translational invariance. In the Dysonian case, everything is in equilibrium and thus invariant in statistics under time-translation and time-reversal. For descent dynamics, these symmetries are trivially broken by the drift. This can easily be handled, but the symmetries are also broken in the noise term. This can be seen if one goes back to earlier analytical approaches [@cugliandolo1993] to the dynamics of the $p$-spin model, where the correlation function $C(t+\Delta t,t)= \langle S_i (t+\Delta t) S_i (t) \rangle$ decays exponentially for $t\gg \Delta t$ with a rate that is inversely proportional to $t$, i.e. we expect $$\begin{aligned} S_i(t+\Delta t) S_i(t) &= 1- \text{const} \, \Delta t/t + \mathcal{O}\left((\Delta t/t)^2\right) \text{.} \end{aligned}$$ Thus, even a temporally coarse-grained version of eq.  that would get rid of correlations in time without drift would not be Markovian, because the strength of noise would be time-dependent. Since we are only interested in first-passage statistics, this is easily mitigated as we can switch to dynamics in a reparametrized time [@cugliandolo1993] $\tau$ with $\mathrm{d}\tau \propto \mathrm{d}t/t$. This logarithmic reparametrization makes the process time-translational invariant. The final missing piece is the energy, which decreases with time during gradient descent. We had to separate this drift from the matrix dynamics to bring the latter into a treatable form, so the relevant process is no longer a first crossing of zero, but the first crossing of a curve given by the energy $\varepsilon(\tau)$. As the fluctuations of the energy (given by the gradient, the first derivative of a Gaussian field) are independent from the fluctuations in eigenvalues (aside from the mean they are given by the random part of the Hessian , the second derivative of a Gaussian field) and of lower order ($\mathcal{O}(N^{-2/3}$ in the eigenvalues, but $\mathcal{O}(N^{-1})$ in the intensive energy) we can replace the actual energy by the asymptotic trajectory $\varepsilon_\infty(\tau)$ for large $N$. This of course not only neglects the dynamical fluctuations along the trajectory, but also the deterministic noise in trajectories from the initial conditions. However, we are interested in the behavior at large times $\tau$, where the effect of the initial configurations is negligible. We can deduce from the structure of the analytical equations [@cugliandolo1993; @franz] that the intensive energy in real-time $\varepsilon_\infty(t)\sim \varepsilon_\text{th}+\text{const} t^{-\gamma}$ asymptotically is a power-law, thus in rescaled time the energy decays exponentially $\varepsilon_\infty(\tau)~\varepsilon_\text{th}+\text{const} {\mathrm{e}^{-\tau/\tau_c}}$. A more pedestrian way to look at this is by directly writing down an equation of motion for the energy $$\begin{aligned} &\partial_t E = \left[\partial_t S_i\right]\left[\partial_{S_i} E\right] \nonumber \\ &= (J_{ijk}J_{ilm} + \ldots) S_j S_k S_l S_m + 3 z J_{ijk} S_i S_j S_k \text{.}\end{aligned}$$ Again changing variables to establish time-translation invariance and performing an average over the disorder we see that $\langle \partial_\tau \varepsilon \rangle = \text{const} + \varepsilon^2$, from which one gathers (using the known asymptotic value) that $\varepsilon_\infty \approx \varepsilon_{th} \operatorname{tanh}(\text{const} \ \tau)$ with the aforementioned asymptotically exponential decay. Putting everything together, we get a model version of the gradient descent as a first-passage process in the standard Dyson dynamics with a time dependent boundary $\varepsilon_\text{model} = \varepsilon_\text{th} \tanh(\tau)$. From this model definition, one can (somewhat [*a posteriori* ]{}admittedly) rationalize the following numerical findings described earlier. (1) It is inherently plausible for the finite-size scaling to be of the same form as the Tracy-Widom distribution because the effective process is indeed one dominated by edge eigenvalue fluctuations; (2) The shape of the distribution (Gaussian on one side and Tracy-Widom on the other) is plausible. The fluctuations of the lowest eigenvalue are of order $N^{-2/3}$ with a hard border to the right, thus there is some time $\tau_0$ where the typical distance is of the same order, so that there is a very high probability for the boundary to not have been crossed. Thus, we know that at $\tau_0$ the fluctuation distribution of the lowest eigenvalue is given by the Tracy-Widom law. However, $\tau_0$ will be close to the actual final time and at small times (up to eigenvalue distances of $\mathcal{O}(N^{-1/2})$) the diffusive part dominates. Thus, it seems within reason that we would see a distribution that effectively looks like a convolution of a Gaussian (the propagator on short times) and the Tracy-Widom law (the fictitious initial condition at $\tau_0$), which would bear the hallmarks we find in the original numerics. Numerical exploration of this effective description is straightforward with various options for sampling this process. One way would be to go back to the initial idea of the Dyson Brownian motion and diagonalize a matrix subject to small noise in time. The non-crossing is manifest in this approach, but diagonalization is a rather costly operation. Alternatively, one might consider event-based Monte-Carlo of the thermal ensemble whose Langevin equation is given by the Dyson Brownian motion. Finally, there is the option to do straightforward integration of the equation of motion with adaptive time-steps that ensure that the trajectories of eigenvalues never cross. Opting for the latter, we find that we can indeed get to satisfying agreement of the numerics by tuning the strength of the white noise inflicted upon the eigenvalues which one can characterize by an effective temperature $T$. A first-principle determination of the specific value of $T$ to be used is beyond the scope of our arguments. A numerical determination is possible, but simulations of spectral trajectories are slow for two reasons: the cost of the diagonalization itself and the need to switch from FIRE to the direct integration to see the dynamics in physical time. We opt for a more pragmatic procedure and simulate the process for a few values (large values of $T$ are slow as increasing the noise induces more collisions); see Fig. \[fig:fpt\] for $N=64$. We see that indeed there is a satisfying agreement between the distributions found from this simple first-passage problem and the real ones in fig. \[fig:combo\_pdf\]. ![Fluctuation distribution as inferred from the first-passage process discussed in the main text for $N=64$ and various values of $T$.[]{data-label="fig:fpt"}](tanh_paper.pdf){width="\linewidth"} We close with a final remark on the contributions to the final energies that we ignored. These are the depth of the final valley after the system has become mechanically stable and then contributions corresponding to energy fluctuations in the first-passage problem. All evidence seems to corroborate that these are of higher order in $1/N$ and almost Gaussian distributed. In this case, they do not contribute to the finite-size scaling and also not to the fluctuation distribution of normalized variables as they would only change the first two cumulants. We are therefore positive that the eigenvalue process outlined here indeed captures the essential mechanism underlying the distribution for the $p$-spin model and, also, the other constraint satisfaction problems discussed earlier. Conclusions =========== We performed gradient descent simulations in several prototypical constraint-satisfaction problems with complex landscapes and found similar asymmetric distributions in the normalized distributions of final energies (fig. \[fig:combo\_pdf\]). These feature a soft tail corresponding to better-than-typical solutions and a hard tail for worse-than-typical solutions. Inspecting in more detail for the spherical $p$-spin spin-glass model, we found that both the finite-size scaling as well as the functional form of the soft tail (fig. \[fig:fs\_distro\]) are reminiscent of the Tracy-Widom distribution, which is usually associated with the fluctuations of extremal eigenvalues in random matrix problems. We made this connection manifest by proposing a novel interpretation of gradient-descent problems as a first-passage process into mechanical stability, i.e. we argue that the energy at which the lowest eigenvalue becomes non-negative is a good proxy for the actual final energy at which the gradient descent terminates with respect to the fluctuation distribution. This is a purely dynamical picture of the out-of-equilibrium gradient-descent process in which typical landscape features such as the basins of attraction are emergent from the random matrix ensemble associated with the dynamical matrix. The very simple nature of the ersatz-process found by reducing the spectral dynamics to their core ingredients could allow for an exact analytical treatment in the future. An open question remains concerning the extent to which the observed phenomenology survives with increasing system size. At least for the $p$-spin model, extrapolation to the large-$N$ limit does lead to a non-Gaussian distribution with the same tail behavior as seen in finite systems. However, this is less clear for the other models studied here, for which it was difficult to obtain comparable statistics. Nevertheless, the perspective of gradient descent as a first-passage process suggests that the highly similar non-Gaussian features seen in the distributions for the other models are not a finite-size effect, and should persist in the thermodynamic limit. The view of the gradient descent process as a first-passage problem could be a rather broadly fruitful one. Most aspects of the (matrix) dynamics of the $p$-spin model are believed to be somewhat general for many complex systems. Additionally, the topological feature that fluctuations towards lower energies (corresponding to minima with atypically soft modes) are substantially easier to find than those towards higher energies (hard modes) should prevail in a vast variety of systems with complex landscapes. This way of thinking should be helpful in understanding phenomenology in experiments such as Ref.  that prominently feature asymmetrical distributions of the fluctuations within the inherent structure landscape. Our reasoning should be applicable to results from finite temperature quenches as long as the initial temperature is sufficiently high that the system is ergodic and the final temperature sufficiently low that the system is confined to a single basin after the quench. Finally, we note that a good understanding of the first-passage into mechanical stability might inspire new ways of tweaking interactions to convert complex landscapes into less rough ones (similar to the methods proposed in ref. ) in order to find better (lower energy) solutions. We end with a caveat: In finite-dimensional models and data, a simple Dysonian random-matrix view as proposed here will necessarily face some issues, one very important one being the existence of sum rules constraining the Hessian, particularly the ones corresponding to mechanical equilibrium. The details of the coordination structure have been argued[@manning2015; @stanifer2018; @benetti2018] to be crucial in understanding essential features of low-dimensional jammed packings such as the scaling of the vibrational density of states. As this is directly linked to the statistics of the extremal eigenvalues, it is very intriguing for future work to study the effect of these constraints (which develop as the system descends in the landscape) on the distributions studied here. Even when such constraints exist, however, the notion is still valid that there is one contributing process to the statistics of descents in disordered landscapes, related to the passage into mechanical stability that we have isolated for the $p$-spin model. We thank S. Teitel for providing us with the data used in Ref.  and S. Ridout for additional isobaric jammed configurations and S. R. Nagel for discussions that inspired this investigation. This work was supported by the “Cracking the glass problem” collaboration grant (348126 to HHB, 454943 to JK and 454945 to AJL), as well as an Investigator grant (327939 to AJL) from the Simons Foundation. [62]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****, ()](\doibase 10.1063/1.1672587) [****,  ()](\doibase 10.1126/science.267.5206.1935) @noop [**]{} (, ) [****,  ()](\doibase 10.1146/annurev.physchem.48.1.545) [****,  ()](\doibase 10.1103/PhysRevLett.78.4051) [****,  ()](\doibase 10.1088/0953-8984/20/37/373101) [****,  ()](\doibase 10.1103/PhysRevE.76.021122) [****, ()](\doibase 10.1103/RevModPhys.83.587) [****,  ()](\doibase 10.1038/ncomms4725) @noop [**]{} (, ) [****,  ()](\doibase 10.1103/PhysRevLett.106.245502) [****,  ()](\doibase 10.1073/pnas.1620497114),  [****,  ()](\doibase 10.1126/science.225.4666.983) [****, ()](\doibase 10.1103/PhysRevB.31.5262) [****,  ()](\doibase 10.1103/PhysRevLett.83.3214) [****,  ()](\doibase 10.1038/35051524) [****,  ()](\doibase 10.1038/35065704) [****,  ()](\doibase 10.1038/nature14539) [****,  ()](\doibase 10.1103/PhysRevE.68.011306) @noop [****,  ()]{} in [**](\doibase 10.1145/800157.805047) (, ) pp. [****,  ()](\doibase 10.21468/SciPostPhys.2.3.019) [****, ()](\doibase 10.1007/BF01309287) [****,  ()](http://stacks.iop.org/0305-4470/29/i=9/a=009) [****,  ()](\doibase 10.1140/epjb/e2003-00325-x) [****, ](\doibase 10.1002/cpa.21422) [****,  ()](\doibase 10.1103/PhysRevE.87.052143) @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevLett.71.173) @noop [ ()]{} [****,  ()](\doibase 10.1002/malq.19670130104) in [**](\doibase 10.1007/978-1-4684-2001-2_9) (, ) pp.  @noop [****,  ()]{} [****, ()](\doibase 10.1007/BF02478259) [****,  ()](\doibase 10.1088/0305-4470/21/1/030) [****,  ()](\doibase 10.1073/pnas.1511134112) [****,  ()](\doibase 10.1146/annurev-conmatphys-070909-104045) [****, ()](\doibase 10.1088/1751-8113/49/14/145001) [****,  ()](\doibase 10.1103/PhysRevE.83.030303) @noop (),  [****,  ()](\doibase 10.1162/neco.2007.19.6.1503) [****,  ()](\doibase 10.1103/PhysRevLett.97.170201) [****,  ()](\doibase 10.1073/pnas.1601858113) [****,  ()](http://www.jstor.org/stable/1970079) [****, ()](\doibase 10.1016/0370-2693(93)91114-3) [****,  ()](\doibase 10.1007/BF02100489) [****,  ()](\doibase 10.1103/PhysRevLett.97.160201) [****,  ()](http://stacks.iop.org/1742-5468/2011/i=04/a=P04001) [****,  ()](http://stacks.iop.org/1742-5468/2014/i=1/a=P01012) in [**](\doibase 10.1016/S0246-0203(99)80010-7), Vol.  (, ) pp. [****,  ()](\doibase 10.1103/PhysRevE.77.041108) [****, ()](\doibase 10.1063/1.1703862) @noop [**]{} (, ) [****,  ()](\doibase 10.1051/jp1:1995201) [****,  ()](\doibase 10.1073/pnas.0600102103) @noop [ ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [^1]: From here on out, we use a summation convention where repeated indices are to summed over. In the spirit of simplification, we also set $J_{ijk}=0$ unless $i<j<k$, wherever it is relevant. [^2]: Note that our conventions are such that the spectrum is intensive. This results in different scalings then the also commonly used extensive spectrum where the width of the semicircle is of order $\sqrt{N}$. [^3]: The Tracy-Widom distribution is usually defined by the fluctuations of the largest eigenvalue, thus $W(x)=F_1(-x)$

--- abstract: | We study classically scale invariant models in which the Standard Model Higgs mass term is replaced in the Lagrangian by a Higgs portal coupling to a complex scalar field of a dark sector. We focus on models that are weakly coupled with the quartic scalar couplings nearly vanishing at the Planck scale. The dark sector contains fermions and scalars charged under dark $SU(2)\times U(1)$ gauge interactions. Radiative breaking of the dark gauge group triggers electroweak symmetry breaking through the Higgs portal coupling. Requiring both a Higgs boson mass of 125.5 GeV and stability of the Higgs potential up to the Planck scale implies that the radiative breaking of the dark gauge group occurs at the TeV scale. We present a particular model which features a long-range abelian dark force. The dominant dark matter component is neutral dark fermions, with the correct thermal relic abundance, and in reach of future direct detection experiments. The model also has lighter stable dark fermions charged under the dark force, with observable effects on galactic-scale structure. Collider signatures include a dark sector scalar boson with mass $\lesssim 250$ GeV that decays through mixing with the Higgs boson, and can be detected at the LHC. The Higgs boson, as well as the new scalar, may have significant invisible decays into dark sector particles. author: - 'Wolfgang Altmannshofer$^1$, William A. Bardeen$^2$, Martin Bauer$^{2,3}$, Marcela Carena$^{2,3,4}$, Joseph D. Lykken$^2$' title: 'Light Dark Matter, Naturalness, and the Radiative Origin of the Electroweak Scale' --- Introduction \[sec:intro\] ========================== The Standard Model (SM) is a renormalizable quantum field theory that makes unambiguous predictions for elementary particle processes over a very large range of energy scales. Apart from a possible metastable vacuum, the SM has no theoretical inconsistencies at least up to the Planck scale at which we expect gravity to become strong and quantum field theories to break down. If this scenario is realized in nature, the Higgs mass parameter seems artificially small compared to the Planck scale. However, in the SM itself the Higgs mass parameter is the only explicit scale in the theory, and therefore it is only multiplicatively renormalized [@Bardeen:1995kv]. An interesting modification of the SM is given by requiring that the Higgs mass term vanishes at some very high energy (UV) scale; in this case it will not be generated by SM radiative corrections at lower scales either. The tree-level potential has only a quartic term, and the full Lagrangian is classically scale invariant. Electroweak symmetry breaking could be triggered, in principle, by the one-loop corrections to the effective potential $$\begin{aligned} \label{eq:veff1} V_\mathrm{eff}(h)=\frac{\lambda}{2} h^4+B\,h^4\,\log(h^2/\mu^2)\,,\end{aligned}$$ in which $\mu$ denotes the renormalization scale and $B$ is a loop suppressed function of the couplings. Such a possibility has been envisioned by Coleman and Weinberg [@Coleman:1973jx]. A very attractive feature of the Coleman-Weinberg (CW) symmetry breaking mechanism is, that for couplings of order 1 at some renormalization scale in the UV, $\mu=\mu_\mathrm{UV}$, the minimum of the potential appears at an exponentially smaller scale $$\begin{aligned} \langle h \rangle \propto \mu_\mathrm{UV}\, e^{-\lambda(\mu_\mathrm{UV})/B}\,.\end{aligned}$$ Therefore, similar to the large disparity between the Planck scale and the confinement scale of QCD, the large disparity between the Planck scale and the electroweak scale is explained through renormalization group running [@Gildener:1976ih]. However, in the SM the CW mechanism is ruled out. The dominant contribution to the effective potential comes from the top quark, which renders it unbounded from below, since it enters the coefficient $B$ in  with a negative sign. In order to overcome the top quark contribution and to reproduce the measured Higgs mass, one would need to extend the SM by bosonic degrees of freedom with sizable couplings to the Higgs [@Dermisek:2013pta; @Hill:2014mqa]. Another motivation for extending the SM is the strong observational evidence for dark matter (DM), plausibly in the form of weakly interacting heavy particles. Even in the absence of a Higgs mass parameter in the UV, such particles will generically introduce additive corrections to the Higgs mass parameter and spoil the CW dynamics in the absence of additional symmetries. This motivates an alternative implementation of the CW mechanism, first proposed by Hempfling [@Hempfling:1996ht]. In this model, the Higgs couples to one extra scalar, which through dynamics of a hidden sector undergoes CW symmetry breaking and communicates the corresponding mass scale through the Higgs portal to the SM. Dark matter can then be given by any of the new hidden sector fields that govern the renormalization group evolution of the scalar potential in the dark sector. There has been a lot of recent interest in models that implement various aspects of these basic ideas [@Foot:2007iy; @Iso:2009ss; @Foot:2010av; @AlexanderNunneley:2010nw; @Iso:2012jn; @Englert:2013gz; @Farina:2013mla; @Heikinheimo:2013fta; @Hambye:2013dgv; @Carone:2013wla; @Farzinnia:2013pga; @Dermisek:2013pta; @Khoze:2013uia; @Tamarit:2013vda; @Gabrielli:2013hma; @Steele:2013fka; @Hashimoto:2013hta; @Holthausen:2013ota; @Hashimoto:2014ela; @Hill:2014mqa; @Radovcic:2014rea; @Khoze:2014xha; @Farzinnia:2014xia; @Pelaggi:2014wba]. Here we will focus on implementations with dark sectors that are fairly simple and thus predictive. In Section \[sec:mot\] we comment on issues of naturalness as applied to classically scale invariant modificiations of the SM, without claiming to resolve these issues. In Section \[sec:TeVDM\] we show, that in extensions of the SM with no explicit mass scales, the combination of a Higgs mass term generated through CW symmetry breaking together with the restriction to have a stable vacuum up to the Planck scale generically sets an upper bound on the dark matter mass scale of the order of a few TeV. Furthermore, the CW mechanism requires sizable couplings for gauge fields in the hidden sector, so that the simplest models in the literature are in addition subject to a lower bound on the DM mass of several hundred GeV. In Section \[sec:model\] we present a model with additional fermions in the hidden sector that can be dark matter candidates with masses at the electroweak scale or below. In Sections \[sec:higgs\], \[sec:DM\] and \[sec:out\], we discuss the collider and dark matter phenomenology of the model. In Section \[sec:out\], we also comment on further implications of this model for the dynamics of galaxy structure formation and a possible first order electroweak phase transition. We conclude in Section \[sec:con\]. The one loop effective potential of the discussed model and the one loop beta functions of the dark sector couplings are collected in Appendices \[sec:Veff\] and \[sec:betafunctions\]. For the beta functions and anomalous dimensions, we follow the methods, conventions and notation of Machacek and Vaughn [@Machacek:1983tz; @Machacek:1983fi; @Machacek:1984zw], with the improvements and extensions introduced by Luo and Xiao [@Luo:2002ey; @Luo:2002ti; @Luo:2002iq]. For the effective potentials, we follow the methods and conventions of Martin [@Martin:2001vx]. There are slight differences of notation in the literature: for example compared to [@Degrassi:2012ry; @Buttazzo:2013uya], our scalar self-coupling is twice as large, and our convention for anomalous dimensions has the opposite sign. Motivation {#sec:mot} ========== A Coleman-Weinberg mechanism as the origin of electroweak symmetry breaking was first considered by Gildener and Weinberg [@Gildener:1976ih]. In the absence of the Higgs mass term, the Lagrangian of the SM exhibits classical scale invariance that is softly broken by quantum effects - the well known scale anomaly. In UV completions of the SM, the physical thresholds associated with new massive states would constitute an explicit breaking of this symmetry. This introduces the need for a fine-tuning of the bare Higgs mass parameter against radiative corrections involving more massive particles. The fact that the Higgs mass parameter is not protected by a symmetry from these radiative corrections is known as the naturalness or hierarchy problem. If the SM is UV completed by a conformal or supersymmetric (SUSY) theory, the Higgs mass parameter is radiatively stable above the scale at which this completion sets in; thus if this scale is not too high, the hierarchy problem is solved. This has led to the expectation that such a UV completion is realized in the vicinity of the electroweak scale. However, the new degrees of freedom predicted by either supersymmetric or conformal UV completions have not been observed, yet. This raises the prospect that the UV scale at which they set in is considerably higher than the electroweak scale, leaving the naturalness problem unresolved. There are a number of experimental observations and theoretical questions, unrelated to the naturalness problem, that point to new high energy scales. Neutrino masses, gauge coupling unification, dark matter, and the expectation of a more fundamental theory of gravity are all expected to introduce new scales and as a consequence introduce an additive renormalization of the Higgs mass parameter. None of these arguments, however, *necessarily* points to a new UV scale relevant to the hierarchy problem. Neutrinos could be Dirac fermions with tiny Yukawa couplings, gauge coupling unification may not occur or may not imply new superheavy states, and dark matter could well be related to the electroweak scale itself. There is still the challenging question of quantum gravity. We know that gravity does not make sense as a fundamental (perturbative) quantum field theory at short distances [@Weinberg:1980kq]. Naively, the Planck scale is expected to correspond at least roughly to a physical threshold where new massive states appear. In string theory this is indeed the case, and one can also argue that the existence of microscopic black holes is enough to require fine-tuning of the Higgs mass parameter [@Dvali:2012wq]. A non-perturbative theory of quantum gravity might avoid this problem, but not if it resembles strongly-coupled gauge theories where new massive states are connected to the scale of strong coupling. At present no mechanism is known that can realize even a toy model for the type of UV completion that would avoid the hierachy problem, despite promising models in 2d [@Dubovsky:2013ira]. On the other hand, all claims about Planckian physics and resulting effects on the renormalization of the Higgs mass parameter are, at this point, speculative. Generic UV completions of the Standard Model certainly have a Higgs naturalness problem [@Tavares:2013dga], but for all we know, spacetime geometry breaks down at the Planck scale, and whether this results in a physical cut-off of relevance to the Higgs mass parameter is an open question. Following the same line of reasoning, it is not clear to what extent the existence of ultra-Planckian Landau poles, as occurs for the hypercharge gauge coupling of the SM, should be regarded as a fundamental issue. In particular, our semi-classical understanding of gravity seems to indicate that such Landau poles are unobservable; the requisite scattering experiments would presumably be dominated by black hole production long before reaching the regime where incipient strong coupling in the hypercharge interactions would show itself [^1]. It is striking that no couplings of the SM run into a Landau pole *below* the Planck scale, which would be an unambigous sign of a new scale and therefore of the need (presumably) to fine-tune the Higgs mass. In this paper, we assume that all explicit mass parameters vanish at the Planck scale, either as the consequence of the UV completed Planckian theory, or in spite of it. In addition we will focus on extensions of the SM that are weakly coupled and have no vacuum instability below the Planck scale. UV Stability and IR Instability from Dark Sectors {#sec:TeVDM} ================================================= UV Stability ------------ It is an intriguing observation about the Standard Model, that it seems to be consistent up to very high mass scales. Below the Planck scale, the only hint for New Physics within the SM itself is a possible instability of the electroweak vacuum triggered by the large top Yukawa. In the SM, the observed Higgs mass of $m_h \simeq 125.5$ GeV implies a Higgs quartic coupling at the electroweak scale of around $\lambda^\text{SM}_H(m_t) \simeq 0.254$ [@Buttazzo:2013uya; @Degrassi:2012ry]. With this infrared boundary condition, assuming central values for $m_t$ and $\alpha_s$, the Higgs quartic coupling runs negative at scales around $10^{10}$ GeV and stays at a small negative value $\lambda_H \simeq -0.02$ up to the Planck scale, rendering the electroweak vacuum unstable [@Buttazzo:2013uya; @Degrassi:2012ry][^2]. The instability can for example be overcome by the extension of the SM by a complex scalar $\Sigma$ with portal coupling to the Higgs $H$, so that the most general scalar potential reads $$\begin{aligned} \label{eq:SLag} V(H,\Sigma)=\mu_H^2\,H^\dagger H + \frac{\lambda_{H}}{2}(H^\dagger H)^2 + \mu_\Sigma^2\Sigma^\dagger \Sigma+ \frac{\lambda_{\Sigma}}{2}(\Sigma^\dagger \Sigma)^2+ \lambda_{\Sigma H}\Sigma^\dagger \Sigma H^\dagger H \,.\end{aligned}$$ The new scalar can affect the stability of the Higgs potential in two ways: (i) by changing the beta function of the Higgs quartic; (ii) by changing the infrared boundary condition of the Higgs quartic. We briefly review both possibilities. \(i) The portal coupling $\lambda_{\Sigma H}$ gives a positive contribution to the beta function of the Higgs quartic. At the one loop level we have $$\beta_{\lambda_H} = \frac{1}{16\pi^2} \left\{ 12\lambda_H^2 -\lambda_H\left(3 (g^\prime)^2 + 9 g^2\right) + \frac{3}{4}(g^\prime)^4 + \frac{3}{2} (g^\prime)^2g^2 + \frac{9}{4} g^4 +12\lambda_H Y_t^2 - 12Y_t^4 + 2 \lambda_{\Sigma H}^2 \right\} ~,$$ where $g^\prime$ and $g$ are the $U(1)$ and $SU(2)$ gauge couplings, $Y_t$ is the top Yukawa coupling and we neglected the contributions from all other Yukawa couplings. In the SM, the top Yukawa contribution, $- 12Y_t^4$, dominates at low scales and drives the Higgs quartic coupling negative. If $\lambda_{\Sigma H}$ is sufficiently large, it can balance the top contribution and stabilize the vacuum. If in addition the vacuum expectation value of the new scalar vanishes, $\langle \Sigma \rangle =0$, the scalar can be stable and is a dark matter candidate [@Hambye:2007vf; @Clark:2009dc; @Lerner:2009xg; @Gonderinger:2009jp]. \(ii) If the new scalar has a non-vanishing vev, $\langle \Sigma \rangle =w/\sqrt{2}$, the tree level scalar mass matrix in the broken phase of reads $$\begin{aligned} \label{eq:mass1} \mathcal{M}^2=\begin{pmatrix} \lambda_H\,v^2& \lambda_{\Sigma H} \,w\, v\\[3pt] \lambda_{\Sigma H} \,w\, v & \lambda_\Sigma\,w^2 \end{pmatrix}\,.\end{aligned}$$ In the limit $\lambda_\Sigma w^2 \gg \lambda_H v^2$, the light Higgs-like mass eigenstate has a mass $$\begin{aligned} \label{eq:lam} m_h^2 =\left( \lambda_H-\frac{\lambda_{\Sigma H}^2}{\lambda_\Sigma} \right) v^2 +\mathcal{O}\left(\frac{v^4}{w^2}\right)\,.\end{aligned}$$ In order to reproduce a Higgs mass of $125.5$ GeV, the value of $\lambda_H$ at the electroweak scale has to be larger than in the SM. In that way the UV instability can be avoided [@EliasMiro:2012ay; @Lebedev:2012zw; @Batell:2012zw]. Indeed, for a Higgs quartic that is about $7\%$ larger than in the SM, $\lambda_H(m_t)\simeq 0.273$, the central value of the Higgs quartic remains positive up to high scales and vanishes around the Planck mass. Taking into account uncertainties in the running of $\lambda_H$ from the top mass and strong gauge coupling, a positive Higgs quartic at the $2\sigma$ level corresponds to $$0.259 \lesssim \lambda_H(m_t) \lesssim 0.288 ~.$$ Interestingly enough, for such a range of boundary conditions not only the Higgs quartic, but also its beta function become zero at scales close to the Planck scale. The required size of the portal coupling to stabilize the potential in the UV is considerably smaller than using mechanism (i). If the scalar quartic $\lambda_\Sigma$ is of the same order of the Higgs quartic $\lambda_H$, a portal coupling of $|\lambda_{\Sigma H}| \sim 0.05$ is sufficient. Although the heavy scalar is unstable in scenario (ii), additional fields which get their mass from couplings to $\Sigma$ could explain dark matter. As pointed out in [@EliasMiro:2012ay], the correction in $\eqref{eq:lam}$ from the heavy scalar persists even in the decoupling limit, $w \to \infty$, so that both mechanisms to mitigate the vacuum instability do not point to a specific scale for the extra sector[^3]. This situation is fundamentally different in models with classical scale invariance in the UV. If the Higgs mass parameter $\mu_H^2$ in is zero it will be generated by the vev of the extra scalar through the portal coupling $$\begin{aligned} \lambda_{\Sigma H}\Sigma^\dagger \Sigma \,H^\dagger H \quad \to\quad \frac{\lambda_{\Sigma H} w^2}{2}\,H^\dagger H\,.\end{aligned}$$ In that case, the ratio of vacuum expectation values is controlled by the portal coupling (note that $\lambda_{\Sigma H}$ has to be negative to trigger a vev for the Higgs boson) $$\begin{aligned} \label{eq:vevs} \frac{v^2}{w^2}= -\frac{\lambda_{\Sigma H}}{\lambda_H} ~,\end{aligned}$$ and the correction in decouple for $w \to \infty$. For a vanishing Higgs mass parameter $\mu_H^2=0$ and for the Higgs being the lightest scalar, the vev of the extra scalar is therefore bounded from above. The requirement that the central value ($2\sigma$ upper bound) of $\lambda_H$ remains positive up to the Planck scale implies $$w \lesssim \left(\lambda_\Sigma\right)^{-\frac{1}{4}} \times 350 ~(470) ~ \text{GeV}~.$$ Here, we worked in the limit $\lambda_\Sigma w^2 \gg \lambda_H v^2$ and neglect the tiny $\lambda_{\Sigma H}$ contributions to the running of $\lambda_H$. In the limit in which $\lambda_{\Sigma} w^2 = \lambda_H v^2$ the bounds become $w \lesssim 3.5 (12.7)$ TeV. This corresponds to the extreme case of maximal mixing between the Higgs and the dark scalar. As we will discuss in Section \[sec:higgs\], the mixing is strongly constrained by collider bounds. IR Instability -------------- We now address the question of how to generate the vev for the scalar $\Sigma$. In particular, if not only the Higgs mass parameter, but all scales in the potential vanish in the UV, $\mu_H^2=\mu_\Sigma^2=0$, the vev of the extra scalar can only be induced radiatively, either through strong dynamics, in which a new condensation scale induces a mass term for the extra scalar [@Holthausen:2013ota; @Heikinheimo:2013fta; @Kubo:2014ova], or by a Coleman-Weinberg mechanism, in which the balance between the quartic and the one-loop corrections to the effective potential determine the vev [@Coleman:1973jx]. We will concentrate on the latter mechanism in the following. In the limit of small Higgs portal coupling $|\lambda_{ \Sigma H}|\ll 1$, we can consider the effective potential $V_\mathrm{eff}$ for the scalar independently from the Higgs boson. Its general one loop form is given by $$\begin{aligned} \label{eq:Veff} V_\mathrm{eff}(s,\mu)= \frac{1}{8}\lambda_\Sigma(\mu)\, s^4+\frac{B(\mu)}{4}\,s^4\,\log\left(\frac{s^2}{\mu^2}\right)\,,\end{aligned}$$ in which the scalar component $s$ in is treated as a background field and $\mu$ is the renormalization scale. Subleading terms in $V_\mathrm{eff}$ that are proportional to the anomalous dimension of $s$ are suppressed. This effective potential has a local minimum if $$\begin{aligned} \label{eq:min1} \frac{\partial V_\mathrm{eff}(s,\mu)}{\partial s }\bigg\vert_{\mu=s=w}=0 \,\qquad \Rightarrow \qquad B=- \lambda_\Sigma\,,\end{aligned}$$ and $$\begin{aligned} \label{eq:cond2} \frac{\partial^2 V_\mathrm{eff}(s,\mu)}{\partial s^2 }\bigg\vert_{\mu=s=w}>0 \,\qquad \Rightarrow \qquad B>0\,.\end{aligned}$$ Since $B$ is a loop-suppressed function, it follows, that in the vicinity of the minimum, the quartic coupling needs to be small and negative for CW symmetry breaking to work. For the full potential, including terms proportional to the portal coupling, $\lambda_{\Sigma H}$, the condition $\lambda_\Sigma<0$ is replaced by [@Sher:1988mj] $$\begin{aligned} \label{eq:fullmin} 4\lambda_H\,\lambda_\Sigma-\lambda_{\Sigma H}^2<0\,.\end{aligned}$$ As long as the portal coupling is small, $\lambda_{\Sigma H}^2 \ll \beta_{\lambda_\Sigma}^{(1)} \lambda_H$, this gives approximately the same constraint[^4]. Further, the coefficient $B$ is related to the beta-function of $\lambda_\Sigma$ by the one-loop renormalization group equation $$\begin{aligned} \mu\frac{\partial}{\partial \mu} V_1(s,\mu) +\beta^{(1)}_{\lambda_\Sigma}\frac{d}{d\lambda_\Sigma} V_0-\gamma_s^{(1)}s\frac{d}{ds}V_0=0~,\end{aligned}$$ in which $\gamma_s^{(1)}$ denotes the one-loop anomalous dimension of the scalar fields $s$ and $\beta_{\lambda_\Sigma}^{(1)}$ the one-loop beta function of the quartic coupling. From the general form  follows $$\begin{aligned} \beta_{\lambda_\Sigma}^{(1)}=4\gamma_s^{(1)}\,\lambda_\Sigma+ 4 B~.\end{aligned}$$ Close to the minimum, the first term can be neglected to good approximation. Therefore the beta function of $\lambda_\Sigma$ has to be positive to induce a vev for the scalar. A natural way to ensure a positive beta function in a region around the minimum is to charge the scalar under a dark gauge symmetry. Loops with dark gauge bosons give a positive contribution to $\beta_{\lambda_\Sigma}$ and lead to the desired IR instability of the scalar potential. At the same time, a positive beta function for the quartic coupling ensures that the scalar potential is stable in the UV. The Scale of Dark Matter ------------------------ Independent on whether the vev of the dark scalar is induced by strong dynamics or by a Coleman-Weinberg mechanism as discussed in the previous section, additional fields with couplings to $\Sigma$ are required, which can provide dark matter candidates. If the new sector does not introduce explicit mass scales, the masses of any new states can only be generated through the vev of the extra scalar. In this context it is very interesting that the range suggested by stability considerations seems to agree with the mass scales suggested by the “WIMP miracle”. We emphasize that this is a generic feature of extensions of the SM with the above properties. Extended models with additional scalars can of course soften this relation. Examples of models in the literature, in which the electroweak vacuum is stabilized through the Higgs portal and dimensionful couplings are absent, reveal the mentioned connection between the dynamical generation of a vacuum expectation value in the IR, the stabilization of the vacuum in the UV, and the dark matter sector: The authors of [@Iso:2012jn] discuss a model with an extra scalar charged under a $U(1)_\mathrm{B-L}$ that gives a Majorana mass to right-handed neutrinos through a CW mechanism. The $Z'$ in this model is not a candidate for DM, because it couples to $B-L$ and the resulting experimental bounds on the $Z'$ push the vev of the extra scalar above 3 TeV. As a consequence, the vacuum cannot be stabilized up to the Planck scale in this model. In [@Hambye:2013dgv], the extra scalar is a doublet under an additional dark $SU(2)$ and breaks it completely. In this case the heavy gauge boson triplet constitutes dark matter, and if vacuum stability is enforced, the vev of the scalar is bound to be at the TeV scale. In addition, the authors of [@Khoze:2014xha] have shown, that the extra gauge couplings that drive the quartic of the extra scalar negative in the IR need to be of order one in these models in order to stabilize the vacuum, so that the masses of the corresponding gauge bosons are bound from below by $m = g\, w/2 \gtrsim 500$ GeV. If the hidden sector in addition to scalars and gauge bosons has also fermionic degrees of freedom, they are generically required to be lighter than the gauge bosons. This can easily be understood from the fact that they enter the effective potential with a negative sign, $$\begin{aligned} \label{eq:Veff2} V_\mathrm{eff}(s,\mu)=V_0(s,\mu)+\frac{1}{64\pi^2}\sum_{i=B,F} n_i m_i^4(s)\,\left[\log\frac{m_i^2(s)}{\mu^2}-C_i\right]\,,\end{aligned}$$ where the sum goes over fermions (F) and bosons (B), $m_i(s)$ denotes the corresponding Higgs dependent masses, $n_i =\mp $ the number of fermionic/bosonic degrees of freedom, and the $C_i$ are renormalization scheme dependent constants. If the fermionic contributions dominate the one loop contributions to the effective potential , the condition (\[eq:cond2\]) cannot be fulfilled. Hence, the effective potential is unbounded from below, i.e. the fermions generate a UV instability instead of a IR instability. Therefore, for the CW mechanism to work, the gauge bosons are generically heavier and fermions constitute dark matter. In the following sections we will discuss in detail an example of a model - that does not contain any explicit mass scales, - that utilizes the Coleman-Weinberg mechanism in a dark sector to induce spontaneous electroweak symmetry breaking through a Higgs portal, - that is weakly coupled below the Planck scale, with all of the running scalar quartic couplings starting near zero at the Planck scale, - that stabilizes the vacuum until the Planck scale, and - that contains fermionic dark matter with masses at or below the electroweak scale. The Model {#sec:model} ========= We consider an extension of the SM by a $SU(2)_X \times U(1)_X$ gauge group, under which all SM fields are uncharged. In addition to the $SU(2)_X \times U(1)_X$ gauge bosons $W^\prime_a$ and $B^\prime$, we introduce a scalar doublet $\Sigma$ under $SU(2)_X$ with $U(1)_X$ charge $Q^X_\Sigma = 1/2$. The fermionic sector consists of two sets of chiral SM singlet fermions: $\psi_i^L$, $\xi_i^R$, $\chi_i^R$, with $i = 1,2$. The left handed fields $\psi_1^L = (\chi_1^L , \xi_1^L)$ and $\psi_2^L = (\xi_2^L , \chi_2^L)$ are $SU(2)_X$ doublets, while the right handed ones are $SU(2)_X$ singlets. We assign the following dark hypercharges that ensure anomaly cancellation: $Q^X_{\psi_1} = +1/2$, $Q^X_{\psi_2} = -1/2$, $Q^X_{\chi_1} = +1$, $Q^X_{\chi_2} = -1$, $Q^X_{\xi_1} = Q^X_{\xi_2} = 0$.\ We denote the field strength tensors of the $SU(2)_X$ and $U(1)_X$ gauge symmetries by $(W^\prime_a)_{\mu\nu}$ and $(B^\prime)^{\mu\nu}$, so that their kinetic terms read $$\begin{aligned} \label{eq:gaugelagrangian} \mathcal{L}_\text{gauge} =\frac{1}{4} (W^\prime_a)_{\mu\nu} (W^\prime_a)^{\mu\nu} + \frac{1}{4} (B^\prime)_{\mu\nu} (B^\prime)^{\mu\nu}~,\end{aligned}$$ where $a = 1,2,3$ is the index of the adjoint of $SU(2)_X$. We assume that there is no kinetic mixing between the $U(1)_X$ gauge boson and the SM hypercharge gauge boson. As our model does not contain fields that are charged under both $U(1)$ symmetries, such a choice is stable under radiative corrections. In the absence of kinetic $U(1)$ mixing, the only renormalizable portal between the dark sector and the SM is the mixing of the dark scalar with the Higgs. Explicit mass terms for the scalars are assumed to vanish, such that $$\label{eq:scalarlagrangian} \mathcal{L}_\text{scalar} = |D\,H|^2 + |D\,\Sigma|^2 - \frac{\lambda_H}{2} |H|^4 - \frac{\lambda_\Sigma}{2} |\Sigma|^4 - \lambda_{\Sigma H} |H|^2 |\Sigma|^2 ~.$$ The covariant derivatives of $H$ and $\Sigma$ are given by (Lorentz indices are suppressed for simplicity) $$D\, H = (\partial - i \frac{g}{2} \sigma^a W_a - i g^\prime Q_H B) H ~,~~ D\,\Sigma = (\partial - i \frac{g_X}{2} \sigma^a W^\prime_a - i g_X^\prime Q^X_\Sigma B^\prime)\Sigma ~.$$ Here, $g$ and $g^\prime$ are the $SU(2)$ and $U(1)$ gauge couplings of the SM, and $g_X$ and $g_X^\prime$ are the corresponding couplings in the dark sector. The Higgs and the dark scalar can be decomposed as follows $$\label{eq:HSig} H = \begin{pmatrix} G^+ \\ \frac{1}{\sqrt{2}}(h + v + iG^0) \end{pmatrix} ~,~~~ \Sigma = \begin{pmatrix} a^+ \\ \frac{1}{\sqrt{2}}(s + w + ia)\end{pmatrix} ~,$$ where $v$ ($w$) is the respective vacuum expectation value that breaks the $SU(2)_{(X)}\times U(1)_{(X)}$ gauge group down to (dark) electromagnetism. The Goldstone bosons $G^\pm$, $G^0$ and $a^\pm$, $a$ provide the longitudinal components of the $W$ and $Z$ boson of the SM, as well as the corresponding $W^\prime$ and $Z^\prime$ in the dark sector. The masses of the dark gauge bosons are given by $$\label{eq:Bmasses} m_{\gamma'}=0~,~~ m_{W^\prime} = \frac{w}{2} g_X ~,~~ m_{Z^\prime} = \frac{w}{2} \sqrt{g_X^2 + g_X^{\prime~2}} ~.$$ Analogous to the photon in the SM, the dark sector contains a massless gauge boson, which we will refer to as dark photon, $\gamma^\prime$. In complete analogy to the SM, we define a dark electromagnetic coupling $e_X$ as well as a dark mixing angle $\theta_X$ $$e_X = \frac{g_X g_X^\prime}{\sqrt{g_X^2 + g_X^{\prime~2}}} ~,~~ c_X = \cos\theta_X = \frac{g_X}{\sqrt{g_X^2 + g_X^{\prime~2}}} ~,~~ s_X = \sin\theta_X = \frac{g_X^\prime}{\sqrt{g_X^2 + g_X^{\prime~2}}} ~.$$ The dark fermions couple to the extra scalar $\Sigma$ through Yukawa couplings, $$\begin{aligned} \mathcal{L}_\text{fermion} &=& i \bar\psi_i^L D\!\!\!\!/ ~\psi_i^L + i \bar\chi^R_i D\!\!\!\!/ ~\chi^R_i + i \bar\xi^R_i \partial\!\!\!/ ~\xi^R_i \nonumber \\ && + (Y_{\chi_1} \bar\psi^L_1 \chi^R_1 \tilde \Sigma + Y_{\chi_2} \bar\psi_2^L \chi_2^R \Sigma + Y_{\xi_1} \bar\psi^L_1 \xi^R_1 \Sigma + Y_{\xi_2} \bar\psi_2^L \xi_2^R \tilde \Sigma ~+~\text{h.c.} )~,\label{eq:fermionlagrangian}\end{aligned}$$ where $\tilde \Sigma = i \sigma_2 \Sigma^*$. As in the scalar sector, we do not consider explicit Majorana mass terms for the fermions that would be allowed given the quantum number assignments for $\psi_i$, $\chi_i$, and $\xi_i$. For simplicity, we also choose flavor diagonal Yukawa couplings for the $\xi_i$ fields. Both the absence of Majorana masses and of flavor off-diagonal Yukawa couplings can for example be enforced by demanding dark fermion number conservation. The covariant derivatives of the fermions are $$D\psi_i^L = (\partial - i \frac{g_X}{2} \sigma^a W^\prime_a - i g_X^\prime Q^X_{\psi_i} B^\prime)\psi_i^L ~,~~ D\chi_i^R = (\partial - i g_X^\prime Q^X_{\chi_i} B^\prime)\chi_i^R ~,$$ and $\xi^R_i$ are total singlets. After breaking of the dark $SU(2)_X \times U(1)_X$ by the vev of $\Sigma$, the fermions become massive and we introduce the Dirac spinors $\chi_i = P_L \chi_i + P_R \chi_i = (\chi^L_i , \chi^R_i)$ and $\xi_i = P_L \xi_i + P_R \xi_i = (\xi^L_i , \xi^R_i)$ with masses $$\label{eq:fmasses} m_{\chi_i} = \frac{Y_{\chi_i}}{\sqrt{2}} w ~,~~ m_{\xi_i} = \frac{Y_{\xi_i}}{\sqrt{2}} w ~.$$ Conservation of dark fermion number and dark electromagnetism implies that both $\chi_i$ and $\xi_i$ can be stable dark matter candidates. The Scalar Spectrum {#sec:CW} ------------------- The one loop effective potential of the model is given in the Appendix \[sec:Veff\]. If the bosonic contributions to the effective potential dominate over the fermionic ones, non-zero scalar vevs will be induced radiatively. In the limit of a small portal coupling, the vevs of the Higgs $v$ and of the dark scalar $w$ are approximately connected by the relation (\[eq:vevs\]). Neglecting the effects of the field anomalous dimensions of the Higgs and the dark scalar as well as the running of both the SM quartic coupling and the portal coupling, while keeping the dominant contribution from the running of $\lambda_\Sigma$, the scalar mass matrix in the minimum of the potential can be written as $$\label{eq:mass2} \mathcal{M}^2 \simeq \frac{v^2}{2} \begin{pmatrix} 2\lambda_H & -2 \sqrt{\lambda_H |\lambda_{\Sigma H}|}\\ -2 \sqrt{\lambda_H |\lambda_{\Sigma H}|} & 2 |\lambda_{\Sigma H}| + \lambda_H \beta_{\lambda_\Sigma}/ |\lambda_{\Sigma H}|\end{pmatrix} ~.$$ This mass matrix can be diagonalized through the rotation $$\label{eq:sina} \begin{pmatrix} h \\ s \end{pmatrix} \rightarrow \begin{pmatrix} c_\alpha & s_\alpha \\ -s_\alpha & c_\alpha \end{pmatrix} \begin{pmatrix} h \\ s \end{pmatrix} ~,~~ \sin2\alpha = \frac{2 \sqrt{\lambda_H|\lambda_{\Sigma H}|} v^2}{m_s^2 - m_h^2} ~,$$ with $s_\alpha = \sin\alpha$ and $c_\alpha = \cos\alpha$. The mass eigenvalues $m_h$ and $m_s$ are given by $$\label{eq:masses} m_h^2 \simeq v^2 \left( \lambda_H - \frac{2 \lambda_{\Sigma H}^2}{\beta_{\lambda_\Sigma} - 2 |\lambda_{\Sigma H}|} \right) ~,~~ m_s^2 \simeq v^2 \left( \frac{\lambda_H \beta_{\lambda_\Sigma}}{2|\lambda_{\Sigma H}|} + \frac{\beta_{\lambda_\Sigma}|\lambda_{\Sigma H}|}{\beta_{\lambda_\Sigma} - 2 |\lambda_{\Sigma H}|} \right) ~,$$ where we expanded to first order in the limit $\lambda_{\Sigma H} , \beta_{\lambda_\Sigma} \ll \lambda_H$. In this limit, the mass of the dark scalar is directly proportional to the beta function of the dark scalar quartic coupling. If the dark scalar beta function is larger than twice the absolute value of the portal coupling, $$\label{eq:scalarmass} \beta_{\lambda_\Sigma} \gtrsim 2 |\lambda_{\Sigma H}| ~,$$ the dark scalar is heavier than the Higgs boson, and the mass of the Higgs boson is reduced compared to the Standard Model expression. Vacuum Stability in the UV {#sec:RGE} -------------------------- We now discuss the renormalization group running of the model parameters up to high scales and demonstrate that the electroweak minimum in the scalar potential can be absolutely stable. The one loop beta functions for all couplings of the dark sector as well as the one loop correction to the beta function of the Higgs quartic are collected in the Appendix \[sec:betafunctions\]. For the SM beta functions we use 2 loop results from [@Machacek:1983tz; @Machacek:1983fi; @Machacek:1984zw; @Luo:2002ey; @Luo:2002ti; @Luo:2002iq]. ![Vacuum stability properties in the $m_s$-$\sin\alpha$ plane. In the shaded region the Higgs quartic is positive up to the Planck scale. Between the two dashed contours the Higgs quartic touches zero close to the Planck scale within $2\sigma$. The dotted lines in the unstable region show the scale at which the Higgs quartic runs negative. The solid lines indicate contours of constant scalar vev, $w$. Note, that large mixing angles $\sin\alpha \gtrsim 0.5$ are phenomenologically strongly constrained by collider bounds, see Section \[sec:higgs\]. \[fig:stability\]](stability.pdf){width="0.6\columnwidth"} As discussed already in Section \[sec:TeVDM\] and as shown in Equation (\[eq:masses\]), the physical Higgs mass is not completely determined by the Higgs quartic, but gets an additional contribution from the mixing with the dark scalar. If the dark scalar is heavier than the Higgs, mixing effects will reduce the Higgs mass and a quartic coupling larger than in the SM is required to accommodate a Higgs mass of $m_h \simeq 125.5$ GeV. If the portal coupling is large enough, the IR boundary condition for $\lambda_H$ is such that $\lambda_H$ stays positive all the way to the Planck scale, or, in the limiting case, “touches” zero close to the Planck scale. The region of the parameter space where this can be achieved is shown in Figure \[fig:stability\] in the plane of the scalar mass $m_s$ and the mixing angle $\sin\alpha$. In the shaded region the Higgs quartic is positive up to the Planck scale. Between the two dashed curves the limiting case where $\lambda_H$ touches zero close to the Planck scale can be realized within $2\sigma$. The dotted lines in the unstable region indicate the scale in GeV where the Higgs quartic runs negative. The solid lines show contours of constant scalar vev $w$, that corresponds to a given scalar mass $m_s$ and mixing angle $\sin\alpha$. As we will discuss in Section \[sec:higgs\], the mixing angle is bounded at the order of $\sin\alpha \lesssim 0.5$. This implies a typical value for $w$ around the TeV scale, and an upper bound of several TeV, as expected from the general discussion in Section \[sec:TeVDM\]. On the other hand, scalar vevs considerably below a TeV can in principle be achieved by increasing $|\lambda_{\Sigma H}|$ (see Equation (\[eq:vevs\])). However, this requires that the beta function $\beta_{\lambda_\Sigma}$ needs to to be increased simultaneously due to the bound (\[eq:scalarmass\]). The beta function $\beta_{\lambda_\Sigma}$ is also bounded from above by perturbativity requirements on the dark gauge couplings. As a result, values for $w$ considerably below the TeV scale are disfavored. In the following, we will concentrate on regions of parameter space with $w=\mathcal{O}$(1 TeV) and a Higgs quartic that touches zero close to the Planck scale. As long as the dark fermion Yukawa couplings are not too large, the beta function of the scalar quartic $\beta_{\lambda_\Sigma}$ is dominated by the dark gauge couplings and stays positive. In such a case, $\lambda_\Sigma$ increases monotonically with the RG scale and is always positive in the UV. Note, however, that sizable dark fermion Yukawas can modify the behavior of $\lambda_\Sigma$ in the UV. In particular, the model allows to accommodate the limiting case where not only the Higgs quartic but also the dark scalar quartic touches zero close to the Planck scale. In the approximation $\beta_{\lambda_\Sigma}\approx 4B$, it is straight forward to compute the leading terms in the beta function of the scalar quartic from comparing with , using and , $$\begin{aligned} \beta_{\lambda_\Sigma}^{(1)}\approx \frac{1}{16\pi^2} \left\{ \frac{9}{4} g_X^4 + \frac{3}{2} g_X^2 g_X^{\prime~2} + \frac{3}{4} g_X^{\prime~4} - 4(Y_{\chi_1}^4 + Y_{\chi_2}^4 + Y_{\xi_1}^4 + Y_{\xi_2}^4) \right\} ~.\end{aligned}$$ The full one loop expression for the beta function can be found in Appendix \[sec:betafunctions\]. The beta function of $\lambda_\Sigma$ receives contributions dominantly from three sources: (i) from $SU(2)_X$ gauge boson loops, (ii) from fermion loops, and (iii) from $U(1)_X$ gauge boson loops. The gauge boson (fermion) loops increase (decrease) $\lambda_\Sigma$ for higher scales. At low scales, the $SU(2)_X$ contribution dominates and leads to the infrared instability in the scalar potential as discussed above. With the given particle content, the $SU(2)_X$ gauge interactions are asymptotically free. Therefore the contribution of the $SU(2)_X$ gauge bosons to the running of the scalar quartic becomes smaller and smaller for higher scales. At sufficiently high scales, the dominant contributions to the beta function can come from the fermion Yukawa couplings, and the scalar quartic will start to decrease again. Finally, at scales close to the Planck scale, the $U(1)_X$ gauge coupling, having a positive beta function, can become large and compensate the effect of the Yukawa couplings. It is possible to adjust parameters such that the scalar quartic as well as its beta function vanish exactly at the Planck scale. ![The renormalization group evolution of the gauge couplings, the Yukawa couplings and the scalar quartic couplings for one example parameter point in the considered model that leads to an almost flat scalar potential at the Planck scale.[]{data-label="fig:RGE"}](running_1.pdf "fig:"){width="0.46\columnwidth"}       ![The renormalization group evolution of the gauge couplings, the Yukawa couplings and the scalar quartic couplings for one example parameter point in the considered model that leads to an almost flat scalar potential at the Planck scale.[]{data-label="fig:RGE"}](running_2.pdf "fig:"){width="0.46\columnwidth"}\ ![The renormalization group evolution of the gauge couplings, the Yukawa couplings and the scalar quartic couplings for one example parameter point in the considered model that leads to an almost flat scalar potential at the Planck scale.[]{data-label="fig:RGE"}](running_3.pdf "fig:"){width="0.46\columnwidth"} The plots of Figure \[fig:RGE\] show the renormalization group evolution of the gauge couplings, Yukawa couplings and the scalar quartic couplings for an example parameter point of the model where such a limiting case is realized[^5]. The shown couplings correspond approximately to the following dark sector spectrum $$\begin{aligned} m_h &\simeq& 125.5~ \text{GeV} ~,~~ m_s \simeq 168~ \text{GeV} ~,~~ m_{W^\prime} \simeq 740~ \text{GeV} ~,~~ m_{Z^\prime} \simeq 850~ \text{GeV} ~, \nonumber \\ m_{\chi_1} &\simeq& 50~ \text{GeV} ~,~~~~ m_{\chi_2} \simeq 50~ \text{GeV} ~,~~~~~ m_{\xi_1} \simeq 160~ \text{GeV} ~,~~ m_{\xi_2} \simeq 700~ \text{GeV} ~. \label{eq:darkmasses}\end{aligned}$$ The values for the dark vev is $$w \simeq 1.1~ \text{TeV}~,$$ and the masses in (\[eq:darkmasses\]) (apart from the Higgs mass) correspond to running $\overline{\text{MS}}$ masses at the scale $\mu = w$. The $SU(2)_X$ gauge coupling is $\mathcal{O}(1)$ at the low scale, while the $U(1)_X$ coupling is $\mathcal{O}(1)$ close to the Planck scale. The $U(1)_X$ gauge coupling develops a Landau pole at around $10^{30}$ GeV, well above the Planck scale. Both the Higgs quartic $\lambda_H$ and the dark scalar quartic $\lambda_\Sigma$ as well as their beta functions are approximately 0 at the Planck scale. The portal coupling $\lambda_{\Sigma H}$ is small and negative at all scales but cannot run to zero at the Planck scale. It is the only link between the SM and the dark sector and is therefore only multiplicatively renormalized. Higgs and Dark Scalar Phenomenology \[sec:higgs\] ================================================= The considered model leads to various testable predictions for Higgs phenomenology. Due to the mixing of the two scalars, the couplings of the Higgs boson $h$ to all SM particles are suppressed by a factor $c_\alpha$ compared to the SM case, resulting in an overall suppression of all Higgs rates by $c_\alpha^2$. The latest results from Higgs rate measurements from ATLAS [@ATLAS-CONF-2014-009] and CMS [@CMS-PAS-HIG-14-009] read $$\mu_\text{ATLAS} = 1.30^{+0.18}_{-0.17} ~,~~ \mu_\text{CMS} = 1.00 \pm 0.09^{+0.08}_{-0.07} \pm 0.07 ~,$$ which we will interpret roughly as a constraint of $c_\alpha \gtrsim 0.9$, equivalent to a $20\%$ reduction of the SM production rate. At the next run of the LHC, the precision of the rate measurements is expected to be improved by around a factor of 3, which will allow to probe deviations of $c_\alpha$ from unity of the order of $5\%$. Moreover, the mixing of the Higgs with the dark scalar also leads to couplings of $h$ to the fermions in the dark sector. If some of these fermions are sufficiently light, the Higgs can decay into them. We find for the corresponding partial decay widths $$\label{eq:hff1} \Gamma(h\to f_i f_i) = \frac{Y_{f_i}^2}{8\pi} ~m_h~ s_\alpha^2 \left( 1 - \frac{4 m_{f_i}^2}{m_h^2} \right)^{\frac 3 2}~,$$ which applies for dark-charged and neutral fermions $f_i= \chi_i, \xi_i$. Analogous to the SM decay of the Higgs into two photons, the Higgs can also decay into two dark photons through loops of dark-charged fermions $\chi_i$ and the dark $W^\prime$ boson. In the limit $m_{W^\prime} \gg m_h$, we find for the $h \to \gamma^\prime \gamma^\prime$ decay width $$\label{eq:hgaga} \Gamma(h\to \gamma^\prime \gamma^\prime) \simeq \frac{1}{16\pi} \, s_\alpha^2 \frac{m_h^3}{w^2} \left(\frac{g_X^2}{16\pi^2}\right)^2 \left| 7 - \sum_i \frac{8 m_{\chi_i}^2}{m_h^2} \left[ 1 + \left( 1 - \frac{4 m_{\chi_i}^2}{m_h^2}\right) f\left(\frac{m_h^2}{4 m_{\chi_i}^2}\right) \right] \right|^2 ~.$$ The loop function $f$ is given in the Appendix \[sec:loop\]. Given that $h \to \gamma^\prime \gamma^\prime$ is loop suppressed, it can only compete with the decay into dark fermions if the dark gauge coupling is large $g_X \gtrsim 1$ and the fermion Yukawas are very small, $Y_{\chi_i} \lesssim 10^{-2}$. ![Top: the invisible branching ratio of the Higgs boson as a function of the charged dark fermion mass, for example choices of the scalar mixing angle. Bottom: the scalar signal strength into SM particles as function of the charged dark fermion mass for example choices of the scalar mixing angle. The scalar mass is fixed to $m_s = 140$ GeV in the left and $m_s = 180$ GeV in the right plot.[]{data-label="fig:decay"}](invisible.pdf "fig:"){width="0.46\columnwidth"}\ ![Top: the invisible branching ratio of the Higgs boson as a function of the charged dark fermion mass, for example choices of the scalar mixing angle. Bottom: the scalar signal strength into SM particles as function of the charged dark fermion mass for example choices of the scalar mixing angle. The scalar mass is fixed to $m_s = 140$ GeV in the left and $m_s = 180$ GeV in the right plot.[]{data-label="fig:decay"}](signal1.pdf "fig:"){width="0.46\columnwidth"}       ![Top: the invisible branching ratio of the Higgs boson as a function of the charged dark fermion mass, for example choices of the scalar mixing angle. Bottom: the scalar signal strength into SM particles as function of the charged dark fermion mass for example choices of the scalar mixing angle. The scalar mass is fixed to $m_s = 140$ GeV in the left and $m_s = 180$ GeV in the right plot.[]{data-label="fig:decay"}](signal2.pdf "fig:"){width="0.46\columnwidth"} Given the tiny total width of the SM Higgs, $\Gamma_h^\text{SM} \simeq 4$ MeV, even for moderate mixing angles $s_\alpha$ the induced invisible branching ratio can be sizable. This is illustrated in the upper plot of Figure \[fig:decay\], that shows for various mixing angles $s_\alpha$ the branching ratio of the Higgs into the charged dark fermions as a function of the charged dark fermion mass, for the example choice $m_{\chi_1} = m_{\chi_2}/2$. The dark vev is set to $w = 1.5$ TeV and the neutral fermions are assumed to be heavier than at least half the Higgs mass. We observe that for moderate mixing angles of $s_\alpha \sim 0.3$, branching ratios into dark fermions of $\mathcal{O}$(10%) are possible. The branching ratio can be even larger for smaller $w$. The branching ratio into dark photons is at most at the percent level and therefore hardly relevant. ATLAS and CMS search for invisible decays of Higgs bosons that are produced in association with a $Z$ boson [@Aad:2014iia; @Chatrchyan:2014tja] and in vector boson fusion [@CMS-PAS-HIG-13-013]. The current best bound reads BR$(h \to \text{invisible}) \lesssim 58 \% ~@95\%$ C.L. [@Chatrchyan:2014tja] (see also [@Zhou:2014dba] where a slightly stronger bound BR$(h \to \text{invisible}) \lesssim 40 \% ~@95\%$ C.L. has been obtained, recasting a CMS stop search [@Chatrchyan:2013xna]). Bounds are expected to be improved down to BR$(h \to \text{invisible}) \lesssim 10 \%$ at the high luminosity LHC [@Dawson:2013bba]. Due to the mixing with the Higgs boson, the dark scalar $s$ acquires in turn couplings to all SM particles that are suppressed by a factor $s_\alpha$ compared to the SM Higgs. Therefore, the dark scalar can be searched for at the LHC in the usual Higgs searches. Particularly strong constraints arise already from current searches in the $WW$ and $ZZ$ channels that exclude a signal strength of the order of $\mu \sim 0.1$ over a very broad range of masses [@Chatrchyan:2013iaa; @Chatrchyan:2013mxa; @ATLASC1; @ATLASC2]. The production cross section of the scalar is suppressed by $s_\alpha^2$ with respect to a SM Higgs boson with the same mass. Therefore, we generically expect a bound on the mixing angle of the order of $s_\alpha \lesssim 0.3$. This is slightly more stringent than the bound obtained from Higgs rate measurements, $c_\alpha \gtrsim 0.9$, discussed above. Note, however, that also the dark scalar can decay into dark sector particles. The corresponding partial width into dark fermions and dark photons are given by the expressions in (\[eq:hff1\]) and (\[eq:hgaga\]) with the replacements $m_h \to m_s$ and $s_\alpha \to c_\alpha$. If the scalar is light, with a mass below the $WW$ threshold, its decay width into SM particles is very small. Therefore, its invisible branching ratio can be sizable, in particular if the decay into dark fermions is kinematically accessible. This can easily reduce the branching ratio into SM particles by a factor of few or more and reduce the scalar signal strength well below $\mu_s = 0.1$ also for mixing angles of $s_\alpha \gtrsim 0.3$. This is illustrated in the lower left plot of Figure \[fig:decay\] that shows the signal strength of a $140$ GeV dark scalar for several choices of $s_\alpha$ as a function of the charged dark fermion masses $m_{\chi_1} = m_{\chi_2}/2$. If the dark scalar has a mass above the $WW$ threshold, its width is dominated by decays into $WW$ and decays into dark fermions tend to give only a small correction. This is illustrated in the lower right plot of Figure \[fig:decay\], where we show the signal strength of the dark scalar for a dark scalar mass of $m_s = 180$ GeV. In summary, we find that in the bulk of parameter space the prospects for detecting the dark scalar at the next run of the LHC are excellent, unless in the case where it dominantly decays into dark fermions. In the latter case, precision measurements of the Higgs signal strength in inclusive Higgs production will provide the strongest constraint on the mixing angle. In the case of sufficiently light dark fermions, a high luminosity LHC could provide sensitivity to the invisible decay of the Higgs boson. Dark Matter and Dark Photon Phenomenology \[sec:DM\] ==================================================== The dark fermion sector of our model contains two charged and two neutral Dirac fermions $\chi_{1,2}$ and $\xi_{1,2}$. If the mass of each fermion is less than the sum of the other three masses, none of the fermions can decay and all four constitute a stable dark matter component. If one of the fermions has a mass that is larger than the sum of the other three masses, it can decay into the lighter three fermions through $W^\prime$ exchange. In that case the dark matter will consist of only the lighter three fermions. None of the other massive particles of the model are stable in the regions of parameter space that we will consider. The heavy dark gauge bosons can decay into a pair of dark fermions, while the dark scalar can decay through the Higgs portal into a pair of SM particles. The massless dark photon can have interesting effects in the early universe. ![Feynman diagrams corresponding to the dominant processes contributing to dark matter annihilation (a), (b), and (c), as well as direct detection (d). In the case of annihilation into dark photons (a) an additional crossed diagram is not shown. \[fig:diagrams\]](annihilation1.pdf "fig:"){width="0.2\columnwidth"}       ![Feynman diagrams corresponding to the dominant processes contributing to dark matter annihilation (a), (b), and (c), as well as direct detection (d). In the case of annihilation into dark photons (a) an additional crossed diagram is not shown. \[fig:diagrams\]](annihilation2.pdf "fig:"){width="0.2\columnwidth"}       ![Feynman diagrams corresponding to the dominant processes contributing to dark matter annihilation (a), (b), and (c), as well as direct detection (d). In the case of annihilation into dark photons (a) an additional crossed diagram is not shown. \[fig:diagrams\]](annihilation3.pdf "fig:"){width="0.2\columnwidth"}       ![Feynman diagrams corresponding to the dominant processes contributing to dark matter annihilation (a), (b), and (c), as well as direct detection (d). In the case of annihilation into dark photons (a) an additional crossed diagram is not shown. \[fig:diagrams\]](directdetection.pdf "fig:"){width="0.225\columnwidth"} Dark Matter Relic Abundance --------------------------- The relic abundance of the charged dark fermions $\chi_i$ is primarily set by annihilation into two massless dark photons $\gamma^\prime$. Annihilation into two dark scalars is p-wave suppressed and typically negligible. Annihilation into SM particles through an s-channel exchange of the dark scalar or the Higgs is strongly suppressed by the small Higgs portal and therefore also negligible[^6]. For the annihilation cross section into dark photons, depicted in diagram (a) of Figure \[fig:diagrams\], we find $$\label{eq:annihilation1} (\sigma v)_{\chi_i} \simeq \frac{e_X^4}{8\pi} \frac{1}{m_{\chi_i}^2} ~.$$ This annihilation cross section decreases for increasing charged dark fermion masses. The relic abundance of stable charged dark fermions is approximately given by $$\label{eq:relic} \Omega_{\chi_i} h^2 \simeq 0.12 \times \left( \frac{2.2 \times 10^{-26} ~cm^3/s}{(\sigma v)_{\chi_i}}\right) ~.$$ A charged dark fermion fraction of the total relic abundance is subject to various constraints [@Ackerman:mha; @Feng:2009mn; @Fan:2013yva]. A component of (strongly) self-interacting dark matter is constrained by halo shapes [@Peter:2012jh] and the observed structure of the Bullet Cluster [@Markevitch:2003at; @Randall:2007ph]. Numerical simulations that account for the observed deviations from spherical halos allow for $\sim 10\%$ interacting dark matter, while simulations of the Bullet Cluster allow for up to $\sim 30\%$ of all dark matter to have arbitrarily strong self-interactions. A more stringent bound comes from possible CMB structure, which constrains the fraction of dark matter coupled to dark radiation to $\lesssim 5\%$ [@Cyr-Racine:2013fsa]. If dark matter forms a disk due to long ranged interactions, the local dark matter density puts a comparable bound on this fraction [@Fan:2013yva]. In the following, we will therefore allow a charged dark matter fraction of at most 5%. This leads to an upper bound on the mass of the charged dark fermions. We find $$m_{\chi_1}^2 + m_{\chi_2}^2 \lesssim (1~ \text{TeV})^2 \times e_X^4 ~.$$ For values of the dark electromagnetic coupling of the order of the electroweak couplings of the SM, $e_X \sim 0.5$, this implies an upper bound on the mass of stable charged dark fermions of a few 100 GeV. It is important to observe, that in the absence of the dark photons, the annihilation cross section of the fermions $\chi_i$ would be strongly suppressed resulting generically in a dark matter relic abundance in excess of the measured value $\Omega_\text{DM} h^2 \simeq 0.12$. Obviously, the neutral dark fermions $\xi_i$ cannot annihilate into the dark photons at tree level. Annihilation into two dark scalars or into SM particles is also suppressed for the same reasons as in the case of the charged dark fermions. The only unsuppressed annihilation of the neutral dark fermions is into the charged dark fermions, which is only an option if the neutral fermions are significantly heavier than the charged ones, such that their freeze out occurs sufficiently earlier. Annihilation into charged dark fermions can proceed through s-channel exchange of a $Z^\prime$ or t-channel exchange of a $W^\prime$ as shown in diagrams (b) and (c) in Figure \[fig:diagrams\]. The s-channel exchange of a dark scalar is suppressed by the charged fermion Yukawa coupling and hardly relevant in the regions of parameter space that we will consider. Even more suppressed is the s-channel annihilation through a Higgs boson. In the limit $m_{\chi_i} \ll m_{\xi_i} \ll m_{Z^\prime}, m_{W^\prime}$, the annihilation cross section is approximately given by $$\label{eq:annihilation2a} (\sigma v)_{\xi_i} \simeq \frac{m_{\xi_i}^2}{2\pi w^4} \big( 4 s_X^4 - 3 s_X^2 + 2 \big) ~.$$ We learn that the annihilation cross section increases for increasing neutral dark fermion mass. For $m_{\xi_i} \sim m_{Z^\prime}/2$, the annihilation cross section is strongly enhanced by the $Z^\prime$ resonance and reaches its maximum. For $m_{\xi_i} \gtrsim m_{Z^\prime}/2$ the annihilation cross section decreases again with increasing mass. Expressions for the annihilation cross section that hold in the general case of arbitrary fermion and gauge boson masses are given in the Appendix \[sec:darkmatter\]. ![The relic density of the light neutral dark fermion species as a function of its mass. In the left (right) plot, the $Z^\prime$ mass is fixed to $m_{Z^\prime} = 1 (2)$ TeV. \[fig:relic\]](relic.pdf){width="\columnwidth"} The relic abundance of the stable neutral dark fermions is given by an expression analogous to (\[eq:relic\]). The relic abundance of the light neutral dark matter species is shown in the plots of Figure \[fig:relic\] as a function of the fermion mass. In the left and right plots the $Z^\prime$ mass is fixed to $m_{Z^\prime} = 1$ TeV and $m_{Z^\prime} = 2$ TeV, respectively. The charged fermion masses are fixed to $m_{\chi_1} = m_{\chi_2} = 50$ GeV and the dark hypercharge gauge coupling is $g_X^\prime = 0.25$. The various curves correspond to different choices of the dark $SU(2)_X$ gauge coupling that ranges from $g_X = 0.5$ up to $g_X = 2$. We observe that the annihilation into charged fermions is very efficient. If the neutral fermion is above the $Z^\prime$ resonance, the requirement of the right relic abundance leads to an upper bound on the $SU(2)_X$ gauge coupling of the order of $g_X \lesssim 1$. Note that a dark matter fermion with mass above the $Z^\prime$ mass implies a large fermion Yukawa coupling and therefore generically leads to a UV instability in the dark scalar quartic $\lambda_\Sigma$. For a dark fermion mass below the $Z^\prime$ resonance as preferred by vacuum stability, gauge couplings over a broad range of values can be made easily compatible with the relic abundance. Dark Matter Direct Detection {#sec:dd} ---------------------------- The dark matter particles couple to SM particles only through the Higgs portal. The direct detection cross section is therefore necessarily suppressed by the mixing between the Higgs and the dark scalar. Working with scalar mass eigenstates and evaluating the relevant diagram in Figure \[fig:diagrams\], we find for the spin-independent cross section for elastic scattering of neutral dark matter particles $\xi_i$ off protons $$\label{eq:directdetection} \sigma_\text{SI} = \frac{Y_{\xi_i}^2}{2\pi} \frac{m_{\xi_i}^2 m_p^4}{v^2 (m_{\xi_i} + m_p)^2} ~f^2~s_\alpha^2 c_\alpha^2 \left(\frac{1}{m_h^2} - \frac{1}{m_s^2}\right)^2 ~,$$ where $m_p$ is the proton mass and $$f = \frac{2}{9} + \frac{7}{9}\Big( f_{T_u} + f_{T_d} + f_{T_s} \Big) \simeq 0.3$$ parametrizes the nuclear matrix element [@Giedt:2009mr]. The dark matter direct detection cross section is suppressed by the scalar mixing angle $s_\alpha^2$ as well as by the destructive interference between the Higgs and dark scalar exchange. We find typical direct detection signals at the level of $\sigma_\text{SI} \simeq 10^{-46}- 10^{-47}\text{cm}^2$, well below the current experimental sensitivities of the XENON100 experiment [@Aprile:2012nq] and the LUX experiment [@Akerib:2013tjd]. The predicted signals are probably also below the sensitivity of XENON1T [@Aprile:2012zx]. They should however be in reach of the planned LZ experiment [@Malling:2011va]. An equation completely analogous to (\[eq:directdetection\]) holds also for the direct detection cross section of the charged dark matter fermions $\chi_i$. However, barring additional structure which radically changes the local density of the charged dark matter component [@Fan:2013yva], the maximal relic density fraction of $5\%$ strongly suppresses sensitivity of direct detection experiments to the $\chi_i$. For charged dark matter masses of $m_{\chi_i} \sim 50$ GeV, there are only corners of parameter space, where the direct detection cross section of the charged fermions might reach $\sigma_\text{SI} \sim \text{few} \times 10^{-48} \text{cm}^2$. Combined with the smaller density of charged dark matter, this results in direct detection rates that are at the border of or even below the atmospheric and supernova neutrino background, and beyond the reach of planned direct detection experiments. Number of Relativistic Degrees of Freedom in the Early Universe --------------------------------------------------------------- The dark photon of our model contributes to the effective number of relativistic degrees of freedom in the early universe. Measurements of the $^4He$ abundance [@Cyburt:2004yc] in the universe and the combination of Planck data with astrophysical measurements of the Hubble constant [@Ade:2013zuv] put constraints on the active degrees of freedom during Big Bang nucleosynthesis (BBN) and at the time at which the Cosmic Microwave Background (CMB) radiation formed, respectively [@Franca:2013zxa; @Fan:2013yva]. The process of scattering of visible photons into dark photons $\gamma \gamma \leftrightarrow \gamma' \gamma'$ can potentially keep the dark and the visible sector in thermal equilibrium. In our model, the Higgs portal is the only connection between the dark and the visible sectors. Therefore $\gamma \gamma \leftrightarrow \gamma' \gamma'$ is induced by a dimension eight operator and suppressed by two loops. As a consequence, this process decouples at very high temperatures. More relevant processes that connect the dark and the visible sector are the annihilation of visible photons into dark fermions, $\gamma\gamma \leftrightarrow \bar \chi \chi$, and of dark photons into SM fermions, $\gamma^\prime\gamma^\prime \leftrightarrow \bar f f$. Such processes are induced by dimension six operators and only suppressed by one loop. Depending on the dark scalar mass, mixing angle and the dark fermion Yukawa couplings, we find that the decoupling temperature is at the order of $T(t_\mathrm{dec})\sim \mathcal{O}(10)$ GeV. Below this temperature, the entropy density should be separately conserved in both sectors, so that the ratio of temperatures $\xi(t)= T_\mathrm{dark}/T_\mathrm{vis}$ in the dark and visible sector at some later time $t$ is given by [@Fan:2013yva] $$\begin{aligned} \xi(t)=\left(\frac{g^\mathrm{dark}_{\ast s}(t_\mathrm{dec})}{g^\mathrm{vis}_{\ast s}(t_\mathrm{dec})} \frac{g^\mathrm{vis}_{\ast s}(t)}{g^\mathrm{dark}_{\ast s} (t)}\right)^{1/3}\,, \end{aligned}$$ where $g_{\ast s}(t)$ denotes the effective number of degrees of freedom at the time $t$. In the SM, all degrees of freedom besides the Higgs boson, the top, and the electroweak gauge bosons are active during decoupling, so that $g^\mathrm{vis}_{\ast s}(t_\mathrm{dec})=86.75$. In the dark sector, the dark photons, and dark fermions can contribute $g^\mathrm{dark}_{\ast s}(t_\mathrm{dec})= 2+\frac{7}{8}\times 4 \times n=2 + n \times 3.5$, where $n$ is the number of dark fermions with masses below the decoupling temperature. At the BBN scale, electrons, neutrinos and photons contribute to the SM entropy density $g^\mathrm{vis}_{\ast s}(t_\mathrm{BBN})=\frac{7}{8} \times 10 + 2= 10.75$, while during formation of the CMB only colder neutrinos and photons remain active $g^\mathrm{vis}_{\ast s}(t_\mathrm{CMB})= \left(\frac{4}{11}\right)^{4/3}\times \frac{7}{8} \times 6 + 2 = 3.36$. In the dark sector, at these times only the dark photon is a relativistic degree of freedom, $g^\mathrm{dark}_{\ast s}(t_\mathrm{BBN})=g^\mathrm{dark}_{\ast s}(t_\mathrm{CMB} )=2$. The temperatures in the dark sector during BBN and CMB are therefore smaller than in the visible sector. We find $$\xi(t_\mathrm{BBN})\approx 0.50/0.70/0.82 ~,~~ \xi(t_\mathrm{CMB})\approx 0.34/0.47/0.56~,$$ where the first/second/third value corresponds to $n=0/1/2$. These temperature ratios can be translated into the change of effective number of neutrinos at these temperatures $$\begin{aligned} \Delta N_{\mathrm{eff},\nu}^\mathrm{BBN}&=\frac{8}{7}~\xi(t_\mathrm{BBN})^4 ~\phantom{\left(\frac{4}{11}\right)^{-\frac{ 4}{3}}}\approx 0.07/0.27/0.53~,\\ \Delta N_{\mathrm{eff},\nu}^\mathrm{CMB}&=\left(\frac{4}{11}\right)^{-\frac{ 4}{3}}~\frac{8}{7}~\xi(t_\mathrm{CMB})^4 \approx 0.06/0.22/0.43~.\end{aligned}$$ In the Standard Model, the effective number of neutrinos is given by $N_{\mathrm{eff},\nu}=N_{\mathrm{eff},\nu}^\mathrm{BBN}=N_{\mathrm{eff},\nu} ^\mathrm{CMB}=3.046$ [@Mangano:2005cc]. Currently, the strongest constraints on the numbers of effective degrees of freedom during BBN [@Cyburt:2004yc] and CMB [@Ade:2013zuv] are $$\begin{aligned} N_{\mathrm{eff},\nu}^\mathrm{BBN} &= 3.24^{+0.61}_{-0.57} \quad \text{at }\quad 68\%~ \text{C.L.}~,\\ N_{\mathrm{eff},\nu}^\mathrm{CMB} &= 3.52^{+0.48}_{-0.45}\quad \text{at }\quad 95\%~ \text{C.L.}~.\end{aligned}$$ This has to be compared with the values for $ N_{\mathrm{eff},\nu}+\Delta N_{\mathrm{eff},\nu}$ in our model, which can be comfortably accommodated within the uncertainties. Future CMB experiments will improve the bounds on $N_{\mathrm{eff},\nu}^\mathrm{CMB}$ significantly [@Feng:2014uja] and might be able to find evidence for the presence of the dark photon in the early universe. Numerical Analysis, Discussion, and Outlook {#sec:out} =========================================== We now analyse the dark scalar and dark matter phenomenology of the model numerically, starting from the underlying model parameters in the Lagrangian. We explore regions of parameter space that are compatible with vanishing Higgs and scalar quartic couplings at the Planck scale. We checked that small non-zero scalar quartics at the Planck scale do not appreciably change any of our findings. In addition, we impose the correct dark matter relic abundance with a $\sim 5\%$ admixture of dark charged dark matter component. The model introduced in Section \[sec:model\] has 9 free parameters: the Higgs quartic $\lambda_H$, the dark scalar quartic $\lambda_\Sigma$, the portal coupling $\lambda_{H\Sigma}$, the $SU(2)_X \times U(1)_X$ gauge couplings $g_X$ and $g_X^\prime$, as well as the Yukawa couplings of the dark fermions $Y_{\chi_1}$, $Y_{\chi_2}$, $Y_{\xi_1}$, and $Y_{\xi_2}$. We consider the one loop effective scalar potential given in the Appendix \[sec:Veff\], with the approximations discussed there. We use 2 loop beta functions for the SM couplings and 1 loop beta functions for the dark sector couplings. For every given set of parameters, we minimize the effective potential numerically and obtain values for the vevs $v$ and $w$, as well as mass eigenvalues for the scalars and their mixing angle. We allow to vary the Higgs quartic coupling at the electroweak scale in the range $0.259 \lesssim \lambda_H(m_t) \lesssim 0.288$ which is compatible with a vanishing $\lambda_H$ at the Planck scale given the current uncertainties on the top mass and the strong gauge coupling. The portal coupling $\lambda_{\Sigma H}$ sets the ratio of the Higgs vev $v$ and the dark vev $w$. For a dark vev at the TeV scale, the portal coupling is small, typically at the order of $|\lambda_{\Sigma H}| \sim 10^{-2}$. We chose a vanishing dark scalar quartic coupling $\lambda_\Sigma$ at the Planck scale. The value of the scalar quartic coupling at the weak scale as well as the value of its beta function are mainly determined by the $SU(2)_X$ gauge coupling $g_X$ and the largest dark fermion Yukawa coupling $Y_{\xi_2}$. We chose these parameters such that we reproduce the Higgs vev of $v=246$ GeV as well as a Higgs mass of $124.5$ GeV $ \lesssim m_h \lesssim 126.5$ GeV. With these boundary conditions the heaviest neutral dark fermion turns out to be unstable. The lighter neutral dark fermion comprises the dominant part of the dark matter relic abundance. Obtaining the right annihilation cross section fixes its Yukawa coupling $Y_{\xi_1}$. We allow to vary $Y_{\xi_1}$ such that its annihilation cross section lies in the range $1.8 \times 10^{-26} cm^3/s < (\sigma v)_{\xi_1} < 2.6 \times 10^{-26} cm^3/s$, reproducing the right dark matter abundance within approximately $20\%$. For simplicity we assume degeneracy among the charged dark matter particles and fix their Yukawa couplings $Y_{\chi_1}$, $Y_{\chi_2}$ such that their masses are $m_{\chi_1} = m_{\chi_2} = 50$ GeV. With this mass, requiring that the charged dark matter is responsible for $\sim 5\%$ of the observed relic abundance fixes the $U(1)_X$ gauge coupling at the TeV scale to be at the order of $g_X^\prime \sim 0.25$. Such small values of $g_X'$ do not significantly impact the running of the dark scalar quartic, and therefore cannot overcome the effect of the largest dark fermion Yukawa coupling $Y_{\xi_2}$. As a consequence the dark scalar quartic will cross zero at the Planck scale. The limiting case where the scalar quartic coupling barely touches zero close to the Planck scale demands a coupling of $g_X' \sim 0.7$, which results in a charged dark matter component well below the percent level. We find that the low energy phenomenology of the dark scalar $s$ and the dominant dark matter component $\xi_1$ hardly depends on these choices. ![Left: Predictions for the signal strength of the dark scalar as function of its mass. The shaded regions are excluded by Higgs searches at the LHC [@Chatrchyan:2013iaa; @Chatrchyan:2013mxa; @ATLASC1; @ATLASC2]. Right: Predictions for the spin independent dark matter nucleon cross section. The region above the solid black line is excluded by the current experimental bound from LUX [@Akerib:2013tjd]. Future sensitivities of XENON1T [@Aprile:2012zx] and LZ [@Malling:2011va] are indicated with the dotted lines. In the region below the dashed line, neutrino background limits the sensitivity of direct detection experiments.[]{data-label="fig:predictions"}](scatterplot.pdf){width="\columnwidth"} Under the discussed boundary conditions, we obtain predictions for the mass and signal strength of the dark scalar, as well as the mass and direct detection cross section of the dominant dark matter component. In Figure \[fig:predictions\], we show the predictions for direct searches for the dark scalar at the LHC (left) as well as dark matter direct detection experiments (right). The red/green/blue points correspond to different choices for the dark scalar vev $w = 1/1.5/2$ TeV as indicated. For a fixed choice of $w$, all parameters of the model are determined by the chosen boundary conditions discussed above. Demanding that central values for $\lambda_H(m_t)$, $m_h$, and, $(\sigma v)_{\xi_1}$ are reproduced, we obtain the predictions indicated by the stars in the plots of Figure \[fig:predictions\]. The dark and light points show the ranges of predictions that can be obtained by varying the Higgs quartic between $0.266 < \lambda_H(m_t) < 0.280$ and $0.259 < \lambda_H(m_t) < 0.288$, which corresponds to the $1\sigma$ and $2\sigma$ ranges for $m_t$ and $\alpha_s$. We find dark scalar masses in the range 140 GeV $\lesssim m_s \lesssim$ 220 GeV. The current sensitivity of Higgs searches in the $ h \to ZZ$ and $h \to WW$ channels at the LHC [@Chatrchyan:2013iaa; @Chatrchyan:2013mxa; @ATLASC1; @ATLASC2] already starts to probe parts of parameter space. Improving the sensitivity down to the percent level over the shown mass range, would probe almost the entire parameter space of the discussed scenario. Note that in this scenario, the charged dark matter mass is fixed to 50 GeV, allowing for a sizable $s \to \chi_i \chi_i$ rate that delutes the signal strength at the LHC. For charged dark matter particles that are heavier than half the dark scalar mass, the dark scalar signal strength increases, especially for dark scalar masses below the $WW$ threshold, $m_s \lesssim 160$ GeV. As expected from the discussion in Section \[sec:higgs\], the prospects for direct detection of the dark scalar at the next run of the LHC are excellent. Typical values for the dark matter mass are at the level of few 100 GeV. As anticipated already in Section \[sec:dd\], the predicted direct detection cross sections are still 1-2 orders of magnitude below the current best experimental sensitivity of the LUX experiment [@Akerib:2013tjd]. The XENON1T experiment [@Aprile:2012zx] might start to probe parts of the parameter space, if the dark scalar vev is around $w \simeq 1$ TeV or smaller, which corresponds to a small dark matter mass of around $m_{\xi_1}\simeq150$ GeV. We expect that future dark matter direct detection experiments like LZ [@Malling:2011va] will be able to detect the dark matter unless the dark vev is far above the TeV scale, in which case the direct searches for the dark scalar become more powerful. The direct detection rates of the charged dark matter component is generically below the neutrino floor. Interestingly, for $m_\chi\leq m_h/2$ the Higgs can decay into the light dark matter candidate with a sizable branching fraction. Improved measurements of the invisible branching ratio of the Higgs can therefore indirectly constrain the mass and fraction of charged dark matter even if direct detection experiments cannot see it.\ Anomaly cancellation enforces at least two generations of charged fermions with dark charges $Q_X=\pm 1$. In the discussed model they are both stable. As argued in [@Fan:2013yva], this can result in dark bound states, which imply cosmological dynamics radically different from cold dark matter. This would provide a new testing ground for our model through measurements of the dark matter distribution within the milky way, for example by precisely mapping the movement of stars as planned by the GAIA survey [@Famaey]. Numerical simulations of galaxy formation are beyond the scope of this work, but we observe that a double disk scenario as discussed in [@Fan:2013yva] can be reproduced within the parameter space of our model.\ If all physics below the Planck scale should be captured in a model without new mass scales, the matter-antimatter asymmetry needs to be addressed. One possibility would be to consider baryogenesis at the electroweak scale. This requires a strongly first order phase transition [@Trodden:1998ym; @Morrissey:2012db]. Since the electroweak scale is induced by a dynamically broken dark gauge symmetry, we expect bubble nucleation to occur through a two step process in the discussed model. A similar scenario has been studied by the authors of [@Patel:2012pi] in an extension of the SM with an additional electroweak scalar triplet. The sphaleron rate for the dark gauge group differs from the one in the SM. A direct comparison to the results of [@Patel:2012pi] is therefore not straight forward and we leave this interesting question for future work. Conclusion \[sec:con\] ====================== After the 8 TeV run of the LHC, the dynamics of the electroweak symmetry breaking mechanism is still a mystery. Natural UV completions of the Standard Model predict new degrees of freedom in the vicinity of the electroweak scale, that have not been discovered so far. If the electroweak scale emerges as a quantum effect from a boundary condition of vanishing mass parameters at the Planck scale, the large disparity of scales can be explained by RGE running similar to the case of the QCD scale. Unlike QCD however, there is no underlying symmetry which protects such a boundary condition at the Planck scale in the SM. We have shown that clasically scale invariant models with a minimal dark sector, that incorporate radiative electroweak symmetry breaking through a Higgs portal, and stabilize the Higgs potential up to the Planck scale put an upper bound of the order of a few TeV on the mass of the dark matter candidate. As discussed previously in the literature, if the dark sector consists of only a scalar charged under dark gauge interactions, the dark gauge bosons that obtain their mass through couplings to the scalar can constitute dark matter. As the dark gauge coupling has to be sizable in order to generate Coleman-Weinberg symmetry breaking, in those scenarios one finds a lower bound on the dark matter mass. As a consequence, dark matter is generically constrained to a window of a few hundred GeV to a few TeV. In this work we have considered the effect of additional fermions in the dark sector. Radiative symmetry breaking dictates that the bosonic contribution to the effective potential needs to be larger than the fermionic contribution. This renders the fermions naturally lighter than the gauge bosons and allows for fermionic dark matter masses at the electroweak scale, or even below. We demonstrated this on the basis of a model that contains a dark sector with a $SU(2)\times U(1)$ gauge group, a dark scalar that is a doublet under the dark $SU(2)$, and two generations of chiral dark fermions. The gauge interactions drive the dark scalar quartic negative at low energies and radiatively induce a vev for the dark scalar. The dark sector gauge symmetry is broken spontaneously by the vev of the scalar doublet, $SU(2)\times U(1)\rightarrow U(1)$, leaving a long-ranged “dark electromagnetism” at low energies. The vev of the scalar doublet also generates a Higgs mass term through a quartic portal coupling and thus triggers breaking of the electroweak symmetry in the visible sector. If this is indeed the origin of electroweak symmetry breaking, the dark scalar mixes with the Higgs and therefore has Higgs-like couplings to Standard Model particles, only suppressed by the portal coupling. We find that if the dark scalar stabilizes the vacuum up to the Planck scale, its mass is constrained to be $m_s\lesssim 250$ GeV. Its signal strength is generically at the level of $\mathcal{O}(10\%)$ of a SM Higgs boson. Current Higgs searches in the $WW$ and $ZZ$ channels already constrain parts of the parameter space of the model, and the prospects for detecting the dark scalar at the next run of the LHC are excellent. Mixing of the dark scalar with the Higgs also leads to a slight reduction of the signal strenghts of the Higgs boson and more precise measurements of the various Higgs signal strengths are equally important to test the discussed framework. For sizable mixing, the Higgs boson can also decay through the scalar portal into the dark charged fermions, if they are kinematically accessible. This can induce an invisible branching ratio of the Higgs of up to $\sim 10\%$ which can be within reach of the high-luminosity LHC. The model has two dark matter components: (i) dark fermions that are charged under the long range dark electromagnetism and with masses typically below the electroweak scale, $m_{\chi_i} \lesssim 100$ GeV; (ii) a neutral dark fermion with mass generically in the range $m_{\xi_1} \sim 100-500$ GeV. The model can easily accommodate a non-negligible fraction of long-range interacting dark matter of the order of a few percent and could have interesting implications for galaxy structure formations. While the neutral dark matter component has a spin-independent scattering cross section with nuclei in reach of future direct detection experiments like XENON1T or LZ, the light dark matter component will be most likely buried in the neutrino background. In addition, the dark radiation present in the model can be independently tested by future measurements of the number of relativistic degrees of freedom in the early universe. Interestingly, the parameter space of our model in which direct detection experiments are least sensitive is the one most strongly constrained by collider searches for the dark scalar. The complementarity of both searches imply excellent prospects to discover or exclude our model in the near future. Finally, we argue, that if the electroweak scale is generated subsequently to the breaking of a dark gauge symmetry, bubble nucleation during the dark and electroweak phase transition becomes a two step process. Previous studies of a similar scenario suggest, that a strong first order phase transition as required by electroweak baryogenesis can be achieved in this setup. Studies in this direction are left for future work. We acknowledge helpful discussions with Prateek Agrawal, Gia Dvali, Steve Giddings, Chris Hill, Pedro Schwaller, Jessie Shelton and Alessandro Strumia. MB and JL acknowledge the hospitality and support of the Theoretical Physics Group at SLAC. MC acknowledges the hospitality of MITP. WA, MC and JL acknowledge the hospitality of the Aspen Center for Physics and partial support by the National Science Foundation Grant No. PHYS-1066293. Fermilab is operated by Fermi Research Alliance, LLC, under contract DE-AC02-07CH11359 with the United States Department of Energy. The research of WA was supported by the John Templeton Foundation. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation. MB acknowledges the support of the Alexander von Humboldt Foundation. Effective Potential \[sec:Veff\] ================================ The one loop effective potential $V_\text{eff}$ of our model is approximately given by $$\begin{aligned} \label{eq:Veff_app} V_\text{eff}(h,s) &\simeq& \frac{1}{8} \lambda_H(\mu_h) h^4 + \frac{1}{4} \lambda_{\Sigma H}(\mu_{sh}) h^2 s^2 + \frac{1}{8} \lambda_\Sigma(\mu_s) s^4 \nonumber \\ &+& \frac{1}{16\pi^2} \Bigg\{ -3 m_t^2 \left[ \log\left(\frac{m_t^2}{\mu_h^2}\right) - \frac{3}{2} \right] \nonumber \\ && \quad\quad\quad+ \frac{3}{2} m_W^2 \left[ \log\left(\frac{m_W^2}{\mu_h^2}\right) - \frac{5}{6} \right] + \frac{3}{4} m_Z^2 \left[ \log\left(\frac{m_Z^2}{\mu_h^2}\right) - \frac{5}{6} \right]\Bigg\} \nonumber \\ &+& \frac{1}{16\pi^2} \Bigg\{ - \sum_i m_{\chi_i}^2 \left[ \log\left(\frac{m_{\chi_i}^2}{\mu_s^2}\right) - \frac{3}{2} \right] - \sum_i m_{\xi_i}^2 \left[ \log\left(\frac{m_{\xi_i}^2}{\mu_s^2}\right) - \frac{3}{2} \right] \nonumber \\ && \quad\quad\quad+ \frac{3}{2} m_{W^\prime}^2 \left[ \log\left(\frac{m_{W^\prime}^2}{\mu_h^2}\right) - \frac{5}{6} \right] + \frac{3}{4} m_{Z^\prime}^2 \left[ \log\left(\frac{m_{Z^\prime}^2}{\mu_h^2}\right) - \frac{5}{6} \right]\Bigg\} ~,\end{aligned}$$ where the field dependent masses are given by $$m_t^2 = Y_t^2 h^2 /2 ~,~~ m_W^2 = g^2 h^2 /4 ~,~~ m_Z^2 = (g^2 + (g^\prime)^2) h^2 /4 ~,$$ $$m_{\chi_i}^2 = Y_{\chi_i}^2 s^2/2 ~,~~ m_{\xi_i}^2 = Y_{\xi_i}^2 s^2/2 ~,~~ m_{W^\prime}^2 = g_X^2 s^2/4 ~,~~ m_{Z^\prime}^2 = (g_X^2 + (g^\prime_X)^2) s^2/4 ~.$$ In (\[eq:Veff\_app\]) we took into account contributions from the top quark, the $W$ and $Z$ bosons, the dark fermions and the dark $W'$ and $Z'$ bosons. Contributions to the effective potential from the Higgs boson $h$, the scalar $s$, and the corresponding Goldstone bosons lead to imaginary parts of the one loop effective potential, whenever the corresponding quartic coupling ($\lambda_H$ or $\lambda_\Sigma$) becomes negative. Such imaginary parts signal the presence of an instability in the potential [@Weinberg:1987vp].[^7] We neglect the contributions from $h$, $s$ and the corresponding Goldstone bosons. We explicitly checked that this leads to shifts in physical observables of a few percent at most. We also do not take into account additional corrections coming from the anomalous dimensions of the Higgs and the scalar field, as they are typically only at the few percent level, as well. All couplings as well as all logarithms in the effective potential depend on a renormalization scale. In (\[eq:Veff\_app\]) we introduced three scales $\mu_h$, $\mu_s$, and $\mu_{hs}$ that cancel separately up to terms suppressed by two loops. In our numerical analysis we set the renormalization scales to the corresponding field values $\mu_h = h$, $\mu_s = s$, and $\mu_{hs} = \sqrt{h s}$, which is expected to keep higher order corrections to the effective potential small. Beta Functions \[sec:betafunctions\] ==================================== The one loop beta functions of the couplings of our framework read ($t = \log\mu$) $$\begin{aligned} \frac{d \lambda_H}{dt} = \beta_{\lambda_H} &=& \beta_{\lambda_H}^{\rm SM} + \frac{1}{16\pi^2} 4 \lambda_{\Sigma H}^2 ~, \\[8pt] \frac{d \lambda_{\Sigma}}{dt} = \beta_{\lambda_\Sigma} &=& \frac{1}{16\pi^2} \Big( 12 \lambda_\Sigma^2 + 4 \lambda_{\Sigma H}^2 - 9 g_X^2 \lambda_\Sigma - 3 (g_X^\prime)^2 \lambda_\Sigma + \frac{9}{4} g_X^4 +\frac{3}{4} (g_X^\prime)^4 +\frac{3}{2} g_X^2 (g_X^\prime)^2 \nonumber \\ && - 4 \sum_i (Y_{\xi_i}^4 + Y_{\chi_i}^4) + 4 \lambda_\Sigma \sum_i (Y_{\xi_i}^2 + Y_{\chi_i}^2) \Big) ~, \\[8pt] \frac{d \lambda_{\Sigma H}}{dt} = \beta_{\lambda_{\Sigma H}} &=& \frac{1}{16\pi^2} \Big[ 4 \lambda_{\Sigma H}^2 + 6 (\lambda_H + \lambda_\Sigma) \lambda_{\Sigma H} - \frac{\lambda_{\Sigma H}}{2} \Big( 3 (g^\prime)^2 + 9 g^2 + 9 g_X^2 + 3 (g_X^\prime)^2 \Big) \nonumber \\ && + \lambda_{\Sigma H} \Big( 6 Y_t^2 + 2 \sum_i (Y_{\xi_i}^2 + Y_{\chi_i}^2) \Big) \Big] ~,\end{aligned}$$ $$\begin{aligned} \frac{d g_X}{dt} = \beta_{g_X} &=& - \frac{1}{16\pi^2} \frac{39}{6} g_X^3 ~, \\ \frac{d g_X^\prime}{dt} = \beta_{g_X^\prime}&=& \frac{1}{16\pi^2} \frac{13}{6} (g_X^\prime)^3 ~,\end{aligned}$$ $$\begin{aligned} \frac{d Y_{\xi_i}}{dt} = \beta_{Y_{\xi_i}} &=& \frac{1}{16\pi^2} Y_{\xi_i} \Big( \frac{3}{2} (Y_{\xi_i}^2 - Y_{\chi_i}^2) + \sum_j (Y_{\xi_j}^2 + Y_{\chi_j}^2) - \frac{9}{4} g_X^2 - \frac{3}{4} (g_X^\prime)^2 \Big) ~, \\ \frac{d Y_{\chi_i}}{dt} = \beta_{Y_{\chi_i}} &=& \frac{1}{16\pi^2} Y_{\chi_i} \Big( \frac{3}{2} (Y_{\chi_i}^2 - Y_{\xi_i}^2) + \sum_j (Y_{\xi_j}^2 + Y_{\chi_j}^2) - \frac{9}{4} g_X^2 - \frac{15}{4} (g_X^\prime)^2 \Big)~.\end{aligned}$$ Loop Function \[sec:loop\] ========================== The loop function that enters the partial width of $h \to \gamma^\prime \gamma^\prime$ given in Section \[sec:higgs\] reads $$f(x) = \left\{ \begin{array}{ll} \arcsin^2 \sqrt{x} & \quad \text{for} ~x \leq 1 ~,\\ -\frac{1}{4} \left( \log\left( \frac{\sqrt{x} + \sqrt{x-1}}{\sqrt{x} - \sqrt{x-1}}\right) - i \pi\right)^2 & \quad \text{for} ~x > 1~. \end{array} \right.$$ Dark Matter Annihilation \[sec:darkmatter\] =========================================== In this appendix we give the annihilation cross section of the lightest neutral dark fermion $\xi_1$ into the charged dark fermions $\chi_1$, $\chi_2$, that are assumed to be lighter than $\xi_1$. Unsuppressed contributions come from s-channel exchange of a $Z^\prime$ boson and t-channel exchange of a $W^\prime$ boson. We find $$\begin{aligned} (\sigma v)_{\xi_1} &\simeq& \frac{1}{2 \pi} \frac{m_{\xi_1}^2}{w^4} \sqrt{1- \frac{m_{\chi_1}^2}{m_{\xi_1}^2}} \left(1 + \frac{m_{\xi_1}^2}{m_{W^\prime}^2} - \frac{m_{\chi_1}^2}{m_{W^\prime}^2}\right)^{-2} \nonumber \\ && + \frac{1}{8 \pi} \frac{m_{\xi_1}^2}{w^4} \sum_{i=1,2} \sqrt{1- \frac{m_{\chi_i}^2}{m_{\xi_1}^2}} \left( 1 - 4s_X^2 + 8 s_X^4 + \frac{m_{\chi_i}^2}{m_{\xi_1}^2} 2 s_X^2(2s_X^2 -1)\right) \nonumber \\ && \qquad \times \left( \left(1 - \frac{4 m_{\xi_1}^2}{m_{Z^\prime}^2}\right)^2 + \frac{\Gamma_{Z^\prime}^2}{m_{Z^\prime}^2} \right)^{-1} \nonumber \\ && + \frac{1}{4 \pi} \frac{m_{\xi_1}^2}{w^4} \sqrt{1- \frac{m_{\chi_1}^2}{m_{\xi_1}^2}} \left( 1 - \frac{4 m_{\xi_1}^2}{m_{Z^\prime}^2} \right) \left( 1 - 2 s_X^2 + \frac{m_{\chi_1}^2}{m_{\xi_1}^2} s_X^2\right) \nonumber \\ && \qquad \times \left(1 + \frac{m_{\xi_1}^2}{m_{W^\prime}^2} - \frac{m_{\chi_1}^2}{m_{W^\prime}^2}\right)^{-1} \left( \left(1 - \frac{4 m_{\xi_1}^2}{m_{Z^\prime}^2}\right)^2 + \frac{\Gamma_{Z^\prime}^2}{m_{Z^\prime}^2} \right)^{-1} ~.\end{aligned}$$ The 1st line is the $W^\prime$ contribution, the 2nd and 3rd lines the $Z^\prime$ contribution and the 4th and 5th line the interference term. 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--- abstract: 'We present the results of special relativistic, adaptive mesh refinement, 3D simulations of gamma-ray burst jets expanding inside a realistic stellar progenitor. Our simulations confirm that relativistic jets can propagate and break out of the progenitor star while remaining relativistic. This result is independent of the resolution, even though the amount of turbulence and variability observed in the simulations is greater at higher resolutions. We find that the propagation of the jet head inside the progenitor star is slightly faster in 3D simulations compared to 2D ones at the same resolution. This behavior seems to be due to the fact that the jet head in 3D simulations can wobble around the jet axis, finding the spot of least resistance to proceed. Most of the average jet properties, such as density, pressure, and Lorentz factor, are only marginally affected by the dimensionality of the simulations and therefore results from 2D simulations can be considered reliable.' author: - 'D. López-Cámara, Brian J. Morsony, Mitchell C. Begelman, Davide Lazzati' title: 'Three-dimensional AMR simulations of long-duration Gamma-Ray Burst jets inside massive progenitor stars' --- Introduction {#sec:intro} ============ Long-duration gamma-ray bursts (GRBs) are produced by collimated relativistic outflows [@sari99] ejected in the core of massive stars at the end of their evolution [@w93; @hjorth03; @s03; @wb06]. Since their relativistic outflows have to propagate through their progenitor star material and exit the star before producing the gamma-ray photons, an outstanding issue with this scenario is to understand the mechanisms that prevent the entrainment of baryons in the light, hot jet [@mw99; @aloy00]. On the other hand, even if the jet-star interaction cannot slow down the jet, it has a strong impact on its dynamics [@mor07] and can supply enough energy to explode the star as a supernova [@khokhlov99; @mwh01; @wheeler02; @maeda03; @laz12]. In most cases, the study of the jet-star interaction has been performed numerically, with analytic models used only for guidance [@aloy02; @gomez04; @mor07; @matzner03; @brom11]. Even so, studying the propagation of a relativistic outflow that is continuously shocked by a much denser environment is not trivial since the length-scale of features in the relativistic material is typically $\sim R/\Gamma$ and therefore a large dynamical range is involved. When possible, adaptive mesh refinement (AMR) codes have been adopted [@mor07; @mor10; @laz09; @laz10; @laz11b; @nag11], and the simulations have been limited to two dimensions [@mw99; @aloy00; @mwh01; @zwm03; @miz06; @mor07; @mor10; @laz09; @laz10; @laz11b; @miz09; @nag11]. These studies have shown that even though the jet material is relativistic, the jet-head propagates sub-relativistically inside the star, thereby allowing causal contact between the bow shock at the head of the jet and the star. The shocked star material therefore drains at the sides of the jet producing a hot cocoon [@rr02; @laz05] instead of being entrained in the jet. Two dimensional (2D) simulations can provide important answers to the outstanding questions listed above. However, they are plagued by artifacts due to the presence of a symmetry axis in the center of the jet. First, a plug of dense material accumulates in front of the jet head, slowing down its propagation and creating plumes of hot plasma at wide angles (see Figure 1 in @laz10 for an example). Second, recollimation shocks coming from the sides of the jet bounce strongly off the jet axis in 2D simulations, while they could dissipate more efficiently in a simulation at the natural dimensionality. Finally, the role of turbulence and instabilities cannot be properly explored in 2D simulations. @wang08 found that in some cases a three dimensional (3D) relativistic jet would break apart and not be able produce a successful GRB (while in 2D it would produce a successful GRB). While 3D simulations of GRB jets have been attempted in the past [@zwh04], they were performed with a fixed grid code, casting doubt on their capability to resolve the required small scales. A 3D test-case with AMR was presented by @wang08, but since the jet-progenitor evolution varied drastically as a function of the numerical resolution (unlike our study), not much could be inferred from their study. Thus, in this paper we present, for the first time, 3D adaptive mesh refinement (AMR) simulations of GRB jets crossing a pre-SN progenitor and then flowing through the interstellar medium. This paper is organized as follows. We first describe the physics, initial setup, and the numerical simulations in Section \[sec:input\], followed by our results and discussion in Section \[sec:results\]. Conclusions are given in Section \[sec:conc\]. Physics, initial setup and simulations {#sec:input} ====================================== Physics and initial setup {#sec:phys&initsetup} ------------------------- As what now seems to be the generic model used for long GRBs [@mor07; @mor10; @laz09; @laz11a; @laz11b; @laz12; @lc09; @lc10; @lind10; @lind12; @nag11 for example], we consider the one-dimensional (1D) pre-supernova 16TI model from @wh06 as our initial stellar configuration. Initially (in the zero-age main sequence) model 16TI is a 16$\,M_\odot$ Wolf-Rayet star with 0.01$\,Z_\odot$ metallicity, and $3.3 \times 10^{52}$ erg s equatorial angular momentum. The final outcome of such model is a pre-SN progenitor with 13.95$\,M_\odot$ and nearly half the size of the sun ($R_0 = 4.1 \times 10^{10}$ cm). Assuming spherical symmetry, the 1D density and pressure profiles were mapped onto a 3D configuration that we assumed to be initially without rotation. The internal energy and the temperature were calculated assuming a relativistic polytropic equation of state ($\gamma$=4/3). The pre-SN progenitor was immersed in an interstellar medium (ISM) with constant density ($\rho_{\rm{{ism}}}=10^{-10}$ g cm$^{-3}$). Even though a wind environment would probably be more appropriate, we note that within the size of our simulation the dynamical role of the ambient medium is negligible and the results are therefore insensitive to the chosen ambient medium profile. A relativistic jet commencing its flow at the center of the pre-SN progenitor was imposed at all times as a boundary inflow condition. The jet was launched at the center of the star (in fact slightly above it), flowing upwards in the polar direction (x=z=0, y=R$_{\rm{{i}}}=$10$^9$cm). The imposed jet had a half-opening angle of $\theta_0$=10$^{{o}}$, a constant luminosity of L$_0=5.33 \times$10$^{50}$ erg s$^{-1}$, an initial Lorentz Factor of $\Gamma_{\rm{{0}}}$=5, and a ratio of internal over rest-mass energy equal to $\eta_0$=80 [@mor07; @mor10; @laz09]. In order to break the 2D axis symmetry, the jet was slightly asymmetric. For the latter, we set the jet with a 1% density and pressure asymmetry on either side of a line in the XZ plane 40 degrees from the X axis. Differently from @wang08 (3D numerical study in which a two dimensional symmetrical initial setup was assumed) our initial setup resembles that from model 3A in @zwh04 enhanced with a small perturbation in the jet. Numerical simulations {#sec:sims} --------------------- In order to follow the temporal evolution of our initial setup, we solved the 3D gas-dynamic equations using the FLASH code (version 2.5) in cartesian coordinates [@fryx00]. The simulation domain covered the top half of the pre-SN progenitor star as well as the ISM it is immersed in (see for example panel $a$ from Figure \[fig1\]). The boundaries were set at y$_{\rm{min}}$=10$^9$cm, y$_{\rm{ {max}}}$=2.4$\times 10^{11}$ cm, x$_{\rm{ {max}}}$=-x$_{\rm{ {min}}}$=6$\times 10^{10}$ cm, and z$_{\rm{ {max}}}$=-z$_{\rm{ {min}}}$=6$\times 10^{10}$ cm. Only the equatorial plane (y=y$_{\rm{ {min}}}$) was set with a reflective boundary condition, all the other boundaries were set with transmission conditions. We used a 10-level binary adaptive grid with square-shaped pixels ($\Delta{x}=\Delta{y}=\Delta{z} \equiv \Delta$). The highest refinement level (also referred to as the finest resolution level) were accessible only at the core of the pre-SN star were the jet is injected and initially propagates. Moving away from the stellar core, the maximum level of refinement was progressively decreased. In practice, the base of the jet had the finest resolution at all times and the three next finest levels followed the jet (and the polar part of the cocoon) as it drilled through the progenitor. Two set of simulations with a different value of $\Delta$ were performed. We will refer to the the hight resolution model as “HR”, and the low resolution as “LR”. The HR model had the finest resolution (covering the core of the star at all times) equal to $\Delta=3.125 \times 10^{7}$ cm, the jet for this case was followed with a resolution of at least $\Delta=1.25 \times 10^{8}$ cm. The LR model had the same setup but the value of the the finest resolution was set equal to $\Delta=6.25 \times 10^{7}$ cm, and the jet was followed with a resolution of at least $\Delta=2.5 \times 10^{8}$ cm. The resolution with which we follow the jet is comparable to that from the 3D collapsar study of @zwh04 (where the maximum resolution was $\Delta \sim$10$^{8}$ cm), and to the most recent 3D GRB jet study from @wang08 (where $\Delta=7 \times 10^{7}$ cm). The resolution with which we resolve the core of the star is comparable to that from previous 2D GRB jet numerical studies [@zwm03; @miz06; @mor07; @nag11]. In order to understand the three-dimensional effects properly, we also ran an extra two dimensional model. Differently from the 3D simulation, the 2D run was performed in cylindrical coordinates, the polar axis being coincident with the jet axis. The 2D model had an initial configuration akin to the XY and the ZY planes of the 3D model. The 2D model had the same input physics and resolution as that of the 3D HR model. A summary with the differences between the numerical models is shown in Table 1. ------- ---------------------- ---------------------- Model $\Delta$ in core $\Delta$ in jet ($\times 10^{7}$ cm) ($\times 10^{8}$ cm) 3D LR 6.250 2.50 3D HR 3.125 1.25 2D HR 3.125 1.25 ------- ---------------------- ---------------------- : Model characteristics \[default\] Results and Discussion {#sec:results} ====================== Global morphology. {#sec:morph} ------------------- In Figure \[fig1\] we show the density stratification maps for the XZ, XY, and ZY planes for the 3D LR model. Each panel shows a different timeframe: a. t$_a = 2.7$ s; b. t$_b = 4.2$ s; c. t$_c = 5.3$ s; d. t$_d = 7.3$ s; and e. t$_e = 9.3$ s. These panels are arranged to illustrate the jet-progenitor-ISM temporal evolution (animations of the density stratification map, Lorentz factor, radial density, radial Lorentz factor, and Schlieren map in the XY plane, are linked to the online version of this manuscript). The morphology of our system is divided into two main phases: when the jet moving inside the progenitor and when the jet has broken out of the star and interacts with the interstellar medium. Such temporal evolution is consistent with what has already been seen in previous numerical studies [@zwm03; @mor07]. Superimposed on the density stratification in Figure \[fig1\], we show the isocontour levels corresponding to 10$^{-4}$, 10$^{-2}$, 1, 10$^{2}$, and 10$^{4}$ (all in g cm$^{-3}$), these isocontour levels are shown in Figure \[fig2\]. {width="70.00000%"} The t$_{\rm{ {bo}}}$=4.2 s breakout time is similar to (but somewhat shorter than) that already seen in previous collapsar studies [@zwm03; @zwh04; @mor07; @mor10]. Depending on the progenitor that one chooses, and the particular characteristics of the jet, it takes 5 to 10 s to cross the stellar envelope. Compared to power-law stellar models, the models from @wh06 are more compact and dense and it takes less time for the jet to cross the realistic progenitors [@miz06]. Our t$_{\rm{ {bo}}}$ is very similar to the breakout time computed with the analytical model from @brom11. Still, it must be stated that since the jet in our numerical simulations is launched at an inner radius which is at least 10$^4$ times the gravitational radius (R$_{\rm{{i}}} \sim$ $10^{4}$R$_g$ for a 1.4M$_\odot$ black hole), the jet from the simulations is somewhat wider than that from the analytical model and thus it propagates slower. The t$_{\rm{ {bo}}}$ for our study implies that the average propagation velocity of the jet inside the star is $\sim0.32c$. The jet, composed of low density material, has its initial opening angle reduced by relativistic hydrodynamic collimation effects. Once the jet crosses the stellar envelope and breaks out of the surface, the cocoon (which surrounds the jet and is present since its formation) expands through the ISM [@rr02; @laz05], differently from when the jet is drilling through the progenitor when the cocoon is bound inside the star and close to the jet. When the jet breaks out of progenitor it becomes uncollimated and the cocoon moves out in the polar direction (moving parallel to the jet), also expanding sideways on top of the stellar surface. Such spreading (see panel c, d, and e from Figures \[fig1\]-\[fig2\]) was predicted by the analytic solution from @brom11. By this time not only does the cocoon present zones where variability is clearly present, but also the jet presents turbulent-like structures. The variability in the cocoon is due to the fact that the jet-cocoon system is at least five orders of magnitude denser than the surrounding ISM. Hence, any instability that forms on the cocoon’s boundary or that travels upwind from the jet into the cocoon is not dissipated. Due to the location of the outer boundaries, we are not able to follow the jet-cocoon-ISM system entirely after approximately 10 s. By this time the cocoon has crossed the outer boundaries (the jet crosses the top boundary at approximately 13 s). Also, it must be noted that as time passes the inner isocontour ($\rho$=10$^{4}$ cm g$^{-3}$) disappears. This is due to the reverse shock, which is expanding and pushing the dense material outwards. Such behavior has already been seen in the study of @laz10. {width="70.00000%"} Symmetry loss {#sec:simloss} ------------- To understand when the cylindrical symmetry is broken, in Figure \[fig3\] we plot the radial density distribution as well as the energy density (U, in erg cm$^{-3}$) for four different paths which move along a cone of 2$^{ {o}}$ half-opening angle (with its origin set at x=y=z=0) for model 3D LR. One of these paths moves radially (R) along the “(+X,+Z)” quadrant; another moves in the “(+X,-Z)” quadrant; another in the “(-X,+Z)” quadrant, and finally a path which moves along the “(-X,-Z)” quadrant. ![Radial profiles for different 2$^{ {o}}$ paths and times for model 3D LR. Panel a through d show different radial density profiles (g cm$^{-3}$), panel e shows different energy density profiles (erg cm$^{-3}$). Each of the radial paths commences in the origin and runs through different quadrants: (+X,+Z) quadrant (red line); (+X,-Z) quadrant (green line); (-X,+Z) quadrant (blue line); (-X,-Z) quadrant (black line). Each panel corresponds to a different timeframe: a. 2.7 s, b. 3.5 s, c. 3.9 s, d. and e. 4.2 s. The forward and reverse shock Mach number for each timeframe are also indicated. A movie of this figure is available in http://www4.ncsu.edu/$\sim$dlopez/Simulations$\underline{\hspace{0.2cm}}$(published).html[]{data-label="fig3"}](f3.jpg){width="45.00000%"} Consistently with previous 2D and 3D collapsar simulations [@zwm03; @zwh04; @miz06; @nag11], we see that as the jet drills through the stellar envelope a complex shock system forms, characterized by a forward and a reverse shock at the head of the jet and by and a series of conical recollimation shocks. The first recollimation shock, visible almost at the base of the jet, seems to be static (at R$\sim$2$\times$10$^9$ cm), but this is due to the fact that it moves at relativistic speed in the rest-frame. Since the study of the shock structure is not our goal, we do not focus on the nature of these shocks, nor do we need to know where the contact discontinuity is set. For the sake of our study all we need to be able to discern is the stellar and jet material that has and has not been shocked. Specifically, the regions which we will be addressing to in the rest of the discussion will be the shocked (SJ) and unshocked (UJ) parts of the jet. The UJ material maintains its initial density profile, while the SJ material breaks the symmetry in the 3D numerical simulations. The density profile can vary up to two orders of magnitude for different locations at the same distance from the progenitor center; on the other hand, the UJ varies less than an order of magnitude. As the jet crosses through the progenitor star its density decreases as a function of time (see Figure \[fig4\]). Before the jet breaks out from the stellar surface, the density profile inside the progenitor follows a quasi-constant profile which for t$_{\rm{ {bo}}}$ is $\sim$10$^{-1}$ g cm$^{-3}$. Then, when the jet breaks out of the stellar surface, it recovers a decaying radial density profile that for 9.3 s reaches density values as low as 10$^{-5}$ g cm$^{-3}$. ![Time evolution (0 s; 5.3 s; 7.3 s; 9.3 s) for the 2$^{ {o}}$ radial density profile (g cm$^{-3}$) from the (+X,+Z) quadrant (black line in Figure \[fig3\]). The stellar surface is indicated by the cyan dashed line. A movie of this figure is available in http://www4.ncsu.edu/$\sim$dlopez/Simulations$\underline{\hspace{0.2cm}}$(published).html[]{data-label="fig4"}](f4.jpg){width="45.00000%"} Lorentz factor evolution {#sec:lor} ------------------------ In Figure \[fig5\] and Figure \[fig6\] we show the temporal evolution for the Lorentz factor (with the velocity field also present); and the radial Lorentz factor profile along the 2$^{o}$ radial path for model 3D LR. Before the breakout time only a relativistic jet (with $\Gamma \sim$10) is present (see panel with t=4.2 s from Figure \[fig5\]). In Figure \[fig6\] we see that the SJ material for t$<$t$_{\rm{{bo}}}$ (blue, red and green lines) reaches values close to $\Gamma=$15; and the UJ Lorentz factor remains practically the same as the initial Lorentz factor ($\Gamma_{\rm{{0}}}$=5). This behavior is consistent with what has already been seen in previous GRB jet numerical simulations where the initial Lorentz factor, prior to t$_{\rm{ {bo}}}$, reaches values close to 10 [@zwh04; @miz06]. Once the jet breaks out of the stellar surface, the jet is accelerated. The high internal energy is able to accelerate material with Lorentz factors values of order $\Gamma \sim$100 in some zones. If accelerated with no energy dissipation, the jet’s maximum Lorentz factor would be 400 ($\Gamma_{\infty}$=$\Gamma_{\rm{{0}}} \eta_{\rm{{0}}}$) [@mor07]. In the panel with t=5.3 s from Figure \[fig5\] (brown line in Figure \[fig6\]) we show how the jet’s forward shock and the recently formed cocoon produce a “mushroom-like” high-$\Gamma$ structure. At later times (t$>$t$_{\rm{ {bo}}}$) (orange, cyan and black lines in Figure \[fig6\]) the mushroom-like structure grows bigger and its Lorentz factor $\Gamma$ increases significantly. To see the high Lorentz factor material in the jet, in Figure \[fig7\] we plot the $\Gamma$ isocontours for t=9.3 s. By this time the mushroom structure is evident and also certain regions in the polar axis reach Lorentz factor values as high as $\Gamma=50$ (pink isocontours). Unfortunately our numerical domain does not permit us to follow such high-$\Gamma$ regions and they escape the top boundary after approximately 10 seconds. This is once more congruent with the results from previous studies where after the t$_{\rm{ {bo}}}$ the cocoon reaches values as high as $\Gamma \sim$15, and the jet values of order $\Gamma \sim$100 [@miz06]. It must also be noted that when the jet breaks out of the stellar surface, a low-speed wind forms. This wind expands isotropically from the point in the stellar surface where the jet drilled through, and moves at an average speed vastly inferior (v$\le$0.01 c) to that of the jet. ![Lorentz factor stratification maps and velocity field for different timeframes (4.2 s; 5.3 s; 7.3 s; 9.3 s) for model 3D LR. The brown dashed line indicates the stellar surface. A movie of this figure is available in http://www4.ncsu.edu/$\sim$dlopez/Simulations$\underline{\hspace{0.2cm}}$(published).html[]{data-label="fig5"}](f5.jpg){width="45.00000%"} ![Same as Figure \[fig4\] but for the Lorentz factor. A movie of this figure is available in http://www4.ncsu.edu/$\sim$dlopez/Simulations$\underline{\hspace{0.2cm}}$(published).html[]{data-label="fig6"}](f6.jpg){width="45.00000%"} ![Lorentz factor isocontour map at t=9.3 s for model 3D LR. The isocontour levels correspond to: 2, 5, 10, 20 and 50. A movie of this figure is available in http://www4.ncsu.edu/$\sim$dlopez/Simulations$\underline{\hspace{0.2cm}}$(published).html[]{data-label="fig7"}](f7.jpg){width="50.00000%"} Resolution effects {#hrvslr} ------------------ In order to be able to evolve the initial setup up to integration times of order $\sim$10 s and to resolve the jet-progenitor with a suitably fine grid an AMR mesh was used. In Figure \[fig8\] we show the fraction of the volume that the three finest resolution levels occupied as a function of time. The finest grid level, the one with which the base of the jet was resolved ($\Delta \sim$10$^7$ cm, red line in Figure \[fig8\]) occupied less than 10$^{-7}$ of the entire volume. Meanwhile, the two next finest levels, which followed the propagation of the jet through the progenitor ($\Delta \sim$10$^8$ cm, blue and green lines in Figure \[fig8\]), occupied less than 10$^{-6}$ and 10$^{-4}$ of the volume. Needless to say, if we had used a fixed mesh with a comparable resolution to that with which the base of the jet was resolved, it would have required $\sim$10$^6$ more computational power (compared the computational power used in our simulations), and thus the benefit from using an AMR scheme. ![Temporal evolution of the fraction of the volume that the two finest level occupied. The red line corresponds to the finest level ($\Delta \sim$10$^7$ cm), and the blue and green lines to the second and third finest ($\Delta \sim$10$^8$ cm).[]{data-label="fig8"}](f8.jpg){width="50.00000%"} To verify that the evolution of the jet from our results is not dependent on the numerical resolution, we ran a new model with the same setup and physics but with a maximum resolution two times finer than for the low resolution (see Section \[sec:sims\] for details). In Figure \[fig9\] we show the density profiles for the 3D HR and 3D LR case. In each case we show the two main phases already discussed in Section \[sec:morph\]: prior to the breakout phase; the breakout; and the post breakout phase. We must note that the selected timeframes for each of the phases were chosen arbitrarily so that the LR and HR cases resemble each other, and hence their basic morphology characteristics can be compared. The main result is that the basic morphology characteristics from each phase (see discussion in Sections \[sec:morph\]$-$\[sec:lor\]), are well reproduced independently of the numerical resolution. Unlike the results from the numerical 3D jet GRB study from @wang08 where high resolution models gave qualitatively different jet dynamics, we obtain consistent jet behavior independently of the resolution. {width="70.00000%"} Among the differences associated with the resolution, are a higher level of turbulence and a slower advance of the jet head in the HR model. The latter’s jet moves $\sim$20% slower than the LR case, hence the breakout time for the HR case is t$_{\rm{ {bo}}}$=5.1 s. The jets velocity resolution difference is due to the fact that the HR case has a wider jet ($\sim$5% wider than the LR case). Since we are powering both jets equally, the narrow-LR jet will move faster. The turbulence resolution difference is due to the fact that the LR simulation has higher diffusion, and thus suppresses the small scale instabilities which are present in the HR model. The higher amount of turbulence in the HR model also slows it down (compared to the LR model), this due to the fact that a larger fraction of the energy is converted into turbulence. Hence, reducing the HR jet’s kinetic energy and ram-pressure. We must note that these two resolution effects are consistent with what has already been seen in previous jet-collapsar simulations, for example @mor07. Even though the latter study is a two-dimensional one, it also presents more vortices in the HR than in the LR case. To illustrate how in the HR model there is more variability than in the LR one, in Figure \[fig10\] we plot the radial density profiles for both resolution models. The upper panel of Figure \[fig10\] is the radial density profile for a path within the jet (specifically a 2$^{ {o}}$ path), while the lower panel is the radial density profile in the edge of the jet (10$^{ {o}}$ path). Inside the jet, there is no major difference between the two resolution models. On the other hand, at the edge of the jet the numerical resolution clearly affects the density profile. Here the HR presents numerous depressions in the density radial profile while the LR case has a radial profile which follows a smoother distribution with less depressions and variability. We also analyzed the effects of numerical resolution on the jet’s Lorentz factor distribution. Before the jet breaks out, apart from the velocity with which the jet evolves, there is no clear difference between the low and high resolution models. But at t$>$t$_{\rm{ {bo}}}$ there are morphological changes due to the resolution. Figure \[fig11\] shows the Lorentz factor map for the 3D LR and the 3D HR models just after breaking out of the stellar surface. The Lorentz factors are comparable but, as expected, the HR case has more turbulent-like structures. For both resolutions the low-density material (situated along the polar axis) has higher Lorentz factors compared to the material in the edge of the jet. The main difference, in qualitative terms, is that the high-$\Gamma$ regions from the LR model are split into smaller regions with lower $\Gamma$ values in the HR model. Another interesting aspect of our results is that the jet Lorentz factor morphology resembles the precessing jet case from @zwh04. In our simulations though, the zig-zagging pattern in the Lorentz factor seems to be due to the ability of the high-$\Gamma$ material to wobble around the star’s rotation axis and propagate through paths of least resistance (see below for more details). The observation of this effect is likely facilitated by the fact that we have a larger dynamic range inside the star ($\delta_R = R_0 / R_{\rm{ {min}}} = 41$) compared to the one from @zwh04 ($\delta_R =8.8$). Thus, there is enough range inside the star for 3D instabilities to develop. In addition, the individual grid pixels in the Zhang et al. (2004) simulations was not square. Rectangular pixels generate more diffusion in the longest of the pixels direction, and results are therefore not as robust as those from square pixel grid simulations (where the diffusion is the same for all three directions). ![Radial density profile (g cm$^{-3}$) for the 3D LR model (red line) and the 3D HR model (black line). For both resolutions the 2$^{ {o}}$ radial paths from the (+X,+Z) are shown in the upper panel, meanwhile the 10$^{ {o}}$ paths are shown in the lower panel.[]{data-label="fig10"}](f10.jpg){width="45.00000%"} ![Lorentz factor stratification maps for the 3D LR (left panel), and the 3D HR model (right panel). For both resolution models, we show a representative timeframe for when the jet has already broken out of the star and is moving across the ISM.[]{data-label="fig11"}](f11.jpg){width="50.00000%"} Finally, we must remark that we do not claim to reach convergency. If we take the number of grid cells across the jet diameter as an estimate of the Reynolds number (Re) of the simulation, we see that at the present time we can only reach Re$\sim$200. Such Reynolds number is approximately two orders of magnitude below the required Reynolds number in which the jet behavior is independent of the resolution [@be72]. Unfortunately, numerical simulations as those presented in this study, with resolutions of at least two orders of magnitude finer are not feasible (due to technical difficulties) to date. 2D vs 3D simulations -------------------- Finally, we checked how the evolution of the jet through the stellar envelope varies in two and three dimensional simulations. For this we ran an extra 2D numerical model with the same resolution ($\Delta$=1.25$\times 10^{8}$ cm), and the same parameter values (luminosity, R$_{\rm{{i}}}$, $\theta_0$, $\Gamma_{\rm{{0}}}$, and $\eta_0$, see section \[sec:sims\] for more details) as the 3D HR case. Apart from the dimension difference from the 3D models, we assumed polar axis symmetry in the 2D simulation; thus in reality we only simulated half of the x-axis domain (i.e x$_{\rm{ {min}}}$=0). The 2D simulation was carried out in cylindrical coordinates, with the polar axis coincident with the jet axis. In Figure \[fig12\] we show the density stratification maps for the 2D model at the two phases (as well as for the breakout time): a. t$<$t$_{\rm{ {bo}}}$; b. t$\approx$t$_{\rm{ {bo}}}$, and c. t$>$t$_{\rm{ {bo}}}$). The timeframes for each of the 2D phases shown in Figure \[fig12\] were chosen so that they resemble the correspondent timeframes from the 3D HR case. The basic morphology in the 2D case resembles that from the 3D model. In both cases we see a collimated jet that manages to drill through the stellar envelope. Apart from the polar axis symmetry imposed in the 2D model, there are many subtle differences between the 2D and 3D results: 1\. The jet moves slower in the 2D model than in the 3D one (congruent with @zwh04). Thus, the 2D breakout time is larger (t$^{ {2D}}_{\rm{ {bo}}}$=7 s) than for its correspondent 3D HR model. The reason for the slower jet’s motion in the 2D model is the imposed symmetry. Not only is the jet axis-symmetric, but also the stellar material in front of the jet has to remain symmetric at all times. The SJ material can only escape from the jets plug sideways, so a lot of energy ends up going into accelerating this stellar material. In 3D models, instead, the jet can deflect slightly and go around the plug (rather than continuing to accelerate it, see below). Finally, the 2D jet model has a SJ material mildly broader ($\approx$20%) than its respective from the 3D model. 2\. Even though the 2D model has the same resolution, the 2D jet presents less turbulent-like morphology than what is present in the 3D HR case. Also, the cocoon is less turbulent and broader in the 2D scenario, as is the case in the numerical study of @zwh04. 3\. Two low-density plumes are present in the 2D simulations (see right panel from Figure \[fig12\]). It must be noted that due to the imposed axis-symmetry, the plumes actually correspond to a low-density torus around the jet head (if it were a three dimensional domain and not two dimensional). Such low-density torus is not present in any of the 3D simulations. This is somewhat similar to the findings from @zwh04 where the head of the jet is noticeably different depending on the simulations dimensionality (either 2D or 3D). 4\. As shown in the upper panel of Figure \[fig14\], where we plot the density profile along the polar axis for the 2D and 3D models, the 2D density radial profile is less turbulent and the depressions are more profound than those in the 3D density radial profile. Also, the SJ material is mode dense by nearly two orders of magnitude in the 2D scenario close to the jet head. {width="90.00000%"} In Figure \[fig13\] we show the Lorentz factor structure for the 2D case (once the jet has just broken out of the stellar surface). Even though the mushroom $\Gamma$ structure forms, the 2D Lorentz factor morphology is noticeably different to that from the 3D HR case. The 2D $\Gamma$ structure presents much less variability. ![Lorentz factor stratification map for the 2D model at t=7.7 s.[]{data-label="fig13"}](f13.jpg){width="45.00000%"} The 2D low-density regions have high $\Gamma$ values of up to 15-20, values which are in agreement with those obtained by other 2D GRB jet studies [@zwh04; @nag11]. The 2D model also has a SJ which is broader than that from the 3D case. As was the case for the density map in the 2D model, the head front of the cocoon has less turbulent-like $\Gamma$ structures. In order to clarify this point, we show the radial Lorentz factor profile (along the polar axis) in Figure \[fig14\] (lower panel). Not only is a smoother radial profile present in the 2D, but also the high-density low-$\Gamma$) relationship is present. The SJ from the 2D case is approximately two orders of magnitude more dense than from the 3D model, but has a rather smaller $\Gamma$ value (of at most 5). ![Radial density profile (g cm$^{-3}$) (upper panel), and the radial Lorentz factor (bottom panel) for both the 2D model (red line) and the 3D model (black line) for the timeframe when the jet has broken out of the star. For both models the path is a polar axis (0$^{ {o}}$) radial paths from the (+X,+Z) quadrant.[]{data-label="fig14"}](f14.jpg){width="45.00000%"} In order to analyze how much the jet changes direction as it drills through the progenitor star (and later through the ISM once the jet has broken out of the star) in a three dimensional domain, we plot in Figure \[fig15\] the energy density ($U$) map in the XZ plane. The XZ planes shown for each timeframe correspond to the position where the $U$ centroid of the forward shock front was located at (see caption of Figure \[fig15\] for more details). Panels a through d show the $U$ map for when the jet is drilling through the stellar progenitor. In these it is noticeable how the centroid of the forward shock (CFS) does not have a gaussian like profile (it may have turbulent like behavior or even multiple spikes) and how the CFS wobbles around the polar axis finding the spot of least resistance to proceed. For example, notice how just before the jet breaks out of the star (panel d), the CFS is located far from the polar axis (x=-0.3$\times$10$^{10}$ cm , z=-0.6$\times$10$^{10}$ cm). Panel e shows how once the jet has broken out of the star and the cocoon has expanded thoroughly around the progenitor star, its correspondent CFS also expands and also remains far from the polar axis. ![Energy density (erg cm$^{3}$) XZ stratification maps for the centroid of the head front of the jet-cocoon structure for different times. Panels a through e correspond to the 3D HR model, panel f correspond to the 3D LR model. Panel a. t=1s, b. t=2s, c. t=4s, d. t=5.1s, e. t=5.7s, f. t=10s; have the XY plane located at Y/(10$^{10}$ cm)=1.3, 3.3, 18.9, 39.9, 57.1 (respectively). In each panel the maximum and minimum values of the forward shock’s energy density centroid (in erg cm$^{3}$) is indicated. Notice how some panels have different scales.[]{data-label="fig15"}](f15.jpg){width="45.00000%"} To further understand the deflection of the jet inside the pre-SN progenitor, we show the temporal evolution of the angle between the CFS and the polar axis ($\theta$, black line in Figure \[fig16\]). For a jet that is well aligned with the polar axis then the CFS displacement angle would yield $\theta =0$, clearly in Figure \[fig16\] this is not the case and the jet wobbles inside the star (with $\theta$ oscillating between 0.1$^{o}$ and 2$^{o}$). Hence, the jet moves faster in 3D than in 2D because it is able to wobble and move along the path with least resistance (apart from having a narrower jet-cocoon). Note that $\theta$ is always within the relativistic collimation angle ($1/\theta$, red line in Figure \[fig16\]), thus the relativistic jet is causally connected at all times. ![Temporal evolution of the CFS (black line), and the relativistic collimation angle (red line).[]{data-label="fig16"}](f16.jpg){width="45.00000%"} Limitations and comparison to other work ---------------------------------------- As with all numerical work, the choices made in carrying out the simulations reflect intentions and biases, and the current investigation lacks in several aspects. For example, similar to @zwh04, @mor07 [@mor10], and @laz09 we assumed that the star was static at all times which is clearly not the real case as the pre-SN for long GRBs have very high angular momentum values (J$>$10$^{15}$ cm$^2$ s) [@wh06]. We justify this by pointing out that the dynamical timescale of the pre-SN is of order close to hours. Then, since the integration time in our numerical simulations was of order 10 s, we were safe to assume that the pre-SN progenitor remained practically static at all times. In a previous study with a similar setup [@laz11b] found that after $10^{2}$ s the pre-SN stellar envelope had only expanded 2% of its original size. Another issue which can be improved is the ISM distribution. The pre-SN progenitor that we use as the initial setup has no hydrogen shell since during its stellar evolution it was lost by the presence of a stellar wind (which will also affect the ISM surrounding the pre-SN star). So, in order to have full consistency the ISM should have a density profile which was affected by the pre-SN wind, i.e. a profile that follows a $\propto$R$^{-2}$ distribution [@zwm03; @zwh04; @can04; @nag11]. But since the jet-cocoon system is an ultra-relativistic flow, the density profile of the ISM will barely affect the jet once this has just broken out of the stellar surface [@mor07]. In fact, the GRB-jet needs to reach $\sim$10$^{14}$cm for the ISM’s profile to play a key role in the flow [@bm76; @dc12]. Thus, we were secure to assume that the ISM density was constant. We use an adiabatic ($\gamma$=4/3) as our equation EOS [@zwm03; @zwh04; @miz06; @mor07; @mor10; @laz09; @nag11]. We do not take into account the neutrino pressure, nor do we take into account the gravitational effects from the central compact object. Even though it has been shown that close to the pre-SN’s progenitor nucleus the neutrinos play an important role [@lc09], since the inner boundary was set so far away, R$_{\rm{{i}}} \sim$10$^9$cm, equivalent to approximately $10^{4}$ gravitational radii, from the region where neutrinos dominate (and where the compacts object relativistic effects must be taken into consideration), the neutrino and relativistic effects were safely ignored. Since the follow up of newly formed elements was not the aim of this study and that the calculation of such new elements would have not permitted us to study the flow both at an adequate resolutions and for the long times desired, nuclear burning was not included. Also, even though magnetic fields will affect the emissivity of the jet [@miz06], and could even give origin to variability in the light curve [@bh98], they were disregarded due to the technical difficulties when following a magnetized relativistic flow with an adaptive mesh code. Conclusions {#sec:conc} =========== We present, for the first time, 3D AMR simulations of GRB jets expanding inside a realistic pre-SN progenitor and then flowing through the interstellar medium. Our numerical simulations, confirm that relativistic jets can propagate and break out of the progenitor star while remaining relativistic. The morphology is divided into two main phases:\ 1. Pre-t$_{bo}$. During this phase the jet head moves at mildly relativistic velocities ($\sim$c/2) inside the progenitor’s stellar envelope.\ 2. Post-t$_{bo}$. Once the jet breaks out of the surface, it accelerates and reaches Lorentz factors of order $\Gamma \sim$ 50. The initial progenitor density profile is reshaped by the forward and reverse shocks. The material between the forward and reverse shocks break the two-dimensional symmetry in the numerical simulations. We obtain similar behavior independently of the numerical resolution. The resolution does not affect in great detail the flow and the morphology in each phase is well reproduced. Still, the amount of turbulence and variability observed in the simulations is higher for higher resolutions. Also, for finer numerical resolutions the jet moves slower; and regions with high Lorentz factors break up into smaller regions with lower $\Gamma$ values. The propagation of the jet head inside the progenitor star is slightly faster in 3D simulations compared to 2D ones at the same resolution. This behavior is due to the fact that the jet in 3D simulations is narrower and can wobble around the jet axis finding the spot of least resistance to proceed. Most of the jet properties, such as density, pressure, and Lorentz factor, are only marginally affected by the dimensionality of the simulations and therefore results from 2D simulations can be considered reliable. If, instead, more detailed properties such as variability are to be investigated, simulations carried out in the proper dimensionality (i.e. 3D) are required. We thank S.E. Woosley and A. Heger for making their pre-SN models available, and the referee for comments, suggestions and constructive criticism which helped improve the original version of the manuscript. The software used in this work was in part developed by the DOE-supported ASC/Alliance Center for Astrophysical Thermonuclear Flashes at the University of Chicago. This work was supported in part by the Fermi GI program grants NNX10AP55G and NNX12AO74G (D.L. and D.L.-C.). B.J.M. is supported by an NSF Astronomy and Astrophysics Postdoctoral Fellowship under award AST-1102796. Aloy, M. A., Ib[á]{}[ñ]{}ez, J. M., Miralles, J. A., & Urpin, V. 2002, , 396, 693 Aloy, M. A., M[ü]{}ller, E., Ib[á]{}[ñ]{}ez, J. M., Mart[í]{}, J. M., & MacFadyen, A. 2000, , 531, L119 Balbus, S. A. & Hawley, J. F. 1998 Rev. Mod. Phys., 70,. 1 Birch, S. F., & Eggers, J. M. 1972, In: Free Turbulent Shear Flows, Conf. Proc. 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--- abstract: 'We consider a class of infinite-dimensional, modular, graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. We identify two subclasses of [*Nottingham Lie algebras*]{} as loop algebras of finite-dimensional simple Lie algebras of Hamiltonian Cartan type. A property of Laguerre polynomials of derivations, which is related to toral switching, plays a crucial role in our constructions.' address: - | Dipartimento di Matematica e Applicazioni\ Università degli Studi di Milano - Bicocca\ via Cozzi 53\ I-20125 Milano\ Italy - | Dipartimento di Matematica\ Università degli Studi di Trento\ via Sommarive 14\ I-38050 Povo (Trento)\ Italy author: - Marina Avitabile - Sandro Mattarei bibliography: - 'References.bib' title: | Nottingham [L]{}ie algebras with diamonds\ of finite and infinite type --- Introduction ============ [*Nottingham Lie algebras*]{} owe their name to one remarkable special case: the graded Lie ring with the lower central series of the Nottingham group, over the field of $p$ elements, with $p>2$ [@Jenn; @John; @Cam]. Although the Nottingham group over ${\mathbb F}_p$ is a very complex object, its lower central series descends in a simple and regular way, the lower central quotients being generally one-dimensional, except for one quotient every $p-1$ being two-dimensional. The essential feature which leads to the generalization considered here is that the Nottingham group is a pro-$p$ group of [*width two*]{} and [*obliquity zero*]{}, in the language of [@KL-GP]. Pro-$p$ groups with those two properties were called [*thin*]{} in [@CMNS], and the properties carry over naturally to graded Lie algebras. The following equivalent but more practical definition is available for Lie algebras. A [*thin Lie algebra*]{} is a graded Lie algebra $L=\bigoplus_{k=1}^{\infty}L_{k}$ over a field ${\mathbb F}$, with $\dim(L_1)=2$ and satisfying the [*covering property*]{} $$\label{eq:covering} L_{i+1}=[u, L_{1}] \quad \textrm{for all $0 \neq u\in L_{i}$, for all $i\geq 1$}.$$ For simplicity in this paper we supplement this definition with the assumption that $L$ has infinite dimension. It follows that a thin Lie algebra $L$ has trivial centre, is generated by $L_1$, and that homogeneous components can only have dimension one or two. Those of dimension two are called [*diamond*]{} (for reasons relating to the lattice of open normal subgroups in the pro-$p$ group case, see [@CMNS]), but there are good reasons to grant the status of [*fake diamonds*]{} to certain one-dimensional components. Diamonds are numbered in the order of occurrence (including the fake ones, properly recognized), starting with the [*first*]{} diamond $L_1$. It may happen that $L_1$ is the only diamond, but then $L$ belongs to the distinguished family of [*graded Lie algebra of maximal class,*]{} which were introduced in [@CMN]. Because those Lie algebras were completely classified in [@CN; @Ju:maximal] (in the infinite-dimensional case), we conveniently exclude them from the definition of thin Lie algebras. Then the degree $k$ where the second diamond occurs becomes the main parameter of a thin Lie algebra. According to [@AviJur] (but see the simpler proof in [@AviMat:A-Z Section 3], where an updated definition accommodates for the peculiarities of characteristic two), the parameter $k$ can only take the values $3$, $5$, $q$ or $2q-1$, for some power $q$ of the characteristic $p$ in case this is positive. Each of the last two cases comprises a large variety of thin Lie algebras, whose features are best described separately. The introduction of [@CaMa:Hamiltonian] contains a detailed exposition of the thin Lie algebras with second diamond in degree $2q-1$. The thin Lie algebras with second diamond in degree $q$ were named [*Nottingham Lie algebras*]{} in [@CaMa:Nottingham]. According to [@CaMa:Nottingham], each diamond past the first in a Nottingham Lie algebra is assigned a [*type,*]{} taking values in the underlying field augmented with infinity, in a unique way which we recall in Section \[sec:nott\]. The second diamond of a Nottingham Lie algebra always has type $-1$. Diamonds of type zero or one are really one-dimensional components, but it is convenient to allow them in certain degrees and dub them [*fake*]{}. Fake diamonds are recognized among the one-dimensional components from certain relations which hold in them. Because $1\equiv -1 \bmod 2$, in the special case of characteristic two the second diamond of a Nottingham Lie algebra would be fake, and so one needs some care to recognize a Nottingham Lie algebra as such. This and more peculiarities of Nottingham Lie algebras of characteristic two are discussed extensively in [@AviMat:A-Z], but in the present paper we conveniently assume $p>2$. We will, however, add separate remarks on how our results need to be modified in characteristic two. Various diamond patterns are possible, and we describe them in Section \[sec:nott\]. There are Nottingham Lie algebras with all diamonds of infinite type, and others with all diamonds of finite type. The goal of this paper is a construction (whence an existence proof) for Nottingham Lie algebras having diamonds of both finite and infinite types. They have diamonds in all degrees congruent to $1$ modulo $q-1$, but only one diamond every $p^s$ diamonds has finite type (starting with the second diamond, of necessity). Furthermore, the finite diamond types follow an arithmetic progression, giving the algebras a periodic structure. A distinction arises according to whether this arithmetic progression is entirely contained in the prime field, or not. Fake diamonds only occur in the former case. This is actually the easier case, and its special case where the arithmetic progression is the constant sequence $-1$ was already considered in [@AviMat:A-Z]. We state and prove our main results in Sections \[sec:prime-field\] and \[sec:big-field\], respectively. As for other examples of thin Lie algebras produced so far (including those with second diamond in degree $2q-1$), those with a periodic structure can be obtained through a [*loop algebra*]{} construction, starting from a finite-dimensional Lie algebra with a suitable cyclic grading. In most cases those finite-dimensional Lie algebras are simple (or close to simple) Lie algebras of the Cartan type $H$ (Hamiltonian). In this paper we need two types of simple Hamiltonian Lie algebras, of dimension a power of $p$ and two less than a power of $p$, whose definitions we recall in Section \[sec:Cartan\]. While the correct Lie algebra to employ is generally easy to guess for dimension reasons (where the absence of fake diamonds, or the presence of one or two in each period, calls for a Lie algebra of dimension a power of $p$, or one or two less), explicitly producing a suitable cyclic grading can be a real challenge without the proper tool. In lucky situations one can obtain the required cyclic gradings starting from the natural gradings of those Hamiltonian Lie algebras. In other cases one needs to pass to new gradings by what we may call a [*grading switching*]{}. This procedure is related to [*toral switching,*]{} a fundamental tool in the theory of modular Lie algebras, but it needs to be more general in one respect, because the cyclic grading of interest may not be associated to a torus in any way. Like toral switching, grading switching is based on taking some version of an exponential of a derivation of the Lie algebra. A toral switching based on [*Artin-Hasse exponentials,*]{} which makes sense for nilpotent derivations $D$ and reduces to [*truncated exponentials*]{} $\sum_{i=0}^{p-1}D^i/i!$ when $D^p=0$, was described in [@Mat:Artin-Hasse]. In the present paper this would only allow one to deal with the case where the arithmetic progression of finite diamond types is contained in the prime field. In the other case we need a special instance of a completely general version of grading switching, valid for arbitrary derivations, which is developed in [@AviMat:Laguerre]. We provide the required details in Section \[sec:Laguerre\]. In his master’s thesis [@Sca:thesis], written under the direction of the first author, Claudio Scarbolo gave a construction for our Nottingham Lie algebras of Section \[sec:big-field\] in the case of characteristic two, based on direct calculations and taking advantage of certain peculiarities of the characteristic (see Remark \[rem:char2-big\]). Nottingham Lie algebras {#sec:nott} ======================= We summarize here the more complete discussion of Nottingham Lie algebras given in [@AviMat:A-Z Section 2], restricting ourselves to information which is essential to our present goals. Thus, suppose $L=\bigoplus_{i=1}^{\infty} L_{i}$ is a thin Lie algebra with second diamond in degree $q>3$, a power of the odd characteristic $p$. As anticipated in the Introduction, we generally call such $L$ a Nottingham Lie algebra (but see [@AviMat:A-Z Remark 3.2] for a motivated exclusion of one case with $q=5$). Choose a nonzero element $Y\in L_1$ with $[L_2,Y]=0$. According to [@CaJu:quotients] we have $$\label{eq:first_chain} C_{L_1}(L_2)=\cdots=C_{L_1}(L_{q-2})=\langle Y \rangle,$$ where $C_{L_1}(L_i)= \{ a \in L_1 \mid [a,b]=0 \textrm{ for every $b \in L_i$} \}$, the centralizer of $L_i$ in $L_1$. It was shown in [@Car:Nottingham] (but see the more accessible proof given in [@AviMat:A-Z Section 2]), that one can choose $X\in L_1\setminus\langle Y\rangle$ such that $$\label{eq:second_diamond_type} [V,X,X]=0=[V,Y,Y], \quad [V,Y,X]=-2[V,X,Y],$$ where $V$ is any nonzero element of $L_{q-1}$. Here we adopt the left-normed convention for iterated Lie brackets, namely, $[a,b,c]$ denotes $[[a,b],c]$. An element $X$ satisfying relations  is determined up to a scalar multiple, and so is $Y$, which is any nonzero element of $C_{L_1}(L_2)$. Relations  also imply that $L_{q+1}$, the homogeneous component immediately following the second diamond, has dimension one. This extends to a more general fact about thin Lie algebras (not necessarily with second diamond in degree $q$): in a thin Lie algebra of arbitrary characteristic with $\dim(L_3)=1$, two consecutive components cannot both be diamonds. Suppose now that $L_{i}$ is a diamond of $L$ in degree $i>1$. Let $V$ be a non-zero element in $L_{i-1}$, which has dimension one according to the previous paragraph, whence $[V,X]$ and $[V,Y]$ span $L_i$. If the relations $$\label{eq:diamond_type} [V,Y,Y]=0=[V,X,X], \quad (1- \mu)[V,X,Y]=\mu [V,Y,X],$$ hold for some $\mu \in {\mathbb F}$, then we say that the diamond $L_i$ has (finite) type $\mu$. In particular, relations  postulate that the second diamond $L_q$ has type $-1$. The third relation in Equation  has a more natural appearance when written in terms of the generators $Z=X+Y$ and $Y$, which were more convenient in [@CaMa:Nottingham], namely, $[V,Z,Y]=\mu [V,Z,Z]$. We also define $L_i$ to be a diamond of type $\infty$ by interpreting the third relation in Equation  as $-[V,X,Y]=[V,Y,X]$ in this case. We stress that the type of a diamond of a Nottingham Lie algebra $L$ is independent of the choice of the graded generators $X$ and $Y$. In fact, we have seen above that our requirement $[L_2,Y]=0$ together with relations  determine $X$ and $Y$ up to scalar multiples, and replacing them with scalar multiples does not affect relations . According to the defining relations , a diamond of type $\mu=0$ should satisfy $[V,X,Y]=0=[V,X,X]$, while a diamond of type $\mu=1$ should satisfy $[V,Y,X]=0=[V,Y,Y]$. But then $[V,X]$, or $[V,Y]$, respectively, would be a central element in $L$ and, therefore, should vanish because of the covering property . Hence, $L_i$ would be one-dimensional after all, and hence not a genuine diamond. Thus, strictly speaking, diamonds of type $0$ or $1$ cannot occur. Nevertheless, we allow ourselves to call [*fake diamonds*]{} certain one-dimensional homogeneous components of $L$ and assign them type $0$ or $1$ if the corresponding relations hold, as given in Equation . This should be regarded as a convenient piece of terminology rather than a formal definition: whenever a one-dimensional component $L_i$ formally satisfies the relations of a fake diamond of type $1$, the next component $L_{i+1}$ satisfies those of a fake diamond of type $0$. However, as a rule, only one of them ought to be called a fake diamond, usually because it fits into a sequence of (genuine) diamonds occurring at regular distances. Before continuing with our description of basic properties of Nottingham Lie algebras we briefly comment on what happens if we temporarily relax our blanket assumptions $p$ odd and $q>3$. Many thin Lie algebras $L$ exist where $L_3$ is the second diamond, and a summary of the known ones is given in [@AviMat:A-Z Remark 3.4]. In particular, in characteristic $p=3$ they include Lie algebras which ought to be included in the class of Nottingham Lie algebras, see [@AviMat:A-Z Remark 3.4]. This is so when generators $X$ and $Y$ can be chosen such that relations  hold, even though uniqueness of $\langle X\rangle$ and $\langle Y\rangle$ fails in this case. With this extended definition the Nottingham Lie algebras which we construct in Sections \[sec:big-field\] and \[sec:prime-field\] will be such also for $q=3$, with the diamond types determined by relations  as usual. The picture for characteristic two is more peculiar, because the second diamond fails to reveal itself as such, being fake of type $1=-1$. It can even occur that all diamonds past the first are fake, and such a Nottingham Lie algebra disguises itself as a graded Lie algebra of maximal class. We refer the reader to the last part of [@AviMat:A-Z Section 3] for a discussion, we safely exclude characteristic two from our statements, but provide an interpretation for them in Remarks \[rem:char2-big\] and \[rem:char2-prime\]. Resuming our general discussion of Nottingham Lie algebras, it was proved in [@CaMa:Nottingham] that $$\label{eq:second_chain} C_{L_1}(L_{q+1})= \cdots = C_{L_1}(L_{2q-3})=\langle Y \rangle,$$ provided $p>5$. In particular, $L$ cannot have a third diamond in degree lower than $2q-1$. The assumption on the characteristic was weakened in [@Young:thesis Proposition 5.1] to $p>3$ and $q\neq 5$ (a necessary exception). We refer the reader to [@AviMat:A-Z Section 2] for a discussion of Nottingham Lie algebras where the third genuine diamond occurs in degree higher than $2q-1$. Assuming that $L$ does have a genuine third diamond of finite type $\mu_3$ (hence different from $0$ and $1$, two exceptional cases which are also discussed in [@CaMa:Nottingham]) in degree $2q-1$, detailed structural information on $L$ was obtained in [@CaMa:Nottingham], for $p>5$. It was shown there that $L$ is uniquely determined as a thin Lie algebra, that the diamonds occur in each degree congruent to $1$ modulo $q-1$, and that their types follow an arithmetic progression (determined by $\mu_2=-1$ and $\mu_3$). The stated uniqueness of $L$ was proved by showing that the relations of degree up to $2q$, in the generators $X$ and $Y$, which $L$ satisfy, actually define a central (or second central in one case) extension of $L$. Diamonds appearing at regular intervals, with types following an arithmetic progression, reveal a periodic structure which in our context is captured by the following definition. \[def:loop\_algebra\] Let $S$ be a finite-dimensional Lie algebra, over the field ${\mathbb F}$, with a cyclic grading $S=\bigoplus_{k\in{\mathbb Z}/N{\mathbb Z}}S_k$. Let $U$ be a subspace of $S_{\bar 1}$ and let $T$ be an indeterminate over ${\mathbb F}$. The [*loop algebra*]{} of $S$ with respect to the given grading and the subspace $U$ is the Lie subalgebra of $S\otimes{\mathbb F}[T]$ generated by $U\otimes T$. The subspace $U$ need not be mentioned when it coincides with $S_{\bar 1}$, as will always be the case in this paper. Loosely speaking, the loop algebra construction produces an infinite-dimensional Lie algebra from $S$ by [*replicating*]{} its structure periodically. Proving that such a loop algebra is thin (in particular, the verification of the covering property ) can then conveniently be done inside the finite-dimensional Lie algebra $S$. Beware that if $S$ is graded over an arbitrary cyclic group $G$ of order $N$ we need to fix a specific isomorphism of $G$ with ${\mathbb Z}/N{\mathbb Z}$ (or, equivalently, a distinguished generator $\bar 1$ for $G$) for the corresponding loop algebra to be defined. Existence results for the Nottingham Lie algebras $L$ whose uniqueness was proved in [@CaMa:Nottingham] have been proved in various articles, always by producing the required algebra as a loop algebra of some finite-dimensional Lie algebra $S$ with respect to a suitable cyclic grading. In each case $L$ has diamonds, possibly fake, in all degrees congruent to $1$ modulo $q-1$. If all diamonds are genuine (not fake) then $S$ is a simple Lie algebra of dimension a power of $p$. If fake diamonds are present then $S$ is simple of dimension two less than a power of $p$, possibly extended by an outer derivation, in the two cases where backwards continuation of the arithmetic progression of diamond types would predict the first diamond $L_1$ to be fake (which it cannot be). The special case $\mu_3=-1$, where the arithmetic progression of diamond types is a constant sequence, was dealt with in [@Car:Nottingham], taking $S=W(1;n)$, a [*Zassenhaus algebra*]{}. When $q=p$ the corresponding loop algebra $L$ is the graded Lie algebra associated with the lower central series of the Nottingham group over the field of $p>2$ elements. For the remaining cases a dichotomy arises according to whether the arithmetic progression of diamond types is contained in the prime field, or not, which will entail the absence or presence of fake diamonds. The Nottingham algebras with $\mu_3\in{\mathbb F}_p\setminus\{-1\}$ were produced in [@Avi], as loop algebras of simple [*graded Hamiltonian algebras*]{} $H(2;(1,n))^{(2)}$ (extended by an outer derivation when $\mu=-2,-3$), and those with $\mu_3\not\in{\mathbb F}_p$ in [@AviMat:A-Z], as loop algebras of [*Albert-Zassenhaus algebras*]{} $H(2:(1,n);\Phi(1))$. Those with $\mu_3\in{\mathbb F}_p$ can also be recovered from those with $\mu_3\not\in{\mathbb F}_p$ through a deformation argument, which is explained in [@AviMat:A-Z Section 7]. The third diamond can also have type $\mu_3=\infty$. In that case a great variety of diamond patterns are possible, most of which are not periodic, as shown in [@Young:thesis Theorem 3.3]. One periodic possible pattern has diamonds occurring in all degrees congruent to $1$ modulo $q-1$, with most of them having infinite type except for one diamond of finite type every $p^s$ diamonds, for some positive integer $s$, and with the finite types following an arithmetic progression. Thus, a diamond of finite type occurs in each degree congruent to $1$ modulo $p^s(q-1)$, and the arithmetic progression of finite diamond types is determined by the earliest such diamond after $L_q$, which occurs in degree $q+p^s(q-1)$. One such algebra was constructed in [@AviMat:A-Z Section 5], as a loop algebra of an Albert-Zassenhaus algebra $H(2:(s+1,n);\Phi(1))$, where all finite types equal $-1$. The main goal of this paper, in Sections \[sec:big-field\] and \[sec:prime-field\], is to provide analogous constructions for the non-constant arithmetic progressions of finite diamond types. Here, too, we have a dichotomy according to whether the progression is entirely contained in the prime field ${\mathbb F}_p$, or not. The loop algebra construction will then employ different finite-dimensional Lie algebras $S$, namely, a simple graded Hamiltonian algebra $H(2;(s+1,n))^{(2)}$ in the former case, and an Albert-Zassenhaus algebra $H(2:(s+1,n);\Phi(1))$ in the latter. Grading switching {#sec:Laguerre} ================= The loop algebra construction described in the Section \[sec:nott\] requires specifying a cyclic grading of a finite-dimensional Lie algebra $S$. As briefly mentioned there, and more extensively discussed in [@CaMa:Hamiltonian; @AviMat:A-Z], the Lie algebras $S$ which have been used to construct thin Lie algebras in this way belong to the classical types $A_1$ and $A_2$ (which are the only ones occurring in characteristic zero as well, see [@CMNS]), or to the generalized Cartan types $W$ and $H$. In this section we discuss a partly new technique to produce suitable cyclic gradings. Gradings over a cyclic group can be obtained as specializations of gradings over abelian groups of larger rank, by passing to cyclic quotients of the latter. One occurrence of gradings for our Lie algebras $S$ is as generalized root space decompositions with respect to tori in $\operatorname{Der}(S)$, but Lie algebras of Cartan type come equipped with natural gradings which are not of that type (because the grading groups do not have exponent $p$). Suitable cyclic specializations of the latter gradings can sometimes produce thin Lie algebras as the corresponding loop algebras (an example from [@AviMat:A-Z] is recalled at the end of this section), but in order to produce thin Lie algebras with different diamond type patterns one may have to pass to new cyclic gradings. In case of gradings associated with tori, one fundamental tool to [*switch gradings*]{} is available. The [*toral switching*]{} technique plays a crucial role in the classification theory of simple modular Lie algebras. The basic idea, originating from [@Win:toral], but then substantially generalized in [@BlWil:rank-two], and finally [@Premet:Cartan], is to apply to a maximal torus of a simple restricted modular Lie algebra some sort of exponential of an inner derivation, in order to produce a new torus with better properties. Associated with the new torus is a new grading of the algebra. Part of the process can be placed in a more general setting, which we describe now, where a new grading can be obtained from a given one without the intervention of any torus. To this purpose it is not restrictive to consider only gradings over a cyclic group. In characteristic zero exponentials of derivations are automorphisms when they are defined (for example when the derivation is nilpotent). This is not the case in positive characteristic, and toral switching takes advantage of this fact. In prime characteristic $p$, evaluating the exponential series $\exp(X)=\sum_{i=0}^{\infty}X^i/i!$ on a derivation $D$ only makes sense under the condition $D^p=0$, allowing one to neglect all terms of the exponential series whose denominator is a multiple of $p$. More rigorously, instead of the ordinary exponential series one considers the [*truncated exponential*]{} $E(X)=\sum_{i=0}^{p-1}X^i/i!$, which can be evaluated on an arbitrary derivation $D$ but is most closely assimilated to $\exp(D)$ when $D^p=0$. Although this condition on $D$ does not ensure that $E(D)$ is an automorphism (only $D^{(p+1)/2}=0$ does, for $p$ odd), it forces $E(D)$ to send a grading into a grading in the following result, which is [@Mat:Artin-Hasse Theorem 2.3]. \[thm:exp\] Let $A$ be a non-associative algebra over a field of prime characteristic $p$, graded over the integers modulo $m$. Suppose that $A$ has derivation $D$ such that $D^{p}=0$ graded of degree $d$, with $m\mid pd$. Then the direct sum decomposition $A=\bigoplus_{i}E(D)A_{i}$ is a grading over the integers modulo $m$. This is the [*grading switching*]{} mentioned earlier. The main result of [@Mat:Artin-Hasse] is that Theorem \[thm:exp\] remains true under the weaker hypothesis that $D$ is nilpotent, provided that $E(D)$ is replaced by $E_p(D)$, where $E_p(X)$ is the [*Artin-Hasse exponential*]{} series. The nilpotency assumption on $D$ is too restrictive for the applications in this paper. In [@AviMat:Laguerre Theorem 5.1] we have further extended Theorem \[thm:exp\] to arbitrary derivations, where the place of $E_p(D)$ is taken by certain maps defined by means of (generalized) Laguerre polynomials. To avoid unnecessary complications we limit ourselves to quoting the special case we need of that general result, where $D$ satisfies a certain equation. The classical (generalized) Laguerre polynomial of degree $n \geq 0$ can be defined as $L_n^{(\alpha)}(X)=\sum_{k=0}^{n}\binom{\alpha+n}{n-k}(-X)^k/k!$. The parameter $\alpha$ is classically a complex number, but the definition allows one to take $\alpha$ in any field where $n!$ is invertible. Let ${\mathbb F}$ be a field of characteristic $p>0$, let $\alpha \in {\mathbb F}$, and consider the Laguerre polynomial $$L_{p-1}^{(\alpha)}(x)=\sum_{k=0}^{p-1}\binom{\alpha+p-1}{p-1-k} \frac{(-x)^k}{k!}.$$ When $\alpha=0$ this coincides with the truncated exponential $E(X)$. \[thm:special\] Let $A=\bigoplus_k A_k$ be a non-associative algebra over a field ${\mathbb F}$ of prime characteristic $p$, graded over the integers modulo $m$. Suppose that $A$ has a graded derivation $D$ of degree $d$ such that $D^{p^2}=\lambda^{(p-1)p}D^p$ for some nonzero $\lambda \in {\mathbb F}$, with $m\mid pd$. Suppose that there exists $\pi \in {\mathbb F}$ with $\pi^p-\pi=\lambda^p$. Let $A=\bigoplus_{a\in{\mathbb F}_p}A^{(a\lambda)}$ be the decomposition of $A$ into a direct sum of generalized eigenspaces for $D$, and let $\mathcal{L}_D:A\to A$ be the linear map whose restriction to $A^{(a\lambda)}$coincides with $L_{p-1}^{(a\pi)}(D)$. Then $A=\bigoplus_k \mathcal{L}_D(A_k)$ is a grading of $A$ over the integers modulo $m$. The conclusion follows from [@AviMat:Laguerre Theorem 5.1] by taking $h(T)=0$ and $g(T)=\pi\lambda^{-p} T^p$, whence $g(a \lambda)=a \pi$. In the special case where $D^p=0$, the value of $\lambda$ is immaterial because $A=A^{(0)}$, and Theorem \[thm:special\] becomes Theorem \[thm:exp\]. Certain Lie algebras of Cartan type {#sec:Cartan} =================================== We briefly recall here the definitions of certain Lie algebras of Cartan type belonging to the Hamiltonian series. A broader discussion of these Lie algebras in the context of thin Lie algebras can be found in [@AviMat:A-Z]. Let ${\mathbb F}$ be a field of prime characteristic $p$. The algebra $\mathcal{O}(1,n)$ of *divided powers* in one indeterminate $x$ of height $n$, is the associative ${\mathbb F}$-algebra with basis elements $x^{(i)}$, for $0\leq i\leq p^{n}$, and multiplication defined by $x^{(i)}\cdot x^{(j)}=\binom{i+j}{i} x^{(i+j)}$. The algebra $\mathcal{O}(2,(n_1, n_2))$ of divided powers in two indeterminates $x$ and $y$ of heights $n_1$ and $n_2$ may be identified with the tensor product algebra $\mathcal{O}(1;n_1)\otimes\mathcal{O}(1;n_2)$. A basis is given by the monomials $x^{(i)}y^{(j)}$, for $0\leq i<p^{n_1}$ and $0\leq j<p^{n_2}$, which are multiplied according to the rule $x^{(i)} y^{(j)}x^{(k)}y^{(l)}=\binom{i+k}{i}\binom{j+l}{j} x^{(i+k)}y^{(j+l)}$. We use the standard shorthands $\bar{x}=x^{(p^{n_{1}}-1)}$ and $\bar{y}=y^{(p^{n_{2}}-1)}$. In the algebra $\mathcal{O}(1,1)$, which is isomorphic with ${\mathbb F}[x]/(x^p)$, we may define a sort of generalized power $(1+x)^{\alpha} = \sum_{i=0}^{p-1} \binom{\alpha}{i}i! x^{(i)}$, where the exponent $\alpha$ is an arbitrary element of ${\mathbb F}$. When $\alpha=0,1,\cdots, p-1$ this expression specializes to the usual binomial theorem expressed in terms of divided powers. The simple graded Hamiltonian algebra $H(2;(n_1,n_2))^{(2)}$ is defined as a subalgebra of the algebra of derivations of $\mathcal{O}(2,(n_1, n_2))$. However, for the present purposes we can identify it with the subspace of $\mathcal{O}(2;(n_1,n_2))$ spanned by its monomials $x^{(i)}y^{(j)}$ with $(i,j)\neq(0,0),(p^{n_1},p^{n_2})$, endowed with the Lie bracket $$\label{eq:Poisson_0} \begin{split} \{x^{(i)} y^{(j)}, x^{(k)} y^{(l)}\} &= x^{(i)} y^{(j-1)} x^{(k-1)} y^{(l)}- x^{(i-1)} y^{(j)} x^{(k)} y^{(l-1)} \\ &= N(i,j,k,l)\, x^{(i+k-1)} y^{(j+l-1)}, \end{split}$$ where $$N(i,j,k,l):= \binom{i+k-1}{i} \binom{j+l-1}{j-1}- \binom{i+k-1}{i-1} \binom{j+l-1}{j}.$$ This Lie algebra is clearly graded (hence its name) over ${\mathbb Z}\times{\mathbb Z}$, by assigning degree $(i,j)$ to the monomial $x^{(i+1)}y^{(j+1)}$. In odd characteristic it is a simple Lie algebra, of dimension $p^{n_1+n_2}-2$, In characteristic two it is simple only when $n_1>1$ and $n_2>1$. The Albert-Zassenhaus algebra $H(2:(n_1,n_2);\Phi(1))$ can be identified with $\mathcal{O}(2;(n_1,n_2))$ with the Lie bracket $$\begin{aligned} &\{x^{(i)} y^{(j)}, x^{(k)} y^{(l)}\} = N(i,j,k,l)\, x^{(i+k-1)} y^{(j+l-1)} \quad\text{if $i+k>0$, and}\label{eq:Poisson_1} \\ &\{y^{(j)}, y^{(l)}\} = \left( \binom{j+l-1}{l}- \binom{j+l-1}{j} \right)\bar x y^{(j+l-1)}.\label{eq:Poisson_exception}\end{aligned}$$ Equations  and  show that $H(2;(n_1,n_2);\Phi(1))$ is graded over the group ${\mathbb Z}/p^{n_1}{\mathbb Z}\times{\mathbb Z}$ by assigning degree $(i+p^{n_1}{\mathbb Z},j)$ to the monomial $x^{(i+1)}y^{(j+1)}$. In odd characteristic this Lie algebra is simple. In characteristic two its derived subalgebra $H(2;(n_1,n_2);\Phi(1))^{(1)}$ is simple, and is spanned by the monomials $x^{(i)}y^{(j)}$ with $(i,j)\neq(p^{n_1},p^{n_2})$. A [*specialization*]{} of this grading of $H=H(2;(n_1,n_2);\Phi(1))$ was considered in [@AviMat:A-Z Section 5], namely, a cyclic grading $H=\bigoplus_{k \in {\mathbb Z}/N {\mathbb Z}}H_k$ obtained by assigning degree $(1-q)i-j+N{\mathbb Z}$ to the monomial $x^{(i+1)}y^{(j+1)}$, where $q=p^{n_2}$ and $N=p^{n_1}(q-1)$. The monomials $x$ and $\bar y$ acquire degree $1$ in this grading, and one easily sees that they generate $H$. According to [@AviMat:A-Z Theorem 5.1], if $p>2$ the loop algebra of $H$ with respect to this grading is a Nottingham algebra with diamonds in all degrees congruent to one modulo $q-1$, all of type $\infty$ except for those in degree congruent to $q$ modulo $p^{n_1}(q-1)$, which have type $-1$. In the next two sections we show that different gradings of $H=H(2;(n_1,n_2);\Phi(1))$ lead to similar Nottingham Lie algebras, with the diamonds of type $-1$ replaced by diamonds whose types follow more general arithmetic progressions. The big field case {#sec:big-field} ================== This section and the next one contain our main results, which are explicit constructions for the new Nottingham Lie algebras announced in the Introduction. They have a diamond in each degree congruent to $1$ modulo $q-1$, with diamonds having infinite type except for one of finite type every $p^s$ diamonds, and the finite types follow an arithmetic progression. In this section we consider the case where the arithmetic progression is not entirely contained in the prime field. A minimal requirement on the base field ${\mathbb F}$ is that it should contain the the various finite diamond types, but we allow ourselves to further enlarge it later as needed. We make the assumption that $p$ is odd and postpone a discussion of the case $p=2$ to Remark \[rem:char2-big\]. Consider the Albert-Zassenhaus algebra $H=H(2;(s+1,n);\Phi(1))$, for some $s,n>0$, and its derivation $D=(\operatorname{ad}y)^{p^s}$. Writing each monomial in $\mathcal{O}(2;(s+1,n))$ in the form $x^{(ap^s)} x^{(k+1)}y^{(j+1)}$, with $0\le a<p$, $-1\leq k<p^s-1$ and $-1\leq j<p^n-1$, we have $$D(x^{(ap^s)} x^{(k+1)}y^{(j+1)})= \begin{cases} x^{((a-1)p^s)}x^{(k+1)}y^{(j+1)}& \textrm{if $a>0$,}\\ -jx^{((p-1)p^s)}x^{(k+1)}y^{(j+1)}& \textrm{if $a=0$.} \end{cases}$$ Consequently, $D^p$ acts semisimply on $H$, with eigenvalues in the prime field, as $$D^{p}(x^{(ap^s)} x^{(k+1)}y^{(j+1)})=-jx^{(ap^s)} x^{(k+1)}y^{(j+1)},$$ and hence $D^{p^2}=D^p$. Consider the cyclic grading $H=\bigoplus_{\ell\in {\mathbb Z}/N {\mathbb Z}}H_\ell$ obtained by assigning degree $(1-q)(ap^s+k)-j+N{\mathbb Z}$ to the monomial $x^{(ap^s)}x^{(k+1)}y^{(j+1)}$, where $q=p^n$ and $N=p^{s+1}(q-1)$. This is the grading from [@AviMat:A-Z Section 5.1] which we recalled at the end of Section \[sec:Cartan\], just in a different notation. Thus, $x$ and $\bar y$ acquire degree one in this grading. The derivation $D$ is graded of degree $N/p$. Furthermore, each homogeneous element in the grading is an eigenvector for $D^p$. Let $\sigma,\pi\in{\mathbb F}$ be such that $(\pi^p-\pi)\sigma^p=1$. We apply the grading switching from [@AviMat:Laguerre] to this grading, in the form recalled earlier as Theorem \[thm:special\] but with the derivation $\sigma^{-1}D$ in place of $D$, and thus obtain another cyclic grading of $H$. Explicitly, consider the elements $$\bar{e}_{j,k,a}=L_{p-1}^{(-j \pi)}\left(\sigma^{-1}D\right) (x^{(ap^{s})}x^{(k+1)}y^{(j+1)}),$$ for $-1\le j<q-1$, $-1\le k<p^{s}-1$, and $a \in {\mathbb F}_{p}$. According to Theorem \[thm:special\] they constitute a basis of $H$, graded over the integers modulo $N$ by assigning $\bar{e}_{j,k,a}$ degree $(1-q)(ap^s+k)-j+N{\mathbb Z}$. Each homogeneous component in this grading has dimension one, except for those in degrees congruent to $1$ modulo $q-1$, which are two-dimensional. For later convenience we multiply $\bar{e}_{j,k,a}$ by the scalar $c_{j,a}=a!\sigma^{a}\tbinom{-j \pi+a}{a}\tbinom{ -j\pi+p-1}{p-1}^{-1}$, obtaining the elements $$\displaystyle{e_{j, k, a}= c_{j,a}\bar{e}_{j,k,a}= (1+ \sigma x^{(p^{s})})^{-j\pi +a}x^{(k+1)}y^{(j+1)}},$$ for $-1\le j<q-1$, $-1\le k<p^{s}-1$, and $a \in {\mathbb F}_{p}$. Products between the basis elements $e_{j,k,a}$ are easily computed, and one finds $$\label{eq:eeq1} \displaystyle{\{e_{j, k, a}, e_{l, h, b}\}=\left( \tbinom{k+h+1}{h} \tbinom{j+l+1}{j}- \tbinom{k+h+1}{k} \tbinom{j+l+1}{l}\right)e_{j+l,k+h,a+b}}$$ for $k+h>-1$ and $$\label{eq:eeq2} \displaystyle{\{e_{j,-1,a}, e_{l,-1,b}\}=\sigma \left( \tbinom{j+l+1}{j}(-l \pi+b)- \tbinom{j+l+1}{l}(-j \pi+a)\right)e_{j+l,p^s-2,a+b-1}}.$$ In fact, because the basis is graded according to Theorem \[thm:special\], the Lie bracket $\{e_{j,k,a}, e_{l,h,b}\}$ is a scalar multiple of $e_{j+l,k+h,a+b}$, for $k+h>-2$. Thus, we only have to compute the scalar factor, for example by computing the coefficient of $x^{(k+h+1)}y^{(j+l+1)}$ in the result, noting that $x^{(k+1)}y^{(j+1)}$ always appears with coefficient $1$ in $e_{j,k, a}$. To do this it suffices to compute the Lie bracket of the only relevant terms, which is $\{x^{(k+1)}y^{(j+1)},x^{(h+1)}y^{(l+1)}\}$. Similarly, in the case $k+h=-2$, the Lie bracket $\{e_{j, -1, a}, e_{l, -1, b}\}$ is a scalar multiple of $e_{j+l,p^s-2,a+b-1}$. The scalar can be recovered by computing the coefficient of $x^{p^s-1}y^{j+l+1}$ in the result. Here the Lie bracket of the relevant terms is $$\sigma (-j \pi +a)\{x^{(p^s)}y^{(j+1)}, y^{(l+1)}\}+ \sigma (-l \pi +b)\{y^{(j+1)}, x^{(p^s)}y^{(l+1)}\}.$$ We are now ready to state and prove the main result of this section. \[thm:big\_field\] Let ${\mathbb F}$ be a field of odd characteristic $p$, let $n,s$ be a positive integer, and set $q=p^n\ge p$. Assume that there are $\sigma,\pi\in{\mathbb F}$ with $(\pi^p-\pi)\sigma^p=1$. Then the elements $$e_{j,k,a}=(1+ \sigma x^{(p^{s})})^{-j\pi +a} x^{(k+1)}y^{(j+1)},$$ for $-1\le j<q-1$, $-1\le k<p^{s}-1$, and $a \in {\mathbb F}_{p}$, form a graded basis of $H(2;(s+1,n);\Phi(1))$ over the integers modulo $(q-1)p^{s+1}$, where $e_{j,k, a}$ has degree $(1-q)(a p^{s}+k)-j \pmod{(q-1)p^{s+1}}$. The corresponding loop algebra $L$ is thin, with second diamond in degree $q$. The diamonds occur in each degree of the form $t(q-1)+1$, with the diamond type being finite exactly when $t\equiv 1\mod p^{s}$. Those finite types follow an arithmetic progression, not entirely contained in the prime field. The diamond in degree $(q-1)(p^s+1)+1$ has type $\nu=-1+1/\pi$. Note that the type $\nu$ of the diamond in degree $(q-1)(p^s+1)+1$, with $\nu\in {\mathbb F}\setminus{\mathbb F}_{p}$, determines $\sigma$ and $\pi$ uniquely via $$\pi^p-\pi=1/\sigma^p, \quad \nu=-1+1/\pi.$$ We have already deduced from Theorem \[thm:special\] that the elements $e_{j,k,a}$ form a graded basis of $H$ with respect to a cyclic grading, with degrees as specified. In particular, the elements $$X= e_{-1, 0, 0}=(1+ \sigma x^{(p^{s})})^{\pi} x \qquad\textrm{and}\qquad Y= e_{q-2, -1, 0}=(1+ \sigma x^{(p^{s})})^{2 \pi} \bar{y}$$ span the homogeneous component of degree $1$ in this grading. Form the tensor product of $H$ with a polynomial ring ${\mathbb F}[T]$. According to Definition \[def:loop\_algebra\] the loop algebra $L$ is the subalgebra of $H\otimes {\mathbb F}[T]$ generated by the elements $X\otimes T$ and $Y\otimes T$. We need to prove that the covering property holds in $L$, and that the diamonds have the types specified. However, it is possible and notationally more convenient to prove those conclusions working inside the Hamiltonian algebra $H$ rather than in its loop algebra $L$. We have already given the full multiplication table of the elements $e_{j,k,a}$ in Equations  and . In particular, for $j \neq -1,0$ we read off that $\{e_{j,k,a},Y\}=0$ and $\{e_{j,k,a},X\}=e_{j-1,k,a}$. Thus, all one-dimensional homogeneous components which do not immediately precede a two-dimensional component are centralized by $Y$, and satisfy the covering property. By this expression we mean that each of them covers the next component, in the sense that for each nonzero element $u$ in any of them, $\{u,L_1\}$ equals the following component. We deal with the remaining components next. Each element $v=e_{0,-1, a}=(1+\sigma x^{(p^{s})})^{a} y$, which has degree $(1-q)(ap^{s}-1)$, spans the homogeneous component just preceding a diamond of finite type. In fact, we have $$\begin{aligned} &\{v,X\}=e_{-1,-1, a}\\ &\{v,Y\}=\sigma(2 \pi + a)e_{q-2,p^s-2,a-1}\\ &\{v,X,X\}=0=\{v,Y,Y \}\\ &\{v,X,Y\}=-\sigma(\pi + a )e_{q-3,p^s-2,a-1}\\ &\{v,Y, X\}=\sigma (2 \pi + a)e_{q-3,p^s-2, a-1} \end{aligned}$$ Besides proving that the covering property holds in the relevant components, these equations show that $\mu\{v,Y,X\}=(1- \mu)\{v,X,Y\}$ where $\mu= -a \nu -a-1$. According to Equation  we are in the presence of a diamond of type $\mu$. In particular, the element $e_{0,-1,0}$, of degree $q-1$, immediately precedes the second diamond, of type $-1$ as always, and the element $e_{0,-1,-1}$, of degree $(q-1)(p^s+1)$, immediately precedes a diamond of type $\nu$. These two diamond types determine an arithmetic progression, which describes all the diamond types considered here. Finally, we show that each element $w=e_{0,k,a}=(1+\sigma x^{(p^{s})})^{a} x^{(k+1)}y$, for $0\le k<p^{s}-1$, which has degree $(1-q)(ap^{s}+k)$, spans the homogeneous components which immediately precedes a diamond of infinite type. In fact, we have $$\begin{aligned} &\{w,X\}=e_{-1,k,a}\\ &\{w,Y\}=-e_{q-2,k-1,a}\\ &\{w,X,X\}=0=\{w,Y,Y\}\\ &\{w,X,Y\}=-e_{q-3,k-1,a}\\ &\{w,Y,X\}=e_{q-3,k-1,a}=-\{w,X,Y\} \end{aligned}$$ We have proved that the loop algebra $L$ is thin, and that the diamonds types are as specified. \[rem:char2-big\] Theorem \[thm:big\_field\] admits an interpretation in the forbidden case of characteristic two. As mentioned in Section \[sec:Cartan\], the algebra $H=H(2;(s+1,n);\Phi(1))$ is not simple in characteristic two, but its derived subalgebra $H^{(1)}$ is. It is spanned by the same monomials except $\bar x\bar y$, and hence has dimension $2^{s+n+1}-1$. All our basis elements $e_{j,k,a}$ belong to $H^{(1)}$ except for $e_{q-2,2^s-2,1}=x^{(p^s-1)}\bar{y}+\sigma \bar{x}\bar{y}$. This is one of the two basis elements of degree $q$ modulo $(q-1)2^{s+1}$, which usually span the second diamond in the loop algebra. The elements $X=e_{-1,0,0}$ and $Y=e_{q-2,-1,0}$ generate $H^{(1)}$. The corresponding loop algebra is still thin, but the second diamond becomes fake of type $1\equiv -1 \bmod 2$. In fact, the homogeneous component of degree $q-1$ is generated by $v=e_{0,-1,0}$, and we have $\{v,Y\}=2 \pi \sigma e_{q-2,2^s-2,1}= 0$. \[rem:s=0\] When $s=0$ the loop algebra of $H(2;(1,n);\Phi(1))$ according to the grading given in Theorem \[thm:big\_field\] is a Nottingham algebra with all diamonds of finite type. This algebra was constructed in [@AviMat:A-Z] as a loop algebra of $H(2;(1,n);\Phi(1))$ with respect to a certain grading, which, however, differs from the grading given here. A very similar discrepancy occurred in [@AviMat:-1], and the detailed explanation for it given in [@AviMat:-1 Remark 4.3] applies here, almost verbatim. The prime field case {#sec:prime-field} ==================== In this final section we construct Nottingham Lie algebras with $p^{s}-1$ diamonds of infinite type separated by single occurrences of a diamond of finite type, with the finite types forming a nonconstant arithmetic progression in the prime field. Assume $p$ odd and let $H=H(2;(s+1,n))^{(2)}$, for some $s>0$, which has dimension $p^{s+n+1}-2$. Set $q=p^{n}$ and $N=p^{s+1}(q-1)$. The derivation $D=(\operatorname{ad}y)^{p^s}$ of $H$ satisfies $D^p=0$, because $$D(x^{(ap^s)} x^{(k+1)}y^{(j+1)})=x^{((a-1)p^s)}x^{(k+1)}y^{(j+1)}.$$ Thus, in this case the exponential of $D$ makes sense on the whole algebra $H$. Let $\pi \in {\mathbb F}_{p}$ with $\pi\neq 0$. We obtain a grading of $H$ over ${\mathbb Z}/N{\mathbb Z}$ by assigning the monomial $x^{(ap^s)}x^{(k+1)}y^{(j+1)}$ degree $(1-q)\bigl((a+j \pi)p^s+k\bigr)-j+N{\mathbb Z}$. Because the derivation $D$ is graded of degree $N/p$ we can apply Theorem \[thm:exp\] and obtain another grading of $H$ over the integers modulo $N$. A new graded basis, with degrees given by the same formula above, consists of the elements $$e_{j,k,a}=a! E(D)(x^{(ap^s)}x^{(k+1)}y^{(j+1)})=(1+x^{(p^s)})^{a}\, x^{(k+1)}y^{(j+1)},$$ where $0\le a<p$, $-1 \leq k <p^s-1$ and $-1 \leq j < q-1$, with $(j,k,a)\not=(-1,-1,0),(q-2,p^{s}-2,p-1)$. For later convenience we set $e_{-1,-1,0}=0=e_{q-2,p^s-2,p-1}$. As in the previous section the products of the elements $e_{j,k,a}$ are easily obtained, and are found to be $$\label{eq1} \{e_{j,k,a},e_{l,h,b}\}= \left( \tbinom{k+h+1}{h} \tbinom{j+l+1}{j}- \tbinom{k+h+1}{k} \tbinom{j+l+1}{l}\right)e_{j+l,k+h, a+ b}$$ for $k+h>-2$, and $$\label{eq2} \{e_{j,-1,a},e_{l,-1,b}\}=\left(b \tbinom{j+l+1}{j}- a \tbinom{j+l+1}{l}\right)e_{j+l,p^s-2,a+ b-1}.$$ \[thm:prime\_field\] Let ${\mathbb F}$ be a field of odd characteristic $p$, let $q=p^n\ge p$, and let $\pi$ be a nonzero element of ${\mathbb F}_p$. Then the elements $$e_{j,k,a}=(1+x^{(p^s)})^a\, x^{(k+1)}y^{(j+1)},$$ for $0\le a<p$, $-1 \leq k <p^s-1$ and $-1 \leq j < q-1$, form a graded basis of $H(2;(s+1,n))^{(2)}$ over the integers modulo $(q-1)p^{s+1}$, where $e_{j,k, a}$ has degree $(1-q)\bigl((a+j \pi)p^s+k\bigr)-j\pmod{(q-1)p^{s+1}}$. The corresponding loop algebra $L$ is thin, with second diamond in degree $q$. The diamonds occur in all degrees of the form $t(q-1)+1$, with the diamond type being finite exactly when $t \equiv 1 \mod p^{s}$. Those finite types follow an arithmetic progression contained in the prime field. The diamond in degree $(q-1)(p^s+1)+1$ has type $\nu=-1+1/\pi$. The preceding discussion shows that the elements $e_{j,k,a}$ form a graded basis of $H$ with respect to a cyclic grading, with degrees as specified. In particular, the elements $$X=e_{-1,0,\pi}=(1+x^{(p^{s})})^{\pi}x \qquad\textrm{and}\qquad Y=e_{q-2,-1,2\pi}=(1+x^{(p^{s})})^{2\pi}\bar y$$ span the homogeneous component of degree $1$ in this grading. As in the proof of Theorem \[thm:big\_field\], we conveniently check the covering property and the diamond types in $H$ rather than in its loop algebra $L$. For $j \neq -1,0$ we have $\{e_{j,k,a},X\}=e_{j-1,k,a+\pi}$ and $\{e_{j,k,a}, Y\}=0$. Next, each element $v=e_{0,-1,a}$, of degree $(1-q)(ap^{s}-1)$, spans the homogeneous component preceding a diamond of finite type $\mu= -a \nu -a-1$, because $$\begin{aligned} &\{v,X\}=e_{-1,-1,a+\pi}\\ &\{v,Y\}=(2\pi+a)e_{q-2,p^s-2,a-1+2\pi}\\ &\{v,X,X\}=0=\{v,Y,Y\}\\ &\{v,X,Y\}=-(\pi+a) e_{q-3,p^s-2,a-1+3\pi}\\ &\{v,Y,X\}=(2\pi+a)e_{q-3,p^s-2,a-1+3\pi}\end{aligned}$$ As in the proof of Theorem \[thm:big\_field\] these diamond types follow the stated arithmetic progression. Finally, each element $w=e_{0,k,a}$, for $0\le k<p^s-1$, of degree $(1-q)(ap^s+k)$, occurs just before a diamond of infinite type, because $$\begin{aligned} &\{w,X\}=e_{-1,k, a+\pi}\\ &\{w,Y\}=e_{q-2,k-1,a+2\pi}\\ &\{w,X,X\}=0=\{w,Y,Y\}\\ &\{w,X,Y\}=-e_{q-3,k-1,a+3\pi}\\ &\{w,Y,X\}=e_{q-3,k-1,a+3\pi}=-\{w,X,Y\}\end{aligned}$$ The proof is complete. The case $\pi=0$, which is excluded above, should correspond to taking $\nu =\infty$. However, the grading in Theorem \[thm:prime\_field\] does not produce a thin Lie algebra because $\{e_{0,-1,0},X,Y\}=0=\{e_{0,-1,0},Y,X\}$. \[rem:char2-prime\] Differently from Theorem \[thm:big\_field\], which needs to be modified in characteristic two as described in Remark \[rem:char2-big\], Theorem \[thm:prime\_field\] remains true as stated when $p=2$, as long as $q>2$. However, except for a brief mention in Section \[sec:nott\] we have not introduced thin Lie algebras of characteristic two in this paper. The discussion in [@AviMat:A-Z Section 3] explains how the relevant terminology is to be interpreted, including the peculiar fact that the second diamond is fake. In fact, in the case under discussion $\pi=1$, and the diamonds of finite type of the loop algebra $L$ are all fake, of alternate types $0$ and $1$.

--- abstract: 'The standard and renormalized coupled cluster methods with singles, doubles, and noniterative triples and their generalizations to excited states, based on the equation of motion coupled cluster approach, are applied to the $^4$He and $^{16}$O nuclei. A comparison of coupled cluster results with the results of the exact diagonalization of the Hamiltonian in the same model space shows that the quantum chemistry inspired coupled cluster approximations provide an excellent description of ground and excited states of nuclei. The bulk of the correlation effects is obtained at the coupled cluster singles and doubles level. Triples, treated noniteratively, provide the virtually exact description.' author: - 'K. Kowalski$^1$, D.J. Dean$^2$, M. Hjorth-Jensen$^3$, T. Papenbrock$^{2,4}$, and P. Piecuch$^{1,5}$' title: Coupled cluster calculations of ground and excited states of nuclei --- The description of finite nuclei requires an understanding of both ground- and excited-state properties based on a given nuclear Hamiltonian. While much progress has been made in employing the Green’s Function Monte Carlo [@pieper02] and no-core shell model [@bruce2] techniques, these methods have apparent limitations to light nuclei. Given that present nuclear structure research facilities and the proposed Rare Isotope Accelerator will continue to open significant territory into regions of medium-mass and heavier nuclei, it becomes imperative to investigate methods that will allow for a description of medium-mass systems. Coupled cluster theory is a particularly promising candidate for such an endeavor due to its enormous success in quantum chemistry . Coupled cluster theory originated in nuclear physics [@coester58; @coester60] around 1960. Early studies in the seventies [@kum78] probed ground-state properties in limited spaces with free nucleon-nucleon interactions available at the time. The subject was revisited only recently by Guardiola [*et al.*]{} [@bishop96], for further theoretical development, and by Mihaila and Heisenberg [@hm99], for coupled cluster calculations using realistic two- and three-nucleon bare interactions and expansions in the inverse particle-hole energy spacings. However, much of the impressive development in coupled cluster theory made in quantum chemistry in the last 15-20 years [@Bartlett95; @Paldus99; @comp_chem_rev00; @Piecuch02a; @Piecuch02b] still awaits applications to the nuclear many-body problem. In this Letter, we apply quantum chemistry inspired coupled cluster methods to finite nuclei. We show that the coupled cluster approach is numerically inexpensive and accurate by comparing our results for $^{4}$He with results from exact diagonalization in a model space consisting of four major oscillator shells. For the first time, we apply coupled cluster theory to excited states in nuclei, exploiting the equation of motion coupled cluster formalism [@Stanton:1993; @Piecuch99]. We discuss several approximations within coupled cluster theory and also compute the ground state of the $^{16}$O nucleus within the same model space. We remind the reader that certain acronyms have become standard in quantum chemistry. For this reason, we use the same abbreviations in this Letter. Coupled cluster theory is based on an exponential ansatz for the ground-state wave function $|\Psi_{0}\rangle=\exp(T) |\Phi\rangle$. Here $T$ is the cluster operator and $|\Phi\rangle$ is the reference determinant. In the CCSD (“coupled cluster with singles and doubles”) method, we truncate the many-body expansion of the cluster operator $T$ at two-body components. The truncated cluster operator $T^{\rm (CCSD)}$, used in the CCSD calculations, has the form [@purvis82]: $T^{\rm (CCSD)} = T_{1} + T_{2}$. Here $T_1=\sum_{i,a} t_a^i a^{a} a_{i}$ and $T_2= \frac{1}{4} \sum_{ij,ab} t_{ab}^{ij} a^{a} a^{b} a_{j} a_{i}$ are the singly and doubly excited clusters, with indices $i,j,k$ ($a,b,c$) designating the single-particle states occupied (unoccupied) in the reference Slater determinant $|\Phi\rangle$ and $a^{p}$ ($a_{p}$) representing the creation (annihilation) operators. We determine the singly and doubly excited cluster amplitudes $t_a^i$ and $t_{ab}^{ij}$, defining $T_1$ and $T_2$, respectively, by solving the nonlinear system of algebraic equations, $\langle \Phi_{i}^{a} | \bar{H}^{\rm (CCSD)}|\Phi\rangle = 0$, $\langle \Phi_{ij}^{ab} | \bar{H}^{\rm (CCSD)}|\Phi\rangle = 0$, where $\bar{H}^{{\rm (CCSD)}} = \exp(-T^{\rm (CCSD)}) \, H \, \exp(T^{\rm (CCSD)})$ is the similarity-transformed Hamiltonian and $|\Phi_{i}^{a}\rangle$ and $|\Phi_{ij}^{ab}\rangle$ are the singly and doubly excited Slater determinants, respectively. Once $T_1$ and $T_2$ amplitudes are determined, we calculate the ground-state CCSD energy $E_{0}^{\rm (CCSD)}$ as $\langle\Phi|\bar{H}^{{\rm (CCSD)}}|\Phi\rangle $. For the excited states $|\Psi_{K}\rangle$ and energies $E_{K}^{\rm (CCSD)}$ ($K > 0$), we apply the EOMCCSD (“equation of motion CCSD”) approximation, in which $$|\Psi_{K}\rangle=R_{K}^{\rm (CCSD)} \exp(T^{\rm (CCSD)}) |\Phi\rangle . \label{eomfun}$$ Here $R_{K}^{\rm (CCSD)} = R_{0}+ R_{1} + R_{2}$ is a sum of the reference ($R_{0}$), one-body ($R_{1}$), and two-body ($R_{2}$) components that are obtained by diagonalizing the similarity-transformed Hamiltonian $\bar{H}^{{\rm (CCSD)}}$ in the same space of singly and doubly excited determinants $|\Phi_{i}^{a}\rangle$ and $|\Phi_{ij}^{ab}\rangle$ as used in the ground-state CCSD calculations [@Stanton:1993; @Piecuch99]. The CCSD and EOMCCSD methods are expected to describe the bulk of the correlation effects with inexpensive computational steps that scale as $n_{o}^{2} n_{u}^{4}$, where $n_{o}$ ($n_{u}$) is the number of occupied (unoccupied) single-particle orbitals. While the inclusion of triply excited clusters $T_{3}$ and three-body excitation operators $R_{3}$ increases the accuracy of the method, the resulting full CCSDT (“T” stands for “triples”) [@Noga:1987a] and EOMCCSDT [@Kowalski:2001d] methods scale as $n_{o}^{3} n_{u}^{5}$ and are rather expensive. For this reason, we add the [*a posteriori*]{} corrections due to triples to the CCSD/EOMCCSD energies, which require $n_{o}^{3} n_{u}^{4}$ noniterative steps. The ground- and excited-state triples corrections, $\delta_{0}$ and $\delta_{K}$ ($K>0$), respectively, are calcultated with the CR-CCSD(T) (“completely renormalized CCSD(T)”) approach [@Piecuch02a; @Piecuch02b; @Kowalski00; @Kowalski03] in which $$\delta_{K} = \mbox{$\frac{1}{36}$} \sum_{ijk,abc} \langle \tilde{\Psi}_{K} | \Phi_{ijk}^{abc} \rangle \, {\cal M}_{abc}^{ijk}(K) / \Delta_{K} \;\; (K \geq 0). \label{deltak}$$ Here $|\Phi_{ijk}^{abc}\rangle$ are the triply excited determinants and ${\cal M}_{abc}^{ijk}(K)$ are the generalized moments of the CCSD ($K=0$) and EOMCCSD ($K > 0$) equations [@Kowalski00; @Kowalski03; @Kowalski01], $${\cal M}_{abc}^{ijk}(K) = \langle \Phi_{ijk}^{abc} | \bar{H}^{{\rm (CCSD)}} S_{K}^{\rm (CCSD)} | \Phi \rangle \;, \label{mk}$$ where $S^{\rm (CCSD)}_0=1$ and $S^{\rm (CCSD)}_K=R_K^{\rm (CCSD)}$ for $K > 0$. They can be calculated using the CCSD and EOMCCSD cluster and excitation operators $T^{\rm (CCSD)}$ and $R_{K}^{\rm (CCSD)}$, respectively. The $\Delta_{K}$ denominators are defined as $$\Delta_{K}= \langle \tilde{\Psi}_{K} | S_{K}^{\rm (CCSD)} \exp(T^{\rm (CCSD)}) |\Phi\rangle \, . \label{denomk}$$ The states $|\tilde{\Psi}_{K}\rangle$ in Eqs. (\[deltak\]) and (\[denomk\]) include the leading triples contributions resulting from the perturbative analysis of the CCSDT and EOMCCSDT equations. We have $\mid\tilde{\Psi}_{0}\rangle=\bar{P}\exp(T^{\rm (CCSD)} + \tilde{T}_{3})|\Phi\rangle$ and $|\tilde{\Psi}_{K}\rangle = \bar{P}(R_{K}^{\rm (CCSD)} + \tilde{R}_{3}) \exp(T^{\rm (CCSD)}) |\Phi\rangle$ for $K > 0$, where $\bar{P}$ is a projection operator on the subspace spanned by the reference $|\Phi\rangle$ and singly, doubly, and triply excited determinants. The most complete forms of $\tilde{T}_{3}$ and $\tilde{R}_{3}$ defining the approximation are [@Piecuch02a; @Kowalski03] $$\begin{aligned} \tilde{T}_{3} &=& \mbox{$\frac{1}{36}$} \sum_{ijk,abc} ({\cal M}_{abc}^{ijk}(0)/D_{ijk}^{abc}(0) ) a^{a} a^{b} a^{c} a_{k} a_{j} a_{i}, \label{t3tilde} \\ \tilde{R}_{3} &=& \mbox{$\frac{1}{36}$} \sum_{ijk,abc} ({\cal M}_{abc}^{ijk}(K)/D_{ijk}^{abc}(K) ) a^{a} a^{b} a^{c} a_{k} a_{j} a_{i}, \label{r3tilde} \end{aligned}$$ where $D_{ijk}^{abc}(K) = E_{K}^{\rm (CCSD)} - \langle \Phi_{ijk}^{abc} | \bar{H}^{\rm (CCSD)} |\Phi_{ijk}^{abc} \rangle$. In the case of the ground-state calculations, we also consider simplified variants of the CR-CCSD(T) theory, termed CR-CCSD(T),a and CR-CCSD(T),b. In the case of CR-CCSD(T),b, the perturbative denominator $D_{ijk}^{abc}(0)$ is replaced by $- \langle \Phi_{ijk}^{abc} | \bar{H}_{1}^{\rm (CCSD)} |\Phi_{ijk}^{abc} \rangle$, where $\bar{H}_{1}^{\rm (CCSD)}$ is the one-body part of $\bar{H}^{\rm (CCSD)}$. For CR-CCSD(T),a we replace $D_{ijk}^{abc}(0)$ by the standard many-body perturbation theory (MBPT) triples denominator ($\epsilon_{i} + \epsilon_{j} + \epsilon_{k} - \epsilon_{a} - \epsilon_{b} - \epsilon_{c}$), where $\epsilon_{i}$ and $\epsilon_{a}$ are the diagonal elements of the Fock matrix. Very accurate results for the excitation energies $E_{K} - E_{0}$ of many-electron systems are obtained if we use the complete CR-CCSD(T),c theory to calculate the energies of excited states and the CR-CCSD(T),b approximation for the ground-state energy [@Kowalski03]. For the ground states, it may sometimes be worthwhile to replace the $\Delta_{0}$ denominator, Eq. (\[denomk\]), which renormalizes the triples correction $\delta_{0}$ by 1, since $\Delta_{0}$ equals 1 plus terms of the second MBPT order or higher [@Kowalski00]. We indicate this by using acronyms, such as CR-CCSD(T),c/$\Delta_{0}=1$ (as opposed to CR-CCSD(T),c, where $\Delta_{0}$ is included). We use the Idaho-A nucleon-nucleon potential [@machleidt02] which was produced using techniques of chiral effective field theory [@weinberg; @vankolck]. Modern two-nucleon interactions, such as Idaho-A, include short-range repulsive cores that require calculations in extremely large model spaces to reach converged results [@hm99]. In order to remove the hard-core part of the interaction from the problem and thereby allow for realistic calculations in manageable model spaces, we renormalize the interactions through a $G$-matrix procedure for use in the $0s$-$0p$-$0d1s$-$0f1p$ oscillator basis. Our Hamiltonian is thus given by $H=t+G(\tilde{\omega})$, where $\tilde\omega$ is the $G$-matrix starting energy. We use a simple procedure described in Ref. [@herbert02] to alleviate the starting-energy dependence of the $G$-matrix in orbitals below the Fermi surface. We also modify the Hamiltonian by adding to it the center-of-mass Hamiltonian times a Lagrange multiplier $\beta_{\rm c.m.}$. Thus, our Hamiltonian becomes $H^{\prime}=H+\beta_{\rm c.m.}H_{\rm c.m.}$. We choose $\beta_{\rm c.m.}$ such that the expectation value of $H_{\rm c.m.}=0.0$ MeV. Details may be found in Ref. [@dean03]. We tested the performance of the above coupled cluster approximations in the context of the nuclear many-body problem by applying them to two closed-shell nuclei, $^{4}$He and $^{16}$O, in the one-particle space of four major oscillator shells. Shell model diagonalization provided an exact answer for a given Hamiltonian in the $^4$He case. Comparing the exact ground- and excited-state energies resulting from the diagonalization of the Hamiltonian in the small model space with the coupled cluster energies obtained in the same model space, we can assess the usefulness of various coupled cluster approximations in calculations for atomic nuclei. In particular, we can learn about the possible role of triply excited clusters in an accurate description of ground and excited states [*without confusing the inaccuracies resulting from the inadequate treatment of the many-body problem by a given coupled cluster approximation with other sources of error*]{}. We report our results for the ground-state energy of $^4$He in Table \[table\_he4\_gs\]. We used two types of reference determinants $|\Phi\rangle$: one constructed from the lowest-energy oscillator states and the Hartree-Fock determinant. Throughout the table we see that the results obtained in the oscillator basis are lower in energy when compared to those obtained in the Hartree-Fock basis. The two best methods in the oscillator basis are CR-CCSD(T),a/$\Delta_{0}=1$ and CR-CCSD(T),c/$\Delta_{0}=1$. They yield results within $40$ keV and $300$ keV of the full configuration interaction (CI) diagonalization problem, respectively. The CR-CCSD(T),a/$\Delta_{0}=1$ approach applied to the oscillator basis overshoots the exact result, which is a consequence of using the standard MBPT denominators in the definition of $\tilde{T}_{3}$, Eq. (\[t3tilde\]). The CR-CCSD(T),a approach, in which the triples correction $\delta_{0}$ is renormalized via the presence of the $\Delta_{0}$ denominator in Eq. (\[deltak\]), is more stable in this regard, providing the upper bound to the energy, although the 600 keV error obtained with the CR-CCSD(T),a method is not as impressive as the $40$ keV error obtained with CR-CCSD(T),a/$\Delta_{0}=1$. In general, the CR-CCSD(T) results are considerably more accurate than the results of the CCSD calculations, in which $T_{3}$ is ignored, although the CCSD approach describes the bulk of the correlation effects, reducing the large 16.273 MeV error obtained by calculating $\langle \Phi | H | \Phi \rangle$ with the oscillator reference $| \Phi \rangle$ to 1.5 MeV. The effectiveness of the CCSD approach can also be illustrated by comparing the CCSD energy with the results of truncated shell-model calculations (CISD), in which the Hamiltonian is diagonalized in the same space of singly and doubly excited determinants as used in the CCSD caculations. The costs of the CISD and CCSD calculations are almost identical (both are $n_{o}^{2} n_{u}^{4}$ procedures), and yet the error in the CISD energy obtained in the oscillator basis is twice as large as the error obtained with CCSD. The noniterative triples corrections defining the CR-CCSD(T) approaches reduce these errors to as little as $40$ keV, which is a lot better than the 1.3 MeV error in the CISDT calculations, where the Hamiltonian is diagonalized in the much larger space of all singly, doubly, and triply excited determinants. This demonstrates the advantages of coupled cluster methods over the diagonalization techniques. Similar observations apply to the Hartree-Fock basis, although the coupled cluster results obtained with this basis are not as good as those obtained with the oscillator basis. For example, the best result in the Hartree-Fock basis, obtained with CR-CCSD(T),c/$\Delta_{0}=1$, is $700$ keV above the exact result. This suggests that we may be better off by using the oscillator basis in coupled cluster calculations. On the other hand, the CR-CCSD(T) results obtained in the Hartree-Fock basis are not unreasonable, allowing us to contemplate the use of the Hartree-Fock basis in coupled cluster calculations for open-shell nuclei. (This would parallel the Hartree-Fock-based coupled cluster calculations for open-shell electronic states in chemistry.) [|ccc|]{} Method & Osc & HF CCSD & -21.978 & -21.385 CR-CCSD(T),a & -22.841 & -22.395 CR-CCSD(T),a/$\Delta_{0}=1$ & -23.524 & -22.711 CR-CCSD(T),b & -22.396 & -22.179 CR-CCSD(T),b/$\Delta_{0}=1$ & -22.730 & -22.428 CR-CCSD(T),c & -22.630 & -22.450 CR-CCSD(T),c/$\Delta_{0}=1$ & -23.149 & -22.783 CISD & -20.175 & -20.801 CISDT & -22.235 & – Exact & -23.484 & -23.484 \[table\_he4\_gs\] We used the EOMCCSD method and its CR-CCSD(T) extension to compute excited states. To our knowledge, this is the first time that nuclear excited states are computed using coupled cluster methods. The results for $^4$He are given in Table \[table\_2\]. The low-lying $J=1$ state most likely results from the center-of-mass contamination which we have removed only from the ground state. The $J=0$ and $J=2$ states calculated using EOMCCSD and CR-CCSD(T) are in excellent agreement with the exact results. For these two states, the EOMCCSD approach provides the relatively small, 0.3-0.4 MeV, errors, which are further reduced by the CR-CCSD(T) triples corrections to $< 0.1$ MeV. Based on the experience with the equation of motion coupled cluster methods in chemistry [@Stanton:1993; @Bartlett95; @Piecuch02a; @Kowalski03; @Kowalski01; @Kowalski:2001d], the very good performance of the EOMCCSD approach for the lowest-energy excited states of $^4$He can be understood if we realize that these states are dominated by single-particle excitations. Again, a comparison of the EOMCCSD and CISD results shows that coupled cluster theory offers much higher accuracies compared to truncated diagonalization of similar numerical effort. According to the experiment, the lowest lying $0^+$ state in $^4$He is a resonance at an excitation of 20.21 MeV and a width of 0.5 MeV, while the first $J=2$ state lies at 21.84 MeV and has a width of 2 MeV. We have not identified the parity of our calculated states, but it seems to us that we will be able to model the excitations in $^4$He and other nuclei using coupled cluster theory. Our results in Table \[table\_2\] are indicative of the accuracies we may expect from such calculations. [|ccccc|]{} State & EOMCCSD & CR-CCSD(T)$^a$ & CISD & Exact J=1 & 11.791 & 12.044 & 17.515 & 11.465 J=0 & 21.203 & 21.489 & 24.969 & 21.569 J=2 & 22.435 & 22.650 & 24.966 & 22.697 $^a$ The difference of the CR-CCSD(T),c energy of excited state and the CR-CCSD(T),b energy of the ground state. \[table\_2\] We also applied the CCSD and CR-CCSD(T) methods to $^{16}$O. Table \[table\_ox16\_gs\] shows the total ground-state energy values obtained with the CCSD and CR-CCSD(T) approaches. As in the $^{4}$He case, coupled cluster methods recover the bulk of the correlation effects, producing the results of the CISDTQ, or better, quality. CISDTQ stands for the expensive shell-model diagonalization in a huge space spanned by the reference and all singly, doubly, triply, and quadruply excited determinants (the most expensive steps of CISDTQ scale as $n_{o}^{4} n_{u}^{6}$). To understand this result, we note that the CCSD $T_1$ and $T_2$ amplitudes are similar in order of magnitude. (For an oscillator basis, both $T_1$ and $T_2$ contribute to the first-order MBPT wave function.) Thus, the $T_1 T_2$ [*disconnected*]{} triples are large, much larger than the $T_3$ [*connected*]{} triples, and the difference between the CISDT (CI singles, doubles, and triples) and CISD energies is mostly due to $T_1 T_2$. The small $T_3$ effects, as estimated by CR-CCSD(T), are consistent with the CI diagonalization calculations. If the $T_3$ corrections were large, we would observe a significant lowering of the CCSD energy, far below the CISDTQ result. The CISDTQ diagonalization is not size-extensive, while the CCSD and CR-CCSD(T)/$\Delta_{0}=1$ methods maintain this property. Moreover, the CCSD and CR-CCSD(T) methods bring the nonnegligible higher-than-quadruple excitations, such as $T_1^3 T_2$, $T_1 T_2^2$, and $T_{2}^{3}$, which are not present in CISDTQ. It is, therefore, quite likely that the CR-CCSD(T) results are very close to the results of the exact diagonalization, which cannot be performed. [|cc|]{} Method & Energy CCSD & -139.310 CR-CCSD(T),a & -139.465 CR-CCSD(T),a/$\Delta_{0}=1$ & -139.621 CR-CCSD(T),b & -139.375 CR-CCSD(T),b/$\Delta_{0}=1$ & -139.440 CR-CCSD(T),c & -139.391 CR-CCSD(T),c/$\Delta_{0}=1$ & -139.467 CISD & -131.887 CISDT & -135.489 CISDTQ & -138.387 \[table\_ox16\_gs\] In summary, we used the quantum chemistry inspired coupled cluster approximations to calculate the ground and excited states of the $^{4}$He and $^{16}$O nuclei. By comparing coupled cluster results with the exact results obtained by diagonalizing the Hamiltonian in the same model space, we demonstrated that relatively inexpensive coupled cluster approximations recover the bulk of the nucleon correlation effects in ground- and excited-state nuclei. These results are a strong motivation to further develop coupled cluster methods for the nuclear many-body problem, so that accurate [*ab initio*]{} calculations for small- and medium-size nuclei become as routine as the molecular electronic structure calculations. Supported by the U.S. Department of Energy under Contract Nos. DE-FG02-96ER40963 (University of Tennessee), DE-AC05-00OR22725 with UT-Battelle, LLC (Oak Ridge National Laboratory), DE-FG02-01ER15228 (Michigan State University), the Research Council of Norway, and the Alfred P. Sloan Foundation. [29]{} natexlab\#1[\#1]{} bibnamefont \#1[\#1]{} bibfnamefont \#1[\#1]{} citenamefont \#1[\#1]{} url \#1[`#1`]{} urlprefix \[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , ****, (). , , , ****, (). , ****, (). , ****, (). , in **, edited by (, ), vol. , pp. . , ****, (). , ****, (). , , , , ****, (). , , , , in **, edited by , , (, ), vol.  of **, pp. . , ****, (). , ****, (). , , , ****, (), and references therein. , ****, (). , ****, (); , ****, (). , ****, (). , ****, (). , ****, (). , ****, (); [**89**]{}, 3401 (1988) (erratum); G.E. Scuseria and H.F. Schaefer III, Chem. Phys. Lett. [**152**]{}, 382 (1988). , ****, (). , ****, (). , ****, (), in press. , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). (), .

--- abstract: 'We propose to synthesize feasible caging grasps for a target object through computing *Caging Loops*, a closed curve defined in the *shape embedding space* of the object. Different from the traditional methods, our approach *decouples* caging loops from the surface geometry of target objects through working in the embedding space. This enables us to synthesize caging loops encompassing multiple topological holes, instead of always tied with one specific handle which could be too small to be graspable by the robot gripper. Our method extracts caging loops through a topological analysis of the distance field defined for the target surface in the embedding space, based on a rigorous theoretical study on the relation between caging loops and the field topology. Due to the decoupling, our method can tolerate incomplete and noisy surface geometry of an unknown target object captured on-the-fly. We implemented our method with a robotic gripper and demonstrate through extensive experiments that our method can synthesize reliable grasps for objects with complex surface geometry and topology and in various scales.' author: - 'Jian Liu$^{1}$, Shiqing Xin$^{1}$, Zengfu Gao$^{1}$, Kai Xu$^{2}$, Changhe Tu$^{1}$ and Baoquan Chen$^{1}$[^1]' bibliography: - 'reference.bib' title: 'Caging Loops in Shape Embedding Space: Theory and Computation' --- Introduction {#sec:intro} ============ As an important type of robot grasping, caging grasps [@Rodriguez-2011FromCT; @Wan-2013AN; @Diankov-2008ManipulationPW], as compared to force-closure grasps [@Zhu-PlanningFG2004; @Ding-Computing3O2000; @Borst-GraspingTD2003; @Ferrari-PlanningOG1992], are advantageous in handling target objects with unknown or uncertain surface geometry and/or friction properties. This makes caging grasps more practically applicable in a wide spectrum of real scenarios. We are especially interested in a simple yet effective type of caging grasp formed by *caging loops*. A caging loop is a closed curve in three dimensional space computed around some part of the target object and used to guide robot grippers to form a caging grasp. Existing methods on 3D caging grasp are based either on the geometric (e.g. [@Zarubin-2013CagingCO]) or the topological (e.g. [@Pokorny-2013GraspingOW]) information of the target surface, or even both [@Kwok-2016RopeCA]. A common issue to these methods is that the computed caging curves seriously depend on topological and geometrical features of objects, while being oblivious to the relative size between the target object and the gripper. Taking the genus-4 Indonesian-Lady model in Fig. \[fig:teaser\] for example. The six handles on the model are all seemingly good candidates for grasping. However, when the size of the model is too small compared to the robot gripper, these handles will no longer be graspable since the holes may be too small for the fingers to pass through. In such case, a more feasible grasp would be enclosing the object with a loop encompassing multiple handles (see Fig. \[fig:teaser\](top) and Fig. \[fig:scale\](a)). Another issue with geometry-based caging curves is that they easily lead to non-convex spatial curves which are not suited for guiding the gripper configuration. The example in Fig. \[fig:scale\](d) demonstrates such case, where the gripper penetrates into the object due to the non-convexity of the caging loop. Estimating a convex hull for the spatial loop still cannot guarantee a penetration-free configuration. Motivation And Contribution --------------------------- ![Grasping a 3D-printed Indonesian-Lady model (the top and middle row) in two sizes. Our method is able to synthesize caging loops (red circles) encompassing multiple topological handles, when the object is too small to be grasped on one handle (top row). When the object is large, our method naturally grasps one handle (middle row). The two cases are integrated seamlessly in method. The bottom row shows how a caging loop computed in the embedding encloses the two handles of a pliers. The 3D objects are acquired by two RGBD cameras and reconstructed on-the-fly (middle column). As a reference, a human grasp is shown to the left for each object.[]{data-label="fig:teaser"}](teaser.png){width="0.95\columnwidth"} These examples motivated us in seeking to “fill up” those small topological holes and “smooth out” the geometric details on the target surface, before computing caging curves. Therefore, we advocate computing caging loops in the *embedding space* of the target surface, through a topological analysis of the distance field defined for the target surface in the embedding space. We conduct a theoretical study on the fundamental relationship between caging loops and Morse singularities (including minimal, maximal and saddle points) of a spatial distance function. Based on that, we develop an algorithm of caging loop extraction through saddle point detection and analysis, within a proper grasping space defined in account of the gripper size. Working with a distance function defined in the embedding space naturally decouples the shape of caging loops from the geometric details of the target surface, while still keeping them aware of the overall shape of the target object. The caging loops are properly placed and scaled based on the relative size of the gripper against the target shape, rather than always tied with a specific handle as in traditional approaches. Another benefit of working in embedding space is the tolerance of incomplete and noisy surface geometry of the target object. This makes our method especially suited for synthesizing grasps for unknown objects which are captured and reconstructed on-the-fly, with a minimal effort of robot observation. In our implementation (see Fig. \[fig:overview\]), two depth cameras are deployed to capture the target object from two (front and back) views. Even with such a sparse capturing and low-quality reconstruction, our method can still synthesize feasible caging loops for robust grasping. We found this simple idea leads to a robust and efficient algorithm, with theoretical guarantees. We implemented our algorithm in a grasping system composed of a Barrett WAM robotic arm with a three-finger gripper and two Xtion Pro RGB-D cameras. Only depth images are used for reconstructing the target surface based on the depth fusion technique [@Newcombe-2011KinectFusionRD]. We have conducted numerous evaluations with both synthetic and real examples to evaluate the performance of our method. We show that our system is able to robustly grasp objects with complex surface geometry and topology and in various scales. Our work makes the following contributions: - We propose a novel method for caging grasp synthesis through topological analysis of shape-aware distance field defined in shape embedding space. The method is able to generate relative-scale-aware caging loops for unknown objects captured on-the-fly. - We provide a rigorous study on the relation between the topology of distance field and caging loops, based on Morse theory, and derive a robust algorithm for caging loop estimation. We also provide a handful of provably effective techniques to reduce the computational cost of our method. - We implement our method in a grasping system using robot gripper, and conduct thorough evaluations and comparisons with both synthetic and real objects. ![By decoupling caging loops from target surfaces, our method synthesizes feasible caging grasps for objects containing tiny topological handles (a) or presenting concave surface geometry (b); see the red circles and the corresponding grasps to the right. In contrast, the loops (yellow circles in c and d) computed over the target surfaces incur gripper-object collision; see the gripper parts in red color in the bottom row.[]{data-label="fig:topologyAndGeometry"}](topologyAndGeometry1.png "fig:"){width="0.95\linewidth"}\ \ ![By decoupling caging loops from target surfaces, our method synthesizes feasible caging grasps for objects containing tiny topological handles (a) or presenting concave surface geometry (b); see the red circles and the corresponding grasps to the right. In contrast, the loops (yellow circles in c and d) computed over the target surfaces incur gripper-object collision; see the gripper parts in red color in the bottom row.[]{data-label="fig:topologyAndGeometry"}](topologyAndGeometry2.png "fig:"){width="0.95\linewidth"}\ \[fig:scale\] Related Work ------------ Robot grasping is a long-standing yet actively studied research topic in the fields of robotics and vision. Force-closure and caging are two typical approaches that have been developed to synthesize grasps. Force-closure methods [@Bicchi-OnTC1995; @Miller-GraspitAV2004; @Liu-QualitativeTA1999; @Howard-OnTS1996; @Ding-Computing3O2000; @Zhu-PlanningFG2004] concentrate on finding a stable grasping configuration for the grippers where a mechanical equilibrium is achieved. The advantage of such approach is that the synthesized grasps are usually physically feasible. The method, however, requires that the 3D shape of the target model is known *a priori* and cannot tolerate much the surface defect such as missing data. Furthermore, the contact area between the gripper and the target surface is often small, leading to unsteady grasps. Caging grasps [@Diankov-2008ManipulationPW], as compared to force-closure ones, seeks for a sufficiently large contact area and thus are deemed to have better stability, although they are not designed to directly reach a mechanical equilibrium. The key benefit of caging [@Diankov-2008ManipulationPW] is that it is robust to surface uncertainty and imperfection. This makes it especially applicable to unknown objects being captured and reconstructed on-the-fly. Some works studied the computation of planar cages in 2D space for planar objects [@Rimon-1999CagingPB; @Pipattanasomporn-2006TwofingerCO; @Pipattanasomporn-2011TwoFingerCO; @Vahedi-2008CagingCP]. Most existing approaches to 3D object caging rely on the topological structure of the target surface [@Pokorny-2013GraspingOW; @Dey-2010ApproximatingLI; @Stork-2013IntegratedMA]. Some further take geometry information into account [@Kwok-2016RopeCA]. However, such approaches cannot compute a caging loop encompassing multiple handles or deal with different relative scales between the gripper and the target object. The Morse theory, as a connection between geometry and topology, has been widely utilized in the graphics and visualization fields [@Bremer-2004ATH; @Ni-2004FairMF]. In our approach, the core algorithmic step is to find a caging loop according to a $p$-based distance field, where $p$ is a point in the grasping space. At this point, the Morse theory is used to build a fundamental relationship between caging loops and Morse saddle points. Theory {#sec:theory} ====== Grasping Strategy ----------------- In order to define a caging loop, we have to [consider at least geometric and mechanical aspects.]{} The geometric considerations include: - A caging loop encompasses the target object - any penetration into the target shape is not allowed. - A caging loop encloses some part of the target object tightly, i.e., cannot be shortened with respect to a slight perturbance (i.e., [*stable grasp*]{}) or at least goes around the target object like a great circle enclosing a sphere (i.e., [*unstable grasp*]{}). - A caging loop should roughly match the real robot gripper size. On the other side, the mechanical considerations include - The center point of a caging loop should be as close as possible to the center of gravity of the target shape so as to minimize the moment of intertia. - A caging loop should be roughly horizontal so that the target object can be taken up steadily. Our strategy is to compute a collection of caging loop candidates in consideration of the above-mentioned geometric principles. For purpose of efficient computation, we also invent a set of filtering techniques to reduce the number of loop candidates as far as possible. Mathematical Formulation ------------------------ Imagine the scenario of a caging grasp where the fingers of the gripper stretch to two opposite directions, roughly forming a loop; See Fig. \[fig:teaser\] and Fig. \[fig:topologyAndGeometry\]. In the following, we shall formally characterize in which space we extract caging loops and systematically establish properties of caging loops. The surface $S$ of the target object, typically represented as a watertight mesh, divides the whole $\mathbb{R}^3$ space into interior parts and exterior parts, where only the visible free space (the outmost surface exterior space) is helpful to determine a real grasp configuration. Rather than constrain caging loops lying on the target surface $S$, we relax caging loops from $S$ to the shape embedding space. The visible free space separated by the target surface $S$ is called the [*grasping space*]{}. Generally speaking, a stable grasp is desired, i.e., the caging loop encloses some part of the target object tightly and cannot be shortened even with a slight perturbance. In some rare cases, however, an unstable grasp like a great circle enclosing a sphere is also acceptable. Both cases imply that there is a fundamental relationship between caging loops and locally shortest loops in the grasping space. \[defn:grasping\_loop\]Suppose the target object $S$ defines a grasping space $\mathbf{G}$. A closed curve $L\in \mathbf{G}$ is called a [*caging loop candidate*]{} if and only if $L$ is [*locally shortest everywhere*]{}, i.e., for any point $p\in L$, any sufficiently short segment of $L$ around $p$ cannot be shortened any more. All such loop candidates constitute a [*caging loop space*]{}, denoted by $\mathbf{L}$. \[property:three\]Each caging loop $L\in{\mathbf{L}}$ touches the target surface at three or more points. [*Proof.*]{} Without loss of generality, we assume that $L$ touches the target surface $S$ at only one point $p$. Then the open curve ${{L}}\backslash p$ lies in the grasping space but doesn’t touch $S$. Considering that a locally shortest curve in $\mathbb{R}^3$ must be a straight line segment, ${{L}}\backslash p$ cannot include a bending point. Furthermore, $p$ is not only the start point of the straight line segment ${{L}}\backslash p$ but also its endpoint. Therefore, $L$ degenerates into a single point under the above assumption, which contradicts to the given condition that $L$ is a caging loop. Similarly, it can be shown that the case of two touching points is impossible. We can further show that each caging loop consists of an alternative sequence of straight line segments in the grasping space and geodesics on the target surface. Let ${S}$ be the target surface. Each caging loop $L$ consists of an alternative sequence of geodesic paths lying on ${S}$ and straight line segments in the grasping space $\mathbf{G}$, where a geodesic segment may degenerate into a single point. In fact, the loop space $\mathbf{L}$ includes all geodesic loops constrained on the surface $S$ and thus cannot be empty, which can be easily verified from the Lusternick-Schnirelmann theorem [@Lusternik1934M]. ${\mathbf{L}}$ is non-empty. However, it is difficult to directly extract a caging loop without any further hint. Therefore, we consider a type of relaxed caging loops. \[defn:relaxed\_loop\]Suppose the target object $S$ defines a grasping space $\mathbf{G}$. Let $p$ be a point in $\mathbf{G}$. A closed curve $L\in \mathbf{G}$ is called a [*$p$-based caging loop candidate*]{} if and only if $L$ is [ locally shortest everywhere]{} except at $p$. When $p$ is taken over all points in $\mathbf{G}$, all such loop candidates constitute a different [caging loop space]{}, denoted by $\widetilde{\mathbf{L}}$. We immediately have the following property. $\widetilde{\mathbf{L}}$ is a superset of ${\mathbf{L}}$. [**Remark:**]{} Each loop $L\in\widetilde{\mathbf{L}}$ carries a base point $p$. If we eliminate those loops that are not locally shortest at the corresponding base point, then $\widetilde{\mathbf{L}}$ becomes ${\mathbf{L}}$. Therefore, the above property implies that we can select the desirable caging loop from $\widetilde{\mathbf{L}}$. Methodology {#sec:methodology} =========== Computing Loop Candidates ------------------------- ![\[fig:morse\_saddle\]An example of Morse-Smale saddle point of the $p$-based distance field restricted in a grasping space $\mathbf{G}$. ](morseSaddle.pdf){width="0.85\columnwidth"} Let $p$ be a point in the grasping space $\mathbf{G}$. For each point $x$ in $\mathbf{G}$, we use $\mathbf{D}_p(x): G \rightarrow \mathbb{R}$ to denote the length of the shortest path connecting $p$ and $x$. $\mathbf{D}_p$ is called the distance field rooted at $p$. Note that the distance is measured in $\mathbf{G}$ rather than on the target surface. Suppose that $L_p$ is a $p$-based loop in the caging loop space $\widetilde{\mathbf{L}}$. It is easy to know that there is a point $q\in L_p$ such that $q$ divides $L_p$ into two equal-length parts. Obviously, both the two sub-curves are locally shortest paths in the grasping space $\mathbf{G}$. In the following, we shall reveal the fact that there is a fundamental relationship between Morse theory and caging loops. \[thm:saddle\]Suppose that the target object defines a grasping space $\mathbf{G}$. Let $\mathbf{D}_p$ be the distance field rooted at $p\in\mathbf{G}$. Each Morse saddle or maximal point of $\mathbf{D}_p$ is able to define a $p$-based caging loop. [*Proof.*]{} Let $q$ be an Morse saddle (or maximal) point of $\mathbf{D}_p$. Since there exists a pair of shortest paths $\Pi_1, \Pi_2$ that go along opposite directions at $q$. By combining $\Pi_1, \Pi_2$, we get a $p$-based caging loop. Although the above discussion is assumed in the continuous setting, Morse-Smale theory is also well defined in the discrete setting; see more details in [@Ni-2004FairMF]. We can inherit the spirit in [@Ni-2004FairMF] and distinguish Morse saddle points, minimal points and maximal points by considering the relative magnitude at a voxel and its neighboring voxels. As Fig. \[fig:morse\_saddle\] shows, we label $p$’s neighbor with a “+” if the neighbor has a higher value and a “-” otherwise. It is easy to know that there are at most $2^6$ possible configurations. Fig. \[fig:morse\_saddle\] gives a typical situation of Morse saddle point, where two opposite neighbors are labeled with “-” while the other four neighbors are labeled with “+”. Similarly, a voxel is classified as a maximal point if all 6 neighbors are labeled with “-”. Based on Theorem \[thm:saddle\], it is natural to devise a naïve algorithm (see Algorithm \[alg:naive\]) to build the $p$-based caging loop space $\widetilde{\mathbf{L}}$. Initialize $\widetilde{\mathbf{L}}$ to be empty.\ Compute a sample set $P$ in the grasping space $\mathbf{G}$.\ Filtering Rules --------------- However, $\widetilde{\mathbf{L}}$ is very large generally. We need to invent a handful of filtering rules to reduce the computational cost. First of all, the reduction of the grasping space $\mathbf{G}$ is much helpful to filter out redundant caging loops. \[thm:convexhull\]Let ${H}$ be the convex hull of the target surface $S$. Any caging loop must lie between $S$ and $H$. [Secondly]{}, Property \[property:three\] asserts that the base point $p$ can be constrained on the target surface $S$, which cannot cause missing any useful caging loop. In fact, the location of $p$ can be more restricted; See the following theorem. \[thm:principal\]For a point $p$ on the target surface $S$, if both the principal curvatures are negative, $p$ cannot determine a caging loop. [*Proof.*]{} Suppose $L_p$ is a caging loop. The sufficiently short segment of $L_p$ around $p$ can be viewed as the intersection between a normal plane at $p$ and the target surface $S$. Since both the principal curvatures at $p$ are negative, the loop can be shortened by moving $p$ toward $\mathbf{G}$ a little bit, leading to a contradiction. [Finally]{}, even if the base point $p$ is given, there is no need to compute the entire distance field $\mathbf{D}_p$ since an overly long loop is no use for grasping. It is sufficient to limit the sweep process in an appropriate range comparable to the gripper size. In practice, it is reasonable to require that the total stretching length of the robot gripper (twice as long as the gripper finger), denoted by $2h$, should be larger than one half of the length of the caging loop. Therefore, during the computation of the $p$-based distance field, we can terminate the sweep process when the sweep radius amounts to $2h$ since at this moment, any $p$-based caging loop longer than $4h$ has been found. Taking the above speedup techniques simultaneously into consideration, we give an advanced algorithm for computing caging loops; See Algorithm \[alg:advanced\] (the difference from Algorithm \[alg:naive\] is underlined). Initialize $\widetilde{\mathbf{L}}$ to be empty.\ Compute the grasping space $\mathbf{G}$ .\ Implementation {#sec:implementation} ============== In a real grasping scenario, the gripper thickness cannot be negligible - a gripper cannot stretch into small topological holes or gaps. A commonly used technique is to filter out those infeasible caging loops by checking interference. Rather than leave it to ex post interference analysis, in this paper, we take each gripper finger as a skeleton curve equipped with a sweep radius $r$. In implementation, we offset the target surface outward in a distance of $r$ and require any caging loop to be lying in the grasping space $\mathbf{G}_r$ separated by the $r$-offset surface $S_r$. Fig. \[fig:overview\] shows an implementation details of our approach. {width="1.0\linewidth"}\ \ {width="1.0\linewidth"}\ Grasping Space $\mathbf{G}_r$ {#subsec:GraspingSpace} ----------------------------- Given a scanned point cloud $\{(x_i,\mathbf{n}_i)\}$ of the target shape, we shall adapt the radial basis function (RBF) technique  [@Carr-2001ReconstructionAR] to represent the $r$-offset surface $S_r$, which is central to define the grasping space $\mathbf{G}_r$. The general RBF with regard to $\{(x_i,\mathbf{n}_i)\}$ is defined as follows: $$f(x) = \sum_{i}^n w_i\phi(\|x-x_i\|)+P(x),$$ where $\phi(t) = t$ is the basis function used in our experiments, and the weighting coefficients $\{w_i\}$ and the low-degree (typically linear) polynomial $P(x)$ is undetermined. Taking $x_j$ into the RBF, we have $$\sum_{i}^n w_i\phi(\|x_j-x_i\|)+(1,x_j)^\text{T}\mathbf{c}=f(x_j) = 0.$$ Furthermore, considering that the point $x_j+r\mathbf{n}_j$ that lies on the $r$-offset surface $S_r$,, we have $$\sum_{i}^n w_i\phi(\|x_j+r\mathbf{n}_j-x_i\|)+(1,x_j)^\text{T}\mathbf{c}=f(x_j+r\mathbf{n}_j) = r,$$ where $\mathbf{c}=(c_0,c_1,c_2,c_3)$ is unknown. At the same time, in the RBF based approach, the four side conditions are $$\sum_{i}^n w_i = \sum_{i}^n w_i x_i = \sum_{i}^n w_i y_i = \sum_{i}^n w_i z_i = 0.$$ The above formulation can be finally transformed into a linear system from which we can immediately compute $\{w_i\}$ and $P(x)$. To this end, we find an implicit surface $f(x)=r$ to represent the $r$-offset surface $S_r$. If the shape embedding space is discretized into voxels, it is very easy to identify outside voxels that meet $f(x)\geq r$, which can be viewed as a discrete representation of the grasping space $\mathbf{G}_r$. Gripper Configuration {#subsec:GripperConfiguration} --------------------- Upon obtaining a desirable caging loop $L$, we need to determine the origin position $o$ of the gripper, as well as an orthogonal frame to set the gripper orientation, which facilitates a real grasp. Suppose that $L$ is represented by a point sequence $\{p_1,p_2,\cdots,p_n\}$. Imagine that there is an inward cone rooted at $p\in\{p_1,p_2,\cdots,p_n\}$ and the center line of the cone coincides with the normal vector at $p$; See Fig. \[fig:gripperConfiguration\]. (If $p$ is not located on $S_r$, the normal vector is discussed later.) We further define $\theta_p$ to be the maximum open angle under the condition that no penetration occurs between the inward cone and the target shape. The origin point $o$ is then selected from $\{p_1,p_2,\cdots,p_n\}$ so as to maximize the opening angle. Let $c=\frac{p_1+p_2+\cdots+p_n}{n}$ be the center point of $L$. We then define the first direction $Dir_1$ as follows: $$Dir_1 = \frac{o-c}{\|o-c\|},$$ which roughly means the forward direction of the gripper. Considering that the loop $L$ is roughly a planar curve, we can fit $L$ using a plane $ \mathbf{n}\cdot x = b, $ where $\mathbf{n}$ is a unit vector. Finally, we define $Dir_2$ as follows: $$Dir_2 = Dir_1 \times \mathbf{n}.$$ If the triple $(o, Dir_1, Dir_2)$ is able to define a valid grasping configuration (no global interference happens), we use it to guide the orientation of the gripper and launch a real grasp. In our experiments, the gripper will spread its fingers and move to the point $o$ first. It then wraps the target object tightly with the hint of $Dir_1$ and $Dir_2$. In Fig. \[fig:gripperConfiguration\], we show some examples on how to find a valid grasping configuration assuming that a desirable caging loop has been found. The key step is to check global interference in the simulation environment of OpenRAVE. It can be observed that our approach can report different gasping configurations on the same model with different sizes. [**Remark:** ]{}If $p$ is not located on $S_r$, $Dir_1$ is given by $\mathbf{t}_p\times \mathbf{n}$, where $\mathbf{t}_p$ is the tangent direction of the loop $L$ at $p$ and $\mathbf{n}$ is the normal to the fitting plane of $L$. ![\[fig:gripperConfiguration\] Inferring the gripper configuration based on a caging loop. In order to define a valid caging configuration, we check global interference in the simulation environment of OpenRAVE. Row 1: Large-size Kitten; Row 2: Small-size Kitten; Row 3: Large-size Yoga; Row 4: Small-size Yoga. ](gripperConfigurationResult.jpg){width="0.95\columnwidth"} Experimental Results {#sec:experimental} ==================== We conducted both simulation and mechanical experiments to validate our approach. [In this section, we first test the effectiveness of our algorithm on a variety of complex objects. We then show that our algorithm can be applied to 3D shapes with various levels of noise and geometric features. After that, we demonstrate the superior caging ability of our approach on real grasping scenarios (the digital models of the target objects are unknown in advance). Finally, we give the timing statistics of the main computational steps.]{} Test on High-genus Models in Various Sizes ------------------------------------------ ![\[fig:adaptive\] Caging loops generated on the Fertility and Yoga models with various sizes. (a) Caging loops (yellow) produced by the method in [@Zarubin-2013CagingCO]. (b) Caging loops (red) computed by our method. ](adaptive.png "fig:"){width="0.95\columnwidth"}\ {width="0.95\linewidth"} There are a number of research works on caging a 3D shape, which largely fall into one of the following three categories: topology guided [@Pokorny-2013GraspingOW], geometry guided [@Zarubin-2013CagingCO] and topology & geometry guided [@Kwok-2016RopeCA; @Varava-2016CagingGO]. Existing approaches, whether topology guided or geometry guided, consider only those loops constrained on the target surface. Therefore, it is hard for them to deal with small topological holes or small gaps. As shown by the comparison in Fig. \[fig:adaptive\], our algorithm handles 3D shapes with small topological holes and is aware of the relative size between the gripper and the shape. More caging loop examples can be found in Fig. \[fig:MoreGraspingLoops\]. Test on Models with Various Levels of Noise and Geometric Feature ----------------------------------------------------------------- ![\[fig:venusbodynoise\] [Top row: Caging loops on the Venus model with varying levels of Gaussian noise (relative to the whole model size): 0.01, 0.03, 0.05, 0.07. Bottom row: Caging loops computed by our method on the Pillar models are insensitive to varying levels of geometric details.]{} ](venusbodynoise.png "fig:"){width="\columnwidth"}\ ![\[fig:venusbodynoise\] [Top row: Caging loops on the Venus model with varying levels of Gaussian noise (relative to the whole model size): 0.01, 0.03, 0.05, 0.07. Bottom row: Caging loops computed by our method on the Pillar models are insensitive to varying levels of geometric details.]{} ](FeatureScale.png "fig:"){width="\columnwidth"} In real grasping scenarios, the target object is often scanned into a point cloud with noise. Therefore, a key criteria to evaluate a caging algorithm is whether it is robust to geometric noise or variations. In Fig. \[fig:venusbodynoise\], the top row shows a group of caging loops on the Venus models with various levels of noise, while bottom row shows a group of caging loops on the Pillar models with various levels of geometric details. Both of them exhibit the robustness of our algorithm against geometric noise and details. Test on Real Objects -------------------- ![\[fig:realDataResult\] Grasping household objects by our system. From left to right: Scanned point clouds, offset surfaces and caging loops, simulation results, progressive demonstration of real grasping. ](RealDataResult.pdf "fig:"){width="1.0\columnwidth"}\ In order to validate our approach on real data, we build a platform with a 7-DoF Barrett WAM arm, a Barrett BH8-282 three-finger gripper and two Xtion Pro depth cameras. For each object shown in Fig. \[fig:realDataResult\], we keep the depth cameras unchanged while rotating the target shape repeatedly for $10$ times. In this way, we recorded of the grasp success rates. A grasp is regarded to be successful if the target shape does not escape from the gripper during a large-scale movement that is about $10$cm off the ground. We report the success rate of grasping in Table \[tab:GraspSuccessfulRate\]. It shows that our approach can synthesize reliable caging grasps, as compared to traditional approaches whose success rate is generally about $85\%$. \[tab:GraspSuccessfulRate\] Performance ----------- In order to accomplish a grasping task, we have to perform a sequence of shape analysis operations and then send a grasp instruction to the robot. Recall that we define a grasping space and search the best caging loop in that space. In our implementation, we discreticize the bounding box enclosing the target shape into $50 \times 50 \times 50$ voxels and label each voxel between the $r$-offset surface $S_r$ and its convex hull $H_r$ with “1”. After that, we extract a set of $500$ uniformly distributed sample points from $S_r$ and eliminate those sample points that do not help determine a caging loop at all (see Theorem \[thm:principal\]). Finally, the loop candidate pool is generated based on Morse theory. Among these steps, the most time-consuming steps include the grasping space computation, a distance field generation for a collection of base points and topological analysis based on Morse theory. The mesh models shown in this paper are discretized into 2K vertices, and the average computation time for each model is about 1.5 seconds. It can be seen that our algorithm runs very fast at this level of resolution. Therefore, if we simultaneously execute the computation task and the move of gripper, it does not introduce a noticeable delay. Conclusion and Discussion ========================= In this work, we propose to synthesize feasible caging grasps in the shape embedding space of the target object. Our caging loops are able to encompass multiple small topological handles and concave regions, which are relatively too small to be grasped, through decoupling their computation from surface geometry of the target object. This also facilitates grasp synthesis for unknown objects which are acquired and reconstructed on-the-fly. Extensive experimental results exhibit that our approach can deal with real objects with complex surface geometry and topology, being aware of the relative size between objects and gripper. Our current solution has several limitations. First, the method used for measuring the physical feasibility is merely a preliminary solution which can definitely be replaced by other alternatives. Our core method for caging loop computation, however, ensures the candidate loops are mostly feasible with respect to the gripper size. Second, the caging loops computed by our method mainly reflect the geometric aspect of graspability and do not account for the high level information of semantics or functionality. For example, an object can be grasped in different ways for different purposes. Synthesizing function related grasps is an interesting venue for future study. [^1]: $^{1}$Jian Liu, Shiqing Xin, Zengfu Gao, Changhe Tu and Baoquan Chen are with School of Computer, Shandong University, 266237 Qingdao, P. R. China. $^{2}$Kai Xu is with School of Computer, National University of Defense Technology, 410073 Changsha, P. R. China. This work was supported by National 973 Program (2015CB352501), and NSFC (61332015, 61772318, 61572507, 61532003, 61622212, 61772016).

--- abstract: 'Non classical rotational inertia observed in rotating supersolid $He^4$ can be accounted for by a gravitomagnetic London moment similar to the one recently reported in rotating superconductive rings.' author: - 'C. J. de Matos[^1]' title: Gravitomagnetic London Moment in Rotating Supersolid $He^4$ --- Non Classical Rotational Inertia (NCRI) was predicted by London 50 years ago [@London]. It was eventually verified experimentally by Hess and Fairbank [@Hess], who set Helium in a suspended bucket into rotation above the critical temperature $T_c$ at which superfluidity sets in, and then cooled it (with the bucket still rotating at angular velocity $\omega$) through $T_c$. They found that, provided $\omega$ is less than a critical value $\omega_c$, the apparent moment of inertia of the helium - that is, the ratio of its angular momentum to $\omega$ - is not given by the classical value $I_{classical}$ but rather by $$I(T)=\frac{L}{\omega}=I_{classical}\Bigg[1-f_s(T)\Bigg]\label{1}$$ where $f_s(T)=\rho^*/\rho$ is the superfluid fraction. In liquid helium, $f_s(T)$ tends to 1 in the zero-temperature limit and to zero when $T=T_c$, $\rho^*$ and $\rho$ being respectively the mass density of superfluid helium, and the mass density of normal helium. NCRI has recently been observed in rotating supersolid $He^4$ by Kim and Chan [@Kim]. They measured the resonance frequency of a torsional oscillator that contains an annulus of solid $He^4$. Below $230 mK$, the frequency experiences a relative increase that depends on the temperature and drive amplitude and reaches a maximum of about four parts in $10^5$. Having excluded by various control experiments, other explanations, they conclude that the data indicate a change in the moment of inertia of the supersolid, which according to Equ.(\[1\]), corresponds to a maximum supersolid fraction ($f_s(T)=\rho^*_{supersolid \, He^4}/\rho_{normal \, solid \, He^4}$) of $~0.017$ (see figure \[chan\]). Tajmar and the author recently [@Tajmar1] [@de; @Matos1] [@Tajmar2] observed a gravitomagnetic London moment, $B_g \, [Rad/s]$, in rotating superconductive rings. $$B_g=2\omega f_s(T)\label{2}$$ Where $\omega$ is the angular velocity of the ring, and $f_s(T)=\rho^*/\rho$ is the Cooper pairs fraction, $\rho^*$ being the Cooper pairs mass density and $\rho$ the superconductor’s bulk density. Assuming that a rotating supersolid $He^4$ crystal also exhibits a gravitomagnetic London moment, $B_g$ proportional to the supersolid fraction, its angular momentum would be given by $$L=I_{classical}\Bigg[\omega-\frac{1}{2} B_g\Bigg]\label{3}$$ Doing Equ.(\[2\]) into Equ.(\[3\]) we find back Equ.(\[1\])! Therefore we conclude that the rotation of supersolid $He^4$ exhibits a gravitomagnetic London moment similar to the one observed in rotating superconductive rings, which can account for the observed NCRI in this physical system. Kim and Chan’s experiment would tend to confirm the existence of the gravitomagnetic London moment in rotating quantum materials. [99]{} London, F., “Superfluids”, wiley New York 1954, Vol II, P. 144 Hess, G. B., Fairbank, W. M., Phys. Rev. Lett., **19** 216 (1967) Kim, E., Chan, W., Science, **305**, 1941 (2004) Tajmar, M., Plesescu, F., Marhold, K., de Matos, C.J., “Experimental Detection of the Gravitomagentic London Moment”, 2006, gr-qc/0603033 de Matos, C. J., “Gravitoelectromagnetism and Dark Energy in Superconductors”, to appear in Int. J. Mod. Phys. D, (2007). (also available gr-qc:/0607004) Tajmar, M., Plesescu, F., Marhold, K. “Measurement of Gravitomagentic and acceleration fields around a rotating superconductor”, 2006, gr-qc/0610015 [^1]: ESA-HQ, European Space Agency, 8-10 rue Mario Nikis, 75015 Paris, France, e-mail: Clovis.de.Matos@esa.int

--- abstract: 'We show that for almost every polynomial $P(x,y)$ with complex coefficients, the difference of the logarithmic Mahler measures of $P(x,y)$ and $P(x,x^n)$ can be expanded in a type of formal series similar to an asymptotic power series expansion in powers of $1/n$. This generalizes a result of Boyd. We also show that such an expansion is unique and provide a formula for its coefficients. When $P$ has algebraic coefficients, the coefficients in the expansion are linear combinations of polylogarithms of algebraic numbers, with algebraic coefficients.' address: | Department of Mathematics\ Amherst College\ Amherst, MA 01002 USA author: - 'John D. Condon' bibliography: - 'mybib.bib' title: Asymptotic expansion of the difference of two Mahler measures --- Mahler measure ,asymptotic expansions ,polylogarithms Introduction {#sec:intro} ============ For a nonzero Laurent polynomial $P\in{\ensuremath{\mathbb{C}}}\bigl[x_1^{\pm 1},\dotsc x_n^{\pm 1}\bigr]$, the *(logarithmic) Mahler measure* of $P$ is defined as $$\label{eq:multivar} \begin{split} m(P) &= \int_{0}^{1}\dotsi \int_{0}^{1}\log\bigl|P \bigl(\exp(2\pi i t_{1}),\ldots,\exp(2\pi i t_{n})\bigr)\bigr| \, d t_{1}\cdots d t_{n} \\[2mm] &= \frac{1}{(2\pi i)^n}\int_{{\mathbb{T}}^n}\log\bigl|P(x_1,\ldots,x_n)\bigr| \,\frac{dx_1}{x_1}\cdots\frac{dx_n}{x_n}, \end{split}$$ where ${\mathbb{T}}$ is the unit circle in ${\ensuremath{\mathbb{C}}}$, oriented counter-clockwise. This integral is always finite, even if the zero set of $P$ intersects ${\mathbb{T}}^n$. When $n=1$, Jensen’s formula implies that if $P(x)=a\prod_{j=1}^d (x-\alpha_j)$, then $$\label{eq:onevar} m(P) = \log(a) + \sum_{j=1}^{d} {\mathop{\log^+\!}}{\lvert\alpha_{j}\rvert},$$ where, for $r>0$, ${\mathop{\log^+\!}}(r){\mathrel{\mathop:}=}\log\bigl(\max\{r, 1\}\bigr)$. The latter construct (or actually, its exponential) was first studied by D. H. Lehmer [@lehmer] in the 1930s. Mahler introduced three decades later [@mahler]. Boyd [@boyd2] established the following connection (generalized by Lawton [@lawton]) between multivariable and single-variable Mahler measure values. \[thm:boyd\] For any nonzero Laurent polynomial $P(x,y)$ with complex coefficients, $$m\bigl(P(x,y)\bigr) = \lim_{n\to\infty} m\bigl(P(x,x^n)\bigr).$$ Also in [@boyd2], Boyd proved the following result, which shows the rate at which the above limit converges in the case of $P(x,y)=1+x+y$: \[prop:boydasymp\] For all positive integers $n$, $$\label{eq:boydexpn} m(1+x+x^n) - m(1+x+y) = \frac{c(n)}{n^2} + {\mathop{O}_{}}\biggl(\frac{1}{n^3}\biggr),$$ where $c(n)$ depends only on $n$ mod 3: $$c(n)=\begin{cases} \phantom{-}\sqrt{3} \pi/18 & \mathrm{if}\ n\equiv 0,1 {\allowbreak\mkern10mu({\operator@font mod}\,\,3)} \\[2 mm] -\sqrt{3} \pi/6 & \mathrm{if}\ n\equiv 2 {\allowbreak\mkern10mu({\operator@font mod}\,\,3)}. \\[2 mm] \end{cases}$$ Motivated by these results, we examine the difference between these Mahler measures. For a nonzero Laurent polynomial $P(x,y)$ and a positive integer $n$, let $\Delta_{n}(P) {\mathrel{\mathop:}=}m\bigl(P(x,x^n)\bigr)-m\bigl(P(x,y)\bigr)$. The right side of could be thought of as the beginning of a formal series for $\Delta_{n}(1+x+y)$ of the form $\sum_{k=2}^\infty c_k(n)/n^k$. We will find such an expression for $\Delta_{n}(P)$, for $P=1+x+y$ as well as many other two-variable polynomials. Such a formal series cannot quite be called an asymptotic power series in $n$, in the sense of [@erdelyi], in that the coefficients in such a series should be independent of $n$. But our coefficients will have a structure that will, in particular, make them bounded as functions of $n$, occasionally depending only on $n {\allowbreak\mkern6mu({\operator@font mod}\,\,m)}$ for some integer $m$. Statement of results ==================== Unless stated otherwise, all variables and functions are complex-valued. ${\ensuremath{\mathbb{N}}}$ will denote the set of positive integers. For $P\in{\ensuremath{\mathbb{C}}}[z_1,\ldots,z_n]$, let $Z(P)$ denote the affine zero set of $P$ in ${\ensuremath{\mathbb{C}}}^{n}$. ${\mathop{\mathrm{Li}_{k}}}(z)$ will denote the principal branch of the $k$-th polylogarithm function [@lewin]. For $k\ge 2$ (which is all we will need) and ${\lvertz\rvert}\le 1$, this is given by $${\mathop{\mathrm{Li}_{k}}}(z) = \sum_{n=1}^{\infty} \frac{z^n}{n^k}.$$ We will say that a function $\omega:{\ensuremath{\mathbb{R}}}\to{\ensuremath{\mathbb{R}}}$ is *quasiperiodic* if it is the sum of finitely many continuous, periodic functions. Quasiperiodic functions are clearly bounded (although this is no longer true if the summand functions are not assumed to be continuous [@keleti]). For a function $f:{\ensuremath{\mathbb{N}}}\to{\ensuremath{\mathbb{R}}}$, we will say that a formal series $\sum_{r=0}^\infty c_r(n)/n^r$ is an *asymptotic pseudo-power series* (or *a.p.p.s.*) *expansion* of $f(n)$ (in powers of $1/n$, as $n\to\infty$) if, for each nonnegative integer $r$, $c_r(n)$ is the restriction to the nonnegative integers of a quasiperiodic function of $n$, and if for all postive integers $n$ and $k$, $$f(n)=\sum_{r=0}^{k-1} \frac{c_r(n)}{n^r} + {\mathop{O_{f,k}}}\biggl(\frac{1}{n^{k}}\biggr).$$ (The subscripts on “${\mathop{O}_{}}$” indicate that the implied constant depends on those subscripts). We will denote this by writing $f(n) {\overset{*}{\sim}}\sum_{r=0}^{\infty} c_r(n)/n^r$. Asymptotic power series expansions in powers of $1/n$, as defined in [@erdelyi], are the same as a.p.p.s. expansions in which the coefficients $c_r(n)$ do not depend on $n$. We will refer to these as *true* asymptotic power series expansions, for contrast. True asymptotic power series expansions for a function are uniquely determined by that function. For our series, quasiperiodicity of the coefficients is enough to rescue uniqueness. \[prop:uniqueness\] If a function $f:{\ensuremath{\mathbb{N}}}\to{\ensuremath{\mathbb{R}}}$ has an a.p.p.s. expansion in powers of $1/n$, that expansion is unique. The following is our main result. \[thm:main\] Let $P(x,y)\in{\ensuremath{\mathbb{C}}}[x,y]$ be such that $P$ and ${{\partial P/\partial y}}$ do not have a common zero on ${\mathbb{T}}\times{\ensuremath{\mathbb{C}}}$. Then $\Delta_{n}(P) = m\bigl(P(x,x^n)\bigr)-m\bigl(P(x,y)\bigr)$ has an a.p.p.s. expansion in powers of $1/n$. Specifically, if $P$ is absolutely irreducible, then $\Delta_{n}(P) {\overset{*}{\sim}}\sum_{r=2}^{\infty} c_r(n)/n^r$, with coefficients $$\label{eq:maincoeffformula} c_r(n) = \frac{1}{\pi}\sum_{(\alpha,\beta)\in \mathcal{E}} s(\alpha,\beta) \sum_{a=2}^{r} {\mathop{\mathfrak{R}_{a+1}}}\bigl(\Omega_{P,r,a}(\alpha,\beta)\bigr) {\mathop{\mathfrak{R}_{a}}}\bigl({\mathop{\mathrm{Li}_{a}}}(\beta/\alpha^n)\bigr),$$ where: - $\mathcal{E} = Z(P)\cap {\mathbb{T}}^{2}$, if this set is finite, or ${\ensuremath{\varnothing}}$ if it is infinite, - $s(x,y)$ is a function from $\mathcal{E}\to \{-1,0,1\}$, independent of $r$, - ${\mathop{\mathfrak{R}_{a}}}(z)={\mathop{\mathrm{Re}}}(z)$ when $a$ is even and ${\mathop{\mathrm{Im}}}(z)$ when $a$ is odd, - $\Omega_{P,r,a}(x,y)$ is a rational function, obtained by taking an integer polynomial $\Psi_{r,a}$ (independent of $P$) in indeterminates $w_{i,j}$, and making the substitutions $w_{i,j} = x^{i} y^{j-1} \bigl(\frac{\partial^{i+j}}{\partial x^{i} \partial y^{j}} P\bigr)/\bigl(\frac{\partial}{\partial y}P\bigr)$. In particular, if $P$ has integer coefficients, then the coefficients $c_{r}(n)$ are $\overline{{\ensuremath{\mathbb{Q}}}}$-linear combination of polylogarithms evaluated at algebraic arguments. (The definition of $s(x,y)$ is given in section \[sec:proofofmain\], and $\Psi_{r,a}$ is defined in .) The only appearance of $n$ in the formula for $c_{r}(n)$ is in ${\mathop{\mathrm{Li}_{a}}}(\beta/\alpha^n)$. Hence, if the $x$-coordinates of the points in $\mathcal{E}$ are all $m$-th roots of unity, then for each $r$, $c_{r}(n)$ depends only on $n {\allowbreak\mkern6mu({\operator@font mod}\,\,m)}$. Because the a.p.p.s. above only sums over $r\ge 2$, $\Delta_{n}(P) = {\mathop{O_{P}}}(n^{-2})$ for all $P$ as in the theorem. This fact was proved by Boyd in [@boyd2]. Indeed, he was able to go further, showing that for every nonzero $P\in{\ensuremath{\mathbb{C}}}[x,y]$, $\Delta_{n}(P) = {\mathop{O_{P}}}(n^{-(1+\delta)})$ for some $\delta>0$. Of course, if $\mathcal{E}$ is empty, the coefficients $c_{r}(n)$ in the theorem are all trivially equal to $0$. It turns out that there is a simple sufficient criterion for when this occurs. Let $P^{\star}(x,y)$ be the reciprocal polynomial of $P$ (defined in section \[sec:notation\]). \[cor:zeroseq\] Let $P(x,y)$ be as in [Theorem \[thm:main\]]{}. If $P^{\star}=cP$ for some constant $c\in {\ensuremath{\mathbb{C}}}$, then $\mathcal{E} = {\ensuremath{\varnothing}}$, hence $\Delta_{n}(P) {\overset{*}{\sim}}0$. Boyd’s [Proposition \[prop:boydasymp\]]{} corresponds to $P(x,y)=1+x+y$. We can be more explicit in that case: \[cor:main\] $\Delta_{n}(1+x+y) {\overset{*}{\sim}}\sum_{r=2}^{\infty} c_r(n)/n^r$, for $$\label{eq:maincoroformula} c_r(n) = \frac{2}{\pi} \!\! \sum_{\substack{a,b \\ a+b=r+1}} \!\! (-1)^{b} {\mathop{\mathfrak{R}_{a}}}\bigl({\mathop{\mathrm{Li}_{a}}}(\xi^{n+1})\bigr) \sum_{j=b}^{r-1} {\Bigl[\genfrac{}{}{0pt}{}{j}{b}\Bigr]}{\Bigl\{\genfrac{}{}{0pt}{}{r-1}{j}\Bigr\}} {\mathop{\mathfrak{R}_{a+1}}}(\xi^{j}),$$ where ${{\textstyle \genfrac{[}{]}{0pt}{}{a}{b}}}$ and ${{\textstyle \genfrac{\{}{\}}{0pt}{}{a}{b}}}$ represent the (unsigned) Stirling numbers of the first and second kind, respectively, and $\xi=\exp(2\pi i/3)$. In particular, the coefficients $c_{r}(n)$ are linear combinations of polylogarithms evaluated at third roots of unity, with coefficients in $\frac{1}{2\pi} {\ensuremath{\mathbb{Z}}}[\sqrt{3}]$. After some preliminary lemmas, we prove the above results in sections \[sec:proofofuniqueness\] through \[sec:proofofcor\]. In section \[sec:earlycoeffs\], we give expressions for the first two nonzero coefficients, $c_{2}(n)$ and $c_{3}(n)$, and verify that the value for $c_2(n)$ in [Corollary \[cor:main\]]{} agrees with Boyd’s $c(n)$ from [Proposition \[prop:boydasymp\]]{}. In section \[sec:numerics\], we presents the results of numerical calculations, giving support for our formulas and analyzing a degenerate case. Section \[sec:signdet\] describes an algebraic procedure for determining the values of the function $s(x,y)$ mentioned in [Theorem \[thm:main\]]{}. Additional Notation and Definitions {#sec:notation} =================================== For a function $f(z_1,\ldots,z_n)$, let $\overline{f}(z_1,\ldots,z_n){\mathrel{\mathop:}=}\overline{f(\overline{z_1},\ldots, \overline{z_n})}$. For a nonzero polynomial $P(z_1,\ldots,z_n)$, its *reciprocal polynomial* is defined as $P^\star(z_1,\ldots,z_n) {\mathrel{\mathop:}=}z_1^{d_1} \cdots z_n^{d_n} \, \overline{P}(z_1^{-1},\ldots, z_n^{-1})$, where $d_i=\deg_{z_i}(P)$. We will say that $P$ is *almost self-reciprocal* (or *a.s.r.*) if $P^{\star}=cP$ for some constant $c$. For $Q_1, Q_2\in{\ensuremath{\mathbb{C}}}[x,y]$, thinking of them as as polynomials in $y$ with coefficients in ${\ensuremath{\mathbb{C}}}[x]$, we let $\operatorname{Res}_y(Q_1,Q_2)$ and $\delta_y(Q_1)$ denote, respectively, the resultant and discriminant with respect to $y$. These are both polynomials in $x$ alone. An *arc* is, for us, a connected subset of ${\mathbb{T}}$, including possibly ${\mathbb{T}}$ itself. We will say that ${\mathbb{T}}$ is an improper arc, and all others are proper. An *open* (respectively, *closed*) arc is an arc that is open (closed) relative to the usual topology on ${\mathbb{T}}$. Arcs will be oriented counter-clockwise. For an open, proper arc $\gamma$, equal to $\bigl\{e^{it} | \alpha<t<\beta\bigr\}$ with $0<\beta-\alpha\le 2\pi$, let ${\left[f(x)\right]_{\partial \gamma}} {\mathrel{\mathop:}=}\lim_{t\to 0^{+}} \bigl(f(e^{i(\beta-t)})-f(e^{i(\alpha+t)})\bigr)$. (Usually, this is just the difference of the values of $f$ at the endpoints of $\gamma$. The limit is most useful when $\beta = \alpha+2\pi$ and $f$ is discontinuous at $e^{i\alpha}=e^{i\beta}$.) For a set $S\subseteq{\ensuremath{\mathbb{C}}}$, let $S^{-1}{\mathrel{\mathop:}=}\{z^{-1}|z\in S\}$ and $\overline{S}{\mathrel{\mathop:}=}\{\overline{z}|z\in S\}$. We will follow the convention of letting the Pochhammer symbol $(x)_k$ denote the “falling factorial” $x(x-1)(x-2)\dotsm (x-k+1)= k! \binom{x}{k}$. As mentioned earlier, ${{\textstyle \genfrac{[}{]}{0pt}{}{n}{k}}}$ and ${{\textstyle \genfrac{\{}{\}}{0pt}{}{n}{k}}}$ will represent the (unsigned) Stirling numbers of the first and second kind, respectively [@wilf]. These are nonnegative integers, and are nonzero if and only if $1\le k\le n$, with the exception that ${{\textstyle \genfrac{[}{]}{0pt}{}{0}{0}}}={{\textstyle \genfrac{\{}{\}}{0pt}{}{0}{0}}}=1.$ For integers $m$ and $k$ with $k\ge 0$, define an operator $H_{k}^{m}$ on $C^{k}$ functions $f(z)$ as follows: $${\mathop{H^{m}_{k}}\!}f {\mathrel{\mathop:}=}\Bigl(z \frac{d}{dz}\Bigr)^k \bigl(f(z)^m\bigr).$$ Let $B_{n,k}(z_1,z_2,\ldots,z_{n-k+1})$ be the exponential, partial Bell polynomial $$\sum \frac{n!}{j_1 ! j_2! \cdots j_{n-k+1}!} \prod_{i=1}^{n-k+1} \Bigl(\frac{z_i}{i!}\Bigr)^{j_i},$$ where the sum is taken over all tuples $(j_1,\dotsc, j_{n-k+1})$ of nonnegative integers such that $\sum_i j_i=k$ and $\sum_i i j_i=n$. These polynomials have integer coefficients and are homogeneous of degree $k$ [@comtet1]. For integers $n,k$, we define $$\Phi_{n,k}(y_0,\ldots,y_{n-k+1}) {\mathrel{\mathop:}=}\sum_{i, j} (-1)^{i-k} {\Bigl[\genfrac{}{}{0pt}{}{i}{k}\Bigr]} {\Bigl\{\genfrac{}{}{0pt}{}{n}{j}\Bigr\}} {y_0}^{n-i} B_{j,i}(y_1,\ldots,y_{j-i+1}).$$ (The summands vanish except when $0\le k\le i\le j \le n$.) This is clearly also an integer polynomial. Like $B_{n,k}$, $\Phi_{n,k}$ is identically zero if $n<k$, $k<0$, or if $k=0$ and $n>0$; otherwise, $\Phi_{n,k}$ is homogeneous of degree $n$. (See Table \[PhiTable\].) These polynomials become unwieldy for large $n$—for instance, $\Phi_{10,1}$ has 138 terms. [|@c@|@c@|p[1.55in]{} |p[1.1in]{}|p[.5in]{}|@c@|]{} $n \,\backslash\, k$ & ------------------------------------------------------------------------ 0 ------------------------------------------------------------------------ & 1 & 2 & 3 & 4\ 0 & ------------------------------------------------------------------------ 1 ------------------------------------------------------------------------ & 0 & 0 & 0 & 0\ 1 & ------------------------------------------------------------------------ 0 ------------------------------------------------------------------------ & $y_1$ & 0 & 0 & 0\ 2 & ------------------------------------------------------------------------ 0 ------------------------------------------------------------------------ & $y_0 y_1 + y_0 y_2 - y_1^2$ & $y_1^2$ & 0 & 0\ 3 & 0 ------------------------------------------------------------------------ & $y_0^2 y_1 + 3 y_0^2 y_2 + y_0^2 y_3$ & $3 y_0 y_1^2 + 3 y_0 y_1 y_2$ & $y_1^3$ & 0\ & & $\,-\, 3 y_0 y_1^2 - {\rule[-2.2mm]{0pt}{0.5mm}} 3 y_0 y_1 y_2 \,+\, 2 y_1^3$ & $\,-\, 3 y_1^3$ & &\ 4 & 0 ------------------------------------------------------------------------ & $y_0^3 y_1 + 7 y_0^3 y_2 + 6 y_0^3 y_3$ & $7 y_0^2 y_1^2 + 18 y_0^2 y_1 y_2$ & $6 y_0 y_1^2 y_2$ & $y_1^4$\ & & $\,+\, y_0^3 y_4 - 7 y_0^2 y_1^2 - 18 y_0^2 y_1 y_2$ & $\,+\, 4 y_0^2 y_1 y_3 + 3 y_0^2 y_2^2$ & $\,+\, 6 y_0 y_1^3$ &\ & & $\,-\, 4 y_0^2 y_1 y_3 - 3 y_0^2 y_2^2$ & $\,-\, 18 y_0 y_1^2 y_2$ & $\,-\, 6 y_1^4$ &\ & & $\,+\, 12 y_0 y_1^3 + 12 y_0 y_1^2 y_2 - 6 y_1^4$ & $\,-\, 18 y_0 y_1^3 + {\rule[-2.2mm]{0pt}{0.5mm}} 11 y_1^4$ & &\ For each integer $n\ge 1$, we define an integer polynomial $Q_{n}$ in indeterminates $w_{i,j}$ (with $i,j\ge 0$ and $i+j\le n$) as follows. Let $E_{n}$ be the set of all doubly-indexed sequences $\mathbf{e} = \{e_{i,j}\}_{i,j\ge 0}$ of nonnegative integers satisfying $e_{0,0}=0$, $\sum_{i,j} i e_{i,j} = n$, $\sum_{i,j} j e_{i,j} = 2n-2$, and $\sum_{i,j} e_{i,j} = 2n-1$. (Each such sequence will clearly have only finitely many nonzero terms.) For each $\mathbf{e}\in E_{n}$, let $w_{\mathbf{e}} {\mathrel{\mathop:}=}\prod_{i,j\ge 0} w_{i,j}^{e_{i,j}}$ and $$b_{n,\mathbf{e}} {\mathrel{\mathop:}=}(-1)^{2n-1-e_{0,1}} \frac{n! (2n-2-e_{0,1})! e_{0,1}!}{\prod_{i,j\ge 0} e_{i,j}! (i!j!)^{e_{i,j}}}.$$ We then define $Q_{n} {\mathrel{\mathop:}=}\sum_{\mathbf{e}\in E_{n}} b_{n,\mathbf{e}} \,w_{\mathbf{e}}$. (See Wilde [@wilde] for a proof that the coefficients $b_{n,\mathbf{e}}$ are integers.) For example, $Q_{1} = -w_{1,0}$ and $Q_{2} = -w_{0,1}^{2} w_{2,0} + 2 w_{0,1} w_{1,0} w_{1,1} - w_{0,2} w_{1,0}^{2}$. For integers $2\le a \le r$, let $$\label{eq:Psidef} \Psi_{r,a} = \Phi_{r-1,r-a+1}\bigl(1, Q_{1}, Q_{2}, \ldots\bigr).$$ This is an integer polynomial in $w_{i,j}$, for $i,j\ge 0$ and $i+j\le a-1$. Some examples of $\Psi_{r,a}$ are shown in Table \[PsiTable\]. [|c|c|p[3.75in]{}|]{} $r$ & $a$ & $\Psi_{r,a}$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ 2 & 2 & $-w_{1,0}$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ 3 & 2 & $w_{1,0}^{2}$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ & 3 & $-w_{0,1}^{2} w_{2,0} + 2 \, w_{0,1} w_{1,0} w_{1,1} - w_{0,2} w_{1,0}^{2} - w_{1,0}^{2} - w_{1,0}$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ 4 & 2 & $-w_{1,0}^{3}$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ & 3 & $3 \, w_{0,1}^{2} w_{1,0} w_{2,0} - 6 \, w_{0,1} w_{1,0}^{2} w_{1,1} + 3 \, w_{0,2} w_{1,0}^{3} + 3 \, w_{1,0}^{3} + 3 \, w_{1,0}^{2}$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ & 4 & $-w_{0,1}^{4} w_{3,0} + 3 \, w_{0,1}^{3} w_{1,0} w_{2,1} + 3 \, w_{0,1}^{3} w_{1,1} w_{2,0} - 3 \, w_{0,1}^{2} w_{1,0}^{2} w_{1,2}$ ------------------------------------------------------------------------ \ & & $ -\, 3 \, w_{0,1}^{2} w_{0,2} w_{1,0} w_{2,0} - 6 \, w_{0,1}^{2} w_{1,0} w_{1,1}^{2} + 9 \, w_{0,1} w_{0,2} w_{1,0}^{2} w_{1,1}$\ & & $+\, w_{0,1} w_{0,3} w_{1,0}^{3} - 3 \, w_{0,2}^{2} w_{1,0}^{3} - 3 \, w_{0,1}^{2} w_{1,0} w_{2,0} + 6 \, w_{0,1} w_{1,0}^{2} w_{1,1}$\ & & $-\, 3 \, w_{0,2} w_{1,0}^{3} - 3 \, w_{0,1}^{2} w_{2,0} + 6 \, w_{0,1} w_{1,0} w_{1,1} - 3 \, w_{0,2} w_{1,0}^{2}$\ & & $ -\, 2 \, w_{1,0}^{3} - 3 \, w_{1,0}^{2} - w_{1,0}$ ------------------------------------------------------------------------ \ Finally, given a polynomial $P\in{\ensuremath{\mathbb{C}}}[x,y]$, an open set $U\subseteq {\ensuremath{\mathbb{C}}}$, and a continuous function $\rho:U\to{\ensuremath{\mathbb{C}}}$, we will say that $\rho(x)$ is a *root function* for $P(x,y)$ on $U$ if $P\bigl(x,\rho(x)\bigr)= 0$ for all $x\in U$. Lemmas {#sec:lemmas} ====== Let $P(x,y)\in{\ensuremath{\mathbb{C}}}[x,y]$. Write $$P(x,y)=\sum_{j=0}^d a_j(x)y^j,$$ where $d{\mathrel{\mathop:}=}\deg_y(P)$ and $a_j(x)\in{\ensuremath{\mathbb{C}}}[x]$ for all $j$. Let $\mathcal{R}(x){\mathrel{\mathop:}=}\operatorname{Res}_y\bigl(P,{{\partial P/\partial y}}\bigr)$, which is also equal to $\pm a_d(x)\delta_y(P)$. For the remainder of this article, we will assume (as in [Theorem \[thm:main\]]{}) that $P$ and ${{\partial P/\partial y}}$ have no common root on ${\mathbb{T}}\times{\ensuremath{\mathbb{C}}}$. This is equivalent to $\mathcal{R}(x)$ not having any roots on ${\mathbb{T}}$. (The roots of $\mathcal{R}(x)$ are sometimes referred to as the *critical points* of $P$.) Also, for this section, we will assume that $P$ is absolutely irreducible, with $\deg_{x}(P)$ and $\deg_{y}(P) >0$. It follows easily that $P^\star$ is also irreducible, with $\deg_x(P^\star)=\deg_x(P)$, $\deg_y(P^\star)=\deg_y(P)$, and $P^{\star \star}=P$. The Implicit Function Theorem guarantees the existence of root functions for $P$, away from the roots of $\mathcal{R}(x)$. More concretely, we have the following [@mark3 Section 45]: \[lem:algebraic\] For any open, simply connected set $U\subset {\ensuremath{\mathbb{C}}}\setminus Z(\mathcal{R})$, there exist exactly $d$ root functions $\rho_1(x),\ldots,\rho_d(x)$ of $P(x,y)$ on $U$. These are all holomorphic, single-valued functions, and for distinct $i, j$, there is no $\alpha\in U$ for which $\rho_{i}(\alpha)=\rho_{j}(\alpha)$. Furthermore, for all $x\in U$, we have the factorization $$P(x,y)=a_d(x)\prod_{j=1}^d \bigl(y-\rho_j(x)\bigr).$$ We now turn to some applications of the reciprocal polynomial, $P^{\star}(x,y)$. Its main utility for us stems from the following two observations: if $P(\alpha,\beta)=0$, then $P^{\star}(1/\overline{\alpha}, 1/\overline{\beta})=0$ as well; and if ${\lvertz\rvert} = 1$, then $z= 1/\overline{z}$. Therefore, $Z(P)\cap {\mathbb{T}}^{2} \subseteq Z(P)\cap Z(P^{\star})$. \[lem:betarootfunction\] Let $U\subset{\ensuremath{\mathbb{C}}}^{\times}$, and suppose that $\rho(x)$ is a root function for $P(x,y)$ on $U$. Let $S$ be the set of all zeros of $\rho(x)$ on $U$. Then the function $\rho^{\star}(x) {\mathrel{\mathop:}=}1/\overline{\rho}(1/x)$ is a root function for $P^{\star}$ on $\overline{(U\setminus S)}^{\,-1}$. For all $x\in\overline{(U\setminus S)}^{\,-1}$, $w{\mathrel{\mathop:}=}1/\overline{x}\in U\setminus S$, and $\overline{\rho}(1/x) = \overline{\rho(w)} \ne 0$, hence $\rho^{\star}(x)$ is defined. Since $w\in U$, $P(w,\rho(w))=0$, hence $0= P^{\star}\bigl(1/\overline{w},1/\overline{\rho}(\overline{w})\bigr) = P^{\star}\bigl(x,\rho^{\star}(x)\bigr)$. If we restrict to $x\in{\mathbb{T}}$, then $\arg \rho^{\star}(x) = \arg \rho(x)$, and ${\lvert\rho^{\star}(x)\rvert} = 1/{\lvert\rho(x)\rvert}$. \[lem:essentialset\] Let $\gamma$ be a proper, open arc in ${\mathbb{T}}$, and $U$ a simply connected, open subset of ${\ensuremath{\mathbb{C}}}\setminus Z(\mathcal{R})$ such that $\gamma = U\cap {\mathbb{T}}$. Let $\rho(x)$ be a root function for $P(x,y)$ on $U$. 1. If there exist infinitely many points $x\in \gamma$ such that ${\lvert\rho(x)\rvert}=1$, then $P$ must be a.s.r. 2. If $P$ is a.s.r. and there exists an $\alpha\in\gamma$ such that ${\lvert\rho(\alpha)\rvert}=1$, then ${\lvert\rho(x)\rvert}\equiv1$ for all $x\in\gamma$. Let $T{\mathrel{\mathop:}=}\rho^{-1}({\mathbb{T}}) \cap\gamma$, let $S$ be the set of all roots of $\rho(x)$ on $U$, and let $V$ be an open, simply connected subset of $U\cap \overline{U}^{-1}$ containing $\gamma$.) Our assumptions on $P$ imply that $y{\mathrel{\centernot\mid}}P$, therefore $S$ is finite. Let $V' = V\setminus (\overline{S}^{\,-1})$. By [Lemma \[lem:betarootfunction\]]{}, $\rho^{\star}(x)$ is a root function for $P^{\star}$ on $V'$. $T\subset V'$, and for all $\alpha\in T$, ${\lvert\rho^{\star}(\alpha)\rvert} = 1/{\lvert\rho(\alpha)\rvert} = 1$ and $\arg \rho^{\star}(\alpha) = \arg\rho(\alpha)$, hence $\rho(\alpha) = \rho^{\star}(\alpha)$. 1. If $T$ is infinite, then then points $\bigl(\alpha,\rho(\alpha)\bigr)$ give infinitely many points in $Z(P)\cap Z(P^{\star})$. Since $P$ and $P^{\star}$ are both irreducible, this implies that $P{\mathrel{\mid}}P^{\star}$ and $P^{\star}{\mathrel{\mid}}P$ (see [@shaf page 2]), hence $P$ must be a.s.r. 2. $P$ being a.s.r. implies that $\rho^{\star}(x)$ is actually a root function for $P$ on $V'$ (indeed, on $U$). By [Lemma \[lem:algebraic\]]{}, distinct root functions of $P$ will never take the same value, but $\rho(\alpha) = \rho^{\star}(\alpha)$, hence $\rho(x)\equiv \rho^{\star}(x)$ on $V'$. Since ${\lvert\rho^{\star}(x)\rvert} = 1/{\lvert\rho(x)\rvert}$ on $\gamma$, we conclude that ${\lvert\rho(x)\rvert}\equiv 1$ on $\gamma$. We will refer to root functions $\rho(x)$ that satisfy ${\lvert\rho(x)\rvert}\equiv 1$ for all $x$ in some arc $\gamma\subset {\mathbb{T}}$ as *toric* root functions, since the map $x\mapsto \bigl(x, \rho(x)\bigr)$, restricted to $\gamma$, parametrizes an infinite subset of $Z(P)\cap {\mathbb{T}}^{2}$. The lemma above asserts that $P$ being a.s.r. (a “global” property of $P$) is a necessary condition for the existence of toric root functions. It is not a sufficient condition, although experiment suggests that most (perhaps almost all?) a.s.r. polynomials in ${\ensuremath{\mathbb{C}}}[x,y]$ have at least one toric root function. Therefore, the lemma shows that the points in $Z(P)\cap{\mathbb{T}}^{2}$ fall into two categories: those lying in the parametrized families coming from toric root functions (which are only possible for $P$ a.s.r.), and finitely many isolated points (which are only possible for $P$ not a.s.r.). Recall from [Theorem \[thm:main\]]{} that we define $\mathcal{E}$ to be $Z(P)\cap {\mathbb{T}}^{2}$ if this is set finite, and $\mathcal{E}={\ensuremath{\varnothing}}$ otherwise. Therefore, if $P$ is not a.s.r., then $\mathcal{E}=Z(P)\cap {\mathbb{T}}^{2}$, and if $P$ is a.s.r., then $\mathcal{E} = {\ensuremath{\varnothing}}$ (whether or not $P$ has toric root functions). This shows that [Corollary \[cor:zeroseq\]]{} follows from [Theorem \[thm:main\]]{}. It also implies that $\mathcal{E}$ coincides with the set of isolated points of $Z(P)\cap{\mathbb{T}}^{2}$. \[lem:algebraiccoords\] If the coefficients of $P$ are algebraic, then the coordinates of the points in $\mathcal{E}$ are algebraic. If $P$ is a.s.r., then $\mathcal{E} = {\ensuremath{\varnothing}}$, so assume otherwise. Therefore $\mathcal{E} = Z(P)\cap {\mathbb{T}}^{2}$, which is $\subseteq Z(P)\cap Z(P^{\star})$, as mentioned before [Lemma \[lem:betarootfunction\]]{}. Since $P$ is not a.s.r., the latter set is finite. The $x$ coordinates of all of its points are roots of the resultant $\operatorname{Res}_{y}(P,P^{\star})\in \overline{{\ensuremath{\mathbb{Q}}}}[x]$; the $y$ coordinates are handled similarly. \[lem:rootfnontorus\] Let $f\colon {\ensuremath{\mathbb{R}}}\to{\ensuremath{\mathbb{R}}}$ be a $C^{\infty}$ function, not identically equal to $nt$ for any integer $n$. Assume that there exist integers $L, M$, with $L>0$, such that $f(t+2\pi L) - f(t) = 2\pi M$ for all $t\in{\ensuremath{\mathbb{R}}}$. Let $I=[0,2\pi L]$. Then for all $k\ge 0$ and all integers $n\ge 1$, $$\int_{I} \log{\lvert1- \exp\bigl(i(f(t)-nt)\bigr)\rvert} \,dt = {\mathop{O_{k}}}(n^{-k}).$$ Our assumptions on $f$ imply that for all $j\ge 1$, $f^{(j)}(t)$ is periodic, hence bounded. Let $C = \sup_{t\in{\ensuremath{\mathbb{R}}}} {\lvertf'(t)\rvert}$. By choosing the implicit constant sufficiently large, the claimed bound is trivially true for $1\le n\le 2C$. We may therefore restrict to integers $n>2C$. Recall that the $k$-th polylogarithm function, ${\mathop{\mathrm{Li}_{k}}}(z)$, is continuous on the closed unit disk for $k\ge 1$, with the exception of a singularity at $z=1$ when $k=1$. (Indeed, ${\mathop{\mathrm{Li}_{1}}}(z) = -\log(1-z)$, using the principal branch of the logarithm for ${\lvertz\rvert}\le 1$.) Further, the functions $G_{k}(t) {\mathrel{\mathop:}=}{\mathop{\mathrm{Li}_{k}}}\bigl(\exp(it)\bigr)$ are differentiable on ${\ensuremath{\mathbb{R}}}\setminus 2\pi{\ensuremath{\mathbb{Z}}}$ for $k\ge 1$, satisfying $G_{k+1}'(t) = i \,G_{k}(t)$. We define a sequence of functions $\{h_{k}(x,t)\}$ for $k\ge 1$ as follows: let $h_{1}(x,t) = 1$, and for $k\ge 2$, let $h_{k}(x,t) = \frac{\partial}{\partial t}\bigl[h_{k-1}(x,t)/(f'(t)-x)\bigr]$. By induction, for each $k\ge 1$, we may write $h_{k}(x,t)=P_{k}(x,t)/(x-f'(t))^{2k-2}$, where $P_{k}(x,t)$ is, with respect to $x$, a polynomial of degree $k-1$ whose coefficients are polynomial expressions in $f'(t), f''(t), \dotsc$. Therefore, substituting $x=n > 2C$, we have $h_{k}(n,t) = {\mathop{O_{k}}}(n^{-(k-1)})$ for $k\ge 1$. Let $b_{n}{\mathrel{\mathop:}=}\int_{I} \log{\lvert1- \exp\bigl(i(f(t)-nt)\bigr)\rvert} \,dt$. (This is finite, as $f(t)\not\equiv nt$.) Then $$b_{n} = -{\mathop{\mathrm{Re}}}\biggl[\int_{I} h_{1}(n,t) G_{1}\bigl(f(t)-nt\bigr) \,dt\biggr].$$ For each $k\ge 1$, $$\int_{I} h_{k}(n,t) G_{k}\bigl(f(t)-nt\bigr) dt = \int_{I} \frac{-i \,h_{k}(n,t)}{\bigl(f'(t) - n\bigr)} \frac{d}{dt}\Bigl[G_{k+1}\bigl(f(t)-nt\bigr)\Bigr] dt.$$ Integrating by parts, $$= \frac{-i \,h_{k}(n,t)}{\bigl(f'(t) - n\bigr)}G_{k+1}\bigl(f(t)-nt\bigr) \biggr|_{0}^{2\pi L} + i \!\int_{I} h_{k+1}(n,t) G_{k+1}\bigl(f(t)-nt\bigr) dt.$$ By the properties of $f$, $-i\, h_{k}(n,t) G_{k+1}\bigl(f(t)-nt\bigr) / \bigl(f'(t) - n\bigr)$ takes on the same values at $t=0$ and at $t=2\pi L$. Therefore, $$\int_{I} h_{k}(n,t) G_{k}\bigl(f(t)-nt\bigr) dt = i \!\int_{I} h_{k+1}(n,t) G_{k+1}\bigl(f(t)-nt\bigr) dt.$$ Hence, for all $k\ge 1$, $$b_{n} = - {\mathop{\mathrm{Re}}}\biggl[i^{k-1} \!\int_{I} h_{k}(n,t) G_{k}\bigl(f(t)-nt\bigr) \,dt\biggr] \\ = {\mathop{O_{k}}}(n^{-(k-1)}). \qedhere$$ \[lem:parts\] Let ${\mathop{\tilde{D}^{}}}=x \frac{d}{dx}$. For $\gamma$ an arc, $k$ a nonnegative integer, $f(x)$ a $C^{k}$ function on a neighborhood of the closure of $\gamma$, and $N$ a nonzero integer, $$\begin{split} \int_\gamma x^N f(x)\,\frac{dx}{x} &\;=\; -\sum_{r=1}^{k} \frac{1}{(-N)^{r}} {\left[x^{N} {\mathop{\tilde{D}^{r-1}}}(f)\right]_{\partial \gamma}} \\ & \quad \;\;\;+\,\frac{1}{(-N)^{k}} \int_\gamma x^{N} {\mathop{\tilde{D}^{k}}}(f)\,\frac{dx}{x}. \end{split}$$ The proof is an easy induction, using integration by parts. \[lem:dibrunocor\] Let $k$ be a nonnegative integer and $f$ a $C^{k}$ function on an open subset $U$ of ${\ensuremath{\mathbb{C}}}$. For all integers $m$, $$\label{eqn:diBrunoLem} {\mathop{H^{m}_{k}}\!} f(z) = f(z)^{m-k}\sum_{j=0}^k {\varphi}_{k,j,f}(z) m^j,$$ where ${\varphi}_{k,j,f}(z){\mathrel{\mathop:}=}\Phi_{k,j}\bigl(f,zf',z^2 f'',\ldots, z^{k-j+1}f^{(k-j+1)}\bigr)$ for $j=0,\dotsc, k$. (In particular, ${\varphi}_{k,j,f}(z)$ is continuous on $U$ for all $j$.) Letting $D=\frac{d}{dz}$, an easy induction shows that $$\label{eqn:zDiterated} (z D)^{k} = \sum_{i}{{\textstyle \genfrac{\{}{\}}{0pt}{}{k}{i}}} z^{i} D^{i},$$ using the recurrence ${{\textstyle \genfrac{\{}{\}}{0pt}{}{k}{i}}} = i {{\textstyle \genfrac{\{}{\}}{0pt}{}{k-1}{i}}}+{{\textstyle \genfrac{\{}{\}}{0pt}{}{k-1}{i-1}}}$ (valid for all $k$ and $i$ except $k=i=0$) [@wilf Equation 1.6.3]. It follows immediately from Faà di Bruno’s formula, written in terms of Bell polynomials [@comtet1 Section 3.4, Theorem A], that $$\label{eqn:diBruno} D^{i} \bigl(f(z)^{m}\bigr) = \sum_{l=0}^i (m)_{l} f(z)^{m-l} B_{i,l}(f',f'',\ldots,f^{(i-l+1)}).$$ Using the identity $(x)_l = \sum_{j=0}^l (-1)^{l-j} {{\textstyle \genfrac{[}{]}{0pt}{}{l}{j}}}x^j$ [@wilf Equation 3.5.2], together with and , we have $${\mathop{H^{m}_{k}}\!} f(z) = \sum_{i,l,j}{{\textstyle \genfrac{\{}{\}}{0pt}{}{k}{i}}} z^{i} (-1)^{l-j} {{\textstyle \genfrac{[}{]}{0pt}{}{l}{j}}} m^j f(z)^{m-l} B_{i,l}(f',f'',\ldots,f^{(i-l+1)}).$$ By definition, every monomial in $B_{i,l}(y_0,\dotsc,y_{i-l+1})$ has “weight” (the sum of the variables’ subscripts, with multiplicity) equal to $i$. Therefore, $$z^{i} B_{i,l}(f',f'',\ldots,f^{(i-l+1)}) = B_{i,l}(z f',z^{2} f'',\ldots,z^{i-l+1} f^{(i-l+1)}).$$ Together with the previous equation, this proves the formula in the Lemma. Since $f$ is $C^{k}$, the continuity of the functions ${\varphi}_{k,j,f}(z)$ is clear, except that ${\varphi}_{k,0,f}(z)$ appears to depend on $f^{(k+1)}$. However, this is illusory, since $\Phi_{k,0}$ is in fact constant, equal to either 0 or 1, for each $k$. \[cor:dibrunoBigO\] Let $k, m$ be integers with $m\ge 1$ and $k\ge 0$, and let $f$ be a $C^k$ function on an open neighborhood of some compact set $S\subset {\ensuremath{\mathbb{C}}}$. Then for $z\in S$ at which $f(z)\ne 0$, $${\mathop{H^{m}_{k}}\!} f(z) = {\mathop{O_{f,S,k}}}\bigl(m^k {\lvertf(z)\rvert}^{m-k}\bigr).$$ \[lem:realanalytic\] Let $I$ be an interval, $\gamma =$ the image of $I$ under the map $t\mapsto e^{it}$, and $f$ a function holomorphic on $\gamma$. Let $g(t){\mathrel{\mathop:}=}|f(e^{it})|$. Then there exists an open interval $J\supseteq I$ such that $g$ is real analytic on $J\setminus\{t\in{\ensuremath{\mathbb{R}}}\,|\,g(t)=0\}$ and continuous on $J$. Let $V$ be an open neighborhood of $\gamma$ on which $f$ is holomorphic, let $U$ be the preimage of $V$ under the map $t\mapsto e^{it}$, and let $J$ be the connected component of $U\cap{\ensuremath{\mathbb{R}}}$ that contains $I$. Since $g(t)^2 = f(e^{it})\overline{f}(e^{-it})$ on $J$ and $\sqrt{x}$ is real analytic on $(0,\infty)$, the claim follows. \[lem:integralbounds\] Let $I$ be an interval, $\gamma =$ the image of $I$ under the map $t\mapsto e^{it}$, and $f$ a function holomorphic on the closure of $\gamma$. Suppose that $|f(z)|\leq 1$ on $\gamma$, with equality at only finitely many points in $\gamma$. There exists a $\delta>0$ such that for all positive integers $m$ and nonnegative integers $k$, $$\int_I \bigl|{\mathop{H^{m}_{k}}\!}(f)(e^{it})\bigr|dt = {\mathop{O_{f,I,k}}}\bigl(m^{k-\delta}\bigr).$$ We may restrict our attention to $m\ge 2k$. Indeed, regardless of what $\delta$ we choose below, by making the implicit constant large enough, we can trivially ensure that the bound is met for $m\in\{1,\dotsc,2k-1\}$. Let $g(t)=|f(e^{it})|$. Since $m-k\ge m/2$, [Corollary \[cor:dibrunoBigO\]]{} implies that $$\begin{split} \int_I \bigl|{\mathop{H^{m}_{k}}\!}(f)(e^{it})\bigr|dt &= {\mathop{O_{f,I,k}}}\biggl(m^k\int_{I} |f(e^{it})|^{m-k} dt \biggr) \\ &= {\mathop{O_{f,I,k}}}\biggl(m^k\int_{I}g(t)^{m/2} dt \biggr). \end{split}$$ It suffices to show that the remaining integral is ${\mathop{O_{f,I}}}(m^{-\delta})$ for some $\delta>0$. By subdividing $I$ at the (finitely many) values of $t$ where $g(t)=1$ and making a linear change of variables on each subinterval, we may assume without loss of generality that $I$ has the form $[0,b]$ for some $b>0$, and that $g(t)<1$ on $(0,b]$. If $g(0)=1$, then by [Lemma \[lem:realanalytic\]]{}, there exists $r>0$ such that $g$ is real analytic on $(-r,r)$. We may take $r<b$. (If $g(0)<1$, let $r=0$.) First consider the interval $[r,b]$. Since $g(t)$ is continuous and $<1$ there, there exists a constant $\lambda<1$ such that $0\le g(t) \le \lambda$ on $[r,b]$. Hence $$\int_{r}^b g(t)^{m/2}dt \le b \lambda^{m/2},$$ which is ${\mathop{o}}(m^{-\delta})$ for every $\delta>0$. If $g(0)\ne 1$, we are finished. Otherwise, it remains to bound the integral on $[0,r]$. Let $\nu$ equal the order of vanishing of $g(t)-1$ at $0$.[^1] There exists a $c>0$ such that $g(t) \le 1-ct^\nu \le \exp(-c t^{\nu})$ for all $t\in [0,r]$. Therefore, $$\int_{0}^{r} g(t)^{m/2}dt \le \int_{0}^{\infty} \! \exp(-cmt^{\nu}/2) \,dt = (cm/2)^{-\frac{1}{\nu}} \Gamma\bigl(1+\tfrac{1}{\nu}\bigr) = {\mathop{O_{f,I}}}(m^{-\frac{1}{\nu}}).$$ Letting $\delta = 1/\nu$, we obtain the desired bound. Later on, when we apply [Lemma \[lem:dibrunocor\]]{}, the function $f$ will be one of the root functions $\rho(x)$ of $P$. As it can be difficult (or impossible) to obtain explicit expressions for such algebraic functions, let alone their higher derivatives, the following generalization of implicit differentiation (see [@wilde], [@comtet1], [@comtet2], [@comtetfiolet]) is useful. Recall the polynomials $Q_{n}$ defined in section \[sec:notation\]. \[lem:implicitderivs\] Let $\rho(x)$ be a root function for $P(x,y)$ on some open subset of ${\ensuremath{\mathbb{C}}}$. For all integers $n\ge 1$, $$\rho^{(n)}(x) = Q_{n}/w_{0,1}^{{2n-1}},$$ after making the substitutions $w_{i,j} = \frac{\partial^{i+j}}{\partial x^{i} \partial y^{j}} P(x,y)$ for all $i,j$ and $y=\rho(x)$. For example, when $n=1$, this gives the familiar fact that $dy/dx = -P_{x}/P_{y}$. Recall the definition of $\Omega_{P,r,a}(x,y)$ from [Theorem \[thm:main\]]{}. \[lem:Psiproperties\] Let $\rho(x)$ be a root function for $P(x,y)$ on some open subset of ${\ensuremath{\mathbb{C}}}$. For integers $2\le a\le r$, $$\Omega_{P,r,a}\bigl(x, \rho(x)\bigr) = {\varphi}_{r-1, r-a+1, \rho}(x)/\rho(x)^{r-1}.$$ Let $\mathbf{v} = \{v_{i,j}\}$ be a doubly-indexed sequence of indeterminates, like the indeterminates $\mathbf{w} = \{w_{i,j}\}$ used by $Q_{n}$ and $\Psi_{r,a}$. By the restrictions on the monomials in the definition of $Q_{n}$, if we substitute $w_{i,j} = x^{i}y^{j-1} v_{i,j}/v_{0,1}$ for all $i,j$, then $Q_{n}(\mathbf{w})= x^{n} Q_{n}(\mathbf{v})/(y \,v_{0,1}^{2n-1})$. Therefore, if we substitute $v_{i,j} = \frac{\partial^{i+j}}{\partial x^{i} \partial y^{j}} P$ and $y=\rho(x)$, [Lemma \[lem:implicitderivs\]]{} implies that $y\, Q_{n}(\mathbf{w}) = x^{n} \rho^{(n)}(x)$. By definition, $\Omega_{P,r,a}(x,y) = \Psi_{r,a}(\mathbf{w}) = \Phi_{r-1,r-a+1}\bigl(1, Q_{1}(\mathbf{w}), Q_{2}(\mathbf{w}), \dotsc\bigr)$. Since $\Phi_{r-1,r-a+1}$ is homogeneous of degree $r-1$, $$\begin{split} \Omega_{P,r,a}(x,y) =\;& \Phi_{r-1,r-a+1}\bigl(y, y\,Q_{1}(\mathbf{w}), y\,Q_{2}(\mathbf{w}), \dotsc\bigr)/y^{r-1} \\ =\; & \Phi_{r-1,r-a+1}\bigl(\rho(x), x \rho'(x), x^{2} \rho''(x),\dotsc\bigr)/y^{r-1} \\ =\; & {\varphi}_{r-1, r-a+1,\rho}(x)/\rho(x)^{r-1}. \qedhere \end{split}$$ Proof of [Proposition \[prop:uniqueness\]]{} {#sec:proofofuniqueness} ============================================ Let $\omega\colon{\ensuremath{\mathbb{R}}}\to {\ensuremath{\mathbb{R}}}$ be a quasiperiodic function, and let $\tilde{\omega} = \omega\bigr|_{{\ensuremath{\mathbb{N}}}}$, the restriction of $\omega$ to ${\ensuremath{\mathbb{N}}}$. If $\tilde{\omega}(n)$ approaches some (finite) limit as $n\to\infty$, then $\tilde{\omega}$ must be constant. Assume that $\tilde{\omega}$ is nonconstant. Choose continuous, periodic functions $\tau_1(t),\dotsc, \tau_k(t)$ such that $\omega(t)=\sum_{i=1}^k \tau_i(t)$. Without loss of generality, each of the summands $\tau_{i}(t)$ is nonconstant. For each $i$, let $\alpha_i$ be the period of $\tau_i$. For real numbers $x$, let ${\lVertx\rVert}$ denote the distance from $x$ to the nearest integer. By a result in simultaneous Diophantine approximation [@cassels Chapter I, Theorem VI], there is an infinite sequence of distinct positive integers $m_j$ such that $\max_{1\le i\le k} {\lVertm_j/\alpha_i\rVert}\to 0$ as $j\to\infty$. Therefore, $m_j\to 0$ simultaneously in each of the metric spaces ${\ensuremath{\mathbb{R}}}/\alpha_i{\ensuremath{\mathbb{Z}}}$ (the natural domain of $\tau_i$). We conclude that for every positive integer $a$, $$\omega(a) =\sum_{i=1}^k \tau_i(a)= \sum_{i=1}^k \lim_{j\to\infty} \tau_i(a+m_j) = \lim_{j\to\infty} \omega(a+m_j) = \lim_{n\to\infty}\omega(n),$$ therefore $\tilde{\omega}$ is constant. It suffices to prove uniqueness for expansions of $0$. That is, if a sequence of quasiperiodic functions $\{c_j(t)\}$ satisfies $0 {\overset{*}{\sim}}\sum_{j=0}^{\infty} c_j(n)/n^j$, we wish to show that $c_{j}(n)=0$ for all integers $n\ge 1$ and $j\ge 0$. We will induct. Let $k$ be a nonnegative integer, and suppose that $c_{j}(n)\equiv 0$ for $0\le j < k$ and $n\in{\ensuremath{\mathbb{N}}}$. (Conveniently, this also applies for our base case, $k=0$, by vacuity.) Then $$\frac{c_{k}(n)}{n^{k}} = \sum_{j=0}^{k} \frac{c_{j}(n)}{n^{j}} = {\mathop{O_{k}}}\biggl(\frac{1}{n^{k+1}}\biggr),$$ hence $c_{k}(n)\to 0$ as $n\to\infty$. Therefore, by the lemma above, $c_{k}(n)=0$ for all positive integers $n$. Proof of [Theorem \[thm:main\]]{} {#sec:proofofmain} ================================= Since $m(P_1 P_2)=m(P_1)+m(P_2)$, we have $\Delta_{n}(P_{1}P_{2}) = \Delta_{n}(P_{1}) + \Delta_{n}(P_{2})$ for all $n$. We may therefore assume without loss of generality that $P$ is absolutely irreducible, at least for the purpose of showing existence of an a.p.p.s. expansion. A change of variables shows that $m\bigl(P(x^{M},y^{N})\bigr) = m\bigl(P(x,y)\bigr)$ for all nonzero integers $M, N$. It follows that if $\deg_{x}(P)$ or $\deg_{y}(P)=0$, then $\mathcal{E} = {\ensuremath{\varnothing}}$ and $\Delta_{n}(P)=0$ for all $n$, hence [Theorem \[thm:main\]]{} is trivially true. We will therefore assume that $\deg_x(P)$ and $\deg_y(P)$ are both $>0$. Also, if there exists an $k\in{\ensuremath{\mathbb{N}}}$ for which $P(x,x^{k})$ is identically zero, then $Z(P)\cap Z(y-x^{k})$ would be infinite, hence $P = c(y-x^{k})$ for some constant $c$, and $\mathcal{E} = {\ensuremath{\varnothing}}$. But $m(y-x^{k}) = 0$, as follows from , and also $m(x^{n}-x^{k})=0$ for all $n\in{\ensuremath{\mathbb{N}}}$ by . Hence, we would have $\Delta_{n}(P) = 0$ for all $n$, and the theorem is again trivially true. We may therefore assume that, for every positive integer $k$, $P(x,x^{k})$ is not identically zero. Since $\mathcal{R}(x)$ has no roots on ${\mathbb{T}}$, we may choose real numbers $0<r_{1} < 1 <r_{2}$ such that the annulus $r_{1} < {\lvertx\rvert} < r_{2}$ contains no roots of $\mathcal{R}(x)$. Let $\mathcal{E}_{x}$ be the set of all $x$ coordinates of the points in $\mathcal{E}$, and fix an arbitrary point ${\alpha_{0}}\in {\mathbb{T}}\setminus \mathcal{E}_{x}$. Define $$W {\mathrel{\mathop:}=}\{x\in {\ensuremath{\mathbb{C}}}\,:\, r_{1} < {\lvertx\rvert} < r_{2} \text{ and } \arg(x) \ne \arg({\alpha_{0}})\},$$ which is the annulus mentioned above sliced along the ray from $0$ through ${\alpha_{0}}$. The set $W$ meets the hypotheses of [Lemma \[lem:algebraic\]]{}. Let $${R}{\mathrel{\mathop:}=}\{\rho_{1}(x),\dotsc, \rho_{d}(x)\}$$ be the set of all root functions of $P$ on $W$. We partition ${R}$ into ${{R}_{\mathrm{tor}}}$, the set of toric root functions on $W$ and ${{R}_{\mathrm{non}}}= {R}\setminus{{R}_{\mathrm{tor}}}$, the non-toric root functions. (The branch cut in the definition of $W$ ensures that the functions in ${R}$ are single-valued, but the break it creates in their domains is unnatural. Each $\rho(x)\in {R}$ may be analytically continued past that ray, becoming a (usually different) element of ${R}$.) We can now define the function $s:\mathcal{E}\to\{-1,0,1\}$ mentioned in [Theorem \[thm:main\]]{}. For each point $(\alpha,\beta)\in\mathcal{E}$, there exists a unique, non-toric root function $\rho(x)\in {{R}_{\mathrm{non}}}$ such that $\beta = \rho(\alpha)$. We define $s(\alpha,\beta)$ to be $+1$ (respectively, $-1$) if ${\lvert\rho(x)\rvert}-1$ changes from negative to positive (respectively, positive to negative) as $x$ moves counterclockwise past $\alpha$ in the unit circle. If ${\lvert\rho(x)\rvert}-1$ does not change sign there, let $s(\alpha,\beta)=0$. In practice, we will usually not be able to obtain explicit formulas for the root functions. In such cases, the signs $s(\alpha,\beta)$ can be rigorously determined by an algebraic method using only on the coordinates $(\alpha,\beta)$ and the coefficients of $P$. This method is described in section \[sec:signdet\]. Subdivide ${\mathbb{T}}$ at the points in $\mathcal{E}_{x}\cup\{{\alpha_{0}}\}$, obtaining a finite set of open, proper arcs $\Gamma = \{\gamma_{1},\dotsc,\gamma_{M}\}$. For each arc $\gamma\in \Gamma$ and each non-toric root function $\rho(x)\in {{R}_{\mathrm{non}}}$, ${\lvert\rho(x)\rvert}$ is either always $>1$ or always $<1$ on $\gamma$. Partition $\Gamma\times {{R}_{\mathrm{non}}}$ into the subsets $$\begin{split} \mathcal{A}_{-} &{\mathrel{\mathop:}=}\bigl\{(\gamma,\rho)\in\Gamma \times {{R}_{\mathrm{non}}}: {\lvert\rho(x)\rvert}>1 \text{ on } \gamma\bigr\}, \\ \mathcal{A}_{+} &{\mathrel{\mathop:}=}\bigl\{(\gamma,\rho)\in\Gamma \times {{R}_{\mathrm{non}}}: {\lvert\rho(x)\rvert}< 1 \text{ on } \gamma\bigr\}. \\ \end{split}$$ (The choice of which is $+$ or $-$ is slightly more convenient this way.) We now extend the domain of $s\colon \mathcal{E}\to \{-1,0,1\}$ by setting $s(\alpha,\beta)=0$ for all $(\alpha,\beta)\notin \mathcal{E}$. Recall the definition of ${\left[f(x)\right]_{\partial \gamma}}$ from section \[sec:notation\]. For a fixed choice of $\pm$, and for any operator $H$ that sends holomorphic functions to continuous functions, careful bookkeeping of signs shows that $$\label{eq:evalsum} \sum_{(\gamma,\rho)\in\mathcal{A}_{\pm}} {\left[H\rho(x)\right]_{\partial \gamma}} = \pm \sum_{\rho\in {{R}_{\mathrm{non}}}}\sum_{\alpha\in \mathcal{E}_{x}} s\bigl(\alpha,\rho(\alpha)\bigr) H \rho(\alpha).$$ (For certain root functions $\rho(x)\in{{R}_{\mathrm{non}}}$, a copy of $\pm \lim_{x\to {\alpha_{0}}} H \rho(x)$ (as $x$ approaches ${\alpha_{0}}$ along ${\mathbb{T}}$, from one direction or the other) may appear on the left side, but this will be cancelled by the contributions from other root functions.) With this setup, we are now ready to examine $\Delta_{n}(P)$. The strategy of the proof is inspired by that used by Boyd in his proof of [Proposition \[prop:boydasymp\]]{}. We first rewrite $m\bigl(P(x,y)\bigr)$, using a standard trick. $$m\bigl(P(x,y)\bigr) = \frac{1}{2\pi i} \int_{{\lvertx\rvert}=1} \biggl[ \frac{1}{2\pi i} \int_{{\lverty\rvert}=1} \log\bigl|P(x,y)\bigr| \frac{dy}{y} \biggr] \frac{dx}{x}$$ The expression inside the brackets is actually a one-variable Mahler measure with respect to $y$, with $x$ viewed there as constant. Therefore, implies that $$\begin{aligned} \label{eqn:PxyMM} \nonumber m\bigl(P(x,y)\bigr) &= \frac{1}{2\pi i} \int_{{\mathbb{T}}} \biggl(\log\bigl|a_{d}(x)\bigr| + \sum_{\rho\in {R}} {\mathop{\log^+\!}}\bigl|\rho(x)\bigr|\biggr) \frac{dx}{x} \\[2mm] &= m(a_{d}) + \frac{1}{2\pi i} \sum_{(\gamma,\rho)\in\mathcal{A}_{-}} \int_{\gamma} \log \bigl|\rho(x)\bigr| \frac{dx}{x}.\end{aligned}$$ We now rewrite $m\bigl(P(x,x^{n})\bigr)$. Recall that for every positive integer $n$, $P(x,x^{n})$ is not identically $0$. $$\begin{split} m\bigl(P(x,x^{n})\bigr) &= \frac{1}{2\pi i} \int_{{\mathbb{T}}} \log\bigl|P(x,x^{n})\bigr| \frac{dx}{x} \\[2mm] &= \frac{1}{2\pi i} \int_{{\mathbb{T}}} \biggl(\log\bigl|a_{d}(x)\bigr| + \sum_{\rho\in {R}} \log\bigl|x^{n}-\rho(x)\bigr|\biggr) \frac{dx}{x} \\[2mm] &= m(a_{d}) + \frac{1}{2\pi i} \int_{{\mathbb{T}}} \sum_{\rho\in {R}} \log\bigl|x^{n}-\rho(x)\bigr|\frac{dx}{x}. \\[2mm] \end{split}$$ Since $P(x,x^{n})\not\equiv 0$, there are, for each $\rho(x)\in {R}$, only finitely many $x\in W$ such that $x^{n} = \rho(x)$. It follows that $\log\bigl|x^{n}-\rho(x)\bigr|$ is integrable on ${\mathbb{T}}$. Therefore, we may interchange the sum and integral above, obtaining $$\label{eqn:PxxnMM} m\bigl(P(x,x^{n})\bigr) = m(a_{d}) + \frac{1}{2\pi i} \sum_{\rho\in{R}} \int_{{\mathbb{T}}} \log\bigl|x^{n}-\rho(x)\bigr|\frac{dx}{x}.$$ We now separately consider those terms in coming from toric root functions. Given one such $\rho(x)\in{{R}_{\mathrm{tor}}}$ and some $t'\in{\ensuremath{\mathbb{R}}}$ such that $\exp(i t')\ne {\alpha_{0}}$, there exists a holomorphic function $f(z)$, defined on a neighborhood of $t'$ in ${\ensuremath{\mathbb{C}}}$, such that $f(t)\in{\ensuremath{\mathbb{R}}}$ for $t\in{\ensuremath{\mathbb{R}}}$ and $\rho\bigl(\exp(i z)\bigr) = \exp\bigl(i f(z)\bigr)$. We analytically continue $f(z)$ to a strip $-\delta<{\mathop{\mathrm{Im}}}(z)<\delta$ for some $\delta>0$. If we let $z$ move to the right in steps of $2\pi$, $\exp\bigl(i f(z)\bigr)$ will cycle through values of $\tilde{\rho}\bigl(\exp(i z)\bigr)$ for various $\tilde{\rho}(x)$ in some subset of ${{R}_{\mathrm{tor}}}$. Letting $L$ equal the size of this orbit, $\exp\bigl(i f(z+2\pi L)\bigr) \equiv \exp\bigl(i f(z)\bigr)$. Therefore, there exists an integer $M$ such that $f(z+2\pi L) = f(z) + 2\pi M$ for all $z$ in the strip. Also, $f(t)\not\equiv nt$ for every integer $n$, as otherwise $\rho(x)\equiv x^{n}$, hence $P(x,x^{n})\equiv 0$. Each function $f(z)$ described above encapsulates some subset of ${{R}_{\mathrm{tor}}}$. Therefore, the terms in coming from the toric root functions can be written as the sum of finitely many expressions of the form $$\frac{1}{2\pi} \int_{0}^{2\pi L} \log{\lvert1-\exp\bigl(i (f(t)-nt)\bigr)\rvert} dt,$$ for some $f$ as described above. Invoking [Lemma \[lem:rootfnontorus\]]{}, this integral is ${\mathop{O_{k}}}(n^{-k})$ for all $k\ge 0$. Therefore, for all $k\ge 0$, $$m\bigl(P(x,x^{n})\bigr) = m(a_{d}) + \frac{1}{2\pi i} \sum_{\rho\in{{R}_{\mathrm{non}}}} \int_{{\mathbb{T}}} \log\bigl|x^{n}-\rho(x)\bigr|\frac{dx}{x} + {\mathop{O_{k}}}(n^{-k}).$$ Split ${\mathbb{T}}$ into the arcs $\gamma\in\Gamma$, and rewrite $\log\bigl|x^{n}-\rho(x)\bigr|$ as $\log\bigl|1- \rho(x)/x^{n}\bigr|$ if $(\gamma,\rho)\in \mathcal{A}_{+}$, and as $\log\bigl|\rho(x)\bigr|+ \log\bigl|1-x^{n}/\rho(x)\bigr|$ if $(\gamma,\rho)\in \mathcal{A}_{-}$. Combining the equation above with and using the shorthand $\sum_{\pm} c_{\pm}$ for $c_{+} + c_{-}$, $$\begin{aligned} \label{eqn:Dnptemp} \nonumber \Delta_{n}(P) &= \frac{1}{2\pi i} \sum_{\pm} \sum_{(\gamma,\rho)\in\mathcal{A}_{\pm}} \int_{\gamma} \log\Bigl|1- \Bigl(\frac{\rho(x)}{x^{n}}\Bigr)^{\pm 1}\Bigr| \frac{dx}{x} + {\mathop{O_{k}}}(n^{-k}) \\[2mm] \nonumber &= \frac{1}{2\pi} \sum_{\pm} \sum_{(\gamma,\rho)\in\mathcal{A}_{\pm}} \int_{\gamma} -{\mathop{\mathrm{Re}}}\biggl(\,\sum_{m=1}^{\infty} \frac{1}{m}\Bigl(\frac{\rho(x)}{x^{n}}\Bigr)^{\pm m} \biggr) d\arg(x) + {\mathop{O_{k}}}(n^{-k}) \\[2mm] &= -\frac{1}{2\pi} {\mathop{\mathrm{Im}}}\biggl(\sum_{\pm} \sum_{(\gamma,\rho)\in\mathcal{A}_{\pm}} \int_{\gamma} \sum_{m=1}^{\infty} \frac{1}{m}\Bigl(\frac{\rho(x)}{x^{n}}\Bigr)^{\pm m} \frac{dx}{x} \biggr) + {\mathop{O_{k}}}(n^{-k}).\end{aligned}$$ By Beppo Levi’s lemma [@apostol1 Theorem 10.26], together with the $k=0$ case of [Lemma \[lem:integralbounds\]]{}, we may interchange the integral and the sum over $m$ in . Therefore, if we define $${I^{\pm}_{n,m}}{\mathrel{\mathop:}=}\sum_{(\gamma,\rho)\in\mathcal{A}_{\pm}} \int_{\gamma} \Bigl(\frac{\rho(x)}{x^{n}}\Bigr)^{\pm m} \frac{dx}{x},$$ then for all $k\ge 0$, $$\label{eqn:Dnpnice} \Delta_{n}(P) = -\frac{1}{2\pi} \sum_{m=1}^{\infty} \frac{1}{m} \sum_{\pm} {\mathop{\mathrm{Im}}}\bigl({I^{\pm}_{n,m}}\bigr) + {\mathop{O_{k}}}(n^{-k}).$$ For integers $n,m, r\ge 1$ and $k\ge 0$, let $${S^{\pm}_{n,m,r}}{\mathrel{\mathop:}=}\pm \!\!\sum_{\rho\in {{R}_{\mathrm{non}}}} \sum_{\alpha\in \mathcal{E}_{x}} s\bigl(\alpha,\rho(\alpha)\bigr)\, \alpha^{\mp nm} {\mathop{H^{\pm m}_{r-1}}\!} \rho(\alpha)$$ and $${J^{\pm}_{n,m,k}}{\mathrel{\mathop:}=}\!\!\!\sum_{(\gamma,\rho)\in \mathcal{A}_{\pm}} \int_{\gamma} x^{\mp nm} {\mathop{H^{\pm m}_{k}}\!} \rho(x) \,\frac{dx}{x}.$$ [Lemma \[lem:parts\]]{}, together with , then implies that $${I^{\pm}_{n,m}}= -\sum_{r=1}^{k} \frac{1}{(\pm nm)^{r}} {S^{\pm}_{n,m,r}}+ \frac{1}{(\pm nm)^{k}} {J^{\pm}_{n,m,k}}.$$ Applying [Lemma \[lem:integralbounds\]]{} to ${J^{\pm}_{n,m,k}}$ (with ${\mathop{H^{\pm m}_{k}}\!} \rho(x)$ rewritten as ${\mathop{H^{m}_{k}}\!} \bigl[\rho(x)^{\pm 1}\bigr]$), there exists a $\delta>0$ such that for $n, m\ge 1$ and $k\ge 0$, ${J^{\pm}_{n,m,k}}= {\mathop{O_k}}(m^{k-\delta})$. Hence $${I^{\pm}_{n,m}}= -\sum_{r=1}^{k} \frac{{S^{\pm}_{n,m,r}}}{(\pm nm)^{r}} + {\mathop{O_k}}\bigl(n^{-k} m^{-\delta}\bigr).$$ By [Corollary \[cor:dibrunoBigO\]]{}, $S^{\pm}_{n,m,k} = {\mathop{O_k}}(m^{k-1})$, hence the $r=k$ term in the sum above may be absorbed into the error term. It follows that $$\sum_{m=1}^\infty \frac{1}{m}{I^{\pm}_{n,m}}= -\sum_{m=1}^\infty \frac{1}{m}\sum_{r=1}^{k-1} \frac{{S^{\pm}_{n,m,r}}}{(\pm nm)^r} \,+\, {\mathop{O_k}}(n^{-k}).$$ (Since ${I^{\pm}_{n,m}}= {J^{\pm}_{n,m,0}}= {\mathop{O_{}}}(m^{-\delta})$, the sums above converge.) Therefore, by , for all $n\ge 1$ and $k\ge 0$, $$\label{eq:almost} \Delta_{n}(P) = \sum_{r=1}^{k-1} \frac{c_r(n)}{n^r} + {\mathop{O_k}}(n^{-k}),$$ where $$c_r(n) {\mathrel{\mathop:}=}\frac{1}{2\pi}\sum_{m=1}^\infty \frac{1}{m^{r+1}} \sum_\pm (\pm 1)^r{\mathop{\mathrm{Im}}}\bigl({S^{\pm}_{n,m,r}}\bigr).$$ We now have a series expansion for $\Delta_{n}(P)$. The coefficients $c_r(n)$ are independent of $k$, as one would hope. To finish the proof of [Theorem \[thm:main\]]{}, we still need to rewrite the coefficients in the form given in and show that they are quasiperiodic. But first, we will prove that $c_{1}(n) = 0$. For all $n, m\ge 1$, $${S^{\pm}_{n,m,1}}= \pm\!\!\sum_{\rho\in {{R}_{\mathrm{non}}}}\sum_{\alpha\in \mathcal{E}_{x}} s\bigl(\alpha,\rho(\alpha)\bigr)\, \Bigl(\frac{\rho(\alpha)}{\alpha^{n}}\Bigr)^{\pm m} = \pm \!\!\!\sum_{(\alpha,\beta)\in\mathcal{E}} s(\alpha,\beta) (\beta/\alpha^{n})^{\pm m}.$$ For $(\alpha,\beta)\in \mathcal{E}$, $(\beta/\alpha^{n})^{-m}=\overline{(\beta/\alpha^{n})^m}$. Therefore, $$\sum_\pm \pm {S^{\pm}_{n,m,1}}= 2 \!\!\sum_{(\alpha,\beta)\in \mathcal{E}} s(\alpha,\beta) {\mathop{\mathrm{Re}}}\bigl((\beta/\alpha^{n})^{m}\bigr).$$ Since this expression is real, $\sum_\pm \pm {\mathop{\mathrm{Im}}}\bigl({S^{\pm}_{n,m,r}}\bigr) = {\mathop{\mathrm{Im}}}\bigl(\sum_\pm \pm {S^{\pm}_{n,m,r}}\bigr) =0.$ It follows that $c_{1}(n)=0$. Consequently, we can restrict to $r\ge 2$ in . Recall the functions ${\varphi}_{k,j,f}(x)$ described in [Lemma \[lem:dibrunocor\]]{}. By that lemma, $${S^{\pm}_{n,m,r}}= \pm\sum_{\rho\in {{R}_{\mathrm{non}}}} \sum_{\alpha\in \mathcal{E}_{x}} s\bigl(\alpha,\rho(\alpha)\bigr) \Bigl(\frac{\rho(\alpha)}{\alpha^{n}}\Bigr)^{\!\pm m} \, \sum_{j=0}^{r-1} \frac{{\varphi}_{r-1,j,\rho}(\alpha)}{\rho(\alpha)^{r-1}} (\pm m)^j.$$ Since $r\ge 2$, ${\varphi}_{r-1,0,\rho}=0$. Therefore, we may restrict to $j\ge 1$. By [Lemma \[lem:Psiproperties\]]{}, $${S^{\pm}_{n,m,r}}= \pm\!\!\!\sum_{(\alpha,\beta)\in \mathcal{E}} s(\alpha,\beta) (\beta/\alpha^{n})^{\pm m} \sum_{j=1}^{r-1} \Omega_{P,r,r-j+1}(\alpha,\beta) (\pm m)^{j}.$$ Therefore, $$\begin{split} c_r(n) &= \frac{1}{2\pi} \sum_{m=1}^\infty \frac{1}{m^{r+1}} \sum_\pm (\pm 1)^{r} {\mathop{\mathrm{Im}}}\bigl({S^{\pm}_{n,m,r}}\bigr) \\ &= \frac{1}{2\pi} \sum_{(\alpha,\beta)\in \mathcal{E}} s(\alpha,\beta) \sum_{j=1}^{r-1} {\mathop{\mathrm{Im}}}\biggl(\Omega_{P,r,r-j+1}(\alpha,\beta) \sum_\pm \sum_{m=1}^\infty \frac{(\beta/\alpha^{n})^{\pm m}}{(\pm m)^{r-j+1}} \biggr). \end{split}$$ Letting $a=r-j+1$, $$c_r(n) = \frac{1}{2\pi} \!\!\sum_{(\alpha,\beta)\in \mathcal{E}} \!\! s(\alpha,\beta) \sum_{a=2}^{r} {\mathop{\mathrm{Im}}}\Bigl(\Omega_{P,r,a}(\alpha,\beta) \sum_\pm (\pm 1)^{a} {\mathop{\mathrm{Li}_{a}}}\bigl((\beta/\alpha^{n})^{\pm 1}\bigr)\Bigr).$$ Recall from [Theorem \[thm:main\]]{} that we define ${\mathop{\mathfrak{R}_{a}}}(z)$ to be ${\mathop{\mathrm{Re}}}(z)$ for $a$ even and ${\mathop{\mathrm{Im}}}(z)$ for $a$ odd. Like before, $(\alpha,\beta)\in\mathcal{E}$ implies ${\mathop{\mathrm{Li}_{a}}}\bigl((\beta/\alpha^{n})^{-1}\bigr) = {\mathop{\mathrm{Li}_{a}}}\bigl(\overline{\beta/\alpha^{n}}\bigr) = \overline{{\mathop{\mathrm{Li}_{a}}}(\beta/\alpha^{n})}$. Therefore, for each $(\alpha,\beta)\in\mathcal{E}$, $$\begin{split} {\mathop{\mathrm{Im}}}\biggl(\Omega_{P,r,a}&(\alpha,\beta) \sum_\pm (\pm 1)^{a} {\mathop{\mathrm{Li}_{a}}}\bigl((\beta/\alpha^{n})^{\pm 1}\bigr) \biggr) \\ =&\, {\mathop{\mathrm{Im}}}\biggl(\Omega_{P,r,a}(\alpha,\beta) \,2{\mathop{\mathfrak{R}_{a}}}\bigl({\mathop{\mathrm{Li}_{a}}}(\beta/\alpha^{n}) \bigr) \begin{cases} 1 & \text{$a$ even} \\[-1.5mm] i & \text{$a$ odd} \end{cases} \biggr) \\[1mm] =&\, 2{\mathop{\mathfrak{R}_{a+1}}}\bigl(\Omega_{P,r,a}(\alpha,\beta)\bigr) {\mathop{\mathfrak{R}_{a}}}\bigl({\mathop{\mathrm{Li}_{a}}}(\beta/\alpha^{n})\bigr), \end{split}$$ which proves . For any given point $(\alpha,\beta)\in \mathcal{E}$, let $\theta$ be some value for $\arg(\alpha)$, so we can write $\beta/\alpha^{n} = \beta \exp(-i\theta n)$. Replacing $n$ with a real parameter $t$, this gives a periodic function of $t$. Since $a\ge 2$, ${\mathop{\mathrm{Li}_{a}}}(z)$ is continuous on ${\mathbb{T}}$. Hence ${\mathop{\mathfrak{R}_{a}}}\bigl({\mathop{\mathrm{Li}_{a}}}(\beta \exp(-i \theta t))\bigr)$ is a continuous, periodic function of $t$. (And if $\alpha$ is a $k$-th root of unity, the function has period dividing $k$.) It follows that $c_{r}(n)$ is the restriction to ${\ensuremath{\mathbb{N}}}$ of a quasiperiodic function. This concludes the proof of [Theorem \[thm:main\]]{}. Proof of [Corollary \[cor:main\]]{} {#sec:proofofcor} =================================== We recall here the formula given in [Corollary \[cor:main\]]{}. Let $\xi = \exp(2\pi i/3)$. Then $\Delta_{n}(1+x+y) {\overset{*}{\sim}}\sum_{r=2}^{\infty} c_{r}(n)/n^{r}$, for $$\label{eq:cororeminder} c_r(n) = \frac{2}{\pi} \sum_{\substack{a,b \\ a+b=r+1}} \!\!\!{\mathop{\mathfrak{R}_{a}}}\bigl({\mathop{\mathrm{Li}_{a}}}(\xi^{n+1})\bigr) \sum_{j=b}^{r-1} (-1)^{b} {\Bigl[\genfrac{}{}{0pt}{}{j}{b}\Bigr]}{\Bigl\{\genfrac{}{}{0pt}{}{r-1}{j}\Bigr\}} {\mathop{\mathfrak{R}_{a+1}}}(\xi^{j}).$$ For this $P$, we have a unique root function: $\rho(x)= -x-1$. Any point $(\alpha,\beta)\in Z(P)\cap {\mathbb{T}}^{2}$ must satisfy ${\lvert\alpha\rvert}={\lvert\alpha+1\rvert}=1$. It follows that $\mathcal{E} =\bigl\{(\xi, \xi^{-1}), (\xi^{-1},\xi)\bigr\}$. The associated signs are $s(\xi^{\mp}, \xi^{\pm}) = \pm 1$. $P$ satisfies the hypotheses of [Theorem \[thm:main\]]{}. Since our root function is so simple, it is easier to calculate $\Omega_{P,r,a}(x,y)$ with [Lemma \[lem:Psiproperties\]]{}, bypassing $\Psi_{r,a}$: $$\begin{split} \Omega_{P,r,a}(x,y) &= {\varphi}_{r-1,r-a+1}(x)/y^{r-1} \\ &=\Phi_{r-1,r-a+1}\bigl(\rho(x), x\rho'(x),x^{2}\rho''(x),\dotsc\bigr)/y^{r-1} \\ &= \Phi_{r-1,r-a+1}\bigl(-x-1, -x,0,0,\dotsc\bigr)/y^{r-1}. \end{split}$$ To calculate this, we need to evaluate the Bell polynomials $B_{j,i}(-x,0,\dotsc,0)$ for various $i,j$. It follows from the definitions that $B_{j,i}(-x,0,\dotsc,0) = 0$ unless $i=j$, and that $B_{j,j}(-x,0,\dotsc,0) = (-x)^{j}$. Therefore, letting $b=r-a+1$ for brevity, $$\Omega_{P,r,a}(x,y) = \frac{1}{y^{r-1}} \sum_{j} (-1)^{j-b} {\Bigl[\genfrac{}{}{0pt}{}{j}{b}\Bigr]} {\Bigl\{\genfrac{}{}{0pt}{}{r-1}{j}\Bigr\}} (-x-1)^{r-j-1}(-x)^{j}.$$ If we substitute a point $(\alpha,\beta)\in\mathcal{E} = \{(\xi^{\mp}, \xi^{\pm})\}$, then since $\beta= -\alpha-1 = 1/\alpha = \alpha^{2}$ for these points, $$\Omega_{P,r,a}(\alpha,\beta) = (-1)^{b} \sum_{j} {\Bigl[\genfrac{}{}{0pt}{}{j}{b}\Bigr]} {\Bigl\{\genfrac{}{}{0pt}{}{r-1}{j}\Bigr\}} \alpha^{-j}.$$ We now apply . Instead of summing over $(\alpha,\beta)\in\mathcal{E}$, we substitute $(\alpha,\beta) = (\xi^{-\sigma},\xi^{\sigma})$ and sum over $\sigma=\pm 1$. (This way, $s(\xi^{-\sigma},\xi^{\sigma}) = \sigma$.) $$\begin{split} c_r(n) &= \frac{1}{\pi}\sum_{\sigma=\pm 1} \sigma \sum_{a=2}^{r} {\mathop{\mathfrak{R}_{a+1}}}\bigl(\Omega_{P,r,a}(\xi^{-\sigma},\xi^{\sigma})\bigr) {\mathop{\mathfrak{R}_{a}}}\bigl({\mathop{\mathrm{Li}_{a}}}(\xi^{\sigma(n+1)})\bigr) \\ &= \frac{1}{\pi}\sum_{\sigma=\pm 1} \,\sigma \!\!\!\sum_{\substack{a+b = r+1,\\ 2\le a \le r}} \!\! {\mathop{\mathfrak{R}_{a}}}\bigl({\mathop{\mathrm{Li}_{a}}}(\xi^{\sigma(n+1)})\bigr) (-1)^{b} \sum_{j} {\Bigl[\genfrac{}{}{0pt}{}{j}{b}\Bigr]} {\Bigl\{\genfrac{}{}{0pt}{}{r-1}{j}\Bigr\}} {\mathop{\mathfrak{R}_{a+1}}}\bigl(\xi^{\sigma j}\bigr) \end{split}$$ The restriction that $2\le a \le r$ may be dropped, since the summands vanish if $a< 1$ or $a> r$. When $\sigma=-1$, the arguments to both ${\mathop{\mathfrak{R}_{a}}}$ and ${\mathop{\mathfrak{R}_{a+1}}}$ above become the conjugates of the values they take when $\sigma=+1$. A careful examination of signs then shows that the $\sigma=-1$ term is identical to the $\sigma=+1$ term. Therefore, we may drop all occurrences of $\sigma$ and double the expression, obtaining . The coefficients $c_{2}(n)$ and $c_{3}(n)$ {#sec:earlycoeffs} ========================================== The formulas for the coefficients in [Theorem \[thm:main\]]{} and [Corollary \[cor:main\]]{} are so complicated that it is difficult to see what they are actually saying, even in simple cases. In this section, we will see in more detail what the first two nontrivial coefficients look like, both for general $P$ and for $P=1+x+y$. For what follows, let $P_{i,j}$ denote $\frac{\partial^{i+j}}{\partial x^{i} \partial y^{j}} P$. \[prop:earlycoeffsgeneral\] Let $P(x,y)$, $\mathcal{E}$, and $s\colon \mathcal{E}\to\{-1,0,1\}$ be as in [Theorem \[thm:main\]]{}. Then for all $n$, $$c_2(n) = \frac{1}{\pi}\sum_{(\alpha,\beta)\in \mathcal{E}} s(\alpha,\beta) {\mathop{\mathrm{Im}}}(F_{P}) {\mathop{\mathrm{Re}}}\bigl({\mathop{\mathrm{Li}_{2}}}(\beta/\alpha^n)\bigr)$$ and $$c_3(n) = \frac{1}{\pi} \!\!\sum_{(\alpha,\beta)\in \mathcal{E}} \!\!s(\alpha,\beta)\Bigl[ {\mathop{\mathrm{Im}}}(F_{P}^{2}) {\mathop{\mathrm{Re}}}\Bigl({\mathop{\mathrm{Li}_{2}}}\Bigl(\frac{\beta}{\alpha^{n}}\Bigr)\Bigr) + {\mathop{\mathrm{Re}}}(G_{P}) {\mathop{\mathrm{Im}}}\Bigl({\mathop{\mathrm{Li}_{3}}}\Bigl(\frac{\beta}{\alpha^{n}}\Bigr)\Bigr) \Bigr],$$ where $$F_{P}(x,y) = -\frac{x P_{1,0}}{y P_{0,1}}$$ and $$G_{P}(x,y) = \Bigl(-\frac{P_{2,0}}{P_{0,1}} + \frac{2P_{1,0} P_{1,1}}{P_{0,1}^{2}} - \frac{P_{0,2} P_{1,0}^{2}}{P_{0,1}^{3}}\Bigr)\frac{x^{2}}{y} + F_{P} - F_{P}^{2}.$$ The formulas follows directly from [Theorem \[thm:main\]]{}, using the values of $\Psi_{2,2}$, $\Psi_{3,2}$, and $\Psi_{3,3}$ found in Table \[PsiTable\]. Things get much nicer when one specializes to a simple polynomial, like $1+x+y$, as then all but a few of the partials vanish. Let $\xi = \exp(2\pi i/3)$. For $P=1+x+y$, $$c_2(n) = -\frac{\sqrt{3}}{\pi} {\mathop{\mathrm{Re}}}\bigl({\mathop{\mathrm{Li}_{2}}}(\xi^{n+1})\bigr)$$ and $$c_3(n) = \frac{1}{\pi}\Bigl(2{\mathop{\mathrm{Im}}}\bigl({\mathop{\mathrm{Li}_{3}}}(\xi^{n+1})\bigr) - \sqrt{3} {\mathop{\mathrm{Re}}}\bigl({\mathop{\mathrm{Li}_{2}}}(\xi^{n+1})\bigr)\Bigr).$$ Before we prove this, note that ${\mathop{\mathrm{Li}_{2}}}(1) = \zeta(2) = \pi^{2}/6$. And because $${\mathop{\mathrm{Li}_{2}}}(1) + {\mathop{\mathrm{Li}_{2}}}(\xi) + {\mathop{\mathrm{Li}_{2}}}(\xi^{2}) = \sum_{n=1}^{\infty} \frac{(1^{n}+\xi^{n}+\xi^{2n})}{n^{2}} = \sum_{m=1}^{\infty} \frac{3}{(3m)^{2}} = \frac{\zeta(2)}{3},$$ it follows that ${\mathop{\mathrm{Re}}}\bigl({\mathop{\mathrm{Li}_{2}}}(\xi^{\pm 1})\bigr) -\pi^{2}/18$. Therefore, the formula for $c_{2}(n)$ above simplifies to the one given in Boyd’s [Proposition \[prop:boydasymp\]]{}. As in the proof of [Corollary \[cor:main\]]{}, the elements of $\mathcal{E}$ are $(\alpha,\beta) = (\xi^{-\sigma},\xi^{\sigma})$, with sign $\sigma$, for $\sigma=\pm 1$. For $P=1+x+y$, using the notation of [Proposition \[prop:earlycoeffsgeneral\]]{}, $F_{P} = -\xi^{\sigma}$. Therefore, ${\mathop{\mathrm{Im}}}(F_{P}) = {\mathop{\mathrm{Im}}}(F_{P}^{2}) = -\sigma\sqrt{3}/2$, and $G_{P} = -(\xi^{\sigma} + \xi^{-\sigma}) = 1$. The claims now follows easily from [Proposition \[prop:earlycoeffsgeneral\]]{}. Numerical Evidence {#sec:numerics} ================== Let $\sum_{r=2}^{\infty} c_{r}(n)/n^{r}$ be the a.p.p.s. expansion for $\Delta_{n}(1+x+y)$. We have calculated the coefficients $c_{r}(n)$ to high precision for $2\le r\le 200$ and $n\in\{0,1,2\}$ (which covers all $n$, since $c_{r}(n)$ depends only on $n {\allowbreak\mkern6mu({\operator@font mod}\,\,3)}$). The values with $r\le 10$ are shown in Table \[BoydCoeffTable\]. We have also calculated $\Delta_{n}(1+x+y)$ up to about $n=300$. [c r@[.]{}l r@[.]{}l r@[.]{}l]{} $r$ & & &\ 2 & $0$&3022998940 & $-0$&9068996821 & $0$&3022998940\ 3 & $-0$&1850879776 & $-0$&9068996821 & $0$&7896877657\ 4 & $-1$&3056972851 & $1$&1934321459 & $0$&1564663299\ 5 & $2$&6830358119 & $3$&2937639738 & $-5$&5860975105\ 6 & $24$&5796866908 & $-26$&3505959270 & $1$&4699140266\ 7 & $-79$&4351285864 & $-87$&7396475567 & $165$&1438848791\ 8 & $-953$&9471505649 & $971$&0686842267 & $-15$&3305415600\ 9 & $4176$&6470338729 & $4302$&0164745649 & $-8460$&9088755602\ 10 & $62932$&4226515189 & $-63281$&1309582098 & $331$&7360326211\ Confirmation of [Corollary \[cor:main\]]{} ------------------------------------------ Let $p_{k}(n){\mathrel{\mathop:}=}\sum_{r=2}^{k} c_{r}(n)/n^{r}$, the $k$-th partial sum of the expansion for $\Delta_{n}(1+x+y)$. By the definition of the expansion, $$\Delta_{n}(1+x+y) = p_{k-1}(n) + \frac{c_{k}(n)}{n^{k}} + {\mathop{O_k}}\Bigl(\frac{1}{n^{k+1}}\Bigr).$$ Therefore, for any fixed $n_{0}\in\{0,1,2\}$, $$\label{eq:approxformula} c_{k}(n_{0}) = \lim_{\substack{\\[-1 pt] n\to\infty, \\ n\equiv n_{0} {\allowbreak\mkern6mu({\operator@font mod}\,\,3)}}} n^{k}\bigl(\Delta_{n}(1+x+y) - p_{k-1}(n)\bigr).$$ With this formula, one may numerically verify each of the coefficients $c_{k}(n_{0})$ obtained from [Corollary \[cor:main\]]{}, assuming the accuracy of the coefficients with smaller $k$. Table \[BoydCoeffVerification\] demonstrates this verification for $n_{0}=1$ and $k=2, 3, 4$. Using $n\approx 300$ and $2\le k\le 30$, we found good agreement between the left and right sides of (with the relative error frequently $< 0.05$ and always $< 0.17$), with the exception of $n_{0}=0$ and $k$ even, when convergence was much slower. Using Richardson extrapolation dramatically improved convergence in all cases. Overall, the data gives strong evidence for the validity of [Corollary \[cor:main\]]{}. Numerical tests with other polynomials were also favorable. [l r@[.]{}l r@[.]{}l r@[.]{}l]{} $n$ & & &\ 1 & $0$&3700812333 & $0$&0677813393 & $0$&2528693169\ 61 & $0$&2989282502 & $-0$&2056702769 & $-1$&2555202582\ 121 & $0$&3006826863 & $-0$&1956821406 & $-1$&2818937239\ 181 & $0$&3012379282 & $-0$&1922158113 & $-1$&2901378960\ 241 & $0$&3015096124 & $-0$&1904578830 & $-1$&2941471883\ 301 & $0$&3016706736 & $-0$&1893953380 & $-1$&2965154617\ Approximation of $\Delta_{n}(1+x+y)$ with partial sums ------------------------------------------------------ Continuing to use $P=1+x+y$, numerics strongly suggest that for each $n$, the power series $\sum_{r=2}^{\infty} c_{r}(n) x^{r}$ has radius of convergence 0, hence $\sum_{r=2}^{\infty} c_{r}(n)/n^{r}$ is divergent. However, as is common with asymptotic expansions, partial sums give good approximations for $\Delta_{n}(1+x+y)$ up to a point, before going wildly off the mark. Let $p_{k}(n){\mathrel{\mathop:}=}\sum_{r=2}^{k} c_{r}(n)/n^{r}$, the $k$-th partial sum, and let $K(n)$ be the value of $k$ for which $p_{k}(n)$ gives the best approximation of $\Delta_{n}(1+x+y)$. The numerical evidence suggests that as $n\to\infty$, $K(n)$ is asymptotic to $n$, and that the relative error $\bigl(p_{n}(n)-\Delta_{n}\bigr)/\Delta_{n} = {\mathop{O_{}}}\bigl(\exp(-n)\bigr)$. For example, for $n=100$, the best approximation of $\Delta_{100}(1+x+y)$ is given by $p_{107}(100)$, which is correct to 46 decimal places—a relative error of $1.5\times 10^{-43}$. A polynomial not meeting the hypothesis {#subsec:deninger} --------------------------------------- Our proof of [Theorem \[thm:main\]]{} does not make it clear what to expect for polynomials $P$ for which $P$ and ${{\partial P/\partial y}}$ do have a common zero on ${\mathbb{T}}^{2}$. We chose such a polynomial—namely, $P = 1+ x + 1/x + y + 1/y$ (famous for Deninger’s conjecture [@deninger], recently proved by Rogers and Zudilin[@rogerszudilin], that its Mahler measure is a value of an elliptic curve $L$-function), for which $P$ and ${{\partial P/\partial y}}$ both vanish at $\bigl(\exp(\pm i \pi/3),-1\bigr)$. We calculated $\Delta_{n}(P)$ for $1\le n\le 1600$. Based on the numerical evidence, $\Delta_{n}(P)$ almost certainly has a main term of the form $c(n)/n^{3/2}$, where $c(n)$ depends only on $n{\allowbreak\mkern6mu({\operator@font mod}\,\,6)}$. This is consistent with the sort of bounds given by Boyd in [@boyd2]. In fact, it appears that the difference $\Delta_{n}(P) - c(n)/n^{3/2}$ vanishes more quickly than $1/n^{r}$ for all $r>0$—that is, $\Delta_{n}(P) {\overset{*}{\sim}}c(n)/n^{3/2}$—but the evidence is less clear for this. Algebraic determination of the signs $s(x,y)$ {#sec:signdet} ============================================= As mentioned earlier, for a point $(\alpha,\beta)$ in $\mathcal{E}$, its sign $s(\alpha,\beta)$ can be determined algebraically, without reference to the root function $\rho(x)$ such that $\beta=\rho(\alpha)$. Let $f(t){\mathrel{\mathop:}=}\operatorname{Log}\bigl(\rho(\alpha \exp(it))/\beta\bigr)$, where $\operatorname{Log}$ is the principal branch of the logarithm. The function ${\mathop{\mathrm{Re}}}\bigl(f(t)\bigr) = \log\bigl|\rho(\alpha \exp(it))\bigr|$ experiences the same sign change at $0$ (with $t$ moving left to right in ${\ensuremath{\mathbb{R}}}$) as ${\lvert\rho(x)\rvert}-1$ does at $\alpha$ (with $x$ moving counterclockwise along ${\mathbb{T}}$). We can determine arbitrarily many of the coefficients in the Maclaurin series for $f(t)$ with the use of [Lemma \[lem:implicitderivs\]]{}. The first of these coefficients whose real part is nonzero will determine the behavior of ${\mathop{\mathrm{Re}}}\bigl(f(t)\bigr)$ at $t=0$, and hence determine $s(\alpha,\beta)$. More specifically, say $f(t) = \sum_{k=1}^{\infty} b_{k} t^{k}$. ($b_{0}=f(0)=0$.) Let $N$ be the smallest positive integer such that ${\mathop{\mathrm{Re}}}(b_{N})\ne 0$. Then $$s(\alpha, \beta) = \begin{cases} 0, & \text{if $N$ even,} \\ \operatorname{sign}({\mathop{\mathrm{Re}}}(b_{N})), & \text{if $N$ odd.} \end{cases}$$ In particular, $b_{1} = i \rho'(\alpha) = -i P_{x}/P_{y}$ (with $P_{x} = \partial P/\partial x$ and $P_{y} = \partial P/\partial y$ evaluated at $(\alpha,\beta)$). Generically, $b_{1}$ will be nonzero. Therefore, it will usually suffice to use the rule of thumb that $s(\alpha, \beta) = \operatorname{sign}({\mathop{\mathrm{Im}}}(P_{x}/P_{y}))$, as long as the right side is nonzero. Incidentally, for $k\ge 1$, it appears that $$b_{k} \stackrel{?}{=} \frac{i^{k}}{k!}\sum_{j=1}^{k} {\Bigl\{\genfrac{}{}{0pt}{}{k}{j}\Bigr\}} \rho^{(j)}(\alpha) \beta^{j-1},$$ although we have not verified this identity. Conclusions =========== It is unclear to what extent it is possible to remove the hypothesis on $P$ and ${{\partial P/\partial y}}$ in [Theorem \[thm:main\]]{}. If $Z(P)$ and $Z({{\partial P/\partial y}})$ have a common solution in ${\mathbb{T}}\times{\ensuremath{\mathbb{C}}}$, then a root function will have an algebraic singularity on ${\mathbb{T}}$. Our current approach would require us to integrate up to that singularity, but this causes havoc with our use of [Lemma \[lem:parts\]]{}. We devoted extensive effort to circumventing this difficulty, for instance with Puiseux expansions and the method of stationary phase [@erdelyi Section 2.9], but those attempts have not been fruitful. The evidence from section \[subsec:deninger\] (including the fractional exponents, echoing those found in Puiseux expansions) might lead one to expect a.p.p.s.’s in powers of $n^{-1/M}$ for some positive integer $M$. However, if similarly complicated formulas hold for the coefficients in such expansions, one would expect to have either all coefficients equal to 0 or infinitely many of them nonzero. This is at odds with the evidence from the example in \[subsec:deninger\]. This discrepency leads us to doubt that [Theorem \[thm:main\]]{} can be extended to all nonzero polynomials $P(x,y)$ without a significant modification of its statement. Acknowledgements ================ I would like to thank Fernando Rodriguez Villegas for first suggesting this problem to me and for valuable suggestions. I am also grateful to Rob Benedetto for providing extensive advice on an earlier draft of this paper, and to David Cox for helpful conversations. I performed a great deal of computer experimentation in the course of this research. I made extensive use of Sage [@sage471] and PARI/GP [@pari243] (both within Sage, and on its own). I also used Mathematica [@mathematica7] for earlier work. [^1]: If $f(z)$ is a rational function, it can be shown that $\nu$ is at most twice the maximum of the degrees of the numerator and denominator of $f$.

--- abstract: 'In this paper, we consider wireless sensor networks (WSNs) with sensor nodes exhibiting clustering in their deployment. We model the coverage region of such WSNs by Boolean Poisson cluster models (BPCM) where sensors nodes’ location is according to a Poisson cluster process (PCP) and each sensor has an independent sensing range around it. We consider two variants of PCP, in particular [Matérn ]{}and Thomas cluster process to form Boolean [Matérn ]{}and Thomas cluster models. We first derive the capacity functional of these models. Using the derived expressions, we compute the sensing probability of an event and compare it with sensing probability of a WSN modeled by a Boolean Poisson model where sensors are deployed according to a Poisson point process. We also derive the power required for each cluster to collect data from all of its sensors for the three considered WSNs. We show that a BPCM WSN has less power requirement in comparison to the Boolean Poisson WSN, but it suffers from lower coverage, leading to a trade-off between per-cluster power requirement and the sensing performance. A cluster process with desired clustering may provide better coverage while maintaining low power requirements.' author: - '[^1]' title: 'On the Coverage Performance of Boolean-Poisson Cluster Models for Wireless Sensor Networks' --- Introduction ============ In WSNs, sensors are deployed over a region such as forest or wetlands, to form a wireless network and exchange mutual data to sense an event. WSN may have a central hub to facilitate the joint detection. There are two essential aspects of WSNs. The first aspect is the coverage aspect to maximize the region covered by sensors, termed the coverage or sensing region. This will ensure that at least one sensor can detect the target event with a certain probability. The second aspect is to minimize energy consumption as wireless sensors have a limited power budget. Sensors can form small clusters with each cluster having one head, which acts as a gateway to the central hub [@iyengar2016distributed]. In this hierarchical network, sensors transmit their sensing data to their local cluster heads, which then communicates it to the central hub to jointly make sensing decision. Such clustering can reduce the power requirement of nodes, but can degrade overall coverage. The deployment of sensors in a WSN is generally random. Hence, the tools of stochastic geometry can be applied to model and analyze WSNs. One popular process to model the coverage area of a WSN is the Boolean-Poisson process. The Boolean-Poisson process is defined as the union of independent random objects with their centers located according to a Poisson point process (PPP) [@haenggibook; @liu2004study; @baek2007spatial]. The random objects denote the individual coverage region of sensors while the centers denote sensors’ locations. Owing to the mathematical tractability of PPPs, Boolean-Poisson process is simple yet powerful to derive performance metrics of WSNs such as the probability that a location is not covered, and the expected area of uncovered region [@BaccelliBook; @chiu2013stochastic; @flint2017wireless]. The capacity function of the Boolean-Poisson process, which characterizes the sensing probability of an event, was studied in [@pandey2018modeling]. As the underlying process of the Boolean Poisson model is PPP, the location of sensors nodes is independent of each other in this model. In some scenarios, the deployment of sensors is not entirely independent and the sensors may exhibit clustering in their deployment. This may be due to the easiness in deploying sensors in small groups or to facilitate the communication between the sensor and its gateway by decreasing their mutual distance. The Poisson cluster process (PCP) can be used to model the locations of sensors in such scenarios [@mekikis2018connectivity]. The two important variants of PCP are the [Matérn ]{}cluster process (MCP) and Thomas cluster process (TCP). The characterization of contact and nearest neighbor distance distribution for these processes is presented in [@pandey2019contact; @afshang2017nearesttcp]. To model the coverage region of WSNs exhibiting such clustering, we propose to use Boolean Poisson cluster models (BPCM) where the underlying process to model sensors’ locations is a PCP and each sensor has an independent sensing region around it. There has been limited work to characterize BPCM [*e.g.*]{} [@last1999empty]. However, coverage and sensing performance of WSNs that are deployed according to BPCM has not been studied in detail.\ In this paper, we consider three WSNs that are deployed according to MCP, TCP, and PPP, respectively. The coverage area of these WSNs can be modeled using Boolean MC, Boolean TC, and Boolean Poisson models (or processes). We first derive the capacity functional of Boolean MC and TC models. Using these expressions, we then compute the sensing probability of an event with a compact spread area. We also provide simple bounds for Boolean MC model to help derive insights for the system. We also derive the power required for each cluster to collect data from all of its sensors for the three considered WSNs. Finally, we perform a comparative analysis of these three deployments. We show that clustering decreases the coverage area and sensing probability, especially in the case of sensors with large individual sensing regions. However, it also reduces the power requirement of sensors. In scenarios where sensors have limited power, clustered deployments can provide better coverage performance. System Model ============ In this paper, we consider a wireless sensor network deployed over $\mathbb{R}^2$ space. The locations of the sensors are modeled by a point process $\Phi$ with density $\lambda$. Each sensor has a sensing range around it denoted by ${\mathsf{S}}_i$ and assumed to be independent of other sensors. The total covered region (the region which falls inside the sensing region of at least one sensor) is given as $$\begin{aligned} \Psi= \bigcup_{{\mathbf{z}}_i \in \Phi} {\mathbf{z}}_i+{\mathsf{S}}_i,\end{aligned}$$ which is known as a Boolean Process/Model and is a special case of Germ-grain model. Each point is termed as a germ with its sensing region as its grain. Sensor network -------------- We assume that sensors follow clusterization where the network is made from many cluster heads with each cluster head responsible to control and communicate with sensors assigned to it. Such network can be modeled using a cluster process. A cluster process consists of daughter point process centered at their parents whose locations are also according to a point process. Let $\Phi_{{\mathrm{p}}}=\{{\mathbf{x}}_i: \forall i \in \mathbb{N}\}$, be a parent point process where ${\mathbf{x}}_i$ is the location of $i$-th parent point (models the location of cluster center or cluster head in WSN) in $\mathbb{R}^2$. For each point ${\mathbf{x}}_i$, there is an associated daughter point process $\Phi^{(i)}_{{\mathrm{d}}}=\{{\mathbf{y}}_j^{(i)}:\forall j\in \mathbb{N}\}$, where ${\mathbf{y}}_j^{(i)}$ is the location of $j$-th daughter point. The absolute location of these points are given as ${\mathbf{z}}_{ij}={\mathbf{x}}_i+{\mathbf{y}}_j^{(i)}.$ Each daughter point process is independent and identically distributed. Now the PP modeling the sensors’ location is the union of all these daughter points $$\begin{aligned} \Phi&=\bigcup_{{\mathbf{x}}_i\in \Phi_{{\mathrm{p}}}}\{{\mathbf{x}}_i+\Phi_{{\mathrm{d}}}^{(i)}\},\\ &=\left\{{\mathbf{z}}_{ij}:{\mathbf{z}}_{ij}={\mathbf{x}}_i+{\mathbf{y}}_j^{(i)},{\mathbf{x}}_i\in\Phi_{\mathrm{p}},{\mathbf{y}}_j^{(i)}\in\Phi_{{\mathrm{d}}}^{(i)} \,\forall i,j \right\},\end{aligned}$$ and known as the cluster process. It is clear from the above discussion that the cluster head is the parent point to all sensors of the cluster. It is intuitive to keep the cluster head at the center of the cluster and as close as possible to the sensors in the cluster to minimize the energy required in communication. We now consider three PPs to model the locations of WSN: ### [Matérn ]{}cluster process In MCP, $\Phi_{{\mathrm{p}}}$ is a homogeneous PPP with intensity $\lambda_{\mathrm{p}}$. Each $\Phi_{{\mathrm{d}}}^{(i)}$ is a finite PPP within a ball ${\mathcal{B}}({\mathrm{o}},r_{\mathrm{d}})$. The mean number of points in each daughter point process is $m$ and therefore the intensity $\lambda_{\mathrm{d}}({\mathbf{y}})$, of each daughter point process will be $\frac{m}{\pi r_{\mathrm{d}}^2}{\mathbbm{1}}(||{\mathbf{y}}||\leq r_{\mathrm{d}})$. Total density of the PP is ${\lambda_{\mathrm{M}}}={\lambda_{\mathrm{p}}}m$. ### Thomas cluster process In TCP, $\Phi_{{\mathrm{p}}}$ is a homogeneous PPP with intensity $\lambda_{\mathrm{p}}$. Each $\Phi_{{\mathrm{d}}}^{(i)}$ is a non-uniform PPP with the intensity $$\begin{aligned} \lambda_{{\mathrm{d}}}({\mathbf{y}})=\frac{m}{2\pi\sigma^2}\exp\left(-\frac{y^2}{2\sigma^2}\right),\end{aligned}$$ where $m$ is the mean number of points in each daughter point process. Total density of the PP is ${\lambda_{\mathrm{T}}}={\lambda_{\mathrm{p}}}m$. **Symbol** **Definition** ----------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------- ${\Ball}({\mathbf{x}},r)$ Ball of radius $r$ centred at location ${\mathbf{x}}$. $\Phi$ 2D process such as MCP or TCP modeling location of sensors. ${\lambda_{{\mathrm{P}}}}$ Intensity of PPP. $\lambda_{\mathrm{p}}$ Intensity of parent point process. $\lambda_{\mathrm{d}}$ Intensity of daughter point process in a cluster process. $r_{\mathsf{K}}$ The size of an event ${\mathsf{K}}$, which may grow with time. $R$ The fixed sensing radius associated with each sensor. $\mathsf{S}_j^{(i)}$ Compact disk of radius $R$ of $j$-th point associted with $i$-th parent and models the sensing zone of sensor. $\Psi$ Occupied region in $\mathbb{R}^2$ by all the points. Represents the area falling under the sensing zone of all sensors. $\oplus$ Minkowski addition. $y=\|{\mathbf{y}}\|$ $L$-2 norm of ${\mathbf{y}}$. $\Phi_{{\mathrm{d}}}^{(i)}$ Daughter point process coressponding to $i$-th parent. $ \areaballintersect{}{x}{r_{\mathrm{d}}}{r}$ Intersecting area between ${\Ball}({\mathrm{o}},r_{\mathrm{d}})$ and ${\Ball}({\mathbf{x}},R)$: $|{\Ball}({\mathrm{o}},r_{\mathrm{d}})\cap{\Ball}({\mathbf{x}},R)|$. : Notation Table[]{data-label="tab:title"} ### Poisson point process In this, sensors are located according to a PPP with intensity ${\lambda_{{\mathrm{P}}}}=\lambda_{\mathrm{p}}m$. For the sake of consistency, another independent PPP with intensity $\lambda_{\mathrm{p}}$ defines the location of cluster heads. Sensors form clusters by selecting the closest cluster head. [.3]{} {width="100.00000%"} [.3]{} {width="100.00000%"} [.3]{} {width="100.00000%"} [.3]{} {width="100.00000%"} [.3]{} {width="100.00000%"} [.3]{} {width="100.00000%"} Now, depending on the deployment of sensors, we consider the following processes to model the total coverage region which is the union of all sensing regions. In each model, sensors have their individual sensing region ${\mathsf{S}}_i$ as ${\mathcal{B}}({\mathrm{o}},R)$ around it independent of other sensors. Here $R$ is the fixed sensing range of each sensor. 1. **Boolean MC process:** The sensors’ locations follow MCP. 2. **Boolean TC process:** The sensors’ locations follow as TCP. 3. **Boolean P process:** The sensors’ location are modeled as PPP. Results for this case are known, however this case is considered for comparison. The important symbols and notations used in the paper are shown in TABLE \[tab:title\]. For simplicity, the same notations are being used to represent similar parameters of cluster processes whenever it is clear from the context. For e.g. $m$ denote the mean number of points in a daughter point process (PP) of TCP as well as in MCP. Sensing Performance ------------------- Sensing performance of a WSN can be measured in terms of Capacity functional. The capacity functional $T_\Psi(\mathsf{K})$ of Boolean process for a compact set $\mathsf{K}$ is the probability that the the set ${\mathsf{K}}$ and $\Psi$ are not disjoint $$\begin{aligned} T_\Psi({\mathsf{K}})=\prob{\Psi\cap {\mathsf{K}}\neq \phi}.\end{aligned}$$ If $\set{K}$ denote any event’s impact area, then $T_\Psi(\mathsf{K})$ denote the coverage/sensing probability probability that the event is sensed by the sensor network. In particular, we will consider $\set{K}$ as $\Ball({\mathrm{o}},r_\set{K})$ in $\mathbb{R}^2$ and denote the capacity functional for this set by ${\mathcal{M}_{}(r_\set{K})}$. Here $r_\set{K}$ denote the size of the event. Since the network is stationary, we have taken the center at the origin ${\mathrm{o}}$. For dynamic events, $\set{K}$ can grow in size with time [@pandey2018modeling]. The capacity functional $T_{\Psi_{{\mathrm{P}}}}({\mathsf{K}})$ for a Boolean P process with intensity ${\lambda_{{\mathrm{P}}}}$ and sensing range $\Ball({\mathrm{o}},R)$ is given as [@chiu2013stochastic] $$\begin{aligned} T_{\Psi_{\mathrm{P}}}({\mathsf{K}})& =1-\expS{- {\lambda_{{\mathrm{P}}}} \left|{\Ball}({\mathrm{o}},R)\oplus\set{K}\right|}.\end{aligned}$$ For circular set $\set{K}=\Ball(0,r_\set{K})$, $T_{\Psi_{\mathrm{P}}}(\Ball(0,r_\set{K}))$ is $$\begin{aligned} {\mathcal{M}_{\Psi_{\mathrm{P}}}(r_\set{K})}&=1-\expS{-m{\lambda_{\mathrm{p}}}\pi(R+r_{\mathsf{K}})^2}.\end{aligned}$$ The probability that an arbitrary point $\set{K=\{{\mathrm{o}}\}}$ is covered is given by: $$\begin{aligned} T_{\Psi_{{\mathrm{P}}}}(\{{\mathrm{o}}\})&=1-e^{(-m{\lambda_{\mathrm{p}}}\pi R^2)}.\end{aligned}$$ Power Requirement ----------------- Power requirement $\reqEnergy{}$ of a system is defined as the sum power required by sensors in a unit area $\mathsf{u}$ to be able to communicate to their cluster head. Assuming a powerlaw path loss with path loss exponent $\alpha$ and the SNR threshold $\SThres$ required for successful communication, the power requirement is given as $$\begin{aligned} \reqEnergy{}&=\expect{\sum_{{\mathbf{z}}_{ij}\in\Phi\cap \set{u}}\tau \|{\mathbf{z}}_{ij}-c_{ij}\|^{\alpha}},\end{aligned}$$ where $c_{ij}={\mathbf{x}}_i$ denotes the cluster head of the sensor ${\mathbf{z}}_{ij}$. Sensing Performance =================== In this section, we will derive the sensing performance of the WSN for the two considered models and provide closed form upper and lower bounds for the same. Boolean MC Process ------------------ \[thm1\] The capacity functional for the Boolean MC Process is given as (See Appendix \[appn A\] for the proof): $$\begin{aligned} \label{MCPCAP1} & T_{\Psi_{\mathrm{M}}}({\mathsf{K}})=1-\nonumber\exp\left(-{\lambda_{\mathrm{p}}}\times \vphantom{\frac{n}{d}}\right. \\ &\left. \int_{\mathbb{R}^2} \left(1-\expS{ -\lambda_{\mathrm{d}}\left| {\Ball}({\mathrm{o}},r_{\mathrm{d}})\cap\left({\Ball}({\mathbf{x}},R) \oplus \set{K}\right) \right| }\right) {\mathrm{d}}{\mathbf{x}}\right).\end{aligned}$$ For circular $\set{K}\equiv{\mathcal{B}}({\mathrm{o}},r_\set{K})$, the Minkowski sum is ${\mathcal{B}}({\mathrm{o}},r_{\mathsf{K}})\oplus{\mathcal{B}}({\mathbf{x}},R)\equiv{\mathcal{B}}({\mathbf{x}},R+r_{\mathsf{K}})$. Hence the expression for the capacity functional is: $$\begin{aligned} \label{MCPcap} &{\mathcal{M}_{{\mathrm{M}}}(r_\set{K})}=1-e^{\left(-2\pi\lambda_{\mathrm{p}}\int_{0}^{r_{\mathrm{d}}+R+r_\set{K}} \left( 1-\expU{-\lambda_{\mathrm{d}}\areaballintersect{{\mathrm{o}}}{x}{r_{\mathrm{d}}}{R+r_\set{K}} }\right) x{\mathrm{d}}x\right)}. \end{aligned}$$ \[thm3\] The upper and the lower bound for ${\mathcal{M}_{{\mathrm{M}}}(r_\set{K})}$ is given as (See Appendix \[apndxb\] for the proof.) $$\begin{aligned} &{\UCFB{{\mathrm{M}}}} =1-\exp(-\pi\lambda_{\mathrm{p}}{\mathbf{A}(r)}) \exp\left(\left( { \frac{\pi\lambda_{\mathrm{p}}}{2\lambda_{\mathrm{d}}^2\beta^2(r)} } \left[ -1 \right.\right.\right.\\ &\left.\left.\left. + 2\lambda_{\mathrm{d}}\beta(r)+e^{- 4\lambda_{\mathrm{d}}\beta^2(r) }\left( {r+r_{{\mathrm{d}}}}{} +|r-r_{{\mathrm{d}}}| e^{- 4\lambda_{\mathrm{d}}\beta^2(r) } \right) \right]\right)\right),\\ &{\LCFB{{\mathrm{M}}}}= 1- \exp\left(- \pi\lambda_{\mathrm{p}}{\mathbf{A}(r)} \right) \exp\left( \frac{4\lambda_{\mathrm{p}}}{{\lambda_{\mathrm{d}}}}\left[-2 \vphantom{\left(\sqrt{a_{n_0}}\right)} \right.\right.\\ &\left.\left.+2e^{-\lambda_{\mathrm{d}}\pi \beta^2(r) }-{\pi}\sqrt{{\lambda_{\mathrm{d}}}}(r+r_{{\mathrm{d}}}) \operatorname{erf}\left( -\sqrt{ \lambda_{\mathrm{d}}\pi } \beta(r) \right) \right] \right),\end{aligned}$$ where $r=R+r_{\mathsf{K}}$, $\beta(r)=\min(r,r_{\mathrm{d}})$, ${\mathbf{A}(r)} =(r-r_{\mathrm{d}})^2(1-\exp(-\lambda_{\mathrm{d}}\pi\beta^2(r)))+4rr_{\mathrm{d}}$. \[thm4\] An another set of bounds for ${\mathcal{M}_{{\mathrm{M}}}(r_\set{K})}$ is $$\begin{aligned} \UUCFB{{\mathrm{M}}}&=1-\exp\left(-\pi\lambda_{\mathrm{p}}(r_{\mathrm{d}}+r)^2\right. \left.\left(1-e^{-\lambda_{\mathrm{d}}\beta^2(r)}\right)\right), \\ \LLCFB{{\mathrm{M}}}&=1-\exp\left(-\pi\lambda_{\mathrm{p}}(r_{\mathrm{d}}-r)^2\right.\left.\left(1-e^{-\lambda_{\mathrm{d}}\beta^2(r)}\right)\right), \end{aligned}$$ which are simpler but less tight than the ones in Theorem \[thm3\]. The upper bound is derived by substituting the intersecting area $\areaballintersect{}{x}{r_{\mathrm{d}}}{r}$ in with its upper bound $\min(\pi r^2,\pi r_{\mathrm{d}}^2)$. For lower bound, we note that from limit $x=0$ to $x=|r-r_{\mathrm{d}}|$ $\areaballintersect{}{x}{r_{\mathrm{d}}}{r}$ is $\min(\pi r^2,\pi r_{\mathrm{d}}^2)$ and for $x=|r-r_{\mathrm{d}}|$ to $x=r+r_{\mathrm{d}}$, it can be lower bounded by $0$. Substituting these bounds in , we get the lower bound. ### Asymptotic behavior of $T_{\Psi _{\mathrm{M}}}$ with $r_{\mathrm{d}}$ while keeping ${\lambda_{{\mathrm{M}}}}$ fixed. By increasing the $r_{\mathrm{d}}$ while keeping ${\lambda_{{\mathrm{M}}}}$ fixed, we can decrease the clustering of points and spread points more in the space. Hence, the asymptotic behavior of $T_{\Psi _{\mathrm{M}}}$ helps us in understanding the impact of clustering (or mutual-attraction of points) on the sensing performance. **When $r_{\mathrm{d}}\rightarrow 0$:** Taking the limit in the Theorem \[thm4\], we get the lower bound $\LLCFB{{\mathrm{M}}}$ $$\begin{aligned} &=1-\lim_{r_{\mathrm{d}}\rightarrow 0}\exp\left(-\pi\lambda_{\mathrm{p}}(r_{\mathrm{d}}-r)^2\left(1-e^{-\lambda_{\mathrm{d}}\min(\pi r_{\mathrm{d}}^2,\pi r^2)}\right)\right),\\ &=1-\exp\left(-\pi\lambda_{\mathrm{p}}r^2\left(1-e^{-m}\right)\right).\end{aligned}$$ Similarly, taking limit of the upper bound in Theorem \[thm3\], we get $\UCFB{{\mathrm{M}}}$ $$\begin{aligned} &=1-\lim_{r_{\mathrm{d}}\rightarrow 0}\exp\left(-\pi\lambda_{\mathrm{p}}(r_{\mathrm{d}}+r)^2\left(1-e^{-\lambda_{\mathrm{d}}\min(\pi r_{\mathrm{d}}^2,\pi r^2)}\right)\right),\\ &=1-\exp\left(-\pi\lambda_{\mathrm{p}}r^2\left(1-e^{-m}\right)\right).\end{aligned}$$ Since, both upper and lower bounds converge to the same function, the capacity functional converges to the same function. As $r_{\mathrm{d}}\rightarrow 0$, ${\mathcal{M}_{{\mathrm{M}}}(r_\set{K})}$ converges to ${\mathcal{M}_{{\mathrm{P}}}(r_\set{K})}$ with intensity $\lambda_{{\mathrm{p}}}(1-e^{-m})$. **When $r_{\mathrm{d}}\rightarrow\infty$:** [Using the expressions from Theorem \[thm3\], we see that both the upper and lower bound converge to $1-\exp(-m\pi\lambda_{{\mathrm{p}}}r^2)$. Note that this is equal to the capacity functional of a Boolean P process with intensity $m\lambda_{{\mathrm{p}}}$.]{} The required power for Boolean MC process is given as: $$\begin{aligned} \reqEnergy{{\mathrm{M}}}&=\expect{\sum_{{\mathbf{z}}_{ij}\in\Phi\cap \set{u}}\tau \|{\mathbf{z}}_{ij}-c_{ij}\|^{\alpha}},\\ &={\lambda_{{\mathrm{M}}}}\int_{\set{u}} \tau \palmexpectx{\|{\mathbf{z}}-c_{{\mathbf{z}}}\|^{\alpha}}{{\mathbf{z}}}\dd{\mathbf{z}}={\lambda_{{\mathrm{M}}}} \tau \palmexpect{\|y\|^{\alpha}},\\ &=m{\lambda_{\mathrm{p}}}\tau \frac1{\alpha/2+1} r_{\mathrm{d}}^\alpha.\end{aligned}$$ Boolean TC Process ------------------ \[thm5\] The capacity functional for a Boolean TC process is given by ${\mathcal{M}_{{\mathrm{T}}}(r_\set{K})}$ (For proof see Appendix \[prf:thm5\] ) $$\begin{aligned} \label{TCPcapacity1} &=1-\exp\left(-2\pi\lambda_{\mathrm{p}}\int_{x=0}^{\infty}\left(1-\vphantom{\frac24}\right.\right.\nonumber\\ &\left.\left.\exp\left(-\frac{m}{2\pi\sigma^2}\int_{\theta=0}^{2\pi}\int_{t=0}^{r}e^{-\frac{x^2+t^2+2xt\cos\theta}{2\sigma^2}}t{\mathrm{d}}t{\mathrm{d}}\theta\right)\right)x{\mathrm{d}}x\right), \end{aligned}$$ where $r=R+r_{\mathsf{K}}$. The required power for Boolean TC process is given as $$\begin{aligned} \reqEnergy{{\mathrm{T}}}&=\expect{\sum_{{\mathbf{z}}_{ij}\in\Phi\cap \set{u}}\tau \|{\mathbf{z}}_{ij}-c_{ij}\|^{\alpha}}={\lambda_{{\mathrm{T}}}}\int_{\set{u}} \tau \palmexpectx{\|{\mathbf{z}}-c_{{\mathbf{z}}}\|^{\alpha}}{{\mathbf{z}}}\dd{\mathbf{z}}, \\ &={\lambda_{{\mathrm{T}}}} \tau \palmexpect{\|y\|^{\alpha}}=m{\lambda_{\mathrm{p}}}\tau \Gamma(\alpha/2+1) (2\sigma^2)^{\alpha/2}.\end{aligned}$$ Boolean Poisson Process ----------------------- In this case, the required power is given as $$\begin{aligned} \reqEnergy{}&=\expect{\sum_{{\mathbf{z}}_{ij}\in\Phi\cap \set{u}}\tau \|{\mathbf{z}}_{ij}-c_{ij}\|^{\alpha}},\\ &={\lambda_{{\mathrm{P}}}}\int_{\set{u}} \tau \palmexpect{\|c_{0}\|^{\alpha}}\dd{\mathbf{z}}={\lambda_{{\mathrm{P}}}} \tau \palmexpect{\|c_{0}\|^{\alpha}}.\end{aligned}$$ Now, $c_0$ is the closest cluster head (which is modeled as PPP with intensity ${\lambda_{\mathrm{p}}}$), hence the distribution of its distance from the origin is given as $$f_{\|c_0\|}(c)=2\pi{\lambda_{\mathrm{p}}}{} c\expS{-\pi{\lambda_{\mathrm{p}}}{} c^2}.$$ Hence, $$\begin{aligned} \reqEnergy{}&= {\lambda_{{\mathrm{P}}}} 2\pi{\lambda_{\mathrm{p}}}\tau \int c^{\alpha} c\expS{-\pi{\lambda_{\mathrm{p}}}c^2}\dd c, \\ &=m{\lambda_{\mathrm{p}}}{(\pi{\lambda_{\mathrm{p}}})}^{-\alpha/2}\SThres \Gamma(\alpha/2+1).\end{aligned}$$ Coverage Analysis with Sensor Power Constraints =============================================== Let the required per-unit area power to be the $\mathcal{E}_{\mathrm{net}}$ which is kept constant for all three deployments. Now, for the provided $\mathcal{E}_{\mathrm{net}}$, we will derive the parameter specifications for the three models to be able to compare their coverage performance. Boolean MC process ------------------ In case of Boolean MC process, the cluster radius should be equal to $$\begin{aligned} r_{\mathrm{d}}&=\left[{(\mathcal{E}_{\mathrm{net}}(1+.5\alpha))}/{m\lambda_{\mathrm{p}}\tau}\right]^{\frac{1}{\alpha}}.\end{aligned}$$ Boolean TC process ------------------ The variance parameter $\sigma$ to achieve required per-unit area power $\mathcal{E}_{\mathrm{net}}$ should be equal to $$\begin{aligned} \sigma&=\left[{(\mathcal{E}_{\mathrm{net}})}/{m\lambda_{\mathrm{p}}\tau\Gamma(1+.5\alpha)2^{.5\alpha}}\right]^{\frac{1}{\alpha}}.\end{aligned}$$ Boolean P process ----------------- The mean number of points per cluster head in Boolean P process should be equal to $$\begin{aligned} m={\mathcal{E}_{\mathrm{net}}}/{(\Gamma(.5\alpha+1)\tau\lambda_{{\mathrm{p}}}(\pi\lambda_{{\mathrm{p}}})^{-.5\alpha})}.\end{aligned}$$ By fixing $m$, we can get the expression for the coverage probability in Boolean P process. ![ Capacity functional for Boolean TC process with $R=5$ and $m=30$. Increasing variance increases the capacity functional as the daughter points are distantly located with each other in each cluster and have higher chance to cover distinct areas. []{data-label="TCPcapacity"}](mcpcapv4.pdf){width=".46\textwidth"} ![ Capacity functional for Boolean TC process with $R=5$ and $m=30$. Increasing variance increases the capacity functional as the daughter points are distantly located with each other in each cluster and have higher chance to cover distinct areas. []{data-label="TCPcapacity"}](tcpcapacity.pdf){width=".46\textwidth"} ![Variation of point coverage probability. [ Boolean-cluster model have better point coverage probability when sensors have limited power. ]{}[]{data-label="pointcoverage"}](compTCPMCPPP1.pdf){width=".46\textwidth"} ![Variation of point coverage probability. [ Boolean-cluster model have better point coverage probability when sensors have limited power. ]{}[]{data-label="pointcoverage"}](tcpmcpfixmrv4.pdf){width=".46\textwidth"} ![Variation of point coverage probability. [ Boolean-cluster model have better point coverage probability when sensors have limited power. ]{}[]{data-label="pointcoverage"}](Point_coverage_prob_thr.pdf){width=".46\textwidth"} \ **Capacity functional/sensing probability of Boolean-TC process:** Fig. \[TCPcapacity\] shows the variation of sensing probability ${\mathcal{M}_{{\mathrm{T}}}(r_\set{K})}$ with respect to the event size $r_{\mathsf{K}}$ for two values of cluster spread $\sigma$. Similar trends as Boolean MC process can be seen here.\ **Impact of sensors’ deployment:** Fig. \[comparision\] shows the comparison among the sensing performance of the three deployments $\Psi_{{\mathrm{M}}}$, $\Psi_{{\mathrm{T}}}$ and $\Psi_{{\mathrm{P}}}$. For the fair comparison, we took $\sigma$ of $\Psi_{{\mathrm{T}}}$ equal to $r_{\mathrm{d}}$ of $\Psi_{{\mathrm{M}}}$. Recall that $\Psi_{{\mathrm{M}}}$ has highest clustering, $\Psi_{{\mathrm{T}}}$ has moderate and $\Psi_{{\mathrm{P}}}$ has no clustering (independence across sensors). It can be observed that $\Psi_{{\mathrm{M}}}$ has the lowest capacity functional. It can be justified by the fact that the confinement of daughter points inside a ball in $\Psi_{{\mathrm{M}}}$ increases the overlap among sensing regions of sensors in the cluster. In case of $\Psi_{{\mathrm{T}}}$, the daughter points are more scattered and can cover a larger region. In the case of $\Psi_{{\mathrm{P}}}$, points are the most scattered which leads to the highest performance. The graph also shows the variation of capacity functional over $\sigma$ and $r_{\mathrm{d}}$. The graph depicts that increasing these parameter, $\Psi_{{\mathrm{M}}}$ or $\Psi_{{\mathrm{T}}}$ will converge to $\Psi_{{\mathrm{P}}}$ (and hence their sensing performance).\ **Trade-off between number of sensors vs sensing radius:** Fig. \[capcityfunvary\] shows the variation of capacity functional with $m$, while keeping $m\times R=150$ fixed. $m\times R$ serves as a proxy to the system cost as increasing any of $m$ and $R$ increases the cost (both- infra stricture and operating). Our analysis shows that for small event size $r_{\mathsf{K}}$, a WSN with larger $R$ provides higher sensing performance. However, as $r_\set{K}$ increases, it is better to have a higher number of sensors than the larger sensing range.\ **Impact on coverage with constrained network power:** The point coverage probability is defined as the capacity functional with $r_{\mathsf{K}}=0$. For the cluster processes we have considered $m=30$ and $\lambda_{{\mathrm{p}}}=20\times10^{-6}$. We now fix the per-cluster power requirement $\mathcal{E}_{\mathrm{net}}$ and compute coverage probability of the three deployments. Fig. \[pointcoverage\] shows the coverage probability of three networks under various power levels. At lower $\mathcal{E}_{\mathrm{net}}$, it can be observed that MCP and TCP Boolean models provide better coverage probability. [As power constraints are less restrictive, the coverage probability increases. With sensors having higher power, the PPP Boolean can provide better coverage. We can observe that clustered deployments can provide better coverage under stricter power constraints.]{} Conclusion ========== This paper performs the coverage analysis of a wireless sensor network when the sensors’ location are according to PPP, MCP and TCP. In these cases, sensor network can be modeled via Boolean-P, Boolean MCP and Boolean TCP processes. We derived the expressions for the capacity function of the two clustered deployments. As far as highest coverage is the goal, PPP performs the best of the three as the rest of two processes are attractive processes and this difference in the coverage area of clustered and PPP deployments reduces with the sensing radius of individual sensors. However, clustered deployments require less energy as their clustered heads are statistically closer than that of Boolean-P deployment. We also derive the average per-cluster required power to achieve a certain coverage area. Raising the average per cluster power allows larger the cluster radius and thus higher point coverage probability. A general trade-off can be see between per-cluster required energy and coverage area when choosing between clustered or Poisson deployments. It is also observed that when sensors have low power levels, deployment, according to a Boolean cluster process, can provide better performance compared to deployment according to a Boolean Poisson process. {#appn A} The probability that ${\mathsf{K}}$ does not intersect with the covered region $\Psi_{\mathrm{M}}$ is given by: $\mathbb{P}(\Psi_{\mathrm{M}}\cap {\mathsf{K}}\neq \phi)=$ $$\begin{aligned} &\mathbb{E}\left[\prod_{{\mathbf{x}}_i\in\Phi_{{\mathrm{p}}}}\prod_{{\mathbf{y}}_j^{(i)}\in \Phi_{{\mathrm{d}}}^{(i)}}\left( {\mathbbm{1}}\left( ({\mathbf{x}}_i+{\mathbf{y}}_j^{(i)}+S_j^{(i)}) \cap {\mathsf{K}}\right)\neq \phi \right)\right],\\ &=1-\mathbb{E}\left[\prod_{{\mathbf{x}}_i\in\Phi_{{\mathrm{p}}}}\prod_{{\mathbf{y}}_j^{(i)}\in \Phi_{{\mathrm{d}}}^{(i)}} {\mathbbm{1}}\left( ({\mathbf{x}}_i+{\mathbf{y}}_j^{(i)}+S_j^{(i)})\cap {\mathsf{K}}= \phi \right) \right],\\ &\stackrel{(a)}=1-\mathbb{E} \left[\prod_{{\mathbf{x}}_i\in\Phi_{{\mathrm{p}}}} \prod_{{\mathbf{y}}_j^{(i)}\in \Phi_{{\mathrm{d}}}^{(i)}} {\mathbbm{1}}\left({\mathbf{y}}_j^{(i)}\notin \left((-{\mathbf{x}}_i+S_{j}^{(i)})\oplus {\mathsf{K}}\right) \right)\right],\\ &\stackrel{(b)}=1-\mathbb{E}_{\Phi_{{\mathrm{p}}}}\left[\prod_{{\mathbf{x}}_i\in \Phi_{{\mathrm{p}}}}\exp(-\lambda_{\mathrm{d}}|{\Ball}({\mathrm{o}},r_{\mathrm{d}})\cap{\Ball}(-{\mathbf{x}}_i,R) \oplus {\mathsf{K}}|) \right]. \end{aligned}$$ Here $(a)$ is due to the definition of Minkowski sum and [$(b)$ is applying the PGFL of $\Phi_{\mathrm{d}}^{(i)}$. Applying the PGFL of $\Phi_{{\mathrm{p}}}$ we get the MCP capacity functional .]{} {#apndxb} To solve the integral expression presented in we are presenting couple of bounds over the intersecting region $\areaballintersect{}{x}{r_{\mathrm{d}}}{r}$. The bounding techniques are similar to [@pandey2019detection Th. 2, App. C]. Fig. \[circleintersection\] depicts the shapes (circle and rectangle) which can bound the area of intersection between the two circle. Let the two circle be ${\mathcal{C}}_1\equiv {\mathcal{B}}({\mathrm{o}},r_{\mathrm{d}})$ and ${\mathcal{C}}_2\equiv{\mathcal{B}}({\mathrm{o}},r_{\mathsf{K}}+R)$, of radius $r_{{\mathrm{d}}}$ and $r_{\mathsf{K}}+R$ respectively. A third circle ${\mathcal{C}}_3$ of radius $\frac{r+r_{{\mathrm{d}}}-x}{2}$ centered at $(\frac{r-r_{{\mathrm{d}}}+x}{2},0)$ will be entirely inside the intersecting region. Similarly, a rectangle of width $r+r_{\mathrm{d}}-x$ and height $2\min(r,r_{\mathrm{d}})$ will completely cover the intersecting area hence acts as an upper bound. For detailed discussion over the bounds readers are advised to refer [@pandey2019detection; @pandey2019contact]. ![ Illustration showing bounds on the intersecting region between the two circle. Geometrical shapes can be used to lower bound (circle) and upper bound (rectangle) the intersecting region.[]{data-label="circleintersection"}](drawingcircleintersection_1.pdf){width=".46\textwidth"} {#prf:thm5} The proof is similar to the proof in Appendix \[appn A\]. For Boolean TCP process, the null probability $\mathbb{P}(\Psi_{{\mathrm{T}}}\cap {\mathsf{K}}\neq \phi)$ is $$\begin{aligned} &=\mathbb{E}\left[\prod_{{\mathbf{x}}_i\in\Phi_{{\mathrm{p}}}}\prod_{{\mathbf{y}}_j^{(i)}\in \Phi_{{\mathrm{d}}}^{(i)}}\left({\mathbbm{1}}\left(({\mathbf{x}}_i+{\mathbf{y}}_j^{(i)}+S_j^{(i)})\cap {\mathsf{K}}\right)\neq \phi\right)\right],\\ &=1-\mathbb{E}_{\Phi_{{\mathrm{p}}}}\left[\prod_{{\mathbf{x}}_i\in \Phi_{{\mathrm{p}}}}\expectop_{{\mathbf{y}}_j^{(i)}|{\mathbf{x}}_i}\left[\prod_{{\mathbf{y}}_j^{(i)}\in \mathbb{R}^2}{\mathbbm{1}}({\mathbf{y}}_j^{(i)}\notin{\Ball}(-{\mathbf{x}}_i,R)\oplus {\mathsf{K}})\right]\right],\\ &\stackrel{(a)}=1-\mathbb{E}_{\Phi_{{\mathrm{p}}}}\left[\prod_{{\mathbf{x}}_i\in \Phi_{{\mathrm{p}}}}\int_{{\mathbf{y}}\in{\Ball}({\mathbf{x}}_i,R+r_{\mathsf{K}})}\frac{m}{2\pi\sigma^2}\exp\left(-\frac{y^2}{2\sigma^2}\right)y{\mathrm{d}}y{\mathrm{d}}\theta\right]. \end{aligned}$$ [Here $(a)$ is acquired by the PGFL of $\Phi_{{\mathrm{d}}}^{(i)}$]{}. Let $r=R+r_{\mathsf{K}}$ and ${\mathbf{y}}={\mathbf{x}}_i+{\mathbf{t}}\implies {\mathrm{d}}{\mathbf{y}}={\mathrm{d}}{\mathbf{t}}$: $$\begin{aligned} \label{last_step} &=1-\mathbb{E}_{\Phi_{{\mathrm{p}}}}\left[\prod_{{\mathbf{x}}_i\in \Phi_{{\mathrm{p}}}}\int_{{\mathbf{t}}\in{\Ball}({\mathrm{o}},r)}\frac{m}{2\pi\sigma^2}e^{\left(-\frac{||{\mathbf{x}}_i+{\mathbf{t}}||^2}{2\sigma^2} \right)}{\mathrm{d}}{\mathbf{t}}\right]. \end{aligned}$$ Without loss of generality, it can be assumed that ${\mathbf{x}}_i\equiv(x_i,0)$ as the parent point process $\Phi_{{\mathrm{p}}}$ is rotation invariant. Thus, replacing $||{\mathbf{x}}+{\mathbf{t}}||^2$ with $x^2+t^2+2xt\cos\theta$. [ Applying the PGFL of $\Phi_{{\mathrm{p}}}$ in , we get the Theorem \[thm5\].]{} [^1]: The authors are with the Modern Wireless Networks Group at the Indian Institute of Technology Kanpur, Kanpur (India) 208016 (Email: kpandey@iitk.ac.in, gkrabhi@iitk.ac.in).

--- abstract: 'The influence of a thermodynamic constraint on the critical finite-size scaling behavior of three-dimensional Ising and XY models is analyzed by Monte-Carlo simulations. Within the Ising universality class constraints lead to Fisher renormalized critical exponents, which modify the asymptotic form of the scaling arguments of the universal finite-size scaling functions. Within the XY universality class constraints lead to very slowly decaying corrections inside the scaling arguments, which are governed by the specific heat exponent $\alpha$. If the modification of the scaling arguments is properly taken into account in the scaling analysis of the data, finite-size scaling functions are obtained, which are [*independent*]{} of the constraint as anticipated by analytic theory.' author: - Michael Krech title: 'Critical finite-size scaling with constraints:Fisher renormalization revisited' --- Introduction ============ The theoretical investigation of classical spin systems has played a key role in the understanding of phase transitions, critical behavior, scaling, and universality [@Amit78; @Parisi88]. In particular, the classical Ising, the XY, and the Heisenberg model are the most relevant spin models in three dimensions. Each of these simple models represents a universality class which, apart from the spatial dimensionality and the range of the interactions, is characterized by the number of components of the order parameter , e.g, the magnetization in the case of ferromagnetic models. Real systems, however, suffer from various kinds of imperfections, e.g., lattice defects, impurities, or vacancies. In an experiment, which is designed to probe critical behavior as a function of temperature, the presence of, say, impurities on the lattice constitutes a thermodynamic constraint, because in a given sample the impurity concentration will remain constant during the temperature scans. According to the concepts of thermodynamics the impurity concentration $n_i$ can be written as the derivative of the grand canonical potential with respect to the chemical potential $\mu_i$ of the impurities, where other parameters like the temperature and the volume of the system are kept fixed. Now the question arises how the critical singularities in the grand canonical potential are affected when the thermodynamic ensemble is changed from ’fixed $\mu_i$’ to ’fixed $n_i$’, where the location of the critical temperature $T_c$ depends on the particular values of $\mu_i$ or $n_i$, respectively. The answer to this question has been given a long time ago by Michael Fisher [@Fisher68]. Provided, that the critical singularites have their usual form in the ’fixed $\mu_i$’ ensemble, then the constraint $n_i = const.$ amounts to a reparameterization of the reduced temperature $t = (T-T_c(n_i))/T_c(n_i)$ of the [*constrained*]{} system in terms of the reduced temperature $\tau = (T-T_c(\mu_i))/T_c(\mu_i)$ of the [*unconstrained*]{} system according to [@Fisher68] $$\label{ttau} t = a \tau + b \tau |\tau|^{-\alpha} + \dots ,$$ where $a$ and $b$ are nonuniversal constants and the dots indicate higher order contributions. Apart from a linear term (\[ttau\]) contains a singular contribution which is characterized by the critical exponent $1 - \alpha$ of the entropy density . Which of the two terms in (\[ttau\]) is the leading one for $t, \tau \to 0$ depends on the sign of $\alpha$. Within the Ising universality class in $d = 3$ dimensions $\alpha \simeq 0.109$ [@LGZJ85] so that $|\tau| \sim |t|^{1/(1-\alpha)}$ to leading order and therefore the critical exponents $\beta$ (order parameter), $\gamma$ (susceptibility), and $\nu$ (correlation length) of the unconstrained system undergo ’Fisher renormalization’ in the constrained system according to [@Fisher68] $$\label{fren} \beta \to \beta' = \beta / (1 - \alpha), \quad \gamma \to \gamma' = \gamma / (1 - \alpha), \quad \nu \to \nu' = \nu / (1 - \alpha) .$$ The specific heat exponent $\alpha$ requires a more careful analysis, because the specific heat is the temperature derivative of the entropy which in addition to the ’renormalization’ displayed in (\[fren\]) causes a sign change $$\label{frena} \alpha \to \alpha' = -\alpha / (1 - \alpha) .$$ Note that analytic background contributions to the entropy of the unconstrained system become [*singular*]{} in the constrained system due to the singularity in the reparameterization given by (\[ttau\]). Within the XY universality class in $d = 3$ the exponent $\alpha$ is negative [@LGZJ85], where probably the best current estimate $\alpha \simeq -0.013$ is obtained from an experiment on $^4$He near the superfluid transition [@LSNCI96]. For negative $\alpha$ the linear term on the r.h.s. of (\[ttau\]) is the dominating one for $\tau \to 0$. However, the XY universality class $\alpha$ is so small, that in practice the singular term in (\[ttau\]) can never be neglected. Instead, the singular contribution to (\[ttau\]) gives rise to very slowly decaying correction terms which must not be confused with Wegner corrections to scaling . These correction terms have to be considered in any scaling analysis in order to obtain correct values for the critical exponents. If the system is finite, which is neccessarily the case for any Monte - Carlo simulation, all critical singularities are rounded, i.e., all quantities are analytic functions of the thermodynamic parameters [@Fisher71] so that a thermal singularity as shown in (\[ttau\]) does not occur. Critical finite-size rounding effects in, e.g., a cubic box $L^d$ are captured by [*universal*]{} finite-size scaling functions [@Fisher71; @Barber83] which restore all critical singularities in the limit $L \to \infty$. Following the line of argument in [@Fisher68], (\[ttau\]) then has to be replaced by $$\label{ttaufL} t = a \tau + \tau |\tau|^{-\alpha} f(\tau L^{1/\nu}) ,$$ where $f(x)$ is the finite-size scaling function of the entropy density and $x = \tau L^{1/\nu}$ is a convenient choice of its scaling argument. For $\tau \to 0$ at finite $L$ the singular prefactor of $f(x)$ in (\[ttaufL\]) must be cancelled so that one has $f(x) = A |x|^{\alpha} + \dots$ in the limit $x \to 0$, where $A$ is a nonuniversal constant such that $f(x)/A$ is a [*universal*]{} function of its argument. To leading order in $\tau$ the reparameterization of the reduced temperature $t$ of the constrained system is therefore [*linear*]{} in the reduced temperature $\tau$ of the unconstrained system and one finds $$\label{ttauL} t = \tau (a + A L^{\alpha/\nu}).$$ According to (\[ttauL\]) the finite-size scaling argument $x$ in the constrained system is given by $$\label{x} x = \tau L^{1/\nu} = t L^{1/\nu} / (a + A L^{\alpha/\nu}),$$ where the [*shape*]{} of the finite-size scaling functions is maintained [@VD], i.e., the presence of the constraint [*only*]{} affects the form of the scaling argument $x$. For $\alpha > 0$ (\[x\]) asymptotically reduces to $x = t L^{1/\nu'}/A$ for large $L$ in accordance with Fisher renormalization (see (\[fren\])). For $\alpha < 0$ (\[x\]) captures the aforementioned slowly decaying corrections to the asymptotic critical behavior in the XY universality class when a thermodynamic constraint is present. Note that $A > 0$ for the Ising universality class and that $A < 0$ for the XY universality class. In the remainder of this paper a simple spin model is introduced which can be efficiently simulated with existing Monte - Carlo algorithms both with and without constraints in three dimensions. For the Ising and the XY version of the model finite-size scaling according to (\[x\]) is tested for the modulus of the order parameter, the susceptibility, and the specific heat. Model and simulation method =========================== The model system which is investigated here can be described as an $O(N-1)$ symmetric classical ’planar’ ferromagnet in a transverse magnetic field. The model Hamiltonian reads $$\label{H} {\cal H} = -J \sum_{\langle i j \rangle} \sum_{x=1}^{N-1} S_i^x S_j^x - h \sum_i S_i^N,$$ where $\langle i j \rangle$ denotes a nearest neighbor pair of spins on a simple cubic lattice in $d = 3$ dimensions. The lattice contains $L$ lattice sites in each direction and in order to avoid surface effects periodic boundary conditions are applied. Each spin ${\bf S}_i$ is a classical spin with $N$ components $\vec{S}_i = \left(S_i^1,S_i^2, \dots,S_i^N\right)$ with the normalization $|\vec{S}_i| = 1$ for each lattice site $i$. The magnetic field $h$ in (\[H\]) only acts on the $N$-components of the spins which are not coupled by the exchange interaction $J$. From the symmetry of the Hamiltonian it is obvious, that the model belongs to the $O(N-1)$ universality class in $d = 3$, where nonuniversal quantities like the critical temperature $T_c = T_c(h)$ depend on the strength of the transverse field $h$. Note that $T_c(h)$ is symmetric around $h = 0$ and decreases with increasing $h$, because the spins become more and more aligned with the $N$-direction as $\pm h$ is increased and due to the normalization condition the typical interaction energy between pairs of spins is decreased. The Hamiltonian given by (\[H\]) defines the unconstrained model. The constraint is imposed on the transverse magnetization $M$ in the form $$\label{Mconst} M \equiv \sum_i S_i^N = const.,$$ where the Hamiltonian of the constrained model is given by $$\label{Hconst} {\cal H}_M = -J \sum_{\langle i j \rangle} \sum_{x=1}^{N-1} S_i^x S_j^x = {\cal H} + hM .$$ The critical temperature of the constrained model is a symmetric and monotonically decreasing function of the prescribed transverse magnetization $M$. The transverse field $h$ here plays the part of the chemical potential $\mu_i$ of impurities (see Sect. 1) and the transverse magnetization $M$ accordingly plays the part of the impurity concentration $n_i$. It is also possible to implement $O(N)$ spin models with impurities or vacancies with diffusion in order to mimic the situation discussed in [@Fisher68]. However, the fact that the Hamiltonians given by (\[H\]) and (\[Hconst\]) only require a single ’species’ with a single coupling constant leads to some simplifications in the algorithms. Note that the symmetric constrained model $(M = 0)$ becomes equivalent to the symmetric unconstrained model $(h = 0)$ for sufficiently large lattices. In particular, both versions of the symmetric model have the same $T_c$. The Monte-Carlo algorithm is chosen as a hybrid scheme, where each hybrid Monte-Carlo step consists of 10 updates each of which can be one of the following: one Metropolis sweep of the whole lattice, one single cluster Wolff update [@Wolff89], or one overrelaxation update of the whole lattice [@CL94], where the latter can only be applied for $N \geq 3$. The Metropolis algorithm updates the lattice sequentially and works in the standard way for the unconstrained model. For the constrained model the constraint $M = const.$ is observed locally by applying a Kawasaki update dynamics for the $N^{th}$ components of the spins. For each lattice site $i$ a nearest neighbor site $j$ is chosen randomly and a random amount of the $N^{th}$ spin component is proposed for exchange such that $S_i^N + S_j^N$ remains constant. Then new spin components $(S_i^1,S_i^2,\dots ,S_i^{N-1})$ are proposed and the spin components $(S_j^1,S_j^2,\dots ,S_j^{N-1})$ are adjusted according to the spin normalization condition $|\vec{S}_i| = |\vec{S}_j| = 1$. The local change $\Delta E$ of the configurational energy is calculated according to (\[Hconst\]). According to detailed balance the proposed update is accepted with probability $p(\beta \Delta E)$, where $\beta = 1/(k_B T)$. For our simulation we have chosen $p(x) = 1 / (\exp(x) + 1)$. Note that all updates must be proposed such that the new spin at lattice site $i$ is taken from the uniform distribution on the unit sphere in $N$ dimensions. The Wolff algorithm also works the standard way [@Wolff89], except that [*only*]{} the first $N-1$ components of the spins are used for the cluster growth, i.e., (\[H\]) and (\[Hconst\]) are treated as planar ferromagnets. This means that a cluster update never changes the $N^{th}$ component of any spin so that the Wolff algorithm is nonergodic in this case. The cluster update is still a valid Monte-Carlo step in the sense that it fulfills detailed balance, however, in order to provide a valid Monte-Carlo algorithm it has to be used together with the Metropolis algorithm described above in a hybrid fashion. The use of Wolff updates allows us to take advantage of improved estimators [@Has90] for magnetic quantities. The overrelaxation part of the algorithm performs a microcanonical update of the configuration in the following way. The local configurational energy has the functional form of a scalar product of the spins, where according to (\[H\]) and (\[Hconst\]) only the first $N-1$ components are involved. With respect to the sum of its nearest neighbor spins each spin has a transverse component in the $(S_i^1,S_i^2,\dots ,S_i^{N-1})$ plane which does not enter the scalar product. The overrelaxation algorithm scans the lattice sequentially, determines this transverse component for each lattice site and flips its sign. This overrelaxation algorithm is similar to the one used in [@CL94] and it quite efficiently decorrelates subsequent configurations over a wider range of temperatures around the critical point than the Wolff algorithm. However, overrelaxation can only be applied for $N \geq 3$. In the following only the cases $N = 2$ (transverse Ising) and $N = 3$ (transverse XY) are considered. In a typical hybrid Monte-Carlo step we use three Metropolis [*(M)*]{}, seven single cluster Wolff [*(C)*]{} updates for $N = 2$ and three Metropolis, five single cluster Wolff, and two overrelaxation updates [*(O)*]{} for $N = 3$ in the critical region of the models. The inividual updates are mixed automatically in the program so that the update sequences [*(M C C M C C M C C C)*]{} for $N = 2$ and [*(M C C M O C M C C O)*]{} for $N = 3$ are generated as one hybrid Monte-Carlo step. The shift register generator R1279 given by the recursion relation $X_n = X_{n-p} \oplus X_{n-q}$ for $(p,q) = (1279,1063)$ is used as the random number generator. Generators like this are known to cause systematic errors in combination with the Wolff algorithm [@cluerr]. However, for lags $(p,q)$ as large as the ones used here these errors will be far smaller than typical statistical errors. They are further reduced by the hybrid nature of our algorithm due to the presence of several Metropolis updates in one hybrid Monte-Carlo step [@AMFDPL]. The hybrid Monte-Carlo scheme described above is employed for lattice sizes $L$ between $L = 20$ and $L = 80$. For each system size and temperature we perform at least 10 blocks of $10^3$ hybrid steps for equilibration followed by $10^4$ hybrid steps for measurements. Each measurement block yields an estimate for all static quantities of interest and from these we obtain our final estimates and estimates of their statistical error following standard procedures. At the critical point (see below) two or three times as many updates have been performed. The integrated autocorrelation time of the hybrid algorithm is determined by the autocorrelation function of the energy or, equivalently, the modulus of the order parameter, which yield the slowest modes for the Wolff algorithm. The autocorrelation times are generally rather short, at the critical point they range from about 5 hybrid Monte-Carlo steps for $L = 20$ to about 10 hybrid Monte-Carlo steps for $L = 80$. The values for the equilibration and measurement periods given above thus translate to roughly 100 and 1000 autocorrelation times, respectively. In order to obtain the best statistics for magnetic quantities a measurement is made after every hybrid Monte-Carlo step. All error bars quoted in the following correspond to one standard deviation. The simulations have been performed on the DEC alpha AXP workstation cluster at the Physics Department and on HP RISC8000 workstations at the Computer Center of the RWTH Aachen. Ising universality class ======================== For $N = 2$ (\[H\]) and (\[Hconst\]) describe a classical Ising model in a (fixed) transverse field or with fixed transverse magnetization, respectively. In the following we will only consider the constrained model with the symmetric constraint $M = 0$ and with the constraint $m \equiv M/L^3 = 1/\sqrt{2}$. The symmetrically constrained model does not show Fisher renormalization [@Fisher68] and we therefore use this case for tests of the algorithm and for the production of data representative of the Ising universality class in $d = 3$. The constraint $m = const. \neq 0$ breaks the $S_i^N \to -S_i^N$ symmetry of the model and Fisher renormatization should become visible within a certain temperature window around $T_c = T_c(m)$. The width of this window is of course a nonuniversal property of the model and in particular one expects this window to widen as $m$ is increased. Due to the spin normalization condition $m$ cannot exceed unity and one therefore also expects, that critical behavior becomes very difficult to resolve numerically if $m$ is too close to its maximum value. Therefore, $m = 1/\sqrt{2}$ is chosen as a compromise between good resolution in the critical regime and a prominent Fisher renormalization effect. The critical temperatures $T_c(m=0)$ and $T_c(m=1/\sqrt{2})$ are determined from temperature scans of the Binder cumulant ratio according to standard procedures [@CFL93]. We obtain the following reduced critical coupling constants $K_c(m) \equiv J / k_B T_c(m)$: $$\label{KcI} K_c(0) = 0.41638 \pm 0.00005 \quad \mbox{and} \quad K_c(1/\sqrt{2}) = 0.6371 \pm 0.0001 .$$ The corresponding estimates for the Binder cumulant ratio obtained for $m = 0$ and $m = 1/\sqrt{2}$ agree with previous estimates obtained for the Ising universality class within two standard deviations [@BLH95], where for the latter choice of $m$ Wegner corrections to scaling are considerable and must be subtracted in order to obtain a reliable estimate. In order to obtain an estimate for the exponent $\nu$ which enters the finite-size scaling argument according to (\[x\]) the cumulant $$\label{X} X \equiv {\partial \over \partial T} \ln \langle \phi^2 \rangle = {1 \over k_B T^2} \left( {\langle \phi^2 {\cal H}_M \rangle \over \langle \phi^2 \rangle} - \langle {\cal H}_M \rangle \right)$$ has been measured, where $\phi = L^{-3}\sum_i S_i^1$ is the order parameter. At the critical temperature $T_c(m)$ the scaling behavior $X \sim x/t$ is expected (see (\[x\])). Corresponding numerical results for $m = 0$ and $m = 1/\sqrt{2}$ are displayed in Fig.\[fig1\] on a double logarithmic scale. ![Cumulant $X$ at the critical point for $m = 0$ ($\times$) and $m = 1/\sqrt{2}$ (+). The solid and dashed lines display power law fits to the data for $30 \leq L \leq 70$ for $m = 0$ and $m = 1/\sqrt{2}$, respectively[]{data-label="fig1"}](fig1.eps){width="80.00000%"} The data are compatible with simple power laws, where the exponents $\nu = 0.622 \pm 0.005$ $(m = 0)$ and $\nu' = 0.714 \pm 0.004$ $(m = 1/\sqrt{2})$ have been obtained. Compared to the best currently known estimate $\nu \simeq 0.630$ [@LGZJ85] the above estimate is too small and only agrees with the theoretical value within two standard deviations. A more thorough analysis shows that the discrepancy can be explained by a mismatch of the order $5 \times 10^{-5}$ between the actual critical temperature and the estimate used here (see (\[KcI\])), which on the other hand is of the same magnitude as the statistical error of $K_c(0)$. The agreement between the above estimate for $\nu' = \nu / (1-\alpha)$ and the theoretical value $\nu' \simeq 0.708$ [@LGZJ85] is better, however, it may again be affected by a mismatch between the actual value of $T_c(1/\sqrt{2})$ and the estimate used here. If the literature values for $\nu$ and $\alpha$ are substituted in (\[x\]), where $a$ and $A$ are used as fit parameters, $a/A \simeq 0.1$ is obtained which is small enough to be ignored in the scaling analysis (see below). The finite-size scaling analysis has been performed for several thermodynamic quantities, in particular, the average modulus of the order parameter $\langle |\phi| \rangle$, the susceptibilities $$\label{Chi} \chi_+ \equiv {L^3 \over k_B T} \langle \phi^2 \rangle , \quad \chi_- \equiv {L^3 \over k_B T} \left( \langle \phi^2 \rangle - \langle |\phi| \rangle^2 \right),$$ and the specific heat $C$. Data will only be shown for $\langle |\phi|\rangle$, $\chi_-$, and $C$, because the finite-size scaling functions for $\langle |\phi| \rangle$ and $\chi_+$ are very similar. According to finite-size scaling theory it must be possible to callapse the data for all $m$ onto one and the same curve, where two nonuniversal scaling factors are required for each quantity. One scaling factor adjusts the magnitude of the scaling argument $x$ (see (\[x\])), the other adusts the absolute normalization of the quantity. Note that the former saling factor must be the same for [*all*]{} quantities. For $m = 0$ the scaling argument $x = t L^{1/\nu}$ is used, whereas for $m = 1/\sqrt{2}$ the choice $x = t L^{(1-\alpha)/\nu} / A$ has been made, where $A \simeq 1.1$ and the coefficient $a$ in (\[x\]) has been neglected. The exponents $\nu$ and $\alpha$ are taken from the literature [@LGZJ85]. ![Scaling plot of $\langle |\phi| \rangle$ for $L = 30$, 40, 50, and 60 for $m = 0$ and $m = 1/\sqrt{2}$. The reduced temperature $t$ has been varied between $-0.007$ and 0.003. Statistical errors are much smaller than the symbol sizes[]{data-label="fig2"}](fig2.eps){width="80.00000%"} The scaling plot of $\langle |\phi| \rangle$ is shown in Fig.\[fig2\], where the abolute normalization of the data for $m = 1/\sqrt{2}$ can be adjusted to the $m = 0$ data by a scale factor of $\sim 0.7$ as one would expect from simple mean field arguments. As shown in Fig.\[fig2\], the espected data collapse can be reproduced rather well, where the literature value for the exponent $\beta / \nu = 0.5168$ [@LGZJ85] has been used. The same holds for the susceptibility $\chi_-$ which is displayed in Fig.\[fig3\], where $\gamma / \nu = 2 - \eta = 1.967$ is also taken form [@LGZJ85]. The absolute magnitudes of $\chi_-/L^{\gamma/\nu}$ for $m = 0$ and $m = 1/\sqrt{2}$ are different by a factor of about $0.5$ which is in accordance with simple mean field arguments. Note that the scaling function of $\chi_-$ has a maximum for $x < 0$ [@Dohm9596]. ![Scaling plot of $\chi_-$ for $L = 30$, 40, 50, and 60 for $m = 0$ and $m = 1/\sqrt{2}$. The reduced temperature $t$ has been varied between $-0.007$ and 0.003. Statistical errors are much smaller than the symbol sizes[]{data-label="fig3"}](fig3.eps){width="80.00000%"} The specific heat $C$ requires a somewhat different treatment due to the fact that unlike the other quantities presented so far the specific heat requires an [*additive*]{} renormalization within renormalized field theory [@Dohm9596]. For the data analysis this means that scaling can only be obtained after a suitable subtraction is applied to the specific heat. One option to obtain scaling is to subtract the bulk specific heat $C^0_b(t)$ at a reference reduced temperature $t = t_0$ which are given by $$\label{Cbt0} C^0_b(t) = {A_\pm \over \alpha} |t|^{-\alpha} + B \quad \mbox{and} \quad t_0 = (L/\xi^0_\pm)^{-1/\nu},$$ where $A_\pm$ and $B$ are nonuniversal constants and $\xi^0_\pm$ is the amplitude of the correlation length. The index $\pm$ refers to temperatures above or below $T_c(m)$, respectively. The reference reduced temperature chosen in (\[Cbt0\]) is positive and therefore only $A_+$ and $\xi^0_+$ are needed. Specifically, the choice $\nu = 0.630$, $\alpha = 0.109$ [@LGZJ85], $A_+ = 0.1552$, $B = -1.697$, and $\xi^0_+ = 0.495$ [@Dohm9596] guarantee scaling of the [*relative*]{} specific heat $\Delta C^0 \equiv C - C^0_b(t_0)$ for the Ising model in $d = 3$. Note that $\xi^0_+$ is measured in units of the lattice constant. For the data to be analyzed here the subtraction defined by (\[Cbt0\]) is only valid for the case $m = 0$, where $\Delta C^0$ scales as $L^{\alpha/\nu}$. For $m = 1/\sqrt{2}$ Fisher renormalization according to (\[ttau\]) must be applied to $C^0_b(t)$ in order to obtain the correct form $C^m_b(t)$ of the subtraction. The result is $$\label{Cbt0m} C^m_b(t) = {A'_+ \over \alpha} |t|^{\alpha/(1-\alpha)} + B' \quad \mbox{and} \quad t_0 = (L/\xi^0_\pm)^{(\alpha-1)/\nu},$$ where $A'_+ = -0.1728$, $B' = 1.598$, and $\xi^0_+ = 0.495$ is not changed. The scaling factor $L^{\alpha/\nu}$, which usually governs the finite-size scaling of the specific heat, is cancelled here, i.e., one expects data collapse for $\Delta C^0 / L^{\alpha/\nu}$ and $\Delta C^m \equiv C - C^m_b(t_0)$ up to an overall scale factor of about $0.5$. The result of the data analysis is shown in Fig.\[fig4\]. ![Relative specific heats $\Delta C^0 / L^{\alpha/\nu}$ for $m = 0$ and $\Delta C^m$ for $m = 1/\sqrt{2}$ for $L = 30$, 40, 50, and 60. The reduced temperature $t$ has been varied between $-0.007$ and 0.003[]{data-label="fig4"}](fig4.eps){width="80.00000%"} The data collapse reasonably well onto a single curve except near the maximum of the scaling function, where also the scatter of the individual data is substantial due to a few bad samples for $L = 50$ and 60. However, there are also systematic deviations from scaling in the data, because the maximum in the scaling functions for the $m = 0$ data is more pronounced than in the $m = 1/\sqrt{2}$ data. These deviations could be due to enhanced Wegner corrections to scaling for $m = 1/\sqrt{2}$ as compared to $m = 0$. XY universality class ===================== For $N = 3$ (\[H\]) and (\[Hconst\]) describe a classical XY model in a (fixed) transverse field or with fixed transverse magnetization, respectively. As for the case $N = 2$ we will only consider the constrained model with the symmetric constraint $m = 0$ and with the constraint $m = 1/\sqrt{2}$ in the following. The symmetrically constrained model is again used for algorithmic tests and data production for the XY universality class. The nonsymmetric constrained XY model does not show Fisher renormalization, however, according to (\[x\]) very slowly decaying corrections to the asymptotic critical behavior are expected, which will be discussed in the following. First, the cumulant $X$ defined by (\[X\]) is evaluated at the critical point, which is given by the reduced coupling constants $$\label{KcXY} K_c(0) = 0.6444 \pm 0.0001 \quad \mbox{and} \quad K_c(1/\sqrt{2}) = 1.1126 \pm 0.0003 ,$$ respectively. The result for $X$ is displayed in Fig.\[fig5\]. ![Cumulant $X$ at the critical point for $m = 0$ (x) and $m = 1/\sqrt{2}$ (+). The solid and dashed lines display fits to the data for $30 \leq L \leq 80$ for $m = 0$ and $m = 1/\sqrt{2}$, respectively (see main text)[]{data-label="fig5"}](fig5.eps){width="80.00000%"} For $m = 0$ the data can be fitted by a power law $\sim L^{1/\nu}$, where $\nu = 0.678 \pm 0.008$ is obtained which agrees with the best current estimate $\nu = 0.671$ [@LGZJ85]. For $m = 1/\sqrt{2}$ the data can also be fitted by a power law, however, the resulting exponent $\nu$ only has the meaning of an effective exponent which does not fit into the XY universality class. As shown in Fig.\[fig5\] the expression $x/t$ according to (\[x\]) also yields a very good representation of the data where the parameter $b$ in Fig.\[fig5\] is given by $b = A/a = -0.941$. The exponents used in the fit (XY universality class) are taken from [@LGZJ85]. The value of the Binder cumulant found here agrees with results reported in the literature for the standard (plane rotator) XY model [@GH93], however, Wegner corrections to scaling become quite substantial for $m = 1/\sqrt{2}$. In the following scaling analysis the finite-size scaling argument for the case $m = 0$ takes its standard form $x = t L^{1/\nu}$ and for $m = 1/\sqrt{2}$ the combination $x = t L^{1/\nu} / (1+b L^{\alpha/\nu})$ for $b = -0.941$ takes care of the slowly decaying correction terms to the asymptotic critical behavior caused by the very small and negative value of $\alpha$ in the XY universality class. As in Sect. 3 we consider $\langle |\phi| \rangle$, $\chi_+$, $\chi_-$, and the specific heat $C$ in the scaling analysis. The scaling functions for $\langle |\phi| \rangle$ and $\chi_+$ again look very similar so that we do not reproduce scaling plots for $\chi_+$ here. The result for $\langle |\phi| \rangle$ is shown in Fig.\[fig6\], where the order parameter $\phi \equiv L^{-3} \sum_i (S^1_i,S^2_i)$ has two components here. ![Scaling plot of $\langle |\phi| \rangle$ for $L = 30$, 40, 50, and 60 for $m = 0$ and $m = 1/\sqrt{2}$. The reduced temperature $t$ has been varied between $-0.005$ and 0.005. Statistical errors are much smaller than the symbol sizes[]{data-label="fig6"}](fig6.eps){width="80.00000%"} The data collapse very well onto a single curve. Note that the absolute magnitudes of $\langle |\phi| \rangle$ for $m = 0$ and $m = 1/\sqrt{2}$ are again related by a factor of $\sim 0.7$ as suggested by mean-field arguments. The values for the critical exponents $\nu = 0.671$ and $\beta = 0.347$ are taken from the literature [@LGZJ85]. The susceptibility $\chi_-$ can be treated essentially as described in Sect.3, where the exponent $\gamma / \nu = 2 - \eta = 1.965$ is taken from [@LGZJ85]. The result of the scaling analysis is displayed in Fig.\[fig7\]. ![Scaling plot of $\chi_-$ for $L = 30$, 40, 50, and 60 for $m = 0$ and $m = 1/\sqrt{2}$. The reduced temperature $t$ has been varied between $-0.005$ and 0.005. Statistical errors are much smaller than the symbol sizes[]{data-label="fig7"}](fig7.eps){width="80.00000%"} The data do not collapse as well as in Fig.\[fig3\]. Especially near the maximum of the scaling function the scatter of the data is substantially larger than in Fig.\[fig3\]. Slight systematic deviations from scaling for $m = 1/\sqrt{2}$ are observed which may again be due to enhanced Wegner corrections to scaling as compared to $m = 0$. Note that contrary to Fig.\[fig3\] the scaling function of $\chi_-$ has a maximum for $x > 0$ in the XY universality class. The specific heat $C$ of the XY model also requires a subtraction before scaling is obtained [@CDE95]. The subtraction $C^0_b(t_0)$ is again used in the form given by (\[Cbt0\]), where $A_+ = 0.42$, $\alpha = -0.013$, $B = -A_+/\alpha$, and $\xi^0_+ = 1.0$ are used here which differ somewhat from the choices made for the standard XY model in [@CDE95]. It turns out, that the quality of the data collapse for the relative specific heat $\Delta C^0 = C - C^0_b(t_0)$ for $m = 0$ is rather insensitive to the choice of $\xi^0_+$. The form of the subtraction $C^m_b(t_0)$ for $m = 1/\sqrt{2}$ requires a little analysis in order to include the slowly varying corrections to the asymptotic behavior coming from (\[ttau\]). One obtains the approximate form $$\label{Cbt0XY} C^m_b(t_0) = {A_+ / \alpha \over 1 + c t_0^{-\alpha}} \left[ \left({t_0 \over 1 + c t_0^{-\alpha}}\right)^{-\alpha} - 1\right], \quad t_0 = L^{-1/\nu} (1 + b L^{\alpha/\nu}),$$ where $b = -0.941$ (see Fig.\[fig5\]), $A_+ = 0.42$ as before, and $c \simeq 2.0$ for optimal data collapse. ![Relative specific heats $\Delta C^{(0,m)} / \left[ L^{\alpha/\nu} / (1 + bL^{\alpha/\nu}) \right]$ with $b = 0$ for $m = 0$ and $b = -0.941$ for $m = 1/\sqrt{2}$ for $L = 30$, 40, 50, and 60. The reduced temperature $t$ has been varied between $-0.005$ and 0.005[]{data-label="fig8"}](fig8.eps){width="80.00000%"} The resulting scaling plot of $\Delta C^0$ and $\Delta C^m \equiv C - C^m_b(t_0)$ is displayed in Fig.\[fig8\]. The overall shape of the scaling function is similar to the one shown in Fig.\[fig4\]. However, the scatter of the data near the maximum is so strong, that data collapse cannot be obtained in this region. In part this deficiency in the data may be due to the presence of ’bad’ samples, e.g, for $L = 50$ and $m = 0$ at $t = -0.003$ and $t = -0.002$ and for $L = 60$ and $m = 1/\sqrt{2}$ at $t = -0.005$ and $t = -0.003$. Apart from that deviations from scaling as in Fig.\[fig4\] may be present which are due to an enhancement of Wegner corrections to scaling for $m = 1/\sqrt{2}$ as compared to the case $m = 0$. However, Figs.\[fig6\] - \[fig8\] confirm, that the choice of the scaling variable $x$ given by (\[x\]) captures the slowly decaying correction term inside the scaling argument in an appropiate way and that furthermore the finite-size scaling behavior of constrained models in the Ising and the XY universality class can be treated on the same footing. Summary and conclusions ======================= The influence of constraints on the critical finite-size scaling behavior of Ising and XY models has been investigated by Monte-Carlo simulations of $O(N-1)$ planar ferromagnets with fixed transverse magnetization. The theoretical idea that only the form of the scaling argument is modified, whereas the shape of the universal scaling functions remains unchanged is verified within the statistical uncertainty of the data for the modulus of the order parameter, the susceptibilites $\chi_+$ (not shown) and $\chi_-$, and the specific heat. The form of the scaling argument used here allows to deal with critical finite-size effects in constrained Ising and XY models on the same footing, where constrained Heisenberg models can be included as well. Within the Ising universality class the finite-size behavior is consistent with the Fisher renormalization of critical exponents. In the XY universality class slowly decaying corrections to the asymptotic critical behavior are generated which are captured systematically by the analytic form of the scaling argument. The treatment of these corrections within the XY universality class may serve as a paradigm for the finite-size scaling analysis of dynamic quantities, where the constraints imposed here reappear as conserved quantities which are statically or dynamically coupled to the order parameter. These corrections may also be important for the interpretation of spin dynamics data for planar ferromagnets, where the energy of the system is conserved during the simulated time evolution of spin models similar to the ones investigated here. Acknowledgments {#acknowledgments .unnumbered} --------------- The author gratefully acknowledges many helpful discussions with V. Dohm and financial support of this work through the Heisenberg program of the Deutsche Forschungsgemeinschaft. [99]{} Amit D. J. 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--- abstract: | [Multilayer relationships among entities and information about entities must be accompanied by the means to analyze, visualize, and obtain insights from such data. We present open-source software ([`muxViz`]{}) that contains a collection of algorithms for the analysis of multilayer networks, which are an important way to represent a large variety of complex systems throughout science and engineering. We demonstrate the ability of [`muxViz`]{} to analyze and interactively visualize multilayer data using empirical genetic, neuronal, and transportation networks. Our software is available at <https://github.com/manlius/muxViz>.]{} [multilayer networks; software; visualization; multiplex networks; interconnected networks]{}\ 2000 Math Subject Classification: 91D30, 05C82, 76M27 author: - | Manlio De Domenico[^1]\ [ *Departament d’Enginyeria Informática i Matemátiques,* ]{}\ [ *Universitat Rovira I Virgili, 43007 Tarragona, Spain* ]{} - | Mason A. Porter\ [ *Oxford Centre for Industrial and Applied Mathematics,* ]{}\ [ *Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK* ]{}\ and\ [ *CABDyN Complexity Centre, University of Oxford, Oxford OX1 1HP, UK* ]{} - | Alexandre Arenas\ [ *Departament d’Enginyeria Informática i Matemátiques,* ]{}\ [ *Universitat Rovira I Virgili, 43007 Tarragona, Spain* ]{} bibliography: - 'muxviz\_final.bib' title: 'MuxViz: A Tool for Multilayer Analysis and Visualization of Networks' --- Introduction ============ Although the study of networks is old, the analysis of complex systems has benefited particularly during the last two decades from the use of networks to model large systems of interacting agents [@newman2010]. Such efforts have yielded numerous insights in many areas of science and technology[@kitano2002computational; @de2002modeling; @barabasi2004network; @sharan2007network; @beyer2007integrating; @sporns2014contributions; @colizza2006role; @gomez2007dynamical; @gomez2008spreading; @eagle2009inferring; @lazer2009life; @balcan2009multiscale; @kitsak2010identification; @aral2012identifying; @vespignani2012modelling]. In the case of biological networks, connections among genes, proteins, neurons, and other biological entities can indicate that they are part of the same biological pathway or exhibit similar biological functions. Network representations focus on connectivity, and they have now become a paradigmatic way to investigate the organization and functionality of cells [@jeong2000large; @jeong2001lethality; @shen2002network; @maslov2002specificity; @tong2004global; @guimera2005functional; @rosenfeld2005gene; @chen2006wiring; @goh2007human; @costanzo2010genetic], synaptic connectivity [@van1996chaos; @sporns2004motifs; @buzsaki2004neuronal; @sporns2004organization; @mantini2007electrophysiological; @bullmore2009complex; @seeley2009neurodegenerative; @bassett2011dynamic; @nicosia2013remote; @nicosia2013phase], and more. There are also myriad applications to other types of systems (e.g., in sociology, transportation, physics, and more)[@Wasserman1994Social; @boccaletti2006complex; @newman2010; @barthelemy2011spatial; @Holme2012Temporal; @kivela2013multilayer]. In parallel, a large variety of computational techniques have been developed to analyze (and visualize) networks and the information that they encode. In biology, for example, such methods have become important tools for attempting to understand and represent cell functionality. However, although the standard network paradigm has been very successful, it has a fundamental flaw: it forces the aggregation of multilayer information to construct network representations that include only a single type of connection between pairs of entities. This can lead to misleading results, and it is becoming increasingly apparent that a more complicated representation is necessary [@kivela2013multilayer]. Recently, a novel mathematical framework to model and analyze multilayer relationships and their dynamics was developed [@mucha2010community; @dedomenico2013mathematical]. In this framework, one represents the underlying network topology and interaction weights as a *multilayer network*, in which entities can exhibit different relationships simultaneously and can exist on different “layers”. Multilayer networks can encode much richer information than what is possible using the individual layers separately (which is what is usually done). This, in turn, provides a suitable framework for versatile and sophisticated analyses that have already been used successfully to reveal multilayer community structure [@mucha2010community] and to measure important nodes and the correlations between them [@dedomenico2013mathematical; @dedomenico2013centrality; @nicosia2013correlations; @battiston2014structural]. However, to meet the requirements of an operational toolbox to be applied to the analysis of [complex]{} systems, it is of paramount importance to also develop open-source software to visualize multilayer networks and represent the results of analyzing such networks in a meaningful way. Multilayer networks have already yielded fascinating insights and are experiencing burgeoning popularity. For example, there have been numerous studies to attempt to understand how interdependencies (e.g., [@buldyrev2010catastrophic; @brummitt2012suppressing]), other multilayer structures (e.g., [@lee2012correlated; @radicchi2013abrupt; @cardillo2013emergence; @dedomenico2013centrality; @cozzo2013clustering; @nicosia2013correlations; @cellai2013percolation; @battiston2014structural]), dynamics (e.g., [@yaugan2012analysis; @gomez2012evolution; @cozzo2012stability; @cozzo2013contact; @dedomenico2014navigability]), and control (e.g., [@mario-review2010]) can improve understanding of complex interacting systems. See the recent review article [@kivela2013multilayer] for extensive discussions and a thorough review of results. [[ The increasing use of more complicated network representations has yielded a new set of challenges: how should one visualize, analyze, and interpret multilayer data. Although there has been progress in numerous applications, many of the key results have concentrated on data from examples like social and transportation networks [@kivela2013multilayer]. Multilayer analysis has rarely been exploited in the investigation of biological networks — even though such a perspective is clearly relevant — and we believe that the lack of appropriate software has contributed to this situation. For example,]{}]{} in a recent study, the genetic and protein-protein interaction networks of *Saccharomyces cerevisiae* were investigated simultaneously [@costanzo2010genetic] to uncover connection patterns. [[ Costanzo *et al* [@costanzo2010genetic] also reported that]{}]{} genetic interactions have an overlap of 10–20% with protein-protein interaction pairs, which is significantly higher than the $3\%$ overlap that they expected based on a random null model. This suggests that many positive and negative interactions occur between — rather than within — complexes and pathways [@costanzo2010genetic] and thereby gives an important example of how exploiting multilayer information might improve understanding of biological structure and functionality. Although the aforementioned overlap is an indication of correlation between a pair of networks, the analysis of multilayer biological data would benefit greatly from techniques and diagnostics that are able to exploit multiplexity (i.e., multiple different ways to interact) in available information. [[ As has been the case in several studies of social and technological systems [@mucha2010community; @dedomenico2013centrality; @nicosia2013correlations; @battiston2014structural; @dedomenico2014navigability], the analysis of multilayer biological data would benefit greatly from techniques and diagnostics that are able to exploit, e.g., multiplexity (i.e., multiple different ways to interact) in available information.]{}]{} Methods ======= The primary contributions of the present work are to address the computational challenge of analysis and visualization of multilayer information by providing a practical methodology, and accompanying software that we call [`muxViz`]{}, for the analysis and the visualization of multilayer networks. In Appendix\[supp:note:5\], we give technical details about the [`muxViz`]{} software. Visualization {#supp:note:1} ------------- [[ In multilayer networks, nodes can exist in several layers simultaneously and entities that exist in multiple layers (such nodes have “replicas” on other layers) are connected to each other via interlayer edges. One can visualize a multilayer network in [`muxViz`]{} either using explicit layers or as an edge-colored multigraph [@kivela2013multilayer], in which edges are “colored” according to the different types of relationships between them (see Fig.\[fig:fig1suppl\] for examples of genetic and neuronal multilayer networks). ]{}]{} [[ The [`muxViz`]{} software focuses predominantly on “multiplex networks”, which refer to networks with multiple relational types and which are arguably the most important (and prevalent) type of multilayer network. A large variety of systems in the biological, social, information, physical, and engineering sciences can be described as multiplex networks. In [`muxViz`]{}, we consider two different types of interlayer connectivity: *ordinal* and *categorical*. In ordinal multilayer networks, interlayer edges exist only between layers that are adjacent to each other with respect to some criterion (e.g., temporal ordering). By contrast, categorical multilayer networks include interlayer edges between replica nodes from every pair of layers. For the sake of simplicity, we illustrate [`muxViz`]{} using interlayer edges of weight 1 in the present paper. In general, how to choose such weights is an open research question. See the discussions in Ref. [@Bassett2013Robust; @kivela2013multilayer]. ]{}]{} ![**Figure\[fig:fig1suppl\]: Multilayer representations of genetic and neuronal networks**. **(A)** Multilayer representation, in which the layers correspond to interaction network of genes in *Saccharomyces cerevisiae* (which was obtained via a synthetic genetic-array methodology) and a correlation-based network in which genes with similar interaction profiles are connected to each other. \[The data comes from Ref.[@costanzo2010genetic].\] In the third layer, we show the corresponding aggregated network. In this visualization, the color of the nodes is their module assignment from multilayer community detection (see the text for further details). **(B)**. Representation of the same network as an edge-colored multigraph. **(C)** Multilayer and **(D)** edge-colored-multigraph representations of the *Caenorhabditis elegans* connectome, where layers correspond to different synaptic junctions: electric (“ElectrJ”), chemical monadic (“MonoSyn”), and polyadic (“PolySyn”). \[The data comes from Ref.[@chen2006wiring].\] In panels **B** and **D**, we color the nodes according to the layer to which they belong. If a node is part of multiple layers simultaneously, then we use an equal distribution of the corresponding colors for the node.[]{data-label="fig:fig1suppl"}](fig1_suppl.pdf){width="16cm"} ![**Multilayer networks embedded in geographical regions**. **(A)** Network of European airports, where each layer represents a different airline [@cardillo2013emergence]. **(B)** Network of mobility and communication in the Ivory Coast, where nodes are geographical districts [@lima2013exploiting]. We used [`muxViz`]{} to visualize these data sets. []{data-label="fig:fig6suppl"}](fig6_suppl.pdf){width="16cm"} [[ For instance, let us examine the genetic-interaction and profile-correlation networks of a cell as different layers for a multilayer network. Such networks were aggregated into a single network in Ref. [@costanzo2010genetic]. In Fig.\[fig:fig1suppl\][**A**]{}, we show multilayer visualizations that we created using [`muxViz`]{}. Other representations are also possible [@kivela2013multilayer]. For example, when representing this data as an edge-colored multigraph, we “color” edges according to the type of relationship that they represent (see Fig.\[fig:fig1suppl\][**B**]{}). In Fig.\[fig:fig1suppl\][**C**]{} and Fig.\[fig:fig1suppl\][**D**]{}, we show the two visualizations for the connectome of *Caenorhabditis elegans*. In this example, each layer corresponds to a different type of synaptic connection [@chen2006wiring]. ]{}]{} [[ In panels [**(A)**]{} and [**(C)**]{} of Fig.\[fig:fig1suppl\], we use a layout in which the positions of the nodes are the same in each layer. We determine the positions of nodes by combining two of the standard force-directed algorithms available in [`muxViz`]{} and applying them to an aggregated network that we obtained by summing the corresponding entries of the adjacency matrices of the individual layers. Specifically, we first apply the Fruchterman-Reingold algorithm [@fruchterman1991graph] to the aggregated network and then use the output of this algorithm as a seed layout for the Kamada-Kawai algorithm [@kamada1989algorithm] to achieve a better spatial separation of nodes in the final layout. The [`muxViz`]{} software also allows other layout choices. For example, the layout of each layer can be independent, or one can determine node locations using any individual layer or an aggregation over any subset of layers. ]{}]{} [[ One can also use [`muxViz`]{} for a large variety of other analyses and visualizations. For example, as we illustrate in Fig.\[fig:fig6suppl\], [`muxViz`]{} can account for spatial information by creating visualizations of multilayer networks that are embedded in geographical regions. ]{}]{} Compression of layers and reducibility dendrogram ------------------------------------------------- \[supp:note:2\] [[ An important open question is the determination of how much information is necessary to accurately represent the structure of multilayer systems and whether it is possible to aggregate some layers without loss of information. It was shown recently that it is possible to [[ compress]{}]{} the number of layers in multilayer networks in a way that minimizes information loss by using an information-theoretic approach [@dedomenico2014reducibility]. The methodology of  [@dedomenico2014reducibility], which we implemented in [`muxViz`]{}, amounts to a tradeoff (which is “optimal” in some sense) between accuracy and complexity. Alternatively, users of [`muxViz`]{} can implement alternative methods based on different notions of “minimal information loss”. ]{}]{} [[ The [[ compression]{}]{} procedure from  [@dedomenico2014reducibility] proceeds as follows. For each pair of layers in the original multilayer network, [`muxViz`]{} calculates the quantum Jensen–Shannon (JS) divergence [@majtey2005jensen]. This estimates the similarity between two networks based on their Von Neumann entropy [@braunstein2006laplacian]. By definition, the quantum Jensen–Shannon divergence is symmetric and its square root, which is usually called the Jensen–Shannon distance, satisfies the properties of a metric [@briet2009properties]. One can use the JS distance to quantify the distance in terms of information gain (or loss) between the normalized Laplacian matrices that are associated to two distinct networks [@dedomenico2014reducibility]. ]{}]{} [[ One places the distances between every pair of layers as the components of a matrix, and one can then perform hierarchical clustering [@dataclustering] using any desired method to produce a dendrogram that indicates the relatedness of the information in the different layers. In [`muxViz`]{}, we have included several methods for hierarchical clustering (e.g., Ward, McQuitty, single, complete, average, median, and centroid linkage clusterings). We show an example of such a “reducibility dendrogram” in panel [**(D)**]{} of Figs.\[fig:fig2-1suppl\], \[fig:fig3-1suppl\], \[fig:fig4-1suppl\], and \[fig:fig5-1suppl\]. A reducibility dendrogram results from a step-by-step merging of a set of layers in a multilayer network, and we calculate a quality function based on the relative Von Neumann entropy to estimate information gain (or loss) at each step [@dedomenico2014reducibility]. To obtain a reduced version of the original multilayer network, we stop the merging procedure at the level of the hierarchy that maximizes the relative entropy. ]{}]{} Annular visualization of multilayer information {#supp:note:3} ----------------------------------------------- [[ It is a challenging problem to represent, visualize, and analyze the wealth of information encoded in the multilayer structure of networks in a compact way. Preserving more information by using multilayer networks rather than ordinary networks complicates the visualization and analysis even further. However, this complication is necessary, because otherwise one might end up with misleading or even incorrect results [@kivela2013multilayer]. We developed the [`muxViz`]{} software to help address these challenges. To summarize all of the information obtained from multilayer-network calculations in a compact way, [`muxViz`]{} includes an annular visualization that facilitates the ability to capture patterns and deduce qualitative information about multilayer data. ]{}]{} [[ To give a concrete example, many researchers are interested in ranking the relative importance of nodes (and other network structures), which traditionally is accomplished using various “centrality” measures. Centralities have been calculated for single-layer networks for several decades [@Wasserman1994Social; @newman2010], and numerous notions of centrality are now also available for multilayer networks [@dedomenico2013centrality; @kivela2013multilayer]. It is therefore necessary to develop visualization tools that make it possible to compare such a wealth of diagnostics to each other in a compact, meaningful way. For example, it is often worthwhile to focus attention on one descriptor and compare the values obtained in each layer separately to the values obtained from the multilayer network and its aggregations. This is easy to do using the [`muxViz`]{} software. ]{}]{} [[ We will now illustrate our annular visualization (see Figs.\[fig:fig2-2suppl\], \[fig:fig3-2suppl\], \[fig:fig4-2suppl\], and \[fig:fig5-2suppl\]) using the example of multilayer centrality measures. Suppose that we have different arrays of information, where one should think of each array as having resulted from the calculation of some centrality diagnostic on a multilayer network. We visualize each array using a ring. The angle indicates node identity (regardless of the layer or layers in which it occurs). We bin the centrality values—e.g., either linearly or logarithmically—and we assign a color to each bin to encode its value. Both the type of binning and the color scheme are customizable in [`muxViz`]{}. We place the rings concentrically, and one can determine both the ring order and ring thicknesses according to any desired criteria. For example, in the visualizations in the present paper, we determine the thickness of each ring according to its information content (which we quantify using the Shannon information entropy of the distribution of the values): thinner rings have less information. Users can customize the order of the rings; in [`muxViz`]{}’s default setting, it is determined automatically via hierarchically clustering. The [`muxViz`]{} software calculates a measure of correlation (e.g., Pearson, Spearman, or JS divergence) between each pair of descriptors to obtain a set of pairwise distances, which we then hierarchically cluster to group the rings. This clustering procedure determines the order of the rings to try to maximize the readability of the annular plot. ]{}]{} [[ One can also use the same principles when fixing some centrality descriptor and letting the rings correspond to the layers in a network, the multilayer network, and an aggregated network (see Section \[anal\]). Such a plot might help to reveal, for instance, if the centrality of nodes in a multilayer network is primarily due to their centrality in a specific layer or if the aggregated network is a good proxy for the multilayer structure. ]{}]{} Analyses of empirical multilayer networks {#anal} ========================================= To demonstrate the ability of [`muxViz`]{} to analyze and visualize multilayer networks, we consider different types of genetic interactions for organisms in the Biological General Repository for Interaction Datasets[@stark2006biogrid] (BioGRID, [thebiogrid.org](thebiogrid.org)), a public database that archives and disseminates genetic and protein interaction data from humans and model organisms. BioGRID currently includes more than 720,000 interactions that have been curated from both high-throughput data sets and individual focused studies using over 41,000 publications in the primary literature. We use BioGRID 3.2.108 (updated 1 Jan 2014). In this section, we focus on *Xenopus laevis* and show a network visualization in Fig.\[fig:fig2-1suppl\]**C**. We give results of computations using [`muxViz`]{} in the other panels of Fig.\[fig:fig2-1suppl\]. See Section\[supp:note:4\] for other examples. ![**Multilayer analysis of a *Xenopus laevis* genetic-interaction network.** See Section\[supp:note:4\] for details about each panel. \[In this figure and all subsequent figures, we have purposely kept font sizes at [`muxViz`]{}’s default level rather than increasing them.\] []{data-label="fig:fig2-1suppl"}](fig2-1_suppl.pdf){width="16cm"} ![**Multilayer analysis of a *Xenopus laevis* genetic-interaction network.** See Section\[supp:note:4\] for details about each panel. []{data-label="fig:fig2-2suppl"}](fig2-2_suppl.pdf){width="16cm"} One can examine the global organization of nodes into modules (i.e., “communities”) through an algorithmic calculation of community structure [@Porter2009; @fortunato2010community]. For example, one can obtain dense communities in multilayer networks by optimizing a multilayer generalization of the modularity quality function [@mucha2010community]. To do this, one takes into account both intralayer and interlayer edges, and one seeks densely connected sets of nodes (i.e., communities) that are sparsely connected to each other as compared to some multilayer random-graph (null) model [@mucha2010community; @Bassett2013Robust; @kivela2013multilayer]. See Fig.\[fig:fig2-1suppl\]**A** for a visualization of communities in *Xenopus laevis* and Section\[supp:note:4\] for other examples. As we discussed previously, one can quantify the importance of a node by using various diagnostics to measure “centrality”. One calculates such a centrality (and a corresponding rank order) for each node by using multilayer generalizations of centrality measures [@dedomenico2013mathematical; @dedomenico2013centrality; @kivela2013multilayer]. The software [`muxViz`]{} has tools for calculating multilayer generalizations of several different types of centrality (e.g., degree, eigenvector [@bonacich1972factoring], hub and authority [@kleinberg1999authoritative], PageRank [@brin1998anatomy], and Katz [@katz1953new]) either for an entire multilayer network or for each layer separately. As we illustrate in Fig.\[fig:fig2-1suppl\]**B**, centrality values (as well as other network measures) can be very different in multilayer networks than in their corresponding aggregations. Such results influence how one should interpret calculations of network measures for, e.g., which genes or proteins are most important for activating or suppressing a given biological processes [[ or which people are most important in social networks.]{}]{} The data in question is multilayer, so the analysis of such data must take multilayer features into account. Researchers are often also interested in considering a “[[ compressed]{}]{} version” of multilayer data sets that preserve as much information as possible without altering the primary descriptors. For such scenarios, it is possible to use the [[ compression procedure]{}]{} discussed in Section\[supp:note:2\] to identify the layers of a multilayer network that are providing redundant information [@dedomenico2014reducibility] [[ (see Fig.\[fig:fig2-1suppl\]**D**)]{}]{}. In Fig.\[fig:fig2-1suppl\]**E**, we show three correlation measures for multilayer networks: (left) mean edge overlap, (center) degree-degree Pearson correlation coefficient, and (right) degree-degree Spearman correlation coefficient. In this example, the degree-degree Pearson and Spearman correlation coefficients between layers quantify the tendency of nodes to be hubs in different layers simultaneously. The [`muxViz`]{} software can include additional correlation measures, and it is easy for users to implement other diagnostics [@nicosia2013correlations]. To summarize all of the information that one obtains from calculations like the ones above in a compact figure, we use an annular visualization (see Section\[supp:note:3\]) that facilitates the ability to capture patterns to deduce qualitative information about multilayer data. In Fig.\[fig:fig2-2suppl\] (see the panel labelled “Multiplex”), we show an example for centrality diagnostics, which measure the importance of nodes in various ways. Each ring indicates a centrality measure, and the angle determines the identity of a node in a network, regardless of the layer(s) in which it exists. One can use the same principles when fixing some centrality descriptor and letting the rings correspond to the layers in a network, the multilayer network, and an aggregated network (see the other panels in Fig.\[fig:fig2-2suppl\]). For the case of layers, one calculates a centrality measure for each layer separately without accounting for multilayer structure. For instance, it is evident that rings 3 (“DirInt” layer) and 5 (“PhAssoc” layer) are negatively correlated in the case of strength centrality because nodes tend to have opposite colors, whereas rings 6 (aggregated network) and 7 (multiplex network) are positively correlated, as expected for strength centrality. Our annular representation makes it easy to see similarity (or dissimilarity) in rank orderings according to different diagnostics. For example, their patterns reveal that physical association and direct interaction are dominant and determine the multilayer strength [[ in the depicted example.]{}]{} In other cases (see Section\[supp:note:4\]), the ranking by some centrality measure in the multilayer network is poorly correlated to the ranking in either an aggregated network or in individual layers separately. [[ This underscores the value of using a multilayer framework for the calculation of the most central proteins (and, more generally, for determining which entities in many complex systems are most important).]{}]{} Analysis of other empirical multilayer networks {#supp:note:4} ----------------------------------------------- In this section, we present multilayer analyses of three additional biological systems to illustrate the power of [`muxViz`]{}. We examine the following examples: - *Caenorhabditis elegans* connectome (see Figs.\[fig:fig3-1suppl\] and \[fig:fig3-2suppl\]); - *Herpes simplex* genetic-interaction network (see Figs.\[fig:fig4-1suppl\] and \[fig:fig4-2suppl\]); - *HIV-1* genetic-interaction network (see Figs.\[fig:fig5-1suppl\] and \[fig:fig5-2suppl\]). As for the case of *Xenopus laevis*, we include two figures for each example. In the first set of figures (see \[fig:fig3-1suppl\], \[fig:fig4-1suppl\], and \[fig:fig5-1suppl\]), we show the following information: - **Panel A**: Multilayer community structure from modularity maximization [@mucha2010community]. The color of each node encodes its community assignment in a multilayer-network visualization. For comparison, we also show the results (and corresponding visualization) of community detection on an aggregated network, which we obtain by summing the corresponding intralayer edge weights of all layers. (In other words, if $A_{ijs}$ gives the edge weight between nodes $i$ and $j$ on layer $s$, then we obtain an aggregated edge weight $W_{ij}$ between nodes $i$ and $j$ by summing over $s$.) - **Panel B:** Multilayer PageRank centrality [@dedomenico2013centrality]. We again use a multilayer-network visualization. We label the top five nodes from a ranking according to multilayer PageRank centrality. For comparison, we also show the results of PageRank centrality calculations on the aforementioned aggregated network. - **Panel C:** Edge-colored multigraph visualization of the network. We color edges according to the layer to which they belong. We color the nodes according to their layer (or layers); if a node exists on multiple layers, then we distribute its corresponding colors evenly. - **Panel D:** [[ Compressibility]{}]{} analysis and corresponding reducibility dendrogram [@dedomenico2014reducibility]. We show the distance matrix and the corresponding dendrogram, which we obtain using Ward hierarchical clustering. - **Panel E:** Measures of correlation between layers: (left) mean edge overlap, (center) degree-degree Pearson correlation coefficient, and (right) degree-degree Spearman correlation coefficient. In the second set of figures (see Figs.\[fig:fig3-2suppl\], \[fig:fig4-2suppl\], and \[fig:fig5-2suppl\]), we show the annular visualization for the centrality descriptors: - In panels titled “Multiplex”, we consider the multilayer network. Each ring corresponds to a different centrality descriptor. - In the other panels, we consider a specific centrality descriptor (which we specify in the title of the panel). Each ring encodes the values of that descriptor, which we calculate in each layer separately. We also include rings for the calculation of the corresponding centrality diagnostic in the multilayer network and in its aggregation to a single-layer weighted network. We specify the order of the rings in the list of labels on the right of each plot. In each case, the top label refers to the innermost ring and the bottom label refers to the outermost ring. ![**Multilayer analysis of a *Caenorhabditis elegans* connectome.** See Section\[supp:note:4\] for details about each panel.[]{data-label="fig:fig3-1suppl"}](fig3-1_suppl.pdf){width="16cm"} ![**Multilayer analysis of a *Caenorhabditis elegans* connectome.** See Section\[supp:note:4\] for details about each panel.[]{data-label="fig:fig3-2suppl"}](fig3-2_suppl.pdf){width="16cm"} ![**Multilayer analysis of a *Herpes simplex* genetic-interaction network.** See Section\[supp:note:4\] for details about each panel.[]{data-label="fig:fig4-1suppl"}](fig4-1_suppl.pdf){width="16cm"} ![**Multilayer analysis of a *Herpes simplex* genetic-interaction network.** See Section\[supp:note:4\] for details about each panel. Note that we do not show eigenvector centrality because one layer consists of a directed acyclic graph (for which eigenvector centrality is unilluminating [@newman2010]). []{data-label="fig:fig4-2suppl"}](fig4-2_suppl.pdf){width="16cm"} ![**Multilayer analysis of *HIV-1* genetic-interaction network.** See Section\[supp:note:4\] for details about each panel.[]{data-label="fig:fig5-1suppl"}](fig5-1_suppl.pdf){width="16cm"} ![**Multilayer analysis of *HIV-1* genetic-interaction network.** See Section\[supp:note:4\] for details about each panel.[]{data-label="fig:fig5-2suppl"}](fig5-2_suppl.pdf){width="16cm"} Conclusion ========== In the current era of “big data”, there is now an intense deluge of multilayer data. To avoid throwing away important information or obtaining misleading results, it is increasingly crucial to use methods that exploit multilayer structure. In this paper, we present new software and associated methodology that exploits the new paradigm of multilayer networks, and we illustrate how it can be used to analyze and visualize several examples. Our software, [`muxViz`]{}, provides an open-source framework for the analysis of multilayer networks. Additionally, the modular structure of [`muxViz`]{} — along with its open-source license — makes it easy to add new methods. [[ Moreover, although we have focused on examples of biological networks, [`muxViz`]{} is clearly also useful for multilayer networks from any other setting.]{}]{} As we illustrate in Fig.\[fig:fig6suppl\], it can even be overlaid over spatial information. Acknowledgements {#acknowledgements .unnumbered} ================ All authors were supported by the European Commission FET-Proactive project PLEXMATH (Grant No. 317614). AA also acknowledges financial support from the Generalitat de Catalunya 2009-SGR-838, the ICREA Academia, and the James S. McDonnell Foundation. MAP acknowledges a grant (EP/J001759/1) from the EPSRC. We thank Serafina Agnello for support with graphics. Technical Details About [`muxViz`]{} {#supp:note:5} ==================================== We developed [`muxViz`]{} using R (<http://www.r-project.org/>), a free and widely-adopted framework for statistical computing, and GNU Octave (<https://www.gnu.org/software/octave/>), an open-source high-level interpreted language that is intended primarily for numerical computations. The Octave language is very similar to the proprietary environment [Matlab]{} (<http://www.mathworks.es/products/matlab/>), and one can import the code to [Matlab]{} in a straightforward manner. The [`muxViz`]{} software requires R 3.0.2 (or above) and Octave 3.4.0 (or above). The [`muxViz`]{} framework is a free and open-source package for the analysis and the visualization of multilayer networks. It is released under GNU General Public License v3 (<https://www.gnu.org/copyleft/gpl.html>) and exploits R to provide an easy and accessible user interface for the visualization of networks, the calculation of network diagnostics, and the visual representation of the results of calculations. Specifically, R allows the construction of a graphical user interface (GUI), which can be used either locally (client-side software) or via the internet (remote Web server), and an Octave library that we developed performs calculations of matrices and tensors. Using [`muxViz`]{} is simple and does not require any programming skill; one can do all computations and visualization via the user interface. Additionally, because of [`muxViz`]{}’s modular structure, users can also create their own modules for calculating new diagnostics and for customizing visual representations. The [`muxViz`]{} framework allows both two-dimensional and three-dimensional visualization of networks. The latter exploits OpenGL technology, so users can interactively change the perspective and navigate the network. We show representative static snapshots of such interactive visualizations in Figs.\[fig:fig1suppl\]**B** and **D** and in panel **C** of Figs.\[fig:fig2-1suppl\], \[fig:fig3-1suppl\], \[fig:fig4-1suppl\], and \[fig:fig5-1suppl\]. [^1]: manlio.dedomenico@urv.cat

--- abstract: 'Using magnetocapacitance data in tilted magnetic fields, we directly determine the chemical potential jump in a strongly correlated two-dimensional electron system in silicon when the filling factor traverses the spin and the cyclotron gaps. The data yield an effective $g$ factor that is close to its value in bulk silicon and does not depend on filling factor. The cyclotron splitting corresponds to the effective mass that is strongly enhanced at low electron densities.' author: - 'V. S. Khrapai, A. A. Shashkin, and V. T. Dolgopolov' title: | Direct measurements of the spin and the cyclotron gaps\ in a 2D electron system in silicon --- A two-dimensional (2D) electron system in silicon metal-oxide-semiconductor field-effect transistors (MOSFETs) is remarkable due to strong electron-electron interactions. The Coulomb energy overpowers both the Fermi energy and the cyclotron energy in accessible magnetic fields. The Landau-level-based considerations of many-body gaps [@ando; @yang], which are valid in the weakly interacting limit, cannot be directly applied to this strongly correlated electron system. In a perpendicular magnetic field, the gaps for charge-carrying excitations in the spectrum should originate from cyclotron, spin, and valley splittings and be related to a change of at least one of the following quantum numbers: Landau level, spin, and valley indices. However, the gap correspondence to a particular single-particle splitting is not obvious [@brener], and the origin of the excitations is unclear. In a recent theory [@iordan], the strongly interacting limit has been studied, and it has been predicted that in contrast to the single-particle picture, the many-body gap to create a charge-carrying (iso)spin texture excitation at integer filling factor is determined by the cyclotron energy. This is also in contrast to the square-root magnetic field dependence of the gap expected in the weakly interacting limit [@ando; @yang]. A standard experimental method for determining the gap value in the spectrum of the 2D electron system in a quantizing magnetic field is activation energy measurements at the minima of the longitudinal resistance [@englert; @klein; @dol88; @usher]. Its disadvantage is that it yields a mobility gap which may be different from the gap in the spectrum. In Si MOSFETs, the activation energy as a function of magnetic field was reported to be close to half of the single-particle cyclotron energy for filling factor $\nu=4$, while decreasing progressively for the higher $\nu$ cyclotron gaps [@englert; @klein; @dol88]. At low electron densities, an interplay was observed between the cyclotron and the spin gaps, manifested by the disappearance of the cyclotron ($\nu=4$, 8, and 12) minima of the longitudinal resistance [@krav00]. On the contrary, for the 2D electrons in GaAs/AlGaAs heterostructures, the activation energy at $\nu=2$ exceeded half the single-particle cyclotron energy by about 40% [@usher]. Another, direct method for determining the gap in the spectrum is measurement of the chemical potential jump across the gap [@smith85; @aristov; @valley]. It was applied to the 2D electrons in GaAs [@smith85] and gave cyclotron gap values corresponding to the band electron mass [@aristov]. Recently, the method has been used to study the valley gap at the lowest filling factors in the 2D electron system in silicon which has been found to be strongly enhanced and increase linearly with magnetic field [@valley; @rem]. The effective electron mass, $m$, and $g$ factor in Si MOSFETs have been determined lately from measurements of the parallel magnetic field of full spin polarization in this electron system and of the slope of the metallic temperature dependence of the conductivity in zero magnetic field [@gm]. It is striking that the effective mass becomes strongly enhanced with decreasing electron density, $n_s$, while the $g$ factor remains nearly constant and close to its value in bulk silicon. This result is consistent with accurate measurements of $m$ at low $n_s$ by analyzing the temperature dependence of the Shubnikov-de Haas oscillations in weak magnetic fields in the low-temperature limit [@us; @rem1]. A priori it is unknown whether or not the so-determined values $g$ and $m$ correspond to the spin and the cyclotron splittings in strong perpendicular magnetic fields. In this paper, we report the first measurements of the chemical potential jump across the spin and the cyclotron gaps in a 2D electron system in silicon in tilted magnetic fields using a magnetocapacitance technique. We find that (i) the $g$ factor is close to its value in bulk silicon and does not change with filling factor, in contrast to the strong dependence of the valley gap on $\nu$; and (ii) the cyclotron splitting is determined by the effective mass that is strongly enhanced at low electron densities. We also verify the systematics of the gaps in that the measured $\nu=4$, 8, and 12 cyclotron gap decreases with parallel magnetic field component by the same amount as the $\nu=2$, 6, and 10 spin gap increases. Measurements were made in an Oxford dilution refrigerator with a base temperature of $\approx 30$ mK on high-mobility (100)-silicon MOSFETs (with a peak mobility close to 2 m$^2$/Vs at 4.2 K) having the Corbino geometry with diameters 250 and 660 $\mu$m. The gate voltage was modulated with a small ac voltage 15 mV at frequencies in the range 2.5 – 25 Hz and the imaginary current component was measured with high precision using a current-voltage converter and a lock-in amplifier. Care was taken to reach the low frequency limit where the magnetocapacitance, $C(B)$, is not distorted by lateral transport effects. A dip in the magnetocapacitance at integer filling factor is directly related to a jump, $\Delta$, of the chemical potential across a corresponding gap in the spectrum of the 2D electron system, and therefore we determine $\Delta$ by integrating $C(B)$ over the dip in the low temperature limit where the magnetocapacitance saturates and becomes independent of temperature [@valley]. Typical magnetocapacitance traces taken at different electron densities, temperatures, and tilt angles are displayed in Fig. \[fig1\] near the filling factor $\nu=hcn_s/eB_\perp=4$ and $\nu=6$. The magnetocapacitance shows narrow minima at integer $\nu$ which are separated by broad maxima, the oscillation pattern reflecting the modulation of the thermodynamic density of states, $D$, in quantizing magnetic fields: $1/C=1/C_0+1/Ae^2D$ (where $C_0$ is the geometric capacitance between the gate and the 2D electrons, and $A$ is the sample area) [@smith85]. As the magnetic field is increased, the maximum $C$ approaches the geometric capacitance indicated by the dashed lines in Fig. \[fig1\]. Since the magnetocapacitance $C(B)<C_0$ around each maximum is almost independent of magnetic field, this results in asymmetric minima of $C(B)$, the asymmetry being more pronounced for $\nu=4$, 8, and 12. The chemical potential jump at integer $\nu=\nu_0$ is determined by the area of the dip in $C(B)$: $$\Delta=\frac{Ae^3\nu_0}{hcC_0}\int_{\text{dip}}\frac{C_{\text{ref}}- C}{C}dB_\perp, \label{Delta}$$ where $C_{\text{ref}}$ is a step function that is defined by two reference levels corresponding to the capacitance values at the low and high field edges of the dip as shown by the dotted line in Fig. \[fig1\]. The so-determined $\Delta$ is smaller than the level splitting by the level width. The last is extracted from the data by substituting $(C_0-C_{\text{ref}})/C$ for the integrand in Eq. (\[Delta\]) and integrating for the case of resolved levels between the magnetic fields $B_1=hcn_s/e(\nu_0+1/2)$ and $B_2=hcn_s/e(\nu_0-1/2)$. Tilting the magnetic field allows us to verify the systematics of the gaps in the spectrum and probe the lowest-energy charge-carrying excitations. As the thickness of the 2D electron system in Si MOSFETs is small compared to the magnetic length in accessible fields, the parallel field couples largely to the electrons’ spins while the orbital effects are suppressed [@simonian]. Therefore, the variation of a gap with $B_\parallel$ should reflect the change in the excitation energy as the Zeeman splitting, $g\mu_BB$, is increased: the excitation energy change is determined by the difference between the spin projections onto magnetic field for the ground and the lowest excited states. Within single-particle picture, e.g., one can expect that with increasing $B_\parallel$ at fixed $B_\perp$, the spin gap will increase, the valley gap will stay constant, and the cyclotron gap, which is given by the difference between the cyclotron splitting and the sum of the spin and the valley splittings, will decrease. In contrast, for spin textures (so-called skyrmions), the dependence of the excitation energy on $B_\parallel$ should be much stronger compared to the single-particle Zeeman splitting [@yang]. In Fig. \[fig2\](a), we show the value of the chemical potential jump, $\Delta_s$, across the $\nu=2$ and $\nu=6$ gaps as a function of magnetic field for different tilt angles. It is insensitive to both filling factor and tilt angle, as expected for spin gaps. The data are best described by a proportional increase of the gap with the magnetic field with a slope corresponding to an effective $g$ factor $g\approx 1.75$. The so-determined value obviously gives a lower boundary for the $g$ factor because both the valley splitting at odd $\nu$ and the level width are disregarded. In Fig. \[fig2\](b), we show how the $\nu=6$ gap changes with $B_\parallel$ at different values of the perpendicular field component. It is noteworthy that the level width contribution, which is indicated by systematic error bars, depends weakly on parallel field, and the valley splitting has been verified to be independent of $B_\parallel$. This, therefore, allows more accurate determination of the $g$ factor as shown by the solid line in Fig. \[fig2\](b). Its slope yields $g\approx 2.6$, which is in agreement with the data obtained for the $\nu=2$ and $\nu=10$ gaps. The fact that this value is close to the $g$ factor $g=2$ in bulk silicon points to the single spin-flip origin of the excitations for the $\nu=2$, 6, and 10 gaps. Unlike spin gaps, the chemical potential jump, $\Delta_c$, across the $\nu=4$, 8, and 12 gaps decreases with parallel magnetic field component, as already seen from Fig. \[fig1\]. In Fig. \[fig3\](a), we compare the behaviors of the $\nu=6$ and $\nu=4$ gaps with $B_\parallel$ at fixed perpendicular field component. For $B_\perp$ between 2.7 and 6.6 T, the absolute values of the slopes of these dependences are equal, within experimental uncertainty, to each other so that the sum of the gaps is approximately constant even if the level width contribution is taken into account. These results lead to two important consequences: (i) the $\nu=4$, 8, and 12 gaps are cyclotron ones, the conventional systematics of the gaps remaining valid in the studied electron density range down to $1.5\times 10^{11}$ cm$^{-2}$; and (ii) the $g$ factor does not vary with filling factor $\nu$. Although our value of $g\approx 2.6$ is in agreement with the previously measured ones [@englert; @gm; @us], we do not confirm the conclusion on oscillations of the $g$ factor with $\nu$ based on activation energy measurements and made in line with theoretical predictions [@ando; @afs] under the assumption of $B_\parallel$-independent level width [@englert]. In Fig. \[fig3\](b), we compare the data for the chemical potential jump across the $\nu=4$, 8, and 12 gaps in perpendicular and tilted magnetic fields including the term $g\mu_B(B-B_\perp)$ that describes the increase of the spin gap with $B_\parallel$. The data coincidence confirms that the changing spin gap is the only cause for the dependence of the cyclotron gap on parallel field component. As is evident from the figure, $\Delta_c$ is considerably smaller than the value ($\hbar\omega_c-2\mu_BB_\perp$) expected within single-particle approach ignoring both valley splitting and level width. To reduce experimental uncertainty related to the inaccurate determination of the level width, we plot in Fig. \[fig4\] the difference, $(\Delta_c-\Delta_s)/2\mu_BB$, of the normalized values of the cyclotron and the spin gaps in a perpendicular magnetic field as a function of electron density. Assuming that the cyclotron splitting is determined by the effective mass $m$, this difference corresponds to ($m_e/m-g$), where $m_e$ is the free electron mass. Using data for $m$ and $g$ obtained in both parallel [@gm] and weak [@us; @smith72] magnetic fields, we find that the value ($m_e/m-g$) is indeed consistent with our data, see Fig. \[fig4\]. The effective mass determined from our high-$n_s$ data using $g=2.6$ is equal to $m\approx 0.23m_e$, which is close to the band mass of $0.19m_e$. As long as our $g$ value is constant, the decrease of the normalized gap difference with decreasing $n_s$ reflects the behavior of the cyclotron splitting, which is in agreement with the conclusion of the strongly enhanced effective mass at low electron densities [@gm; @us]. We now discuss comparatively the results obtained for the valley and the spin gaps. According to Ref. [@valley], the enhanced valley gap at the lowest filling factors $\nu=1$ and $\nu=3$ in Si MOSFETs is comparable to the single-particle Zeeman splitting. As our data for the spin gap correspond to the single-particle Zeeman splitting, this may lead to a different systematics of the gaps in the spectrum compared to the single-particle picture. Such a possibility has been supported by a recent theory [@brener] which shows the importance of the Jahn-Teller effect for the ground state of a 2D electron system in bivalley (100)-Si MOSFETs in quantizing magnetic fields. At $\nu=2$, due to static and dynamic lattice deformations, the valley degeneracy is predicted to lift off giving rise to a complicated phase diagram including three phases: spin-singlet, canted antiferromagnet, and ferromagnet. In our experiment, over the studied range of magnetic fields down to 3 T, we observe at $\nu=2$ a spin-ferromagnetic ground state only. This gives an estimate of the strength of the suggested mechanism for the valley splitting enhancement. The fact that we do not observe oscillations of the $g$ factor as a function of $\nu$ is not too surprising, because our value of $g$ is close to the $g$ factor in bulk silicon so that those oscillations may be small. At the same time, our data for the $g$ factor allow us to arrive at a conclusion that at $\nu=2$, the valley gap is small compared to the spin gap. Therefore, the valley splitting does oscillate with filling factor [@ando], the conclusion being valid, at least, for the strongly enhanced gaps at $\nu=1$ and $\nu=3$. We stress that this effect occurs in the strongly correlated electron system, which is beyond the conventional theory of exchange-enhanced gaps [@ando]. Let us finally discuss the results obtained for the cyclotron gap. The data of Fig. \[fig4\] indicate unequivocally that the origin of the small $\Delta_c$ value in Fig. \[fig3\](b) is not related to valley splitting and level width. Instead, it is renormalization of the effective mass and $g$ factor due to electron-electron interactions: the observed decrease of the gap difference with decreasing $n_s$ in Fig. \[fig4\] as well as the systematics of the gaps are in agreement with both the decrease of the ratio of the cyclotron and the spin gaps with decreasing $n_s$ [@krav00] and the sharp increase of the effective mass at low electron densities [@gm; @us]. Needless to say that the conventional theory [@ando] yields an opposite sign of the interaction effect on the cyclotron splitting. We gratefully acknowledge discussions with I. L. Aleiner, S. V. Iordanskii, A. Kashuba, and S. V. Kravchenko. This work was supported by the RFBR, the Russian Ministry of Sciences, and the Programme “The State Support of Leading Scientific Schools”. 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--- abstract: 'In this paper, we study the mathematical structure and numerical approximation of elliptic problems posed in a (3D) domain $\Omega$ when the right-hand side is a (1D) line source $\Lambda$. The analysis and approximation of such problems is known to be non-standard as the line source causes the solution to be singular. Our main result is a splitting theorem for the solution; we show that the solution admits a split into an explicit, low regularity term capturing the singularity, and a high-regularity correction term $w$ being the solution of a suitable elliptic equation. The splitting theorem states the mathematical structure of the solution; in particular, we find that the solution has anisotropic regularity. More precisely, the solution fails to belong to $H^1$ in the neighbourhood of $\Lambda$, but exhibits piecewise $H^2$-regularity parallel to $\Lambda$. The splitting theorem can further be used to formulate a numerical method in which the solution is approximated via its correction function $w$. This approach has several benefits. Firstly, it recasts the problem as a 3D elliptic problem with a 3D right-hand side belonging to $L^2$, a problem for which the discretizations and solvers are readily available. Secondly, it makes the numerical approximation independent of the discretization of $\Lambda$; thirdly, it improves the approximation properties of the numerical method. We consider here the Galerkin finite element method, and show that the singularity subtraction then recovers optimal convergence rates on uniform meshes, i.e., without needing to refine the mesh around each line segment. The numerical method presented in this paper is therefore well-suited for applications involving a large number of line segments. We illustrate this by treating a dataset (consisting of $\sim 3000$ line segments) describing the vascular system of the brain.' address: - 'Department of Mathematics, University of Bergen, Norway. ' - 'Department of Mathematics, Karlstad University, Sweden. ' - 'Department of Mathematics, Technical University of Munich, Germany. ' author: - 'Ingeborg G. Gjerde' - Kundan Kumar - 'Jan M. Nordbotten' - Barbara Wohlmuth date: 'October 25, 2018' title: Splitting method for elliptic equations with line sources --- [^1] Introduction ============ Decomposition and regularity properties ======================================= Numerical methods ================= Numerical results ================= **Acknowledgements** The authors thank J. Reichenbach and A. Deistung for bringing our attention to the data used in section \[sec:num-brain\] [@brain], and E. Hanson and E. Hodneland for providing us with the data segmentation and tree extraction. Conclusions =========== We studied an elliptic equation having line sources in a 3D domain. The line sources act as Dirac measure defined on a line causing the solutions to be singular on the line itself. Central to this work is the result that the solution admits a split into a singular and a regular part. This allows us to study the nature of the solution as well as to develop a numerical algorithm for solving the problem. Mathematically, we see that the solution has anisotropic regularity, it is smooth along the line source and the line singularity acts as a Dirac point measure in a 2D domain. Our numerical approach solves for the regular part only and therefore obtains optimal convergence rates. We illustrate our approach for several numerical examples including a data set describing the vascular system of a human brain. Our solution approach is mesh-independent and can be adapted to a variety of discretizations. \[sec:conclusions\] [^1]: This work was partially supported by the Research Council of Norway, project number 250223, and Deutsche Forschungsgemeinschaft, grant number WO-671/11-1.

--- abstract: 'Small non-autonomous perturbations around an equilibrium of a nonlinear delayed system are studied. Under appropriate assumptions, it is shown that the number of $T$-periodic solutions lying inside a bounded domain $\Omega\subset \R^N$ is, generically, at least $|\chi \pm 1|+1$, where $\chi$ denotes the Euler characteristic of $\Omega$. Moreover, some connections between the associated fixed point operator and the Poincaré operator are explored.' author: - 'P. Amster' - 'M. P. Kuna' - 'G. Robledo' title: Multiple solutions for periodic perturbations of a delayed autonomous system near an equilibrium --- Introduction ============ Let $\Omega\subset \R^N$ be a bounded domain with smooth boundary. An elementary result from the theory of ODEs establishes that if a smooth function $G:\overline\Omega\to \R^N$ is inwardly pointing over $\partial\Omega$, that is $$\label{hart-weak} \langle G(x),\nu(x)\rangle <0 \qquad x\in \partial\Omega,$$ where $\nu(x)$ denotes the outer normal at $x$, then the solutions of the autonomous system of ordinary differential equations $$u'(t)=G(u(t))$$ with initial data $u(0)=u_0\in \overline \Omega$ are defined and remain inside $\Omega$ for all $t>0$. [Now, let us denote the space of $T$–periodic continuous functions as $$C_T:=\{u\in C(\R,\R^N):u(t+T)=u(t)\}$$ and, for given $p\in C_{T}$, consider the non-autonomous system $$u'(t)=G(u(t)) + p(t).$$]{} [If $\overline\Omega$ has the fixed point property, then the above system has at least one $T$-periodic orbit, provided that $\|p\|_\infty$ is small.]{} This is a straightforward consequence of the fact that the time-dependent vector field $G(x)+ p(t)$ is still inwardly pointing for all $t$; hence, the set $\overline \Omega$ is invariant for the associated flow and thus the Poincaré operator given by $Pu_0:=u(T)$ is well defined for $u_0\in \overline\Omega$ and satisfies $P(\overline\Omega)\subset \overline\Omega$. More generally, observe that, when (\[hart-weak\]) is assumed, the homotopy defined by $h(x,s):= sG(x) - (1-s)\nu(x)$ with $s\in [0,1]$ does not vanish on $\partial\Omega$; whence $$deg_B(G,\Omega,0) = deg_B(-\nu,\Omega,0),$$ where $deg_B$ stands for the Brouwer degree. Thus, it follows from [@hopf] that $deg_B(G,\Omega,0)=(-1)^N\chi(\Omega)$, where $\chi(\Omega)$ denotes the Euler characteristic of $\Omega$. It is worthy to recall (see *e.g.*,[@wecken]) that if $\overline \Omega$ has the fixed point property, then $\chi(\Omega)$ is different from $0$. This follows easily in the present setting from the fact that if $\chi(\Omega)=0$ then one can construct a field $G$ satisfying (\[hart-weak\]) that does not vanish in $\Omega$. If $\overline\Omega$ has the fixed point property, then there exist (non-constant) $T$-periodic solutions of all periods which, in turn, implies that $G$ vanishes, a contradiction. Interestingly, the converse of the result in [@wecken] is not true; that is, one can easily find $\Omega$ with nonzero Euler characteristic such that $\overline \Omega$ has not the fixed point property. For such a domain, the Poincaré map has obviously a fixed point (because $G$ vanishes in $\Omega$). This yields the conclusion that a fixed point-free map in $C(\overline \Omega,\overline\Omega)$ cannot belong to the closure of the set of all the Poincaré maps associated to the homotopy class of $-\nu$. Now suppose, independently of the value of $\chi(\Omega)$, that $G$ vanishes at some point $e\in \Omega$, namely, that $e$ is an equilibrium point of the autonomous system. It is well known that if $M:=DG(e)$ is nonsingular, then the degree of $G$ over any small neighbourhood $V$ of $e$ is well defined and coincides with $s(M)$, where $$\label{sM} s(M):= sgn ({\rm det}(M)).$$ Thus, if $s(M)$ is different from $(-1)^N\chi(\Omega)$, then the excision property of the degree implies that the system has at least another equilibrium point in $\Omega\setminus \overline V$. Furthermore, it follows from Sard’s lemma that, for almost all values $\overline p$ in a neighbourhood of $0\in \R^N$, the mapping $G + \overline p$ has at least $\Gamma$ different zeros in $\Omega$, with $$\label{Gamma} \Gamma=\Gamma(M):=|\chi(\Omega)- (-1)^{N} s(M)| + 1.$$ Thus, one might expect that if $p\in C(\R,\R^N)$ is $T$-periodic and $\|p\|_\infty$ is small, then the number of $T$-periodic solutions of the non-autonomous system is generically greater or equal to $\Gamma$. Here, ‘generically’ should be understood in the sense of Baire category, that is, the property is valid for all $p$ (close to the origin) in the space of continuous $T$-periodic except for a meager set. It can be shown, indeed, that the fixed point index of the Poincaré map $P$ at $e$ is equal to $(-1)^Ns(M)$ and, moreover, a homotopy argument shows that the degree of $P$ over $\Omega$ is equal to $\chi(\Omega)$. Details are omitted because the result follows from the main theorem of the present paper. For several reasons, the situation is different for the delayed system $$\label{ec} u'(t) = g(u(t),u(t-\tau))$$ where, for simplicity, we shall assume that $g:\overline\Omega\times \overline\Omega\to \R^N$ is continuously differentiable. In the first place observe that, due to the delay, the condition that the field $G(x):=g(x,x)$ is inwardly pointing does not necessarily avoid that solutions with initial data $x_0:=\phi\in C([-\tau,0],\overline\Omega)$ may eventually abandon $\overline\Omega$. However, taking into account that $$|u(t_0-\tau)- u(t_0)| \le \tau \max_{t\in [t_0-\tau,t_0]} |u'(t)|,$$ it follows that the flow-invariance property, now over the set $C([-\tau,0],\overline\Omega)$, is retrieved under the stronger assumption $$\label{hart} \langle g(x,y),\nu(x)\rangle < 0 \qquad (x,y)\in \mathcal A_\tau (\Omega)$$ where $$\mathcal A_\tau (\Omega):= \{ (x,y)\in \partial\Omega\times \overline\Omega: |y-x|\le \tau\|g\|_{\infty}\}.$$ In the second place, the previous considerations regarding the Poincaré map become less obvious, since the latter is now defined not over $\overline\Omega$ but over the metric space $C([-\tau,0],\overline\Omega)$. In connection with this fact, we recall that the characteristic equation for the autonomous linear delayed systems is transcendental (also called quasipolynomial equation), so there exist typically infinitely many complex characteristic values. Throughout the paper, we shall assume as before that system (\[ec\]) has an equilibrium point $e\in \Omega$, that is, such that $g(e,e)=0$. This necessarily occurs when $\chi(\Omega)\neq 0$, although this latter condition shall not be imposed. Denote by $A,B\in \R^{N\times N}$ the respective matrices $D_xg(e,e)$ and $D_yg(e,e)$. Again, if $A+B$ is nonsingular and $s(A+B)$ is different from $(-1)^N\chi(\Omega)$, then the system has at least one extra equilibrium point in $\Omega$; furthermore, the number of equilibria in $\Omega$ is generically greater or equal to $\Gamma$. This is readily verified by writing the set of all the functions $g\in C^1(\overline\Omega\times\overline\Omega,\R^N)$ satisfying (\[hart\]) as the union of the closed sets $$X_n:=\left\{g\in C^1(\overline\Omega\times\overline\Omega,\R^N): \langle g(x,y),\nu(x)\rangle \le -\frac 1n \quad\hbox{for $(x, y)\in \mathcal A_\tau (\Omega)$} \right\}$$ and noticing that $X_n\cap \mathcal C$ is nowhere dense, where $ \mathcal C$ denotes the set of those functions $g$ such that $0$ is a critical value of the corresponding $G$. Our goal in this work is to extend the preceding ideas for non-autonomous periodic perturbations of (\[ec\]), namely the problem $$\label{nonaut} u'(t) = g(u(t),u(t-\tau)) + p(t)$$ with [$p\in C_{T}$]{}. As a basic hypothesis, we shall assume that the linearisation at the equilibrium, that is, the system $$\label{linear} u'(t) = Au(t)+ Bu(t-\tau)$$ has no nontrivial $T$-periodic solutions. This clearly implies, in particular, the above-mentioned condition that $A+B$ is invertible. From the Floquet theory for DDEs, it is known that the latter condition is also sufficient for nearly all positive values of $T$ (*i.e.*, except for at most a countable set). For the sake of completeness, this specific consequence of the Floquet theory shall be shown below (see Remark \[remark1\]). Our main result reads as follows. \[main\] Let the equilibrium $e$ and the matrices $A$ and $B$ be as before and assume that the linear system (\[linear\]) has no nontrivial $T$-periodic solutions. Then: - There exists $r>0$ such that [for any $p\in C_{T}$]{} with $\|p\|_\infty<r$ the non-autonomous problem (\[nonaut\]) has at least one $T$-periodic solution. - If moreover (\[hart\]) holds and $ s(A+B) \neq (-1)^N\chi(\Omega) $ with $s$ defined as in (\[sM\]), then (\[nonaut\]) has at least two $T$-periodic solutions. - Furthermore, there exists a residual set $\Sigma_r\subset C_T$ such that if $p\in \Sigma_r\cap B_r(0)$, then the number of $T$-periodic solutions is at least $\Gamma(A+B)$, where $\Gamma$ is given by (\[Gamma\]). The next result is an immediate consequence of [Theorem \[main\] combined with the preceding comments]{}. \[corol\] Let $e, A$ and $B$ be as before and assume that $A+B$ is invertible. Then for nearly all $T>0$ there exists $r=r(T)>0$ such that if [$p\in C_{T}$]{} with $\|p\|_\infty<r$ then the non-autonomous problem (\[nonaut\]) has at least one $T$-periodic solution. If moreover (\[hart\]) holds and $ s(A+B) \neq (-1)^N\chi(\Omega), $ then the number of $T$-periodic solutions is at least $2$ and generically $\Gamma(A+B)$. [For small delays, the condition that (\[linear\])]{} has no nontrivial $T$-periodic solutions can be formulated explicitly in terms of the matrix $A+B$ : \[smalldelay\] Let $e, A$ and $B$ be as before and assume that $\frac{2k\pi}Ti$ is not an eigenvalue of the matrix $A+B$ for all $k\in\N_0$. Then for each $\tau$ small enough there exists $r=r(\tau)$ such that the non-autonomous problem (\[nonaut\]) has at least one $T$-periodic solution for any [$p\in C_{T}$]{} with $\|p\|_\infty<r$. If moreover (\[hart-weak\]) holds for $G(x):=g(x,x)$ and $s(A+B)\ne (-1)^N\chi(\Omega)$, then (\[nonaut\]) has at least two $T$-periodic solutions and generically $\Gamma(A+B)$. It is worthy mentioning that if $\Omega$ is for example a ball, then the condition $s(A+B)\neq (-1)^N\chi(\Omega)$ implies that the equilibrium is unstable. As we shall see, this can be regarded as a consequence of the fact that the Leray-Schauder index of the fixed point operator defined in the proof of our main theorem is $(-1)^{N+1}$. This connection can be deduced from a version of the Krasnoselskii relatedness principle, which implies that the mentioned index coincides except for a $(-1)^N$ factor with that of the Poincaré operator. As shown in Proposition \[poinc-stab\], this implies, in turn, that the equilibrium cannot be stable. The paper is organised as follows. In the next section, we prove some basic facts concerning the linearised problem (\[linear\]); in particular, we give a necessary and sufficient condition in order to ensure that it has no nontrivial $T$-periodic solutions. In section \[dem\] we present a proof of Theorem \[main\] by means of an appropriate fixed point operator. The next two sections are devoted to a proof In section \[sec-delay\], we give a proof Corollary \[smalldelay\]. In section \[poincare\], we make some considerations on the stability of the equilibrium and the indices, on the one hand, of the fixed point operator defined in section \[dem\] and of the Poincaré map, on the other hand. Finally, a simple application of the main results for a singular system is introduced in section \[exam\]. Linearised system ================= In this section, we shall prove some basic facts concerning the linear system (\[linear\]). To this end, let us introduce some notation. For $k\in \mathbb N_0$, define $$\lambda_k:= \frac{2k\pi}T$$ and $$\varphi_k(t):= \cos(\lambda_k t) \qquad \psi_k(t):= \sin (\lambda_k t).$$ It is readily verified that $$\varphi_k(t-\tau)= \varphi_k(t)\varphi_k(\tau) + \psi_k(t)\psi_k(\tau)$$ $$\psi_k(t-\tau)= \psi_k(t)\varphi_k(\tau) - \varphi_k(t)\psi_k(\tau)$$ and $$\varphi_k'= -\lambda_k \psi_k,\qquad \psi_k'=\lambda_k\varphi_k.$$ For an element $u\in C_T$, we may consider its Fourier series, namely $$u = a_0 + \sum_{k=1}^\infty (\varphi_k a_k +\psi_k b_k)$$ in the $L^2$ sense, with $a_k, b_k\in \R^N$. Furthermore, recall that if $u$ is smooth (*e.g.*, of class $C^2$) then the series and its term-by-term derivative converge uniformly to $u$ and $u'$ respectively. \[lema\] Let $u\in C_T$ and define $$\label{matrices} X_k:=A+\varphi_k(\tau)B, \qquad Y_k:=\lambda_kI + \psi_k(\tau) B.$$ Then $u$ is a solution of (\[linear\]) if and only if $$\label{matr-ident} \left(\begin{array}{cc} X_k & -Y_k\\ Y_k & X_k \end{array} \right) \left(\begin{array}{c} a_k\\ b_k \end{array} \right) = \left(\begin{array}{c} 0\\ 0 \end{array} \right)$$ for all $k\in \mathbb N_0$. Since $\varphi_k'(t), \varphi_k(t-\tau), \psi_k'(t)$ and $\psi_k(t-\tau)$ belong to ${\rm\bf span}\{ \varphi_k(t),\psi_k(t)\}$, it follows that $u$ is a solution of of (\[linear\]) if and only if $$(A+B)a_0=0$$ and $$\varphi_k'(t) a_k +\psi_k'(t) b_k = A(\varphi_k(t) a_k +\psi_k(t) b_k) + B(\varphi_k(t-\tau) a_k + \psi_k(t-\tau) b_k)$$ for all $k>0$. The latter identity, in turn, is equivalent to $$\begin{array}{ccc} \lambda_k b_k & = & [A+ \varphi_k(\tau) B] a_k - \psi_{k}(\tau) Bb_k \\ {} \\ -\lambda_k a_k & = & \psi_{k}(\tau)Ba_k + [A+ \varphi_k(\tau) B] b_k, \end{array}.$$ that is, $$X_ka_{k} -Y_kb_{k}= Y_ka_{k} + X_kb_{k}= 0.$$ Because $X_0=A+B$ and $Y_0=0$, we deduce that $u$ is a solution of (\[linear\]) if and only if (\[matr-ident\]) holds for all $k\in \mathbb N_0$. \[no-nontrivial\] (\[linear\]) has no nontrivial $T$-periodic solutions if and only if $$\label{nec-suf} h_k:={\rm det}\left(\begin{array}{cc} X_k & -Y_k\\ Y_k & X_k \end{array} \right)\neq 0$$ for all $k\in \mathbb N_0$. \[remark1\] 1. Because $A+B$ is invertible, it is clear that for nearly all $T>0$ condition (\[nec-suf\]) is satisfied for all $k$. Indeed, it suffices to observe that $h_k$, regarded as a function of $T\in (0,+\infty)$, is an analytic function and, consequently, it has at most a countable number of zeros. 2. It can be shown that $h_k\ge 0$; in particular, its roots have even multiplicity. The proof is straightforward when $A$ and $B$ commute, since in this case $${\rm det}\left(\begin{array}{cc} X_k & -Y_k\\ Y_k & X_k \end{array} \right) = {\rm det}(X_k ^2+Y_k ^2).$$ The conclusion then follows, because for any pair of square real matrices $X, Y$ such that $XY=YX$ it is verified that $${\rm det}(X ^2+Y^2)= {\rm det}[(X+iY)(X-iY)] = {\rm det}(X+iY)\overline{{\rm det}(X+iY)}\ge 0.$$ [A proof for the non-commutative case is given below in section \[dem\], step \[directo-fourier\]. ]{} It is noticed that (\[nec-suf\]) may hold for non-invertible matrices $X_k$ and $Y_k$: for instance, observe that $$\left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right)^2 + \left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}\right)^2 = I.$$ 3. \[k\_0\] Since $\lambda_k\to +\infty$ it follows, for $k$ large, that $$h_k={\rm det}(Y_k){\rm det}(Y_k + X_kY_k^{-1}X_k)\simeq \lambda_k^{2N} > 0.$$ In particular, there exists $k_0$ such that if $u$ is a $T$-periodic solution of (\[linear\]) then $a_k=b_k=0$ for $k>k_0$. This means that $u$ is a (vector) trigonometric polynomial. Incidentally, observe that, because the family $\{\varphi_k, \psi_k\}$ is uniformly bounded, the constant $k_0$ may be chosen independent of $\tau$. [ In other words, if we consider the linear operator $L:C_T\to C_T$, given by $Lu(t):=u'(t) - Au(t) - Bu(t-\tau)$, then $ {\rm ker}(L)\subset {\rm \bf span}\{\varphi_k,\psi_k\}_{0\le k\le k_0}$. Observe furthermore that ${\rm Im}(L)$ consists of all the Fourier series $a_0+ \sum_{k>0}(\varphi_ka_k + \psi_kb_k)$ such that $a_0\in {\rm Im}(A+B)$ and $(a_k,b_k)\in {\rm Im}(M_k)$, where $M_k$ is the matrix defined in (\[matr-ident\]). This yields a direct proof of the well-known fact that $L$ is a zero-index Fredholm operator. Moreover, it is verified that $(a_k,b_k)\in {\rm ker}(M_k)\iff (-b_k,a_k) \in {\rm ker}(M_k)$, a fact that will be of relevance in the proofs of our results.]{} Proof of the main theorem {#dem} ========================= For convenience, a little of extra notation shall be introduced. For a function $u\in C_T$, let us write $$\mathcal Iu(t):= \int_0^t u(s)\, ds, \qquad \overline u:= \frac 1T {\mathcal Iu (T)}.$$ Moreover, denote by $\mathcal N$ the Nemitskii operator associated to the problem, namely $$\mathcal Nu(t):= g(u(t),u(t-\tau)).$$ Without loss of generality we may assume $e=0$ and fix $T>0$ such that (\[linear\]) has no nontrivial $T$-periodic solutions. For simplicity, we shall assume from the beginning that all the assumptions are satisfied; it shall be easy for the reader to deduce the existence of one solution near the equilibrium when (\[hart\]) is not satisfied. Define the open bounded set $U=\{u\in C_T:u(t)\in \Omega\,\hbox{ for all $t$} \}$ and the compact operator $K:\overline U\to C_T$ defined by $$Ku(t):= \overline u -t\, \overline {\mathcal Nu} + \mathcal I\mathcal Nu(t) - \overline{\mathcal I\mathcal Nu}.$$ We shall prove that the Leray-Schauder degree of $I-K$ is equal to $(-1)^N\chi(\Omega)$ over $U$ and to $s(A+B)$ over $B_\rho(0)$ for small values of $\rho>0$. To this end, let us proceed in several steps: 1. Let $K_0u:= \overline u -\frac T2 \overline {\mathcal Nu}$ and define, for $s\in [0,1]$, the operator given by $K_s:=s K +(1-s)K_0$. We claim that $K_s$ has no fixed points on $\partial U$. Indeed, for $s>0$ it is clear that $u\in\overline U$ is a fixed point of $K_s$ if and only if $u'(t)=s\mathcal Nu(t)$, that is: $$u'(t)= sg(u(t),u(t-\tau)).$$ Suppose there exists $t_0$ such that $u(t_0)\in\partial\Omega$, then we deduce, as before, $$|u(t_0-\tau)- u(t_0)| \le \tau \max_{t\in [t_0-\tau,t_0]} |u'(t)| \le \tau \|g\|_ \infty$$ and by (\[hart\]) we obtain $$0= \langle u'(t_0),\nu(u(t_0))\rangle = s\langle g(u(t_0),u(t_0-\tau)),\nu(u(t_0))\rangle <0,$$ a contradiction. On the other hand, we observe that the range of $K_0$ is contained in the set of constant functions, which can be identified with $\R^N$; thus, the Leray-Schauder degree of $I-K_0$ can be computed as the Brouwer degree of its restriction to $\overline U\cap \R^N = \overline \Omega$. Furthermore, for $u(t)\equiv u\in \overline \Omega$ it is clear that $K_0u= u - \frac T2 G(u)$, which does not vanish on $\partial \Omega=\partial U\cap \R^N$. By the homotopy invariance of the degree, we conclude that $$deg(I-K,U,0)=deg \left(\frac T2G,\Omega,0\right)=(-1)^N\chi(\Omega).$$ 2. Let $K_L$ be the operator associated to the linearised problem, defined by $$K_Lu(t):= \overline u -t\,\overline {\mathcal N_Lu} + \mathcal I\mathcal N_Lu(t) - \overline{\mathcal I\mathcal N_Lu},$$ with $\mathcal N_Lu(t):= Au(t) + Bu(t-\tau).$ As before, it is seen that $K_Lu=u$ if and only if $u$ is a solution of (\[linear\]); hence, it follows from the assumptions that $K_L$ has no nontrivial fixed points. Furthermore, the degree of $I-K_L$ coincides with the degree of $I-K$ on $B_\rho(0)$ when $\rho$ is small. This is a well-known fact but, for the reader’s convenience, a simple proof is sketched as follows. Since the degree is locally constant, we may assume that $g$ is of class $C^2$ near $(0,0)$, then [for some $C>0$,]{} $$\|Kv-K_Lv\|_{\infty} \le C\|\mathcal Nv-\mathcal N_Lv\|_\infty = o(\rho).$$ Because $K_L$ is compact, it is verified that, for some $\theta>0$, $$\|v-K_Lv\|_{\infty}\ge \theta \rho$$ for all $v\in \partial B_\rho(0)$. Indeed, due to linearity, it suffices to prove the claim for $\rho=1$. By contradiction, suppose there exists a sequence $\{v_n\}\subset \partial B_1(0)$ such that $\|v_n-K_Lv_n\|_{\infty}\to 0$, then passing to a subsequence we may assume that $\{K_Lv_n\}$ converges to some $v$. Then $v_n\to v$ which, in turn, implies that $\|v\|_{\infty}=1$ and $v=K_Lv$, a contradiction. It follows that if $\rho>0$ is small then $sK + (1-s)K_L$ has no fixed points on $\partial B_\rho(0)$ for $s\in [0,1]$ because $$\|v - sKv - (1-s)K_Lv\|_{\infty} \ge \|v - K_Lv\|_{\infty} - \|K_Lv-Kv\|_{\infty} \ge \theta \rho - o(\rho)>0$$ for $v\in \partial B_\rho(0)$. Thus, the degree of $I-K$ is well defined and coincides with the degree of $I-K_L$ over $B_\rho(0)$. 3. Claim: $deg(I-K_L,B_\rho(0),0) = s(A+B)$. \[directo-fourier\] Indeed, for $u$ as before it is seen by direct computation that $$u-K_Lu=\tilde a_0 + \sum_{k\ge 1} (\varphi_k\tilde a_k + \psi_k\tilde b_k)$$ where $$\tilde a_0= \mathcal M_0 a_0$$ and $$\left( \begin{array}{c} \tilde a_k \\ \tilde b_k \end{array}\right) = \mathcal M_k \left( \begin{array}{c} a_k \\ b_k \end{array}\right)$$ with $$\mathcal M_0:= \frac T2(A+B)\qquad \hbox{and }\,\, \mathcal M_k:= \frac 1{\lambda_k} \left( \begin{array}{cc} Y_k & X_k \\ -X_k & Y_k \end{array}\right)\quad \hbox{for}\, k>0.$$ Hence, the degree coincides with the sign of the determinant of the block matrix $$\left( \begin{array}{ccccc} \mathcal M_0 & 0 & 0 & \ldots & 0 \\ 0 & \mathcal M_1 & 0 & \ldots & 0 \\ 0 & 0 & \mathcal M_2 & \ldots & 0\\ \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & 0 & 0 & \ldots & \mathcal M_J \end{array}\right)$$ for $J$ sufficiently large. Thus, the proof follows in a straightforward manner from the fact that ${\rm det}(\mathcal M_k) >0$ for all $k>0$. We remark that the latter property holds even when $A$ and $B$ do not commute (see Remark \[remark1\]). Indeed, identifying the pairs $(a,b)\in \R^N\times \R^N$ with vectors $a+ib\in \mathbb C^N$, a matrix of the form $\left( \begin{array}{cc} X & -Y \\ Y & X \end{array}\right)$ may be called a [$\mathbb C$-linear matrix]{}. Thus, we need to prove that if $\mathcal M$ is an arbitrary invertible $\mathbb C$-linear matrix, then the algebraic multiplicity of each eigenvalue $\sigma<0$ of $\mathcal M$ is even. It is known that this value can be computed as the dimension of the kernel of the matrix $(\mathcal M-\sigma I)^m$, where $m$ is the minimum integer such that ${\rm ker}(\mathcal M-\sigma I)^m = {\rm ker}(\mathcal M-\sigma I)^{m+1}$. Now observe that the set of $\mathbb C$-linear matrices is a subring of $\R^{2N\times 2N}$; thus, $(\mathcal M-\sigma I)^m$ is again a $\mathbb C$-linear matrix. In particular, if $(a,b)\in {\rm ker}(\mathcal M-\sigma I)^m$ then $(-b,a)\in {\rm ker}(\mathcal M-\sigma I)^m$ and the result follows. 4. *Existence of two solutions for small $p$*. From the previous steps and the fact that the degree is locally constant we deduce that $$deg(I-K,U,\hat p)=(-1)^N\chi(\Omega),\qquad deg(I-K,B_\rho(0),\hat p)=s(A+B)$$ when $\|\hat p\|_\infty$ is small. Now the excision property of the Leray-Schauder degree implies $$deg(I-K,B_\rho(0),\hat p)=s(A+B)\ne 0,$$ and $$deg(I-K,U\backslash B_\rho(0),\hat p)=(-1)^N\chi(\Omega)- s(A+B) \ne 0.$$ Thus, there exists $\hat r>0$ such that the equation $(I-K)u=\hat p$ has at least two solutions for $\|\hat p \|_\infty <\hat r$. Finally, for each $p\in C_T$ define $$\hat p (t):= \mathcal I p(t) - \overline{\mathcal Ip} - t\overline p,$$ then clearly $\|\hat p\|_\infty\le c\|p\|_\infty$ for [some $c>0$]{}. The result is then deduced from the fact that if $u-Ku=\hat p$, then $u$ is a $T$-periodic solution of [(\[nonaut\])]{}. $$u'(t)=g(u(t),u(t-\tau))+p(t).$$ 5. *Genericity.* The last part of the proof follows as a consequence of the following particular case of the Sard-Smale Theorem [@smale]: Let $\mathcal F:X \to Y$ be a $C^1$ Fredholm map of index $0$ between Banach manifolds, i.e. such that $D\mathcal F(x):T_x X \to T_{\mathcal F(x)} Y$ is a Fredholm operator of index $0$ for every $x\in X$. Then the set of regular values of $\mathcal F$ is residual in $Y$. At this point, we notice that the argument is a bit subtle: when applied to $\mathcal F:=I-K$, the Sard-Smale Theorem implies the existence of a residual set $\Sigma\subset C_T$ such that the mapping $\mathcal F-\hat p$ has at least $\Gamma - 1$ zeros in $U\setminus B_\rho(0)$ for $\hat p\in \Sigma\cap B_{ \hat r}(0)$. [Indeed, it is readily seen that $K$ is of class $C^1$ and $DK(u)$ is compact for all $u$. Thus, $\mathcal F=I-K$ is a zero-index Fredholm operator. If $\hat p$ is a regular value, that is, $D\mathcal F(u)$ is surjective for every preimage $u \in \mathcal F^{-1}(\hat p)$ then, since the index is $0$, it is also injective and from the open mapping theorem we conclude that $D\mathcal F(u)$ is an isomorphism. Hence, the number of such preimages in $U\setminus B_{\rho}(0)$ is greater or equal than $|deg(I-K,U\setminus B_{\rho}(0),0)|$. This follows by taking small neighbourhoods $N_u$ around each of these values $u$ such that $\mathcal F:N_u\to \mathcal F(N_u)$ is a diffeomorphism. Because there are no other zeros of $\mathcal F -\hat p$ in $U\setminus B_{\rho}(0)$, the degree is the sum of the degrees $d_u$ over each of these neighbourhoods. The claim then follows from the fact that $d_u=\pm 1$ for each $u$.]{} However, although the mapping $p\mapsto \hat p$ defined before establishes an isomorphism $J:C_T\to C_T^1$, it might happen that $J^{-1}(\Sigma\cap C^1_T)$ is not a residual set. The difficulty is overcome for example by considering the same operator $K$ as before, now defined over the set $$\hat U:= \{u\in C^1_T: u(t)\in \Omega,\, \|u'\|_\infty < \|g\|_{\infty} + 1\} \subset C^1_T.$$ Details are left to the reader. [Notice that]{} 1. The existence of a solution near the equilibrium can be also proved in a direct way by the Implicit Function Theorem. 2. Condition (\[hart\]) alone implies the existence of generically $|\chi(\Omega)|$ solutions. 3. Analogous conclusions are obtained if the sign of (\[hart\]) is reversed. In this case, $G$ is homotopic to $\nu$ and hence $deg(I-K,U,0)= \chi(\Omega)$. However, in this latter case the considerations about the Poincaré operator become less clear, because it is not guaranteed that solutions with initial values $\phi$ with $\phi(t)\in \overline \Omega$ remain inside $\Omega$. Small delays {#sec-delay} ============ As mentioned in the introduction, condition (\[hart\]) implies that the vector field $G(x)=g(x,x)$ is inwardly pointing over $\partial \Omega$, although the converse is not true; the need of a condition stronger than (\[hart-weak\]) is due to the presence of the delay. However, if only (\[hart-weak\]) is assumed, then Theorem \[main\] is still valid for all $\tau<\tau ^*$, where $\tau ^*$ depends only on $\|g\|_\infty$. More precisely, by continuity we may fix $\ee>0$ such that (\[hart\]) holds for all $x\in \partial \Omega$ and all $y\in\overline\Omega$ with $|y-x|<\ee$ and take $\tau* := \frac \ee{\|g\|_\infty}$. In this section, we show that the problem for small $\tau$ can be seen as a perturbation of the non-delayed case, thus giving the explicit sufficient condition for the non-existence of nontrivial $T$-periodic solutions of (\[linear\]) expressed in Corollary \[smalldelay\]. We shall make use of the following lemmas: \[lambdas\] $1$ is a Floquet multiplier of the system $u'(t)=Mu(t)$ if and only if $-\lambda_k^2$ is an eigenvalue of $M^2$ for some $k\in\N_0$, that is, if and only if $\pm i\lambda_k$ are eigenvalues of $M$ for some $k$. The result follows by direct computation, or from Lemma \[lema\] with $\tau=0$. For example, when $M$ is triangularizable (or, equivalently, when all its eigenvalues are real), $1$ is not an eigenvalue of the system $u'(t)=Mu(t)$ if and only if $M$ is nonsingular; in this particular case, the conclusion follows directly, because the system uncouples and the result is obviously true for a scalar equation. \[Floq\] Assume that $1$ is not a Floquet multiplier of the linear ODE system $u'(t)=(A+B)u(t)$. Then the DDE system (\[linear\]) has no nontrivial $T$-periodic solutions, provided that $\tau$ is small. Suppose that $u_n\in C_T$ is a nontrivial solution for $\tau_n\to 0$. Without loss of generality, it may be assumed that $\|u_n\|_\infty=1$ and hence $\|u_n'\|_\infty\le C$ for some constant $C$. Thus, we may assume that $u_n$ converges uniformly to some $u\in C_T$ with $\|u\|_\infty=1$. Because $\|u_n(t-\tau_n)-u_n(t)\|\le C\tau_n\to 0$, it becomes clear that $u_n'$ converges uniformly to $(A+B)u$ which, in turn, implies $u'=(A+B)u$, a contradiction. A more direct proof of Lemma \[Floq\] follows just by considering Remark \[remark1\].\[k\_0\] and Lemma \[lambdas\]. Indeed, in the context of Lemma \[lema\] it suffices to check that $h_k\ne 0$ only for a finite number of values of $k$. By continuity, this is true for small $\tau$, because ${\rm det} [(A+B)^2 + \lambda_k^2 I]\neq 0$ for all $k$. However, the previous proof has an interest in its own because it can be extended in a straightforward manner to the non-autonomous case. : As a consequence of the preceding lemma, the conclusions of Theorem \[main\] hold for small $\tau$, provided that the linearisation has no nontrivial $T$-periodic solutions for the non-delayed case. Thus, in view of Lemma \[lambdas\], the proof is complete. $\square$ Poincaré operator {#poincare} ================= In this section, we shall make some considerations regarding the Poincaré operator associated to the system. Let us firstly observe that if $\chi(\Omega)=1$ (for example, if $\Omega$ is homeomorphic to a ball), then the condition $s(A+B) \neq (-1)^N\chi(\Omega)$ in Theorem \[main\] simply reads $(-1)^N {\rm det}(A+B)<0$. This, in turn, implies that the equilibrium is unstable. [Indeed, consider the characteristic function $h(\lambda)= {\rm det}\left(\lambda I - A - Be^{-\lambda\tau} \right)$, then $h(0)= (-1)^N {\rm det}(A+B)<0$ and $h(\lambda) =\lambda^N$ for $|\lambda|\gg 0$. In particular, this implies the existence of a characteristic value $\lambda>0$. ]{} We shall show that, in the present context, the instability of the equilibrium when $(-1)^N {\rm det}(A+B)<0$ is due to the fact, proved in section \[dem\], that the index of the fixed point operator $K$ at $e$ (i. e. the degree of $I-K$ over small balls around $e$) is equal to $(-1)^{N+1}$. When $\tau=0$, this can be regarded as a direct consequence of the following properties: 1. $deg(I-K, B_\rho(e),0)$ with $B_\rho(e)\subset C_T$ is equal to $(-1)^Ndeg_B(I-P, B_\rho(e),0)$ with $B_\rho(e)\subset\R^N$, where $P$ is the Poincaré map. 2. If the equilibrium is stable, then the index of $P$ is $1$. The first property is a particular case of a *relatedness principle* due to Krasnoselskii (see [@krasno]). The second property is well-known and can be found for example in [@K]. For more details see [@rafa], where sufficient conditions for the validity of the converse statement are also obtained. Our goal in this section consists in understanding the connections between the instability of the equilibrium and the index of the fixed point operator defined in the proof of the main theorem. With this aim, let us define the Poincaré operator for the delayed case as follows. Let $\tau\le T$ and consider a general autonomous system $$\label{general} u'(t)=F(u_t)$$ with $F: C([-\tau,0])\to \R^N$ locally Lipschitz, *i.e.*: for all $R>0$ there exists a constant $L$ such that $$\|F(\phi)-F(\psi)\|\le L\|\phi-\psi\|_\infty$$ for all $\phi,\psi\in \overline {B_R(0)}\subset C([-\tau,0],\mathbb R^n)$. The notation $u_t$ expresses, as usual, the mapping defined by $u_t(\theta):=u(t+\theta)$ for $\theta\in [-\tau,0]$. Denote by ${\rm dom}(P)\subset C([-\tau,0])$ the set of those functions $\phi$ such that the unique solution $u=u(\phi)$ of the problem with initial condition $\phi$ is defined up to $t=T$, then $P:{\rm dom}(P)\to C([-\tau,0])$ is defined by $$P\phi(s):=u(T+s).$$ Clearly, the $T$-periodic solutions of the problem can be identified with the fixed points of $P$. We shall see that, as in the non-delayed case, if the linearisation has no nontrivial $T$-periodic solutions then the index $i(P)$ of the operator $P$ at a stable equilibrium is equal to $1$. To this end, assume without loss of generality that $e=0$ and observe that stability implies that ${\rm dom}(P)$ is a neighbourhood of $0$. It is worth noticing that, in the general setting, extra conditions are required in order to prove the compactness of $P$ (see *e.g*. [@liu]), so the Leray-Schauder degree may be not well defined; however, it is verified that the stability assumption implies that $P$ is compact over small neighbourhoods of $0$. More precisely: \[compact\] Let $F$ be as before and assume that for some open $U\subset C([-\tau,0])$ there exists $R>0$ such that if $\phi\in U$ then the solution $u$ with initial condition $\phi$ is defined and satisfies $|u(t)| <R$ for all $t\in [0,T]$. Then $P$ is well defined and compact over $U$. Let $B\subset U$ be bounded and observe, in the first place, that $P(B)$ is bounded. Moreover, if $u$ is a solution with initial condition $\phi\in B$, then $$u(t)= \phi(0) + \int_0^t F(u_s)\, ds.$$ Enlarging $R$ if necessary, we may assume $B\subset B_R(0)$, then $\|u_s\|_\infty <R$ for all $s\in [0,T]$. Given $t_1<t_2$ in $[-\tau,0]$, since $\tau\le T$ it is verified that $$|P\phi(t_2)-P\phi(t_1)| \le \int_{T+t_1}^{T+t_2} |F(u_s)|\, ds.$$ Let $L$ be the Lipschitz constant corresponding to $R$, then $$|F(\phi)|\le |F(0)| + L\|\phi\|_\infty\le C + LR,$$ where $C:=|F(0)|$. Hence $|P(t_2)-P(t_1)|\le (C+LR)(t_2-t_1)$ and the result follows from the Arzelà-Ascoli Theorem. For example, the assumptions of the previous lemma are satisfied if $F$ has linear growth, that is $$|F(\phi)|\le \gamma\|\phi\|_\infty + \delta.$$ Furthermore, extra assumptions are required to ensure the non-existence of nontrivial periodic solutions near $0$; this is why we shall impose this fact as an extra condition (see Proposition \[poinc-stab\] below), which is clearly satisfied for example when the stability is asymptotic. For simplicity, we shall also assume that $F$ is Fréchet differentiable at $0$, that is, $$F(\phi)= D_\phi(0)\phi + \mathcal R(\phi)$$ with $\|\mathcal R(\phi)\|_\infty \le o(\|\phi\|_\infty)\|\phi\|_\infty$. Thus, it is readily verified that the linearisation of $P$ at the origin coincides with the Poincaré operator associated to the linearised system $u'(t)=D_\phi(0)u_t$. \[poinc-stab\] In the previous setting, assume that $0$ is a stable equilibrium of (\[general\]) such that its linearisation has no nontrivial $T$-periodic solutions. Then $i(P)=1$. Without loss of generality, we may assume that $P$ is compact on $\overline V$ for some neighbourhood $V$ of $0$. It follows from the assumptions that the index of $P$ is well defined and coincides with the index of its linearisation $P_L$. According to Theorem 13.8 in [@brown], $deg(I-P_L,B_\rho(0),0)$ is equal to $(-1)^\alpha$, where $\alpha$ is the sum of the (finite) algebraic multiplicities of the (finitely many) eigenvalues $\sigma$ of $P_L$ satisfying $\sigma>1$. If $deg(I-P_L,B_\rho(0),0) = -1$, then $P_L$ has an eigenfunction $\phi$ with eigenvalue $\sigma>1$. If $u$ is the corresponding solution of the linearised problem with initial condition $u=\phi$ on $[-\tau,0]$ then $u$ can be extended to $\R$ in a $(T,\sigma)$-periodic fashion, that is, with $u(t+T)=\sigma u(T)$ for all $t$ (see [@pinto]). In particular, $u(t)$ is unbounded for $t>0$. In other words, $0$ is unstable for the linearised problem which, in turn, implies that it cannot be stable for the original problem (see *e.g.* [@hale]). In order to complete the picture for system (\[ec\]), it would be interesting to prove that, indeed, the index of the Poincaré operator at the equilibrium when the linearisation has no nontrivial solutions is $(-1)^Ns(A+B)= (-1)^Ni(K)$. Here, we shall simply verify that the claim holds when the delay is small; the analysis of the general case and [a version of the Krasnoselskii relatedness principle for delayed systems shall be the subject of a forthcoming paper.]{} To this end, let us start with a direct computation for the non-delayed case: \[degP\] Let $M\in \mathbb R^{N\times N}$ and let $P_M$ be the Poincaré operator associated to the linear ODE system $u'(t)=Mu(t)$ for some fixed $T$. If $1$ is not a Floquet multiplier, then $$deg_B(I-P_M,V,0) = (-1)^Ns(M)$$ for any neighbourhood $V\subset \R^N$ of the origin. By definition, $$(I-P_M)(u)= \left(I-e^{TM}\right)u.$$ Write $M$ in its (possibly complex) Jordan form $M=C ^{-1}JC$, where $J$ is upper triangular. Then $${\rm det}\left(I- e^{TM}\right) = {\rm det}\left(I- e^{TJ}\right) = \prod_{j=1}^N \left(1-e ^{\lambda_jT}\right),$$ where $\lambda_j$ are the eigenvalues of $M$. Now observe that if $\lambda=a+ib\notin \R$, then $$\left(1-e ^{\lambda T}\right)\left(1-e ^{\overline{\lambda}T}\right) = 1 + e ^{aT}\left(e^{aT} -2\cos (bT)\right) >0.$$ Thus, complex eigenvalues do not affect the sign of ${\rm det}\left(I- e^{TM}\right)$, as well as it happens with the sign of ${\rm det}(M)$ because $\lambda\overline\lambda =|\lambda|^2$. The result follows now from the fact that, for $\lambda\in\R$, $$sgn\left(1 - e^{\lambda T}\right) = -sgn (\lambda).$$ An alternative (somewhat exotic) proof follows from the relatedness principle. Indeed, we may consider the operator $K_L$ in the proof of Theorem \[main\] with $A=M$ and $B=0$, then $deg_B(I-P,V,0)=(-1)^Ndeg(I-K_L, V,0) = (-1)^Ns(M)$. The conclusion for small $\tau$ is obtained now by a continuity argument. Indeed, fix $r>0$ and $P_L$ as before. The solutions of (\[linear\]) with initial value $\phi\in B_r(0)$ are uniformly bounded; thus, by Gronwall’s lemma we deduce that $\|P-P_0\| = O(\tau)$, where the operator $P_0$ is defined by $P_0(\phi)(t)\equiv v(T)$, with $v$ the unique solution of the system $v'(t)=(A+B)v(t)$ satisfying $v(0)=\phi(0)$. Moreover, recall that if $\tau$ is small then $P_L$ is homotopic to $P_0$; thus, the result follows from Lemma \[degP\]. Example: a system of DDEs with singularities {#exam} ============================================ A simple example is presented here in order to illustrate our main results. Let $0\le J_0\le J \ne 0$ and $$g(x,y):= -dx + |y|^2\left( \sum_{j=1}^{J_0} a_j\frac{x-v_j}{|x-v_j|^{\alpha_j}} + \sum_{j=J_0+1}^{J} a_j\frac{y-v_j}{|y-v_j|^{\alpha_j}} \right)$$ where $d,a_j>0$, $\alpha_j>2$ and $v_j\in \R^N\backslash\{0\}$ are pairwise different vectors. A simple computation shows that $$\langle g(x,x),x\rangle < 0 \qquad |x|\gg 0$$ and $$\langle g(x,x),v_j-x\rangle < 0 \qquad |x-v_j|\ll 1$$ for $j=1,\ldots, J$. Moreover, $g(0,0)=0$ and $$A=D_xg(0,0)=-dI, \quad B=D_yg(0,0)=0.$$ Thus, taking $\Omega:=B_R(0)\backslash \cup_{j=1}^J B_\eta(v_j)$ where $R\gg 0$ and $\eta\ll 1$, Corollary \[smalldelay\] applies. Since $\chi(\Omega)= 1-J < 1 = (-1)^Ns(A+B)$, we conclude that the number of $T$-periodic solutions of (\[nonaut\]) for small $\tau$ and $\|p\|_\infty$ is generically $J+1$. Acknowledgements {#acknowledgements .unnumbered} ================ The first two authors were partially supported by projects CONICET PIP 11220130100006CO and UBACyT 20020160100002BA. The first author wants to thank Prof. J. Barmak for his thoughtful comments regarding the fixed point property and the Euler characteristic. 1 R. F. Brown, *A Topological Introduction to Nonlinear Analysis*. First edition, Birkhäuser (2004). J. K. Hale and S. M. Verduyn Lunel, *Introduction to Functional Differential Equations*, Springer, New York (1993). H. Hopf, [*Vektorfelder in $n$-dimensionalen Mannigfaltigkeiten*]{}. Math. Ann. 96 (1926/1927), pp. 225–250. J. Liu, G. N’Guérékata and Nguyen Van Minh, [*Topics on Stability and Periodicity in Abstract Differential Equations*]{}. World Scientific, Singapore (2008). M. A. Krasnoselskii, [*The operator of translation along the trajectories of differential equations*]{}. Amer. Math. Soc., Providence, RI, (1968). M. A. Krasnoselskii, P.P Zabreiko, [*Geometrical methods of nonlinear analysis*]{}. Springer-Verlag, Berlin (1984). R. Ortega, [*Topological degree and stability of periodic solutions for certain differential equations*]{}, J. London Math. Soc. (2) 42 (1990), pp. 505–516. M. Pinto, [*Pseudo-almost periodic solutions of neutral integral and differential equations with applications*]{}. Nonlinear Anal. 72(12) (2010), pp. 4377–-4383. S. Smale, [*An infinite dimensional version of Sard’s theorem*]{}. American Journal of Mathematics 87 (1965), pp. 861–866. H. Smith, [*An Introduction to Delay Differential Equations with Applications to the Life Sciences*]{}, Springer-Verlag, New York (2011). F. Wecken, [*Fixpunktklassen*]{}. Ill, Math. Ann. 118-119 (1941-1943), pp. 544–577. Pablo Amster and Mariel Paula Kuna *E-mails*: pamster@dm.uba.ar – mpkuna@dm.uba.ar. Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Buenos Aires, Argentina and IMAS-CONICET. Gonzalo Robledo *E-mail*: grobledo@uchile.cl. Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653 Santiago, Chile.

--- abstract: 'We study various aspects of extracting spectral information from time correlation functions of lattice QCD by means of Bayesian inference with an entropic prior, the maximum entropy method (MEM). Correlator functions of a heavy-light meson-meson system serve as a repository for lattice data with diverse statistical quality. Attention is given to spectral mass density functions, inferred from the data, and their dependence on the parameters of the MEM. We propose to employ simulated annealing, or cooling, to solve the Bayesian inference problem, and discuss practical issues of the approach.' author: - H Rudolf Fiebig title: Spectral density analysis of time correlation functions in lattice QCD using the maximum entropy method --- \[sec:intro\]Introduction ========================= Numerical simulations of quantum chromodynamics (QCD) on a Euclidean space-time lattice provides access to mass spectra of hadronic systems through the analysis of time correlation functions. In theory the latter are linear combinations of exponential functions $$C(t,t_0)=Z_1e^{-E_1(t-t_0)}+Z_2e^{-E_2(t-t_0)}+\ldots\,, \label{exp2}$$ where the $E_n$ are the excitation energies of the system and the strength coefficients $$Z_n=|\langle n|\hat{\Phi}(t_0)|0\rangle|^2 \label{Zp}$$ are matrix elements of some vacuum-subtracted operator $\hat{\Phi}(t_0)=\Phi(t_0)-\langle 0|\Phi(t_0)| 0\rangle$ between the vacuum $|0\rangle$ and a ground or excited state $|n\rangle, n>0$. In practice the exponential model (\[exp2\]) is fitted to noisy numerical simulation ‘data’. The statistical quality of simulation data rarely is good enough for the two-exponential fit (\[exp2\]) to succeed. It is common practice to look at the large-$t$ behavior of the correlation function $C(t,t_0)$ in a $t$-interval where it is dominated by only one exponential, with the lowest energy, and then make a one-parameter fit to a plateau of the effective-mass function $\mu_{\rm eff}(t,t_0)=-{\partial}\ln C(t,t_0)/{\partial t}$. Possible discretizations are $$\begin{aligned} \mu_{\rm eff,0}(t,t_0)&=&-\ln\left(\frac{C(t+1,t_0)}{C(t,t_0)}\right) \simeq m_{\rm eff,0}\label{eff0}\\ \mu_{\rm eff,1}(t,t_0)&=&\frac{C(t+1,t_0)}{C(t,t_0)}\simeq e^{-m_{\rm eff,1}}\label{eff1}\\ \mu_{\rm eff,2}(t,t_0)&=&\frac{C(t+1,t_0)-C(t-1,t_0)}{2C(t,t_0)}\label{eff2}\\ & &\simeq -\sinh(m_{\rm eff,2})\nonumber\\ \mu_{\rm eff,3}^2(t,t_0)&=&\frac{C(t+1,t_0)+C(t-1,t_0)-2C(t,t_0)}{C(t,t_0)}\nonumber\\ & &\simeq 2(\cosh(m_{\rm eff,3})-1)\label{eff3}\,.\end{aligned}$$ The expressions after the $\simeq$ are the values of $\mu_{\rm eff}$ for a pure plateau of mass $m_{\rm eff}$. The procedure implies the selection of consecutive time slices $t=t_1\ldots t_2$ for which $\mu_{\rm eff}={\rm const}$, within errors, and an appropriate fit. The selection of this, so-called, plateau is a matter of judgment. A condition for reliable results is that the correlation function (\[exp2\]) is dominated by just one exponential term, usually the ground state. The latter can be enhanced by the use of smeared operators [@Alexandrou:1994ti] and fuzzy link variables [@Alb87a]. This analysis procedure discourages consideration of excited states. In fact it will only produce reliably results if those are suppressed. Workarounds involve diagonalization of a correlation matrix of several operators or variational techniques [@Morningstar:1999rf]. Those however, still rely on plateau selection without utilizing the information contained in the entire available time-slice range of a correlation function. As lattice simulations of QCD now aim at excited hadron states, $N^\ast$’s for example [@Lee:1998cx; @Gockeler:2001db; @Sasaki:2001nf], this situation is unsatisfactory. Alternative methods employing Bayesian inference [@Jar96] are a viable option. The maximum entropy method (MEM), which involves a particular choice of the Bayesian prior probability, falls in this class. Bayesian statistics [@Box73] is a classic subject with a vast range of applications. However, application within the context of lattice QCD is relatively new [@Nakahara:1999vy; @Nakahara:1999bm; @Asakawa:2000pv; @Asakawa:2000tr; @Lepage:2001ym]. In this work we report on our experience using the MEM for extracting spectral mass density functions $\rho(\omega)$ from lattice-generated time correlators $$C(t,t_0)=\int d\omega\,\rho(\omega) e^{-\omega(t-t_0)}\,, \label{Crho}$$ where a discrete set of time slices $t$ is understood. Discretization of the $\omega$-integral with reasonably fine resolution leads to an ambiguous problem where the number of parameters values $\rho(\omega)$ is (typically much) larger than the number of lattice data $C(t,t_0)$. In the MEM an entropy term involving the spectral density is used as a Bayesian prior to infer $\rho(\omega)$ from the data. We here apply MEM analysis to sets of lattice correlation functions of a meson-meson system. Those particular simulations are aimed at learning about mechanisms of hadronic interaction. This will be discussed separately [@Fie02c]. The lattice data generated within that project involve local and nonlocal operators. They exhibit a wide range of statistical quality from ‘very good’ to ‘marginally acceptable’. Our focus here is to utilize those data as a testing ground for Bayesian MEM analysis. In contrast to other works we employ simulated annealing to the solution of the Bayesian inference problem. The main aim of this work is to explore the feasibility of this approach for extracting masses from a lattice simulation using realistic lattice data, including excitations. For the most part this translates into studying the sensitivity of the method to to its native parameters. \[sec:BayesCF\]Bayesian Inference for Curve Fitting =================================================== From a Bayesian point of view the spectral density function $\rho$ in (\[Crho\]) is a random variable subject to a certain probability distribution functional ${\cal P}[\rho]$. Solution of the curve fitting problem consists in finding the function $\rho$ which maximizes the conditional probability ${\cal P}[\rho\leftarrow C]$, the [*posterior probability*]{}, given a ‘measured’ data set $C$. Computation of $\rho$ is then based on Bayes’ theorem [@Jar96] $${\cal P}[\rho\leftarrow C]\, {\cal P}[C] ={\cal P}[C\leftarrow \rho]\, {\cal P}[\rho]\,, \label{BayesT}$$ also known as ‘detailed balance’ in a different context. The functional ${\cal P}[C]$, the [*evidence*]{}, gives the probability of measuring a data set $C$. The conditional probability ${\cal P}[C\leftarrow \rho]$, the [*likelihood function*]{}, determines the probability of measuring $C$ given a spectral function $\rho$. Finally ${\cal P}[\rho]$, the Bayesian [*prior*]{}, defines a constraint on the spectral density function $\rho$. Its choice is a matter of judgment. Ideally, the prior should reflect the physics known about the system, for example an upper limit on the hadronic mass scale. The posterior probability is the product of the likelihood function and the prior ${\cal P}[\rho\leftarrow C] = {\cal P}[C\leftarrow \rho]\, {\cal P}[\rho]/{\cal P}[C]$, where the [*evidence*]{} merely plays the role of a normalization constant [@Jar96]. Indeed, the normalization condition $\int[d\rho]{\cal P}[\rho\leftarrow C]=1$ applied to (\[BayesT\]) gives ${\cal P}[C] = \int[d\rho]{\cal P}[C\leftarrow \rho]\, {\cal P}[\rho]$. Thus, for a fixed $C$, we have $${\cal P}[\rho\leftarrow C]\propto{\cal P}[C\leftarrow \rho]\, {\cal P}[\rho]\,. \label{BayesT3}$$ The curve fitting problem requires the product of the [*likelihood function*]{} and the [*prior*]{} function. \[sec:spectralD\]Spectral density --------------------------------- Our lattice data come from correlation functions built from heavy-light meson-meson operators $$\Phi_v=v_1\Phi_1+v_2\Phi_2\,, \label{Phiv}$$ where $\Phi_1$ and $\Phi_2$ involve local and non-local meson-meson fields, respectively, at relative distance $r$, and $v$ are some coefficients [@Fiebig:2001mr; @Fiebig:2001nn]. On a finite lattice the corresponding correlator $C_v(t,t_0)=\langle\hat{\Phi}^\dagger_v(t)\hat{\Phi}_v(t_0)\rangle$, where $\hat{\Phi}=\Phi-\langle\Phi\rangle$, has a purely discrete spectrum $$C_v(t,t_0)=\sum_{n\neq 0} |\langle n|\Phi_v(t_0)|0\rangle|^2 e^{-\omega_n(t-t_0)}\,. \label{Cvn}$$ Here $|n\rangle$ denotes a complete set of states with energies $\omega_n$, some of which may be negative due to periodic lattice boundary conditions and operator structure. Our normalization conventions for forward and backward going propagators are determined by defining $$\exp_T(\omega,t)=\Theta(\omega)e^{-\omega t} +\Theta(-\omega)e^{+\omega(T-t)}\,, \label{expT}$$ where $0\leq t< T$, and $\Theta$ denotes the step function. We then expect the lattice data to fit the following model $$F(\rho_T|t,t_0)=\int_{-\infty}^{+\infty}d\omega\/ \rho_T(\omega)\exp_T(\omega,t-t_0)\,, \label{Fc}$$ where $\rho_T(\omega)$ is a spectral density function, defined for positive (forward) and negative (backward) frequencies. The requirement that the model be exact, $F(\rho_T|t,t_0)=C_v(t,t_0)$, leads to $$\begin{aligned} \rho_T(\omega)&=&\sum_{n\neq 0}\delta(\omega-\omega_n)\, |\langle n|\Phi_v(t_0)|0\rangle|^2\times\nonumber\\ & &[\Theta(\omega_n)+\Theta(-\omega_n)e^{-\omega_n T}]\,.\label{rhoT}\end{aligned}$$ Thus a discrete sum over $\delta$-peaks is the theoretical form of the spectral function. Our objective is to compute $\rho_T(\omega)$ from lattice data using Bayesian inference. \[sec:likely\]Likelihood function --------------------------------- Toward this end we proceed to construct the likelihood function. The lattice data come in the form of an average over $N_U$ gauge configurations $$C_v(t,t_0)=\frac{1}{N_U}\sum_{n=1}^{N_U}C_v(U_n|t,t_0)\,, \label{CNU}$$ where $C_v(U_n|t,t_0)$ is the value of an operator, in this case $\hat{\Phi}^\dagger_v(t)\hat{\Phi}_v(t_0)$, in one gauge field configuration $U_n$. Correlation function data on different time slices are stochastically dependent. Their errors are described by the covariance matrix $$\begin{aligned} \Gamma_v(t_1,t_2)=\frac{1}{N_U}\sum_{n=1}^{N_U} &&\left(\rule{0mm}{4mm}C_v(t_1,t_0)-C_v(U_n|t_1,t_0)\right)\times\nonumber\\ &&\left(\rule{0mm}{4mm}C_v(t_2,t_0)-C_v(U_n|t_2,t_0)\right).\label{Ecov}\end{aligned}$$ The $\chi^2$-distance of the spectral model (\[Fc\]) from the lattice data then is $$\begin{aligned} \chi^2=\sum_{t_1,t_2} &&\left(\rule{0mm}{4mm}C_v(t_1,t_0)-F(\rho_T|t_1,t_0)\right)\Gamma_v^{-1}(t_1,t_2) \times\nonumber\\ &&\left(\rule{0mm}{4mm}C_v(t_2,t_0)-F(\rho_T|t_2,t_0)\right).\label{Chi2}\end{aligned}$$ For numerical work a discretization scheme of the $\omega$-integral in (\[Fc\]) is required. Our choice is $$F(\rho_T|t,t_0)\simeq\sum_{k=K_-}^{K_+} \rho_k\,\exp_T(\omega_k,t-t_0) \label{Fd}$$ where $\omega_k=\Delta\omega k$, $\Delta\omega$ is an appropriate (small) interval, $\rho_k=\Delta\omega \rho_T(\omega_k)$, and $K_- < 0 < K_+$. The likelihood function ${\cal P}[C\leftarrow \rho]$ describes the probability distribution of the data $C$ given a certain parameter set $\rho$. If we imagine that the data are obtained by a large number of measurements, at fixed $\rho$, then the probability distribution for $C$ is Gaussian by virtue of the central limit theorem, $${\cal P}[C\leftarrow \rho]\propto e^{-\chi^2/2}\,. \label{Pchi}$$ This is the standard argument for employing the above form of the likelihood function in the context of Bayesian inference [@Jar96; @Bra76]. \[sec:prior\]Entropic prior --------------------------- In case some information is available about the physics of the system it can be used to constrain the parameter space of the model. This is the role of the Bayesian prior. In the standard approach plateau methods are a severe form of imposing restrictions. A two-exponential fit (\[exp2\]), if feasible, is less constraining. In a Bayesian context it is possible to gradually increase the number of exponentials until convergence is reached. This is a strategy advocated in [@Lepage:2001ym], see also [@Morningstar:2001je]. There, the model for the correlation function is $\sum_n A_n e^{-E_nt}$, initially with small number of terms, which is then constrained by the Bayesian prior $e^{-\sum_n[(A_n-\bar{A}_n)^2/2\bar{\sigma}_{A_n}^2 +(E_n-\bar{E}_n)^2/2\bar{\sigma}_{E_n}^2]}$. The quantities $\bar{A}_n,\bar{\sigma}_{A_n},\bar{E}_n,\bar{\sigma}_{E_n}$ are input. Their choice is inspired by prior knowledge about the physics of the system. On the other hand, there is usually no [*a priory*]{} information about the location and the strengths of the peaks in the mass spectrum. The view that only [*minimal information*]{} is available about the spectral density function can also be implemented in the Bayesian prior. The information content, in the sense of [@Sha49; @Jay57a; @Jay57b], is measured by the entropy $S=-\sum_k \rho_k\ln(\rho_k/m)$, on some scale $m$. Rather, a commonly used variant is the Shannon-Jaynes entropy [@Jar96] $${\cal S}[\rho]=\sum_{k=K_-}^{K_+}\left(\rho_k-m_k-\rho_k\ln\frac{\rho_k}{m_k}\right)\,. \label{Smem}$$ Note that $\rho_k\ge 0$, according to (\[rhoT\]). The configuration $m=\{m_k : K_- \leq k \leq K_+\}$ is called the default model. We have ${\cal S}\leq 0$, $\forall\rho$, while ${\cal S}=0 \iff \rho=m$. The default model is a unique absolute maximum of ${\cal S}$. Choosing the prior probability as $${\cal P}[\rho]\propto e^{\alpha{\cal S}} \label{Pent}$$ entails that ${\cal P}[\rho]$ is maximal in the absence of information about $\rho$. An argument for (\[Pent\]) can be found in [@Jar96]. The entropy strength $\alpha$ and the default model $m$ are parameters. \[sec:compute\]Computing the spectral density --------------------------------------------- With (\[Pchi\]) and (\[Pent\]) the posterior probability (\[BayesT3\]) becomes $${\cal P}[\rho\leftarrow C]\propto e^{-(\chi^2/2-\alpha{\cal S})}\,. \label{Ppost}$$ We wish to maximize ${\cal P}[\rho\leftarrow C]$ with respect to $\rho$, at fixed $C$. It can be shown that both $\chi^2[\rho]$ and $-{\cal S}[\rho]$ are convex functions of $\rho=\{\rho_k : K_- \leq k \leq K_+\}$. Thus $$W[\rho]=\chi^2/2-\alpha S \label{Wrho}$$ has a unique absolute minimum. The functional $W[\rho\/]$ is nonlinear and maximally nonlocal since all degrees of freedom $\rho_k$ are coupled via the covariance matrix (\[Ecov\]) in (\[Chi2\]). To find the minimum of $W[\rho]$ one option is to use singular value decomposition (SVD), see [@Asakawa:2000tr]. In keeping with the Bayesian probabilistic interpretation of $\rho$ an attractive alternative is to employ stochastic methods to solve the optimization problem $W[\rho\/]=\min$. In this work we employ simulated annealing [@Kir84], equivalently known as cooling. The algorithm is based on the partition function $$Z_W=\int [d\rho\/] e^{-\beta_W W[\rho\/]}\,. \label{Zmem}$$ It involves the generation of equilibrium configurations $\rho$ while gradually increasing $\beta_W$ from an initially small value, following some annealing schedule. The latter is subject to experimentation. We have used the power law $$\beta_W(n)=(\beta_1-\beta_0)\left(n/N\right)^\gamma+\beta_0 \label{powerlaw}$$ with annealing steps $n=0\ldots N$ between an initial $\beta_0$ and a final $\beta_1$. A standard Metropolis algorithm was used to generate configurations $\rho$ with the distribution in (\[Zmem\]). In consecutive sweeps local updates were done by multiplying the spectral parameters with positive random numbers, $\rho_k\rightarrow x\rho_k$. Some experimenting showed that $\Gamma$-distributed random deviates of order two, $p_a(x)=x^{a-1}e^{-x}/\Gamma(a), a=2$, work quite efficiently at an acceptance rate centered at about 50%. \[sec:results\]Results ====================== All simulations were done on an $L^3\times T=10^3\times 30$ lattice. The gauge field and fermion actions are both anisotropic, with bare aspect ratio of $a_s/a_t=3$, and tadpole improved. The gauge field action is that of [@Morningstar:1999rf] with $\beta=2.4$, leading to a spatial lattice constant of $a_s\simeq 0.25{\rm fm}, a_s^{-1}\simeq 800{\rm MeV}$. For the light fermions we use a clover improved Wilson action. The hopping parameter $\kappa=0.0679$ results in a mass ratio $m_\pi/m_\rho \simeq 0.75$. Following [@Morningstar:1999rf] only spatial directions are improved with spatial tadpole renormalization factors $u_s=\langle\, \framebox(5,5)[t]{}\,\rangle^{1/4}$, while $u_t=1$ in the time direction. Clover terms involving time directions are omitted. Some guidance for a reasonable $\omega$-discretization (\[Fd\]), of (\[Fc\]), may be derived from the physical value of the lattice constant $a_t$, and the time extent $Ta_t$ of the lattice. Admissible lattice energies thus lie approximately between $\pi/a_t\approx 7.5{\rm GeV}$ and $\pi/Ta_t\approx 250{\rm MeV}$, or $\approx 3$ and $\approx 0.1$ in units of $a_t^{-1}$. In practice these are somewhat extreme bounds. Typical hadronic excitation energies are much less than $\pi/a_t\approx 7.5{\rm GeV}$. The lower bound, on the other hand, may well be ignored as a criterion for choosing the discretization interval $\Delta\omega$, because the theoretical form of $\rho$ is a superposition of $\delta$-peaks. Thus the resolution $\Delta\omega$ should be small, in fact much smaller than $\approx 0.1$. A reasonable lower bound is the likely statistical error on spectral masses. For most of the results presented here $\Delta\omega=0.04$, and $ K_-=-40, K_+=+80$, leading to $-1.6\leq\omega\leq +3.2$, were used with (\[Fd\]). With the annealing schedule (\[powerlaw\]), we have used $N=2048$ cooling steps, at 128 sweeps per temperature, starting at $\beta_0=1.0\times 10^{-5}\beta$ and ending at $\beta_1=1.0\times 10^{+5}\beta$, with a geometric average of $\beta=1.0\times 10^{+3}$. These choices are an outcome of experimentation. With $\gamma\simeq 16.61$ in (\[powerlaw\]) about half of the cooling steps operate in the regions $\beta_W(n)<\beta$ and $\beta_W(n)>\beta$, respectively. The average value $\beta$ is such that $\beta_W W[\rho\/]$ fluctuates about one at around $N/2$ cooling steps. With the final annealing temperature kept constant, $\beta_W=\beta_1$, an additional 1024 steps were done keeping 16 configurations $\rho$ in order to measure cooling fluctuations. Results are robust within reasonable changes of the annealing schedule parameters, they were used throughout this work. \[sec:alpha\]Entropy weight dependence -------------------------------------- The extent to which the spectral density $\rho$ depends on the value of the entropy weight parameter $\alpha$, in (\[Wrho\]), is a primary concern. We are interested in testing the $\alpha$ dependence for a case where both ground and excited states are prominently present in a time correlation function. For this reason we have constructed a mock correlator $C_{\rm X}(t,t_0)$. Its building blocks were the eigenvalues of the $2\times 2$ correlation matrix $C_{ij}(t,t_0) = \langle\hat{\Phi}_i^\dagger(t)\hat{\Phi}_j(t_0)\rangle$ using the above mentioned local and non-local meson fields. Pieces of those were arbitrarily matched and enhanced in order to exhibit a multi-exponential correlation function. While $C_{\rm X}(t,t_0)$ bears no physical significance, its rich structure provides a useful laboratory for testing the $\alpha$ dependence of the spectral density function. In Fig. \[fig1\] we show a sequence of six pairs of Bayesian fits to the mock correlator $C_{\rm X}(t,t_0)$ and the corresponding spectral densities $\rho$ for a wide range of entropy weights $\alpha$. The stability of the global structure of $\rho$ while $\alpha$ changes from $1.4\times 10^{-2}$ to $1.4\times 10^{+7}$ is most notable[^1]. As $\alpha$ becomes larger entire peaks vanish starting with the smallest one. The reason is that the annealing action (\[Wrho\]) gradually loses memory of the data, contained in $\chi^2$, in favor of the entropy. The fit at $\alpha=1.4\times 10^{+7}$ exhibits the onset of a smoothing of the micro structure, starting with the largest peak. This is the signature of emerging entropy dominance over the data. In practice this situation should be avoided. In our case entropy strengths in the region $\alpha < 10^{+6}$ over eight orders of magnitude give stable consistent results. It has been proposed that spectral functions be integrated over $\alpha$ to avoid the parameter dependence [@Jar96]. Inspection of our results clearly indicates that averaging over $\alpha$ would be without consequence to the gross structure of $\rho$, only the micro structure would be affected. Even the region $\alpha>10^{+6}$ could be included, since the magnitude of $\rho$ quickly becomes insignificant. {width="84mm"}{width="84mm"}\ {width="84mm"}{width="84mm"}\ {width="84mm"}{width="84mm"} In order to decide on a tuning criterion for $\alpha$ it is useful to monitor quantities like $$\begin{aligned} Y_{S/W}&=&\frac{\langle -\alpha S\rangle_{\beta_W\rightarrow\infty}} {\langle W\rangle_{\beta_W\rightarrow\infty}}\label{SW}\\ Y_{S/\chi^2}&=&\frac{\langle -\alpha S\rangle_{\beta_W\rightarrow\infty}} {\langle \chi^2/2\rangle_{\beta_W\rightarrow\infty}}\label{SC}\,,\end{aligned}$$ where $\langle\ldots\rangle_{\beta_W\rightarrow\infty}$ refers to the annealing average measured at the final cooling temperature, $\beta_1$. We will refer to the above quantities as entropy loads. Those are shown in Fig. \[fig2\]. It turns out that $\log(Y)$ depends linearly on $\log(\alpha)$ in the regions $\log(\alpha)<+1$ and $\log(\alpha)<+4$, for $Y_{S/W}$ and $Y_{S/\chi^2}$, respectively. (In fact $Y\approx 6.2\times10^{-4}\alpha$.) Beyond the linear region too much entropy is loaded into the annealing action $W$, leading to a smoothing of peaks, as seen in Fig. \[fig1\]. Empirically, the criterion emerging from this observation is to tune the entropy weight such that $\log(Y)\approx -2\pm 1$ within the linear region. The precise value of $\log(Y)$ is not important, also $Y=Y_{S/W}$ and $Y=Y_{S/\chi^2}$ work equally well. As is evident from Fig. \[fig1\] results are extremely robust against varying $\alpha$. ![\[fig2\]Empirical dependence of the entropy loads $Y_{S/W}$ and $Y_{S/\chi^2}$ on the entropy weight parameter $\alpha$, see (\[SW\], \[SC\]). These results are for the mock correlator $C_X(t,t_0)$. The lines indicate the extent of linear relationships.](specdens-fig2a.eps "fig:"){width="42mm"} ![\[fig2\]Empirical dependence of the entropy loads $Y_{S/W}$ and $Y_{S/\chi^2}$ on the entropy weight parameter $\alpha$, see (\[SW\], \[SC\]). These results are for the mock correlator $C_X(t,t_0)$. The lines indicate the extent of linear relationships.](specdens-fig2b.eps "fig:"){width="42mm"} \[sec:single\]Single-meson spectrum ----------------------------------- The correlation function $c(t,t_0)=\langle\hat{\phi}^\dagger(t)\hat{\phi}(t_0)\rangle$ of a single pseudoscalar heavy-light meson operator $\phi(t)=\sum_{\vec{x}}\overline{Q}_A(\vec{x}t)\gamma_5 q_A(\vec{x}t)$ delivers high quality data in this simulation. We use these to compare with plateau methods and make some observations relevant to the present stochastic approach to the MEM. In Fig. \[fig3\] plots of the mass function discretizations (\[eff0\]–\[eff3\]), built from $c(t,t_0)$, and the corresponding plateau fits are displayed. Plateau fits were made directly to $\mu_{\rm eff,i=0\ldots 3}$. The resulting masses, other than $m_{\rm eff,0}$, are from solving (\[eff1\]–\[eff3\]). Table \[tab1\] shows that those are consistent within statistical (jackknife) errors. ![\[fig3\]Effective mass functions (\[eff0\]–\[eff3\]) for a single heavy-light meson. The horizontal lines are plateau fits in the time slice range $6\leq t\leq 18$.](specdens-fig3.eps){width="84mm"} ----------------- ----------------- ----------------- ----------------- ----------- ------------ $m_{\rm eff,0}$ $m_{\rm eff,1}$ $m_{\rm eff,2}$ $m_{\rm eff,3}$ $E_1$ $\Delta_1$ 0.468(8) 0.468(7) 0.468(3) 0.47(2) 0.471(15) 0.017(6) ----------------- ----------------- ----------------- ----------------- ----------- ------------ : \[tab1\]Plateau masses derived from (\[eff0\]–\[eff3\]) on the time slice range $6\leq t\leq 18$. The entry $E_1$ is the Bayesian result with $\Delta_1$ being the peak width (standard deviation) computed from the spectral density function $\rho$. Statistical errors are derived from a gauge configuration jackknife analysis. Figure \[fig4\] gives a sense of the annealing dynamics. Beside (\[SW\]) and (\[SC\]) also shown are $$\begin{aligned} Y_{S}&=&\langle -\alpha S\rangle_{\beta_W\rightarrow\infty}\label{YS}\\ Y_{\chi^2}&=&\langle \chi^2/2\rangle_{\beta_W\rightarrow\infty}\,.\label{YC}\end{aligned}$$ In [@Fiebig:2001nn] and [@Fiebig:2001mr] the use of $Y_{S/W}$ was advocated as a tuning criterion. In view of Fig. \[fig4\] $Y_{S/\chi^2}$ appears to be a better choice given its monotonic nature. A target entropy load of $Y_{S/\chi^2}\approx 10^{-1\pm 1}$ is a safe tuning criterion, provided the cooling algorithm runs in the (upper) linear region, see Fig. \[fig2\]. ![\[fig4\]Annealing dynamics in terms of the tuning functions $Y_{S/W}$, $Y_{S/\chi^2}$, and $Y_{S}$, $Y_{\chi^2}$, versus the cooling parameter $\beta_W$. The graphs are labeled with reference to the entropy loads (\[SW\], \[SC\]), and (\[YS\], \[YC\]). This example is for the single-meson correlator, with entropy strength $\alpha=5.0\times 10^{-5}$ and a constant default model $m=1.0\times 10^{-12}$.](specdens-fig4.eps){width="56mm"} The Bayesian analysis of the time correlation function $c(t,t_0)$ is shown in Fig. \[fig5ab\]. The solid line in Fig. \[fig5ab\](a) derives from the computed spectral density $\rho$, via (\[Fd\]). With the exception of $t_0=0$ all available time slices were used. Parameters are $\alpha=5.0\times 10^{-5}$, for the entropy strength, a constant default model $m=1.0\times 10^{-12}$, and a random annealing start about $m$. The graph of $\rho$ in Fig. \[fig5ab\](b) exhibits a global structure consisting of distinct peaks, some broad, and a micro structure of fluctuations on the scale of $\Delta\omega$. The micro structure depends on details of the annealing process, particularly the start configuration. Clearly, it makes no sense to infer the micro structure from the data. The reason is that only $T-1=29$ data points do not contain enough information to determine $K_{+}-K_{-}+1=K=121$ spectral parameters (with any sizable probability). ![\[fig5ab\]Time correlation function for a single heavy-light meson together with a Bayesian fit (a), and the corresponding spectral density function (b). This result stems from a single random start, with entropy weight $\alpha=5.0\times 10^{-5}$, and a constant default model $m=1.0\times 10^{-12}$.](specdens-fig5a.eps "fig:"){width="42mm"} ![\[fig5ab\]Time correlation function for a single heavy-light meson together with a Bayesian fit (a), and the corresponding spectral density function (b). This result stems from a single random start, with entropy weight $\alpha=5.0\times 10^{-5}$, and a constant default model $m=1.0\times 10^{-12}$.](specdens-fig5b.eps "fig:"){width="42mm"} On the other hand the global structure is a stable feature. In the region $\omega>0$ three peaks can be distinguished in Fig. \[fig5ab\](b). By way of inspection we loosely define $$\delta_n=\{\omega:\omega\in{\rm peak}\ \#n\}\quad n=1,2\ldots\,. \label{deltan}$$ Then, for each peak $n$, we may calculate the volume $Z_n$, the mass $E_n$, and the width $\Delta_n$, according to $$\begin{aligned} Z_n&=&\int_{\delta_n}d\omega\/\rho_T(\omega)\label{Zn}\\ E_n&=&Z_n^{-1}\int_{\delta_n}d\omega\/\rho_T(\omega)\omega\label{En}\\ \Delta_n^2&=&Z_n^{-1}\int_{\delta_n}d\omega\/\rho_T(\omega)\left(\omega-E_n\right)^2\,. \label{Dn}\end{aligned}$$ These integrated, low moment, quantities are evidently insensitive to the micro structure. They constitute the information that reasonably can be expected to flow from the Bayesian analysis. The spectral density of Fig. \[fig5ab\](b) is replotted in Fig. \[fig5cd\] on linear scales. The tall narrow peak in Fig. \[fig5cd\](c) corresponds to the plateau masses of Fig. \[fig3\], as listed in Tab. \[tab1\]. There, the entries $E_1$ and $\Delta_1$ are the Bayesian results. Their statistical errors are derived from a jackknife analysis selecting four subsets of gauge configurations. (Note that the uncertainties in Fig. \[fig5cd\] are standard deviations from eight annealing starts.) Cold starts from the default model $m$ were used to suppress the dependence on the annealing start configuration. The peak width $\Delta_1$ is comparable to the gauge configuration statistical error. This is the exception. With correlation function data of lesser quality (like with the two-meson operators below) the size of the peak width is typically larger than the statistical error. It appears that the peak width $\Delta_n$ is related to the size $\Theta_n$ of the corresponding effective mass function plateau, like in Fig. \[fig3\], or the size of the $\log$-linear stretch in a plot like in Fig. \[fig5ab\](a). As a very coarse description $\Delta_n \Theta_n\approx {\rm const}$ comes to mind. Using $\Theta_1=12$ and $\Delta_1=0.017$ we have ${\rm const}\approx 0.2$. The peaks $n=2$ and $n=3$ seen in Fig. \[fig5cd\](d) would thus appear to originate from $\Theta_n\approx 0.2/\Delta_n$, or 1.3 and 0.8 time slices, respectively. (By inspection of Fig. \[fig3\] as many as 5 time slices appear involved, however.) The physical relevance of, at least, peak $n=3$ is therefore questionable. On the other hand it is remarkable that the maximum entropy method is sensitive to the slightest details in the correlation function data. ![\[fig5cd\]Spectral density $\rho$ for a single heavy-light meson, same as in Fig. \[fig5ab\](b), but on linear scales, emphasizing the ground and the excites states (c) and (d), respectively. The uncertainties of $Z_n$, $E_n$, and $\Delta_n$ are standard deviations from eight annealing runs.](specdens-fig5c.eps "fig:"){width="41.2mm"} ![\[fig5cd\]Spectral density $\rho$ for a single heavy-light meson, same as in Fig. \[fig5ab\](b), but on linear scales, emphasizing the ground and the excites states (c) and (d), respectively. The uncertainties of $Z_n$, $E_n$, and $\Delta_n$ are standard deviations from eight annealing runs.](specdens-fig5d.eps "fig:"){width="42.8mm"} \[sec:defaultm\]Default model dependence ---------------------------------------- The Shannon-Jaynes entropy (\[Smem\]) implies the possible dependence of the computed spectral density $\rho$ on the default model $m=\{m_k : K_- \leq k \leq K_+\}$. We explore the $m$ dependence using as an example the time correlation function $C_v$ with $v_1=1$ $v_2=0$, in the notation of (\[Phiv\]), at relative distance $r=4$. Figure \[fig6d7d\] shows the time correlation function data together with the Bayesian fit, and the corresponding spectral density $\rho$. The latter is the average over eight random annealing start configurations. This has the effect of smoothing out the micro structure of $\rho$. We have used a constant default model $m_k=1.0\times 10^{-12}$, all $k$. ![\[fig6d7d\]Correlation function $C_{11}=\langle\hat{\Phi}^\dagger_1(t)\hat{\Phi}_1(t_0)\rangle$ of a heavy-light meson-meson operator at relative distance $r=4$. The Bayesian fit (solid line) is from the spectral density $\rho$ shown on the right. At $\alpha=2\times 10^{-6}$ and constant default model $m=1.0\times 10^{-12}$ the spectral density $\rho$ is obtained from an average over eight random annealing start configurations. The average entropy load is $Y_{S/\chi^2}=0.477$ for these runs.](specdens-fig6d.eps "fig:"){width="42.2mm"} ![\[fig6d7d\]Correlation function $C_{11}=\langle\hat{\Phi}^\dagger_1(t)\hat{\Phi}_1(t_0)\rangle$ of a heavy-light meson-meson operator at relative distance $r=4$. The Bayesian fit (solid line) is from the spectral density $\rho$ shown on the right. At $\alpha=2\times 10^{-6}$ and constant default model $m=1.0\times 10^{-12}$ the spectral density $\rho$ is obtained from an average over eight random annealing start configurations. The average entropy load is $Y_{S/\chi^2}=0.477$ for these runs.](specdens-fig7d.eps "fig:"){width="41.8mm"} The stability of this result is tested by varying the default model through 15 orders of magnitude, $m=10^{-12}\ldots 10^{+3}$, as shown in Fig. \[fig18\]. To keep effects of the annealing start configuration small cold starts from $\rho=m$, using the same random seed, were employed for all values of $m$. In each case the entropy strength parameter $\alpha$ was tuned such that the entropy load $Y_{S/\chi^2}$ remained constant. Aside from the familiar micro structure fluctuations, the global (physical) features are stable within the range of, a remarkable, fifteen orders of magnitude. Numerical experiments with non-constant $m$ do not change this assessment. In Tab. \[tab2\] are listed the three integral quantities (\[Zn\])–(\[Dn\]) averaged over the six default models together with the corresponding standard deviations. Their smallness (0.3–3%) attests to the default model independence of the Bayesian fits. Given the huge variation of the default model the stability of $\rho$ is remarkable. ![\[fig18\]A sequence of spectral densities $\rho$ obtained from a wide range of constant default models $m$, see inserts. The entropy strength parameter $\alpha$ was tuned to keep the entropy load constant, $Y_{S/\chi^2}\approx 0.045$. The operator is the same as in Fig. \[fig6d7d\].](specdens-fig18A.eps "fig:"){width="42mm"} ![\[fig18\]A sequence of spectral densities $\rho$ obtained from a wide range of constant default models $m$, see inserts. The entropy strength parameter $\alpha$ was tuned to keep the entropy load constant, $Y_{S/\chi^2}\approx 0.045$. The operator is the same as in Fig. \[fig6d7d\].](specdens-fig18B.eps "fig:"){width="42mm"}\ ![\[fig18\]A sequence of spectral densities $\rho$ obtained from a wide range of constant default models $m$, see inserts. The entropy strength parameter $\alpha$ was tuned to keep the entropy load constant, $Y_{S/\chi^2}\approx 0.045$. The operator is the same as in Fig. \[fig6d7d\].](specdens-fig18C.eps "fig:"){width="42mm"} ![\[fig18\]A sequence of spectral densities $\rho$ obtained from a wide range of constant default models $m$, see inserts. The entropy strength parameter $\alpha$ was tuned to keep the entropy load constant, $Y_{S/\chi^2}\approx 0.045$. The operator is the same as in Fig. \[fig6d7d\].](specdens-fig18D.eps "fig:"){width="42mm"}\ ![\[fig18\]A sequence of spectral densities $\rho$ obtained from a wide range of constant default models $m$, see inserts. The entropy strength parameter $\alpha$ was tuned to keep the entropy load constant, $Y_{S/\chi^2}\approx 0.045$. The operator is the same as in Fig. \[fig6d7d\].](specdens-fig18E.eps "fig:"){width="42mm"} ![\[fig18\]A sequence of spectral densities $\rho$ obtained from a wide range of constant default models $m$, see inserts. The entropy strength parameter $\alpha$ was tuned to keep the entropy load constant, $Y_{S/\chi^2}\approx 0.045$. The operator is the same as in Fig. \[fig6d7d\].](specdens-fig18F.eps "fig:"){width="42mm"} ------------ ---------- ------------ ----------------- $Z_1$ $E_1$ $\Delta_1$ $m_{\rm eff,0}$ 3923.(18.) 0.972(3) 0.100(3) 0.94(1) ------------ ---------- ------------ ----------------- : \[tab2\]Averages of volume, energy, and width of the dominant peak seen in Fig. \[fig18\] over the six default model choices $m=10^{-12}\ldots 10^{+3}$ at fixed entropy load $Y_{S/\chi^2}\approx 0.045$. The uncertainties are the corresponding standard deviations. The entry $m_{\rm eff,0}$ is the plateau mass (\[eff0\]) from Fig.\[fig3Rab\] with the statistical (jackknife) error, see Sect. \[sec:plateau\]. \[sec:anneal\]Annealing start dependence ---------------------------------------- The annealing algorithm starts with some initial spectral configuration $\rho_{\rm ini}$. Depending on the purpose we have used cold starts from the default model, $\rho_{\rm ini}=m$, or random starts from the default model, $\rho_{{\rm ini},k}=x_km_k$, where the $x_k$ are drawn from a gamma distribution of order two, $p_a(x)=x^{a-1}e^{-x}/\Gamma(a), a=2$. The global features of the final spectral density are of course independent of the start configuration, but the micro structure of $\rho$ is not. The reason is that in practice the annealing process is neither infinitely slow nor is the final cooling temperature $\beta_1^{-1}$ exactly zero. Therefore the annealing result for $\rho$ settles close to the global minimum, say $\rho_{\rm min}$, of $W[\rho]$. Considering annealing (thermal) fluctuations only, we expect the deviation $|\rho-\rho_{\rm min}|$ to be large in directions (of $\rho$ space) where the minimum is shallow. Thermal fluctuations are easily controlled, however. Those were kept negligible in the present study. More importantly, there may be local minima close to $\rho_{\rm min}$ which are only slightly larger than $W[\rho_{\rm min}]$. This situation invites computing a set of spectral densities from different, say random, initial configurations. The averages and standard deviations of the $\rho_k$ then gives some insight into the structure of the peak and the nature of the minimum of $W$ and its neighborhood. To present an example we have selected an excited state time correlation function $C_2(t,t_0)$ of the meson-meson system at relative distance $r=4$. $C_2(t,t_0)$ is the smaller of the eigenvalues of the $2\times 2$ correlation matrix $C_{ij}(t,t_0) = \langle\hat{\Phi}_i^\dagger(t)\hat{\Phi}_j(t_0)\rangle$, on each time slice. The reason for selecting this operator is to see how the MEM responds to a data set that is marginally acceptable, at best. Figure \[fig15d16d\] shows the correlator and the corresponding spectral density obtained from an average over eight Bayesian fits based on different random annealing start configurations. The same spectral density is displayed in the first frame of Fig. \[fig19\] on a linear scale. The dotted lines represent the limits within one standard deviation. The remaining three frames of Fig. \[fig19\] show spectral functions from selected single start configurations. They illustrate the micro structure fluctuations. ![\[fig15d16d\]Excited state correlation function $C_2$ of a heavy-light meson-meson operator at relative distance $r=4$. The Bayesian fit (solid line) is from the spectral density $\rho$ shown on the right. At $\alpha=5\times 10^{-7}$ and constant default model $m=1.0\times 10^{-12}$ the spectral density $\rho$ is obtained from an average over eight random annealing start configurations.](specdens-fig15d.eps "fig:"){width="42.2mm"} ![\[fig15d16d\]Excited state correlation function $C_2$ of a heavy-light meson-meson operator at relative distance $r=4$. The Bayesian fit (solid line) is from the spectral density $\rho$ shown on the right. At $\alpha=5\times 10^{-7}$ and constant default model $m=1.0\times 10^{-12}$ the spectral density $\rho$ is obtained from an average over eight random annealing start configurations.](specdens-fig16d.eps "fig:"){width="41.8mm"} ![\[fig19\]Spectral densities $\rho$ of the excited state correlation function of Fig. \[fig15d16d\]. The sequence of four frames shows the average (ave) over a sample of eight random annealing start configurations including the bounds (dotted lines) of one standard deviation ($\pm$sig), and three selected examples of spectral functions making up that sample.](specdens-fig19.eps "fig:"){width="42mm"} ![\[fig19\]Spectral densities $\rho$ of the excited state correlation function of Fig. \[fig15d16d\]. The sequence of four frames shows the average (ave) over a sample of eight random annealing start configurations including the bounds (dotted lines) of one standard deviation ($\pm$sig), and three selected examples of spectral functions making up that sample.](specdens-fig19c.eps "fig:"){width="42mm"}\ ![\[fig19\]Spectral densities $\rho$ of the excited state correlation function of Fig. \[fig15d16d\]. The sequence of four frames shows the average (ave) over a sample of eight random annealing start configurations including the bounds (dotted lines) of one standard deviation ($\pm$sig), and three selected examples of spectral functions making up that sample.](specdens-fig19e.eps "fig:"){width="42mm"} ![\[fig19\]Spectral densities $\rho$ of the excited state correlation function of Fig. \[fig15d16d\]. The sequence of four frames shows the average (ave) over a sample of eight random annealing start configurations including the bounds (dotted lines) of one standard deviation ($\pm$sig), and three selected examples of spectral functions making up that sample.](specdens-fig19h.eps "fig:"){width="42mm"} We argue that the micro structure, on a fine discretization scale $\Delta\omega$, is extraneous information. On the basis that the number of measured data points, as supplied by the time correlation function $C_v(t,t_0)$, is much smaller than the number of inferred parameters $\rho_k$, exact knowledge of $\rho$ would actually constitute an information gain not supported by the data. Rather, only averages of suitable observables based on the inferred spectral density, like (\[Zn\])–(\[Dn\]) for example, are relevant information that can be extracted from the Bayesian analysis. Whether or not the $\rho$ average of a certain observable is relevant information supported by the data may possibly be decided by the criterion that the standard deviation with respect to different annealing starts be small. From Tab. \[tab3\] we see that the standard deviations for the small-moment averages (\[Zn\],\[En\],\[Dn\]) are comparable to typical gauge configuration statistical errors, for example those in Tab. \[tab1\]. This should be an acceptable test, certainly high resolution operators would fail it. ------------ ---------- ------------ ----------------- $Z_1$ $E_1$ $\Delta_1$ $m_{\rm eff,0}$ 2156.(11.) 2.012(7) 0.214(11) 1.92(3) ------------ ---------- ------------ ----------------- : \[tab3\]Averages of volume, energy, and width of the dominant peak seen in Figs. \[fig19\] over eight random annealing start configurations, at fixed $\alpha=5.0\times 10^{-7}$ and constant default model $m=10^{-12}$. The entry $m_{\rm eff,0}$ is the plateau mass (\[eff0\]) from Fig.\[fig3Rab\] with the statistical (jackknife) error, see Sect. \[sec:plateau\]. \[sec:plateau\]Relation to plateau methods ------------------------------------------ Aside from the obvious differences in algorithm and philosophy it is important to understand that the traditional plateau method and the celebrated Bayesian approach also are distinctly different in the way they utilize the lattice correlator data. First, the former uses data on only a (subjectively) truncated contiguous set of time slices while completely ignoring the rest, whereas the latter utilizes the data on all available time slices without bias. Second, in the plateau method the stochastic dependence of the data between the plateau time slices is often ignored[^2] whereas in the Bayesian approach the dependence is fully accounted for through the covariance matrix (\[Ecov\]). Hence, the traditional plateau method and the Bayesian inference approach cannot be compared on an equal footing. In particular, their systematic errors are in principle different. A comparison of those methods is thus reduced to observing their responses to the same data sets. If the numerical quality of data is very good both methods (in fact any two methods) will of course give the same answers. An example is the single-meson case discussed above, see Tab. \[tab1\]. In case of imperfect numerical data, however, the two methods should be expected to give different results. We illustrate this point by showing in Fig. \[fig3Rab\] the effective mass functions (\[eff0\]) of the correlators $C_{11}$ and $C_2$ displayed in Figs. \[fig6d7d\] and \[fig15d16d\], respectively. While the $C_{11}$ data are somewhat level within 9 time slices, the $C_2$ data are extreme in the sense that only 2 data points are available to the plateau method. Bayesian inference, as illustrated by Fig. \[fig15d16d\] and also Fig. \[fig19\], has no problem responding with a distinct peak. The reason, of course, is that the entire set of correlator data including their correlations is available to the Bayesian approach. ![\[fig3Rab\]Effective mass functions $\mu_{\rm eff,0}$, see (\[eff0\]), of the correlator examples $C_{11}$ and $C_2$ shown in Figs.\[fig6d7d\] and \[fig15d16d\]. The plateaus are shown as horizontal lines extending over 9 and 2 time slices, respectively.](specdens-fig3Ra.eps "fig:"){width="42.0mm"} ![\[fig3Rab\]Effective mass functions $\mu_{\rm eff,0}$, see (\[eff0\]), of the correlator examples $C_{11}$ and $C_2$ shown in Figs.\[fig6d7d\] and \[fig15d16d\]. The plateaus are shown as horizontal lines extending over 9 and 2 time slices, respectively.](specdens-fig3Rb.eps "fig:"){width="42.0mm"} In Tabs. \[tab2\] and \[tab3\] we compare the plateau masses $m_{\rm eff,0}$ obtained from (\[eff0\]) to the Bayesian results $E_1$. The numbers differ by about 3–5%. Note that the statistical (jackknife) errors on the plateau masses are much smaller. Because of the data truncation the method has no way of ‘knowing’ about the poor quality of the correlator data, particularly in the $C_2$ case of the exited state correlator. The Bayesian method, on the other hand, is fully ‘aware’ of this fact and conveys this information by responding with a sizable peak width $\Delta_1$, which easily encompasses the plateau masses. This raises the question whether Bayesian peak widths or plateau mass statistical errors are a better measure for the uncertainty of masses extracted from lattice simulations. The answer is beyond the scope of this work. \[sec:conclusion\]Summary and conclusion ======================================== We have reported on our experience using Bayesian inference with an entropic prior, the maximum entropy method, to extract spectral information from lattice generated time correlation functions. The latter were taken from a simulation aimed at studying hadronic interaction, but used here only as a repository of simulation data of diverse quality. In contrast to other works the method of choice for extracting spectral densities was simulated annealing. Between the maximum entropy method and simulated annealing there were three major concerns about the parameter and algorithm dependence of the results: Dependence on (i) the entropy weight, (ii) the default model, and (iii) the annealing start configuration. Besides suggesting strategies for parameter tuning, independence of the Bayesian inferred spectral density $\rho$ on (i) the entropy weight, and (ii) the default model could be demonstrated within a range of eight and fifteen orders of magnitude of the parameters, respectively. Concerning the annealing start configuration dependence (iii) we argued that only spectral density averages of certain operators are acceptable. From an information theory point of view [@Sha49], those should be operators insensitive to the micro structure of the inferred spectral density. In particular, keeping in mind that the theoretical structure of the lattice spectral function is a superposition of distinct peaks, those operators include the spectral peak volume $Z_n$, or normalization, the peak energy $E_n$, or mass, and the peak width $\Delta_n$, or standard deviation. Bayesian inference has too long been ignored by the lattice community as an analysis tool. It has an advantage over conventional plateau methods for extracting hadron masses from lattice simulations because the entire information contained in the correlator function, or matrix, is utilized. This aspect is particularly important where excited state masses are desired, since the noise contamination of their signal can be significant. The maximum entropy method is very robust with respect to changing its parameters. Simulated annealing is practical for obtaining spectral density functions. The method should be given serious consideration as an alternative for conventional ways. This material is based upon work supported by the National Science Foundation under Grant No. 0073362. Resources made available through the Lattice Hadron Physics Collaboration (LHPC) were used in this project. [24]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , , , ****, (), . , ****, (). , ****, (), . , ****, (), . () (), . , , , ****, (), . , ****, (). , ** (, , ). , , , ****, (), . , , , ****, (), . , , , ****, (). , , , ****, (), . , , , , , , , ****, (), . , **, . (), ****, (), . (), ****, (), . , ** (, , ). (), . , ** (, , ). , ****, (). , ****, (). , ****, (). , ****, (), . , ****, (), . [^1]: In [@Fiebig:2001nn] and [@Fiebig:2001mr] a different normalization of the covariance matrix (\[Ecov\]) was used. This can be accounted for by a rescaling of the entropy weight $\alpha=(N_U-1)\bar{\alpha}$, where $\bar{\alpha}$ refers to the above references and $N_U=708$. [^2]: Uncorrelated fits to a mass function may be justified if the number of gauge configurations $N$ is large compared to the number of plateau times slices $D$, see [@Michael:1994yj; @Michael:1995sz]. There the condition $N>\max(D^2,10(D+1))$ applied to the situation of Fig. \[fig3\] gives $708>169$.

--- author: - Christopher Deninger date:   title: Number theory and dynamical systems on foliated spaces --- Introduction ============ In this paper we report on some developments in the search for a dynamical understanding of number theoretical zeta functions that have taken place since my ICM lecture [@D2]. We also point out a number of problems in analysis that will have to be solved in order to make further progress. In section 2 we give a short introduction to foliations and their cohomology. Section 3 is devoted to progress on the dynamical Lefschetz trace formula for one-codimensional foliations mainly due to Álvarez López and Kordyukov. In section 4 we make the comparison with the “explicit formulas” in analytic number theory. Finally in section 5 we generalize the conjectural dynamical Lefschetz trace formula of section 3 to phase spaces which are more general than manifolds. This was suggested by the number theoretical analogies of section 4. This account is written from an elementary point of view as far as arithmetic geometry is concerned, in particular motives are not mentioned. In spirit the present article is therefore a sequel to [@D1]. There is a different approach to number theoretical zeta functions using dynamical systems by A. Connes [@Co]. His phase space is a non-commutative quotient of the adèles. Although superficially related, the two approaches seem to be deeply different. Whereas Connes’ approach generalizes readily to automorphic $L$-functions [@So] but not to motivic $L$-functions, it is exactly the opposite with our picture. One may wonder whether there is some kind of Langlands correspondence between the two approaches. I would like to thank the Belgium and German mathematical societies very much for the opportunity to lecture about this material during the joint BMS–DMV meeting in Liège 2001. Foliations and their cohomology =============================== A $d$-dimensional foliation ${{\mathcal F}}= {{\mathcal F}}_X$ on a smooth manifold $X$ of dimension $a$ is a partition of $X$ into immersed connected $d$-dimensional manifolds $F$, the “leaves”. Locally the induced partition should be trivial: Every point of $X$ should have an open neighborhood $U$ diffeomorphic to an open ball $B$ in ${{\mathbb{R}}}^a$ such that the leaves of the induced partition on $U$ correspond to the submanifolds $B \cap ({{\mathbb{R}}}^d \times \{ y \})$ of $B$ for $y$ in ${{\mathbb{R}}}^{a-d}$. One of the simplest non-trivial examples is the one-dimensional foliation of the two-dimensional torus $T^2 = {{\mathbb{R}}}^2 / {{\mathbb{Z}}}^2$ by lines of irrational slope $\alpha$. These are given by the immersions $${{\mathbb{R}}}\hookrightarrow T^2 \; , \; t \mapsto (x + t \alpha , t) {\;\mathrm{mod}\;}{{\mathbb{Z}}}^2$$ parametrized by $x {\;\mathrm{mod}\;}{{\mathbb{Z}}}+ \alpha {{\mathbb{Z}}}$. In this case every leaf is dense in $T^2$ and the intersection of a global leaf with a small open neighborhood $U$ as above decomposes into countably many connected components. It is the global behaviour which makes foliations complicated. For a comprehensive introduction to foliation theory, the reader may turn to [@Go] for example. To a foliation ${{\mathcal F}}$ on $X$ we may attach its tangent bundle $T {{\mathcal F}}$ whose total space is the union of the tangent spaces to the leaves. By local triviality of the foliation it is a sub vector bundle of the tangent bundle $TX$. It is integrable i.e. the commutator of any two vector fields with values in $T {{\mathcal F}}$ again takes values in $T{{\mathcal F}}$. Conversely a theorem of Frobenius asserts that every integrable sub vector bundle of $TX$ arises in this way. Differential forms of order $n$ along the leaves are defined as the smooth sections of the real vector bundle $\Lambda^n T^* {{\mathcal F}}$, $${{\mathcal A}}^n_{{{\mathcal F}}} (X) = \Gamma (X, \Lambda^n T^* {{\mathcal F}}) \; .$$ The same formulas as in the classical case define exterior derivatives along the leaves: $$d^n_{{{\mathcal F}}} : {{\mathcal A}}^n_{{{\mathcal F}}} (X) \longrightarrow {{\mathcal A}}^{n+1}_{{{\mathcal F}}} (X) \; .$$ They satisfy the relation $d^{n+1}_{{{\mathcal F}}} {\mbox{\scriptsize $\,\circ\,$}}d^n_{{{\mathcal F}}} = 0$ so that we can form the leafwise cohomology of ${{\mathcal F}}$: $$H^n_{{{\mathcal F}}} (X) = {\mathrm{Ker}\,}d^n_{{{\mathcal F}}} / {\mathrm{Im}\,}d^{n-1}_{{{\mathcal F}}} \; .$$ For our purposes these invariants are actually too subtle. We therefore consider the reduced leafwise cohomology $${\bar{H}}^n_{{{\mathcal F}}} (X) = {\mathrm{Ker}\,}d^n_{{{\mathcal F}}} / \overline{{\mathrm{Im}\,}d^{n-1}_{{{\mathcal F}}}} \; .$$ Here the quotient is taken with respect to the topological closure of ${\mathrm{Im}\,}d^{n-1}_{{{\mathcal F}}}$ in the natural Fréchet topology on ${{\mathcal A}}^n_{{{\mathcal F}}} (X)$. The reduced cohomologies are nuclear Fréchet spaces. Even if the leaves are dense, already ${\bar{H}}^1_{{{\mathcal F}}} (X)$ can be infinite dimensional. The cup product pairing induced by the exterior product of forms along the leaves turns ${\bar{H}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}} (X)$ into a graded commutative ${\bar{H}}^0_{{{\mathcal F}}} (X)$-algebra. The Poincare Lemma extends to the foliation context and implies that $$H^n_{{{\mathcal F}}} (X) = H^n (X, {{\mathcal R}}) \; .$$ Here ${{\mathcal R}}$ is the sheaf of smooth real valued functions which are locally constant on the leaves. In particular $${\bar{H}}^0_{{{\mathcal F}}} (X) = H^0_{{{\mathcal F}}} (X) = H^0 (X, {{\mathcal R}})$$ consists only of constant functions if ${{\mathcal F}}$ contains a dense leaf. For the torus foliation above with $\alpha \notin {{\mathbb{Q}}}$ we therefore have ${\bar{H}}^0_{{{\mathcal F}}} (T^2) = {{\mathbb{R}}}$. Some Fourier analysis reveals that ${\bar{H}}^1_{{{\mathcal F}}} (T^2) \cong {{\mathbb{R}}}$. The higher cohomologies vanish since almost by definition we have $$H^n_{{{\mathcal F}}} (X) = 0 \quad \mbox{for all} \; n > d = \dim {{\mathcal F}}\; .$$ For a smooth map $f : X \to Y$ of foliated manifolds which maps leaves into leaves, continuous pullback maps $$f^* : {{\mathcal A}}^n_{{{\mathcal F}}_Y} (Y) \longrightarrow {{\mathcal A}}^n_{{{\mathcal F}}_X} (X)$$ are defined for all $n$. They commute with $d_{{{\mathcal F}}}$ and respect the exterior product of forms. Hence they induce a continuous map of reduced cohomology algebras $$f^* : {\bar{H}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}_Y} (Y) \longrightarrow {\bar{H}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}_X} (X) \; .$$ A (complete) flow is a smooth ${{\mathbb{R}}}$-action $\phi : {{\mathbb{R}}}\times X \to X , (t,x) \mapsto \phi^t (x)$. It is called ${{\mathcal F}}$-compatible if every diffeomorphism $\phi^t : X \to X$ maps leaves into leaves. If this is the case we obtain a linear ${{\mathbb{R}}}$-action $t \mapsto \phi^{t*}$ on ${\bar{H}}^n_{{{\mathcal F}}} (X)$ for every $n$. Let $$\Theta : {\bar{H}}^n_{{{\mathcal F}}} (X) \longrightarrow {\bar{H}}^n_{{{\mathcal F}}} (X)$$ denote the infinitesimal generator of $\phi^{t*}$: $$\Theta h = \lim_{t\to 0} \frac{1}{t} (\phi^{t*} h -h ) \; .$$ The limit exists and $\Theta$ is continuous in the Fréchet topology. As $\phi^{t*}$ is an algebra endomorphism of the ${{\mathbb{R}}}$-algebra ${\bar{H}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}} (X)$ it follows that $\Theta$ is an ${{\mathbb{R}}}$-linear derivation. Thus we have $$\label{eq:1} \Theta (h_1 \cup h_2) = \Theta h_1 \cup h_2 + h_1 \cup \Theta h_2$$ for all $h_1 , h_2$ in ${\bar{H}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}} (X)$. For arbitrary foliations the reduced leafwise cohomology does not seem to have a good structure theory. For Riemannian foliations however the situation is much better. These foliations are characterized by the existence of a “bundle-like” metric $g$. This is a Riemannian metric whose geodesics are perpendicular to all leaves whenever they are perpendicular to one leaf. For example any one-codimensional foliation given by a closed one-form without singularities is Riemannian. The graded Fréchet space ${{\mathcal A}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}} (X)$ carries a canonical inner product: $$(\alpha , \beta) = \int_X \langle \alpha , \beta \rangle_{{{\mathcal F}}} {\mathrm{vol}}\; .$$ Here $\langle , \rangle_{{{\mathcal F}}}$ is the Riemannian metric on $\Lambda^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}} T^* {{\mathcal F}}$ induced by $g$ and ${\mathrm{vol}}$ is the volume form or density on $X$ coming from $g$. Let $$\Delta_{{{\mathcal F}}} = d_{{{\mathcal F}}} d^*_{{{\mathcal F}}} + d^*_{{{\mathcal F}}} d_{{{\mathcal F}}}$$ denote the Laplacian using the formal adjoint of $d_{{{\mathcal F}}}$ on $X$. Since ${{\mathcal F}}$ is Riemannian the restriction of $\Delta_{{{\mathcal F}}}$ to any leaf $F$ is the Laplacian on $F$ with respect to the induced metric [@AK1] Lemma 3.2, i.e. $$(\Delta_{{{\mathcal F}}} \alpha) \, |_F = \Delta_F (\alpha \, |_F) \quad \mbox{for all} \; \alpha \in {{\mathcal A}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}} (X) \; .$$ We now assume that $T {{\mathcal F}}$ is orientable. Via $g$ the choice of an orientation determines a volume form ${\mathrm{vol}}_{{{\mathcal F}}}$ in ${{\mathcal A}}^d_{{{\mathcal F}}} (X)$ and hence a Hodge $*$-operator $$*_{{{\mathcal F}}} : \Lambda^n T^*_x {{\mathcal F}}{\stackrel{\sim}{\longrightarrow}}\Lambda^{d-n} T^*_x {{\mathcal F}}\quad \mbox{for every} \; x \; \mbox{in} \; X \; .$$ It is determined by the condition that $$v \wedge *_{{{\mathcal F}}} w = \langle v,w \rangle_{{{\mathcal F}}} \, {\mathrm{vol}}_{{{\mathcal F}}, x} \quad \mbox{for} \; v,w \; \mbox{in} \; \Lambda^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}} T^*_x {{\mathcal F}}\; .$$ These fibrewise star-operators induce the leafwise $*$-operator on forms: $$*_{{{\mathcal F}}} : {{\mathcal A}}^n_{{{\mathcal F}}} (X) {\stackrel{\sim}{\longrightarrow}}{{\mathcal A}}^{d-n}_{{{\mathcal F}}} (X) \; .$$ We now list some important properties of leafwise cohomology. [**Properties**]{} Assume that $X$ is compact, ${{\mathcal F}}$ a $d$-dimensional oriented Riemannian foliation and $g$ a bundle-like metric for ${{\mathcal F}}$. Then the natural map $$\label{eq:2} {\mathrm{Ker}\,}\Delta^n_{{{\mathcal F}}} {\stackrel{\sim}{\longrightarrow}}{\bar{H}}^n_{{{\mathcal F}}} (X) \; ,\; \omega \longmapsto \omega {\;\mathrm{mod}\;}\overline{{\mathrm{Im}\,}d^{n-1}_{{{\mathcal F}}}}$$ is a topological isomorphism of Fréchet spaces. We denote its inverse by ${{\mathcal H}}$. This result is due to Álvarez López and Kordyukov [@AK1]. It is quite deep since $\Delta_{{{\mathcal F}}}$ is only elliptic along the leaves so that the ordinary elliptic regularity theory does not suffice. For non-Riemannian foliations (\[eq:2\]) does not hold in general [@DS1]. All the following results are consquences of this Hodge theorem. The Hodge $*$-operator induces an isomorphism $$*_{{{\mathcal F}}} : {\mathrm{Ker}\,}\Delta^n_{{{\mathcal F}}} {\stackrel{\sim}{\longrightarrow}}{\mathrm{Ker}\,}\Delta^{d-n}_{{{\mathcal F}}}$$ since it commutes with $\Delta_{{{\mathcal F}}}$ up to sign. From (\[eq:2\]) we therefore get isomorphisms for all $n$: $$\label{eq:3} *_{{{\mathcal F}}} : {\bar{H}}^n_{{{\mathcal F}}} (X) {\stackrel{\sim}{\longrightarrow}}{\bar{H}}^{d-n}_{{{\mathcal F}}} (X) \; .$$ For the next property define the trace map $${\mathrm{tr}}: {\bar{H}}^d_{{{\mathcal F}}} (X) \longrightarrow {{\mathbb{R}}}$$ by the formula $${\mathrm{tr}}(h) = \int_X *_{{{\mathcal F}}} (h) {\mathrm{vol}}:= \int_X *_{{{\mathcal F}}} ({{\mathcal H}}(h)) {\mathrm{vol}}\; .$$ It is an isomorphism if ${{\mathcal F}}$ has a dense leaf. Note that for [*any*]{} representative $\alpha$ in the cohomology class $h$ we have $${\mathrm{tr}}(h) = \int_X *_{{{\mathcal F}}} (\alpha) {\mathrm{vol}}\; .$$ Namely $\alpha - {{\mathcal H}}(h) = d_{{{\mathcal F}}} \beta$ and $$\begin{aligned} \int_X *_{{{\mathcal F}}} (d_{{{\mathcal F}}} \beta) {\mathrm{vol}}& = & \pm \int_X d^*_{{{\mathcal F}}} (*_{{{\mathcal F}}} \beta) {\mathrm{vol}}\\ & = &\pm ( 1 , d^*_{{{\mathcal F}}} (*_{{{\mathcal F}}} \beta)) \\ & = & \pm ( d_{{{\mathcal F}}} (1) , *_{{{\mathcal F}}} \beta)\\ & = & 0 \; .\end{aligned}$$ Alternatively the trace functional is given by $${\mathrm{tr}}(h) = \int_X \alpha \wedge *_{\perp} (1)$$ where $*_{\perp} (1)$ is the transverse volume element for $g$ c.f. [@AK1], §3. It is not difficult to see using (\[eq:2\]) that we get a scalar product on ${\bar{H}}^n_{{{\mathcal F}}} (X)$ for every $n$ by setting: $$\begin{aligned} \label{eq:4} (h,h') & = & {\mathrm{tr}}(h \cup *_{{{\mathcal F}}} h') \\ & = & \int_X \langle {{\mathcal H}}(h) , {{\mathcal H}}(h') \rangle_{{{\mathcal F}}} {\mathrm{vol}}\; . \nonumber\end{aligned}$$ It follows from this that the cup product pairing $$\label{eq:5} \cup : {\bar{H}}^n_{{{\mathcal F}}} (X) \times {\bar{H}}^{d-n}_{{{\mathcal F}}} (X) \longrightarrow {\bar{H}}^d_{{{\mathcal F}}} (X) \xrightarrow{tr} {{\mathbb{R}}}$$ is non-degenerate. Next we discuss the Künneth formula. Assume that $Y$ is another compact manifold with a Riemannian foliation ${{\mathcal F}}_Y$. Then the canonical map $$H^n_{{{\mathcal F}}_X} (X) \otimes H^m_{{{\mathcal F}}_Y} (Y) \longrightarrow H^{n+m}_{{{\mathcal F}}_X \times {{\mathcal F}}_Y} (X \times Y)$$ induces a topological isomorphism [@M]: $$\label{eq:6} {\bar{H}}^n_{{{\mathcal F}}_X} (X) \hat{\otimes} {\bar{H}}^m_{{{\mathcal F}}_Y} (Y) {\stackrel{\sim}{\longrightarrow}}{\bar{H}}^{n+m}_{{{\mathcal F}}_X \times {{\mathcal F}}_Y} (X \times Y) \; .$$ Since the reduced cohomology groups are nuclear Fréchet spaces, it does not matter which topological tensor product is chosen in (\[eq:6\]). The proof of this Künneth formula uses (\[eq:2\]) and the spectral theory of the Laplacian $\Delta_{{{\mathcal F}}}$. Before we deal with more specific topics let us mention that also Hodge–Kähler theory can be generalized. A complex structure on a foliation ${{\mathcal F}}$ is an almost complex structure $J$ on $T {{\mathcal F}}$ such that all restrictions $J \, |_F$ to the leaves are integrable. Then the leaves carry holomorphic structures which vary smoothly in the transverse direction. A foliation ${{\mathcal F}}$ with a complex structure $J$ is called Kähler if there is a hermitian metric $h$ on the complex bundle $T_c {{\mathcal F}}= (T {{\mathcal F}}, J)$ such that the Kähler form along the leaves $$\omega_{{{\mathcal F}}} = - {\frac{1}{2}}{\mathrm{im}\,}h \in {{\mathcal A}}^2_{{{\mathcal F}}} (X)$$ is closed. Note that for example any foliation by orientable surfaces can be given a Kählerian structure by choosing a metric on $X$, c.f. [@MS] Lemma A.3.1. Let $$L_{{{\mathcal F}}} : {\bar{H}}^n_{{{\mathcal F}}} (X) \longrightarrow {\bar{H}}^{n+2}_{{{\mathcal F}}} (X) \; , \; L_{{{\mathcal F}}} (h) = h \cup [\omega_{{{\mathcal F}}}]$$ denote the Lefschetz operator. The following assertions are consequences of (\[eq:2\]) combined with the classical Hodge–Kähler theory. See [@DS3] for details. Let $X$ be a compact orientable manifold and ${{\mathcal F}}$ a Kählerian foliation with respect to the hermitian metric $h$ on $T_c {{\mathcal F}}$. Assume in addition that ${{\mathcal F}}$ is Riemannian. Then we have: $$\label{eq:7} {\bar{H}}^n_{{{\mathcal F}}} (X) \otimes {{\mathbb{C}}}= \bigoplus_{p+q=n} H^{pq} \; , \quad \mbox{where} \; \overline{H^{pq}} = H^{qp} \; .$$ Here $H^{pq}$ consists of those classes that can be represented by $(p,q)$-forms along the leaves. Moreover there are topological isomorphisms $$H^{pq} \cong {\bar{H}}^q (X , \Omega^p_{{{\mathcal F}}})$$ with the reduced cohomology of the sheaf of holomorphic $p$-forms along the leaves. Furthermore the Lefschetz operator induces isomorphisms $$\label{eq:8} L^i_{{{\mathcal F}}} : {\bar{H}}^{d-i}_{{{\mathcal F}}} (X) {\stackrel{\sim}{\longrightarrow}}{\bar{H}}^{d+i}_{{{\mathcal F}}} (X) \quad \mbox{for} \; 0 \le i \le d \; .$$ Finally the space of primitive cohomology classes ${\bar{H}}^n_{{{\mathcal F}}} (X)_{{\mathrm{prim}}}$ carries the structure of a polarizable ${\mathrm{ind}}\, {{\mathbb{R}}}$-Hodge structure of weight $n$. After this review of important properties of the reduced leafwise cohomology of Riemannian foliations we turn to a specific result relating flows and cohomology. \[t21\] Let $X$ be a compact $3$-manifold and ${{\mathcal F}}$ a Riemannian foliation by surfaces with a dense leaf. Let $\phi^t$ be an ${{\mathcal F}}$-compatible flow on $X$ which is conformal on $T{{\mathcal F}}$ with respect to a metric $g$ on $T{{\mathcal F}}$ in the sense that for some constant $\alpha$ we have: $$\label{eq:9} g (T_x \phi^t (v) , T_x \phi^t (w)) = e^{\alpha t} g (v,w) \; \mbox{for all} \; v,w \in T_x {{\mathcal F}}, x \in X \; \mbox{and} \; t \in {{\mathbb{R}}}\; .$$ Then we have for the infinitesimal generator of $\phi^{t*}$ that: $$\Theta = 0 \; \mbox{on} \; {\bar{H}}^0_{{{\mathcal F}}} (X) = {{\mathbb{R}}}\quad \mbox{and} \quad \Theta = \alpha \; \mbox{on} \; {\bar{H}}^2_{{{\mathcal F}}} (X) \cong {{\mathbb{R}}}\; .$$ On ${\bar{H}}^1_{{{\mathcal F}}} (X)$ the operator $\Theta$ has the form $$\Theta = \frac{\alpha}{2} + S$$ where $S$ is skew-symmetric with respect to the inner product $( , )$ above. For the bundle-like metric on $X$ required for the construction of $(,)$ we take any extension of the given metric on $T {{\mathcal F}}$ to a bundle-like metric on $TX$. Such extensions exist. [**\[t21\]**]{} Because we have a dense leaf, ${\bar{H}}^0_{{{\mathcal F}}} (X) = H^0_{{{\mathcal F}}} (X)$ consists only of constant functions. On these $\phi^{t*}$ acts trivially so that $\Theta = 0$. Since $$*_{{{\mathcal F}}} : {{\mathbb{R}}}= {\bar{H}}^0_{{{\mathcal F}}} (X) {\stackrel{\sim}{\longrightarrow}}{\bar{H}}^2_{{{\mathcal F}}} (X)$$ is an isomorphism and since $$\phi^{t*} (*_{{{\mathcal F}}} (1)) = e^{\alpha t} (*_{{{\mathcal F}}} 1)$$ by conformality, we have $\Theta = \alpha$ on ${\bar{H}}^2_{{{\mathcal F}}} (X)$. For $h_1 , h_2$ in ${\bar{H}}^1_{{{\mathcal F}}} (X)$ we find $$\label{eq:10} \alpha (h_1 \cup h_2) = \Theta (h_1 \cup h_2) \stackrel{\rm (\ref{eq:1})}{=} \Theta h_1 \cup h_2 + h_1 \cup \Theta h_2 \; .$$ By conformality $\phi^{t*}$ commutes with $*_{{{\mathcal F}}}$ on ${\bar{H}}^1_{{{\mathcal F}}} (X)$. Differentiating, it follows that $\Theta$ commutes with $*_{{{\mathcal F}}}$ as well. Since by definition we have $$(h, h') = {\mathrm{tr}}(h \cup *_{{{\mathcal F}}} h') \quad \mbox{for} \; h , h' \in {\bar{H}}^1_{{{\mathcal F}}} (X) \; ,$$ it follows from (\[eq:10\]) that as desired: $$\alpha ( h , h' ) = ( \Theta h , h' ) + ( h , \Theta h' ) \; .$$ Dynamical Lefschetz trace formulas ================================== The formulas we want to consider in this section relate the compact orbits of a flow with the alternating sum of suitable traces on cohomology. A suggestive but non-rigorous argument of Guillemin [@Gu] later rediscovered by Patterson [@P] led to the following conjecture [@D2] §3. Let $X$ be a compact manifold with a one-codimensional foliation ${{\mathcal F}}$ and an ${{\mathcal F}}$-compatible flow $\phi$. Assume that the fixed points and the periodic orbits of the flow are non-degenerate in the following sense: For any fixed point $x$ the tangent map $T_x \phi^t$ should have eigenvalues different from $1$ for all $t > 0$. For any closed orbit $\gamma$ of length $l (\gamma)$ and any $x \in \gamma$ and integer $k \neq 0$ the automorphism $T_x \phi^{kl (\gamma)}$ of $T_x X$ should have the eigenvalue $1$ with algebraic multiplicity one. Observe that the vector field $Y_{\phi}$ generated by the flow provides an eigenvector $Y_{\phi,x}$ for the eigenvalue $1$. Recall that the length $l (\gamma) > 0$ of $\gamma$ is defined by the isomorphism: $${{\mathbb{R}}}/ l (\gamma) {{\mathbb{Z}}}{\stackrel{\sim}{\longrightarrow}}\gamma \; , \; t \longmapsto \phi^t (x) \; .$$ For a fixed point $x$ we set[^1] $$\varepsilon_x = {\mathrm{sgn}\,}\det (1 - T_x \phi^t {\, | \,}T_x {{\mathcal F}}) \; .$$ This is independent of $t > 0$. For a closed orbit $\gamma$ and $k \in {{\mathbb{Z}}}{\setminus}0$ set$^1$ $$\varepsilon_{\gamma} (k) = {\mathrm{sgn}\,}\det (1 - T_x \phi^{kl (\gamma)} {\, | \,}T_x X / {{\mathbb{R}}}Y_{\phi,x}) = {\mathrm{sgn}\,}\det ( 1 - T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}}) \; .$$ It does not depend on the point $x \in \gamma$. Finally let ${{\mathcal D}}' (J)$ denote the space of Schwartz distributions on an open subset $J$ of ${{\mathbb{R}}}$. \[t31\] For $X , {{\mathcal F}}$ and $\phi$ as above there exists a natural definition of a ${{\mathcal D}}' ({{\mathbb{R}}}^{>0})$-valued trace of $\phi^*$ on the reduced leafwise cohomology ${\bar{H}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}} (X)$ such that in ${{\mathcal D}}' ({{\mathbb{R}}}^{>0})$ we have: $$\label{eq:11} \sum\limits^{\dim {{\mathcal F}}}_{n=0} (-1)^n {\mathrm{Tr}}(\phi^* {\, | \,}{\bar{H}}^n_{{{\mathcal F}}} (X)) = \sum\limits_{\gamma} l (\gamma) \sum\limits^{\infty}_{k=1} \varepsilon_{\gamma} (k) \delta_{kl (\gamma)} + \sum\limits_x \varepsilon_x |1 - e^{\kappa_x t}|^{-1} \; .$$ Here $\gamma$ runs over the closed orbits of $\phi$ which are not contained in a leaf and $x$ over the fixed points. For $a \in {{\mathbb{R}}}, \delta_a$ is the Dirac distribution in $a$ and $\kappa_x$ is defined by the action of $T_x \phi^t$ on the $1$-dimensional vector space $T_x X / T_x {{\mathcal F}}$. That action is multiplication by $e^{\kappa_x t}$ for some $\kappa_x \in {{\mathbb{R}}}$ and all $t$. The conjecture is not known (except for $\dim X = 1$) if $\phi$ has fixed points. It may well have to be amended somewhat in that case. The analytic difficulty in the presence of fixed points lies in the fact that in this case $\Delta_{{{\mathcal F}}}$ has no chance to be transversally elliptic to the ${{\mathbb{R}}}$-action by the flow, so that the methods of transverse index theory do not apply directly. In the simpler case when the flow is everywhere transversal to ${{\mathcal F}}$, Álvarez López and Kordyukov have proved a beautiful strengthening of the conjecture. Partial results were obtained by other methods in [@Laz], [@DS2]. We now describe their result in a convenient way for our purposes: \[t32\] Assume $X$ is a compact oriented manifold with a one codimensional foliation ${{\mathcal F}}$. Let $\phi$ be a flow on $X$ which is everywhere transversal to the leaves of ${{\mathcal F}}$. Then ${{\mathcal F}}$ inherits an orientation and it is Riemannian [@Go] III 4.4. Fixing a bundle-like metric $g$ the cohomologies ${\bar{H}}^n_{{{\mathcal F}}} (X)$ acquire pre-Hilbert structures (\[eq:4\]) and we can consider their Hilbert space completions $\hat{H}^n_{{{\mathcal F}}} (X)$. For every $t$ the linear operator $\phi^{t*}$ is bounded on $({\bar{H}}^n_{{{\mathcal F}}} (X) , \| \; \|)$ and hence can be continued uniquely to a bounded operator on $\hat{H}^n_{{{\mathcal F}}} (X)$ c.f. theorem \[t34\]. By transversality the flow has no fixed points. We assume that all periodic orbits are non-degenerate. \[t33\] Under the conditions of (\[t32\]), for every test function $\varphi \in {{\mathcal D}}({{\mathbb{R}}}) = C^{\infty}_0 ({{\mathbb{R}}})$ the operator $$A_{\varphi} = \int_{{{\mathbb{R}}}} \varphi (t) \phi^{t*} \, dt$$ on $\hat{H}^n_{{{\mathcal F}}} (X)$ is of trace class. Setting: $${\mathrm{Tr}}(\phi^* {\, | \,}{\bar{H}}^n_{{{\mathcal F}}} (X)) (\varphi) = {\mathrm{tr}}A_{\varphi}$$ defines a distribution on ${{\mathbb{R}}}$. The following formula holds in ${{\mathcal D}}' ({{\mathbb{R}}})$: $$\label{eq:12} \sum^{\dim {{\mathcal F}}}_{n=0} (-1)^n {\mathrm{Tr}}(\phi^* {\, | \,}{\bar{H}}^n_{{{\mathcal F}}} (X)) = \chi_{{\mathrm{Co}}} ({{\mathcal F}}, \mu) \delta_0 + \sum_{\gamma} l (\gamma) \sum_{k \in {{\mathbb{Z}}}{\setminus}0} \varepsilon_{\gamma} (k) \delta_{kl (\gamma)} \; .$$ Here $\chi_{{\mathrm{Co}}} ({{\mathcal F}}, \mu)$ denotes Connes’ Euler characteristic of the foliation with respect to the transverse measure $\mu$ corresponding to $*_{\perp} (1)$. (See [@MS].) It follows from the theorem that if the right hand side of (\[eq:12\]) is non-zero, at least one of the cohomology groups ${\bar{H}}^n_{{{\mathcal F}}} (X)$ must be infinite dimensional. Otherwise the alternating sum of traces would be a smooth function and hence have empty singular support. By the Hodge isomorphism (\[eq:2\]) one may replace cohomology by the spaces of leafwise harmonic forms. The left hand side of the dynamical Lefschetz trace formula then becomes the ${{\mathcal D}}' ({{\mathbb{R}}})$-valued transverse index of the leafwise de Rham complex. Note that the latter is transversely elliptic for the ${{\mathbb{R}}}$-action $\phi^t$. Transverse index theory with respect to compact group actions was initiated in [@A]. A definition for non-compact groups of a transverse index was later given by Hörmander [@Si] Appendix II. As far as we know the relation of (\[eq:12\]) with transverse index theory in the sense of Connes–Moscovici still needs to be clarified. Let us now make some remarks on the operators $\phi^{t*}$ on $\hat{H}^n_{{{\mathcal F}}} (X)$ in a more general setting: \[t34\] Let ${{\mathcal F}}$ be a Riemannian foliation on a compact manifold $X$ and $g$ a bundle like metric. As above $\hat{H}^n_{{{\mathcal F}}} (X)$ denotes the Hilbert space completion of ${\bar{H}}^n_{{{\mathcal F}}} (X)$ with respect to the scalar product (\[eq:4\]). Let $\phi^t$ be an ${{\mathcal F}}$-compatible flow. Then the linear operators $\phi^{t*}$ on ${\bar{H}}^n_{{{\mathcal F}}} (X)$ induce a strongly continuous operator group on $\hat{H}^n_{{{\mathcal F}}} (X)$. In particular the infinitesimal generator $\Theta$ exists as a closed densely defined operator. On ${\bar{H}}^n_{{{\mathcal F}}} (X)$ it agrees with the infinitesimal generator in the Fréchet topology defined earlier. There exists $\omega > 0$ such that the spectrum of $\Theta$ lies in $- \omega \le {\mathrm{Re}\,}s \le \omega$. If the operators $\phi^{t*}$ are orthogonal then $T = - i \Theta$ is a self-adjoint operator on $\hat{H}^n_{{{\mathcal F}}} (X) \otimes {{\mathbb{C}}}$ and we have $$\phi^{t*} = \exp t \Theta = \exp it T$$ in the sense of the functional calculus for (unbounded) self-adjoint operators on Hilbert spaces. [**Sketch of proof**]{} Estimates show that $\| \phi^{t*} \|$ is locally uniformly bounded in $t$ on ${\bar{H}}^n_{{{\mathcal F}}} (X)$. Approximating $h \in \hat{H}^n_{{{\mathcal F}}} (X)$ by $h_{\nu} \in {\bar{H}}^n_{{{\mathcal F}}} (X)$ one now shows as in the proof of the Riemann–Lebesgue lemma that the function $t \mapsto \phi^{t*} h$ is continuous at zero, hence everywhere. Thus $\phi^{t*}$ defines a strongly continuous group on $\hat{H}^n_{{{\mathcal F}}} (X)$. The remaining assertions follow from semigroup theory [@DSch], Ch. VIII, XII, and in particular from the theorem of Stone. We now combine theorems \[t21\], \[t33\] and \[t34\] to obtain the following corollary: \[t35\] Let $X$ be a compact $3$-manifold with a foliation ${{\mathcal F}}$ by surfaces having a dense leaf. Let $\phi^t$ be a non-degenerate ${{\mathcal F}}$-compatible flow which is everywhere transversal to ${{\mathcal F}}$. Assume that $\phi^t$ is conformal (\[eq:9\]) with respect to a metric $g$ on $T{{\mathcal F}}$. Then $\Theta$ has pure point spectrum ${\mathrm{Sp}}^1 (\Theta)$ on $\hat{H}^1_{{{\mathcal F}}} (X)$ which is discrete in ${{\mathbb{R}}}$ and we have the following equality of distributions on ${{\mathbb{R}}}$: $$\label{eq:13} 1 - \sum_{\rho \in {\mathrm{Sp}}^1 (\Theta)} e^{t\rho} + e^{t\alpha} = \chi_{{\mathrm{Co}}} ({{\mathcal F}}, \mu) \delta_0 + \sum_{\gamma} l (\gamma) \sum_{k \in {{\mathbb{Z}}}{\setminus}0} \varepsilon_{\gamma} (k) \delta_{k l (\gamma)} \; .$$ In the sum the $\rho$’s appear with their geometric multiplicities. All $\rho \in {\mathrm{Sp}}^1 (\Theta)$ have ${\mathrm{Re}\,}\rho = \frac{\alpha}{2}$. [**Remarks**]{} 1) Here $e^{t\rho} , e^{t \alpha}$ are viewed as distributions so that evaluated on a test function $\varphi \in {{\mathcal D}}({{\mathbb{R}}})$ the formula reads: $$\label{eq:14} \Phi (0) - \sum\limits_{\rho \in {\mathrm{Sp}}^1 (\Theta)} \Phi (\rho) + \Phi (\alpha) = \chi_{{\mathrm{Co}}} ({{\mathcal F}}, \mu) \varphi (0) + \sum\limits_{\gamma} l (\gamma) \sum\limits_{k \in {{\mathbb{Z}}}{\setminus}0} \varepsilon_{\gamma} (k) \varphi (k l (\gamma)) \; .$$ Here we have put $$\Phi (s) = \int_{{{\mathbb{R}}}} e^{ts} \varphi (t) \, dt \; .$$ 2) Actually the conditions of the corollary force $\alpha = 0$ i.e. the flow must be isometric with respect to $g$. We have chosen to leave the $\alpha$ in the fomulation since there are good reasons to expect the corollary to generalize to more general phase spaces $X$ than manifolds, where $\alpha \neq 0$ becomes possible i.e. to Sullivan’s generalized solenoids. More on this in section 5.\ 3) One can show that the group generated by the lengths of closed orbits is a finitely generated subgroup of ${{\mathbb{R}}}$ under the assumptions of the corollary. In order to achieve an infinitely generated group the flow must have fixed points. [**\[t35\]**]{} By \[t21\], \[t33\] we need only show the equation $$\label{eq:15} {\mathrm{Tr}}(\phi^{t*} {\, | \,}{\bar{H}}^1_{{{\mathcal F}}} (X)) = \sum_{\rho \in {\mathrm{Sp}}^1 (\Theta)} e^{t \rho}$$ and the assertions about the spectrum of $\Theta$. As in the proof of \[t21\] one sees that on $\hat{H}^1_{{{\mathcal F}}} (X)$ we have $$( \phi^{t*} h , \phi^{t*} h') = e^{\alpha t} (h , h') \; .$$ Hence $e^{-\frac{\alpha}{2} t} \phi^{t*}$ is orthogonal and by the theorem of Stone $$T = -iS$$ is selfadjoint on $\hat{H}^1_{{{\mathcal F}}} (X) \otimes {{\mathbb{C}}}$, if $\Theta = \frac{\alpha}{2} + S$. Moreover $$e^{-\frac{\alpha}{2} t} \phi^{t*} = \exp it T \; ,$$ so that $$\label{eq:16} \phi^{t*} = \exp t \Theta \; .$$ In [@DS2] proof of 2.6, for isometric flows the relation $$-\Theta^2 = \Delta^1 \, |_{\ker \Delta^1_{{{\mathcal F}}}}$$ was shown. Using the spectral theory of the ordinary Laplacian $\Delta^1$ on $1$-forms it follows that $\Theta$ has pure point spectrum with finite multiplicities on ${\bar{H}}^1_{{{\mathcal F}}} (X) \cong \ker \Delta^1_{{{\mathcal F}}}$ and that ${\mathrm{Sp}}^1 (\Theta)$ is discrete in ${{\mathbb{R}}}$. Alternatively, without knowing $\alpha = 0$, that proof gives: $$- \left( \Theta - \frac{\alpha}{2} \right)^2 = \Delta^1 \, |_{\ker \Delta^1_{{{\mathcal F}}}} \; .$$ This also implies the assertion on the spectrum of $\Theta$ on $\hat{H}^1_{{{\mathcal F}}} (X)$. Now (\[eq:15\]) follows from (\[eq:16\]) and the fact that the operators $A_{\varphi}$ are of trace class. In more general situations where $\Theta$ may not have a pure point spectrum of $\hat{H}^1_{{{\mathcal F}}} (X)$ but where $e^{-\frac{\alpha}{2}} \phi^{t*}$ is still orthogonal, we obtain: $$\langle {\mathrm{Tr}}(\phi^* {\, | \,}{\bar{H}}^1_{{{\mathcal F}}} (X)) , \varphi \rangle = \sum_{\rho \in {\mathrm{Sp}}^1 (\Theta)_{\mathrm{point}}} \Phi (\rho) + \int^{\frac{\alpha}{2} + i \infty}_{\frac{\alpha}{2} - i \infty} \Phi (\lambda) m (\lambda) \, d \lambda$$ where $m (\lambda) \ge 0$ is the spectral density function of the continuous part of the spectrum of $\Theta$. Comparison with the “explicit formulas” in analytic number theory ================================================================= Consider a number field $K / {{\mathbb{Q}}}$. The explicit formulas in analytic number theory relate the primes of $K$ to the non-trivial zeroes of the Dedekind zeta function $\zeta_K (s)$ of $K$. \[t41\] For $\varphi \in {{\mathcal D}}({{\mathbb{R}}})$ define $\Phi (s)$ as in the preceeding section. Then the following fomula holds: $$\begin{aligned} \label{eq:17} \lefteqn{\Phi (0) - \sum_{\rho} \Phi (\rho) + \Phi (1) = - \log |d_{K / {{\mathbb{Q}}}}| \varphi (0)} \nonumber \\ & & + \sum_{{\mathfrak{p}}\nmid \infty} \log N {\mathfrak{p}}\left( \sum_{k \ge 1} \varphi (k \log N {\mathfrak{p}}) + \sum_{k \le -1} N {\mathfrak{p}}^k \varphi (k \log N {\mathfrak{p}}) \right) \nonumber \\ & & + \sum_{{\mathfrak{p}}{\, | \,}\infty} W_{{\mathfrak{p}}} (\varphi) \; .\end{aligned}$$ Here $\rho$ runs over the non-trivial zeroes of $\zeta_K (s)$ i.e. those that are contained in the critical strip $0 < {\mathrm{Re}\,}s < 1$. Moreover ${\mathfrak{p}}$ runs over the places of $K$ and $d_{K / {{\mathbb{Q}}}}$ is the discriminant of $K$ over ${{\mathbb{Q}}}$. For ${\mathfrak{p}}{\, | \,}\infty$ the $W_{{\mathfrak{p}}}$ are distributions which are determined by the $\Gamma$-factor at ${\mathfrak{p}}$. If $\varphi$ has support in ${{\mathbb{R}}}^{> 0}$ then $$W_{{\mathfrak{p}}} (\varphi) = \int^{\infty}_{-\infty} \frac{\varphi (t)}{1 - e^{\kappa_{{\mathfrak{p}}} t}} \, dt$$ where $\kappa_{{\mathfrak{p}}} = -1$ if ${\mathfrak{p}}$ is complex and $\kappa_{{\mathfrak{p}}} = -2$ if ${\mathfrak{p}}$ is real. If $\varphi$ has support on ${{\mathbb{R}}}^{< 0}$ then $$W_{{\mathfrak{p}}} (\varphi) = \int^{\infty}_{-\infty} \frac{\varphi (t)}{1 - e^{\kappa_{{\mathfrak{p}}} |t|}} \, e^t \, dt \; .$$ There are different ways to write $W_{{\mathfrak{p}}}$ on all of ${{\mathbb{R}}}$ but we will not discuss this here. See for example [@Ba] which also contains a proof of the theorem for much more general test functions. Formula (\[eq:17\]) implies the following equality of distributions on ${{\mathbb{R}}}^{> 0}$: $$\label{eq:18} 1 - \sum_{\rho} e^{t\rho} + e^t = \sum_{{\mathfrak{p}}\nmid \infty} \log N{\mathfrak{p}}\sum^{\infty}_{k=1} \delta_{k \log N {\mathfrak{p}}} + \sum_{{\mathfrak{p}}{\, | \,}\infty} (1 - e^{\kappa_{{\mathfrak{p}}} t})^{-1} \; .$$ This fits rather nicely with formula (\[eq:11\]) and suggests the following analogies: ------------------------------------------------------------------------------ --------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ${\mathrm{spec}\,}{\mathfrak{o}}_K \cup \{ {\mathfrak{p}}{\, | \,}\infty \}$ ${\;\widehat{=}\;}$ $3$-dimensional dynamical system $(X , \phi^t)$ with a one-codimensional foliation ${{\mathcal F}}$ satisfying the conditions of conjecture \[t31\] finite place ${\mathfrak{p}}$ ${\;\widehat{=}\;}$ closed orbit $\gamma = \gamma_{{\mathfrak{p}}}$ not contained in a leaf and hence transversal to ${{\mathcal F}}$ such that $l (\gamma_{{\mathfrak{p}}}) = \log N {\mathfrak{p}}$ and $\varepsilon_{\gamma_{{\mathfrak{p}}}} (k) = 1$ for all $k \ge 1$. infinite place ${\mathfrak{p}}$ ${\;\widehat{=}\;}$ fixed point $x_{{\mathfrak{p}}}$ such that $\kappa_{x_{{\mathfrak{p}}}} = \kappa_{{\mathfrak{p}}}$ and $\varepsilon_{x_{{\mathfrak{p}}}} = 1$. ------------------------------------------------------------------------------ --------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- In order to understand number theory more deeply in geometric terms it would be very desirable to find a system $(X , \phi^t , {{\mathcal F}})$ which actually realizes this correspondence. For this the class of compact $3$-manifolds as phase spaces has to be generalized as will become clear from the following discussion. Formula (\[eq:17\]) can be written equivalently as an equality of distributions on ${{\mathbb{R}}}$: $$\begin{aligned} \label{eq:19} \lefteqn{1 - \sum_{\rho} e^{t\rho} + e^t = - \log |d_{K / {{\mathbb{Q}}}}| \delta_0} \nonumber \\ & & + \sum_{{\mathfrak{p}}\nmid \infty} \log N{\mathfrak{p}}\left( \sum_{k \ge 1} \delta_{k \log N {\mathfrak{p}}} + \sum_{k \le 1} N {\mathfrak{p}}^{k} \delta_{k \log N {\mathfrak{p}}} \right) \nonumber \\ & & + \sum_{{\mathfrak{p}}{\, | \,}\infty} W_{{\mathfrak{p}}} \; .\end{aligned}$$ Let us compare this with formula (\[eq:13\]) in Corollary \[t35\]. This corollary is the best result yet on the dynamical side but still only a first step since it does not allow for fixed points which as we have seen must be expected for dynamical systems of relevance for number fields. Ignoring the contributions $W_{{\mathfrak{p}}}$ from the infinite places for the moment we are suggested that $$\label{eq:20} - \log |d_{K/ {{\mathbb{Q}}}}| {\;\widehat{=}\;}\chi_{Co} ({{\mathcal F}}, \mu) \; .$$ There are two nice points about this analogy. Firstly there is the following well known fact due to Connes: \[t42\] Let ${{\mathcal F}}$ be a foliation of a compact $3$-manifold by surfaces such that the union of the compact leaves has $\mu$-measure zero, then $$\chi_{Co} ({{\mathcal F}}, \mu) \le 0 \; .$$ Namely the non-compact leaves are known to be complete in the induced metric. Hence they carry no non-zero harmonic $L^2$-functions, so that Connes’ $0$-th Betti number $\beta_0 ({{\mathcal F}}, \mu) = 0$. Since $\beta_2 ({{\mathcal F}}, \mu) = \beta_0 ({{\mathcal F}}, \mu)$ it follows that $$\begin{aligned} \chi_{Co} ({{\mathcal F}}, \mu) & = & \beta_0 ({{\mathcal F}}, \mu) - \beta_1 ({{\mathcal F}}, \mu) + \beta_2 ({{\mathcal F}}, \mu) \\ & = & - \beta_1 ({{\mathcal F}}, \mu) \le 0 \; .\end{aligned}$$ The reader will have noticed that in accordance with \[t42\] the left hand side of (\[eq:20\]) is negative as well: $$- \log |d_{K / {{\mathbb{Q}}}}| \le 0 \quad \mbox{for all} \; K / {{\mathbb{Q}}}\; .$$ The second nice point about (\[eq:20\]) is this. The bundle-like metric $g$ which we have chosen for the definition of $\Delta_{{{\mathcal F}}}$ and of $\chi_{Co} ({{\mathcal F}}, \mu)$ induces a holomorphic structure of ${{\mathcal F}}$ [@MS], Lemma A3.1. The space $X$ is therefore foliated by Riemann surfaces. Let $\chi_{Co} ({{\mathcal F}}, {{\mathcal O}}, \mu)$ denote the holomorphic Connes Euler characteristic of ${{\mathcal F}}$ defined using $\Delta_{\overline{\partial}}$-harmonic forms on the leaves instead of $\Delta$-harmonic ones. According to Connes’ Riemann–Roch Theorem [@MS] Cor. A.2.3, Lemma A3.3 we have: $$\chi_{Co} ({{\mathcal F}}, {{\mathcal O}}, \mu) = {\frac{1}{2}}\chi_{Co} ({{\mathcal F}}, \mu) \; .$$ Therefore $\chi_{Co} ({{\mathcal F}}, {{\mathcal O}}, \mu)$ corresponds to $-\log \sqrt{|d_{K / {{\mathbb{Q}}}}|}$. For completely different reasons this number is defined in Arakelov theory as the Arakelov Euler characteristic of $\overline{{\mathrm{spec}\,}{\mathfrak{o}}_K} = {\mathrm{spec}\,}{\mathfrak{o}}_K \cup \{ {\mathfrak{p}}{\, | \,}\infty \}$: $$\label{eq:21} \chi_{Ar} ({{\mathcal O}}_{\overline{{\mathrm{spec}\,}{\mathfrak{o}}_K}}) = - \log {\textstyle \sqrt{|d_{K / {{\mathbb{Q}}}}|}} \; .$$ See [@N] for example. Thus we see that $$\chi_{Ar} ({{\mathcal O}}_{\overline{{\mathrm{spec}\,}{\mathfrak{o}}_K}}) \quad \mbox{corresponds to} \; \chi_{Co} ({{\mathcal F}}, {{\mathcal O}}, \mu) \; .$$ It would be very desirable of course to understand Arakelov Euler characteristics in higher dimensions even conjecturally in terms of Connes’ holomorphic Euler characteristics. Note however that Connes’ Riemann–Roch theorem in higher dimensions does not involve the $R$-genus appearing in the Arakelov Riemann–Roch theorem. The ideas of Bismut [@Bi] may be relevant in this connection. He interpretes the $R$-genus in a natural way via the geometry of loop spaces. Further comparison of formulas (\[eq:13\]) and (\[eq:19\]) shows that in a dynamical system corresponding to number theory we must have $\alpha = 1$. This means that the flow $\phi^{t*}$ would act by multiplication with $e^t$ on the one-dimensional space ${\bar{H}}^2_{{{\mathcal F}}} (X)$. As explained before this would be the case if $\phi^t$ were conformal on $T{{\mathcal F}}$ with factor $e^t$: $$\label{eq:22} g (T_x \phi^t (v) , T_x \phi^t (w)) = e^t g (v,w) \quad \mbox{for all} \; v,w \in T_x {{\mathcal F}}\; .$$ However as mentioned before, this is not possible in the manifold setting of corollary \[t35\] which actually implies $\alpha = 0$.\ An equally important difference between formulas (\[eq:13\]) and (\[eq:19\]) is between the coefficients of $\delta_{kl (\gamma)}$ and of $\delta_{k \log N{\mathfrak{p}}}$ for $k \le -1$. In the first case it is $\pm 1$ whereas in the second it is $N{\mathfrak{p}}^k = e^{k \log N {\mathfrak{p}}}$ which corresponds to $e^{kl (\gamma)}$. Thus it becomes vital to find phase spaces $X$ more general than manifolds for which the analogue of corollary \[t35\] holds and where $\alpha \neq 0$ and in particular $\alpha = 1$ becomes possible. In the new context the term $\varepsilon_{\gamma} (k) \delta_{k l (\gamma)}$ for $k \le -1$ in formula (\[eq:13\]) should become $\varepsilon_{\gamma} (k) e^{\alpha k l (\gamma)} \delta_{kl (\gamma)}$. The next section is devoted to a discussion of certain laminated spaces which we propose as possible candidates for this goal. Remarks on dynamical Lefschetz trace formulas on laminated spaces ================================================================= In this section we extend the previous discussion to more general phase spaces than manifolds. The class of spaces we have in mind are the foliated spaces with totally disconnected transversals in the sense of [@MS]. We will call them laminated spaces for short. They also go by the name of (generalized) solenoids c.f. [@Su]. \[t51\] An $a$-dimensional laminated space is a second countable metrizable topological space $X$ which is locally homeomorphic to the product of a non-empty open subset of ${{\mathbb{R}}}^a$ with a totally disconnected space. Then $a$ is the topological dimension of $X$. Transition functions between local charts $\varphi_1$ and $\varphi_2$ have the following form locally: $$\label{eq:23} \varphi_2 {\mbox{\scriptsize $\,\circ\,$}}\varphi^{-1}_1 (x,y) = (F_1 (x,y) , F_2 (y)) \; .$$ Here $x,y$ denote the euklidean resp. totally disconnected components. This is due to the fact, that continuous functions from connected subsets of ${{\mathbb{R}}}^a$ into a totally disconnected space are constant. Because of (\[eq:23\]) the inverse images $\varphi^{-1} (\, , *)$ patch together, to give a partition ${{\mathcal L}}$ of $X$ into $a$-dimensional topological manifolds. These [*leaves*]{} of the laminated space $X$ are exactly the path components of $X$. The classical solenoid $$\label{eq:24} {{\mathbb{S}}}^1_p = {{\mathbb{R}}}\times_{{{\mathbb{Z}}}} {{\mathbb{Z}}}_p = \lim_{\leftarrow} (\ldots \to {{\mathbb{R}}}/ {{\mathbb{Z}}}\xrightarrow{p} {{\mathbb{R}}}/ {{\mathbb{Z}}}\xrightarrow{p} \ldots )$$ is an example of a compact connected one-dimensional laminated space with dense leaves homeomorphic to the real line. A $C^{\infty , 0}$-structure on a laminated space is a maximal atlas of local charts whose transition functions are smooth in the euklidean component and continuous in the totally disconnected one. Furthermore all derivatives in the euklidean directions should be continuous in all components. The leaves then become $a$-dimensional smooth manifolds. $C^{\infty , 0}$-laminated spaces are examples of foliated spaces in the sense of [@MS] Def. 2.1. A stronger structure that may exist on a laminated space was introduced by Sullivan \cite{}. A $C^{\infty , \infty}$- or $TLC$-structure on a laminated space $X$ is given by a maximal atlas whose transition functions are smooth in the euklidean component and uniformly locally constant in the totally disconnected one. That is, locally they have the form: $$\label{eq:25} \varphi_2 {\mbox{\scriptsize $\,\circ\,$}}\varphi^{-1}_1 (x,y) = (F_1 (x) , F_2 (y))$$ with $F_1$ smooth and $F_2$ locally constant. Every $C^{\infty , \infty}$-structure gives rise to a $C^{\infty , 0}$-structure. It is clear that ${{\mathbb{S}}}^1_p$ is naturally a $C^{\infty , \infty}$-laminated space. For a $C^{\infty , 0}$-laminated space $X$ let $TX = T {{\mathcal L}}$ denote its tangent bundle in the sense of [@MS] p. 43. For a point $x \in X$, the fibre $T_x X$ is the ordinary tangent space to the leaf through $x$. A Riemannian metric on $X$ is one on $TX$. Morphisms between $C^{\infty , 0}$-laminated spaces are continuous maps which induce smooth maps between the leaves of the lamination. They induce morphisms of tangent bundles. The two most prominent places in mathematics where laminated spaces occur naturally are in number theory e.g. as adelic points of algebraic groups and in the theory of dynamical systems as attractors. \[t52\] We now introduce foliations of laminated spaces. Let $X$ be an $a$-dimensional laminated space. For our purposes, a foliation ${{\mathcal F}}$ of $X$ by laminated spaces is a partition of $X$ into $d$-dimensional laminated spaces. The foliation is supposed to be locally trivial with euclidean transversals. More precisely ${{\mathcal F}}$ is given by a maximal atlas of local charts on $X$ $$\varphi : U {\stackrel{\sim}{\longrightarrow}}V_1 \times V_2 \times Y$$ with $V_1 \subset {{\mathbb{R}}}^d , V_2 \subset {{\mathbb{R}}}^{a-d}$ open and $Y$ totally disconnected, having the following property: The transition maps have the form: $$\varphi_2 {\mbox{\scriptsize $\,\circ\,$}}\varphi^{-1}_1 (x_1 , x_2 , y) = (G_1 (x_1 , x_2 , y) , G_2 (x_2 , y) , G_3 (y))$$ where $G_1 , G_2$ are smooth in the $x_1 , x_2$ components and $G_3$ and all $\partial^{\alpha_1 , \alpha_2}_{x_1 , x_2} G_1$ and $\partial^{\alpha_2}_{x_2} G_2$ are continuous. Setting $x = (x_1 , x_2)$, $$F_1 (x,y) = (G_1 (x_1 , x_2 , y_2) , G_2 (x_2 , y))$$ and $F_2 = G_2$ this induces a $C^{\infty , 0}$-structure on $X$ which is supposed to agree with the given one. The leaves of ${{\mathcal F}}$ are obtained by patching together the sets $\varphi^{-1} (V_1 \times \{ * \} \times Y )$. They are $d$-dimensional $C^{\infty , 0}$-laminated spaces with local transition functions given by: $$(x_1 , y) \longmapsto (G_1 (x_1 , * , y) , G_3 (y)) \; .$$ Their leaves are $d$-dimensional manifolds which foliate the $a$-dimensional manifolds which occur as leaves of the lamination on $X$. Thus $X$ is partitioned into $a$-dimensional manifolds each of which carries a $d$-dimensional foliation in the usual sense. Besides ${{\mathcal L}}$ and ${{\mathcal F}}$ there is a third foliated structure denoted ${{\mathcal F}}{{\mathcal L}}$ on $X$. The space $X$ is foliated with leaves the $d$-dimensional manifolds that occur as path components of the ${{\mathcal F}}$-leaves. Here the transverse space is of the form open subspace of ${{\mathbb{R}}}^{a-d} \times$ totally disconnected. The local transition maps are given by: $$(x_1 , z) \longmapsto (H_1 (x_1 , z) , H_2 (z))$$ with $z = (x_2 , y) , H_1 (x_1 , z) = G_1 (x_1 , x_2 , y)$ and $H_2 (z) = (G_2 (x_2 , y) , G_3 (y))$. Of the three foliated structures ${{\mathcal L}}, {{\mathcal F}}$ and ${{\mathcal F}}{{\mathcal L}}$ on $X$ the first and the last fit into the context of [@MS] but not the second. \[t53\] We now turn to cohomology. The foliation ${{\mathcal F}}{{\mathcal L}}$ gives rise to the integrable rank $d$ subbundle $T{{\mathcal F}}{{\mathcal L}}$ of $TX = T {{\mathcal L}}$. These bundles are tangentially smooth in the sense of [@MS], p. 43 with respect to ${{\mathcal L}}$. The leafwise cohomology of $X$ along ${{\mathcal F}}{{\mathcal L}}$ is by definition the cohomology of the sheaf ${{\mathcal R}}_{{{\mathcal F}}{{\mathcal L}}}$ of real valued smooth functions on $X$ wich are locally constant along the ${{\mathcal F}}{{\mathcal L}}$ leaves. Here a continuous function or form on an open subset of $X$ is called smooth if its restrictions to the ${{\mathcal L}}$-leaves are smooth. The sheaf ${{\mathcal R}}_{{{\mathcal F}}{{\mathcal L}}}$ is resolved by the de Rham complex of smooth forms along ${{\mathcal F}}{{\mathcal L}}$. Using the natural Fréchet topology on the spaces of global differential forms, one defines the reduced version ${\bar{H}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}{{\mathcal L}}} (X)$ of leafwise cohomology $H^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}{{\mathcal L}}} (X)$ as its maximal Hausdorff quotient. By $H^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}} (X)$ we denote the cohomology of the sheaf ${{\mathcal R}}_{{{\mathcal F}}}$ of real valued smooth functions on $X$ which are locally constant along the ${{\mathcal F}}$-leaves. \[t54\] A flow $\phi$ on $X$ is a continuous ${{\mathbb{R}}}$-action such that the induced ${{\mathbb{R}}}$-actions on the leaves of ${{\mathcal L}}$ are smooth. It respects ${{\mathcal F}}$ if every $\phi^t$ maps leaves of ${{\mathcal F}}$ into leaves of ${{\mathcal F}}$. It follows that $\phi^t$ maps ${{\mathcal F}}{{\mathcal L}}$-leaves into ${{\mathcal F}}{{\mathcal L}}$-leaves. Thus $(X , {{\mathcal F}}, \phi^t)$ is partitioned into the foliated dynamical systems $(L, {{\mathcal F}}_L , \phi^t \, |_L)$ for $L \in {{\mathcal L}}$. Here $${{\mathcal F}}_L = {{\mathcal F}}{{\mathcal L}}\, |_L = \{ S \in {{\mathcal F}}{{\mathcal L}}{\, | \,}S \subset L \} \; .$$ Any ${{\mathcal F}}$-compatible flow $\phi^t$ induces pullback actions $\phi^{t*}$ on $H^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}{{\mathcal L}}} (X)$ and ${\bar{H}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}{{\mathcal L}}} (X)$. \[t55\] We now state as a working hypotheses a generalization of the conjectured dynamical trace formula \[t31\]. We allow the phase space to be a laminated space. Moreover we extend the formula to an equality of distributions on ${{\mathbb{R}}}^*$ instead of ${{\mathbb{R}}}^{>0}$. After checking various compatibilities we state a case where our working hypotheses can be proved and give a number theoretical example. [**Working hypotheses:**]{} \[t56\] Let $X$ be a compact $C^{\infty , 0}$-laminated space with a one-codimensional foliation ${{\mathcal F}}$ and an ${{\mathcal F}}$-compatible flow $\phi$. Assume that the fixed points and the periodic orbits of the flow are non-degenerate. Then there exists a natural definition of a ${{\mathcal D}}' ({{\mathbb{R}}}^*)$-valued trace of $\phi^{t*}$ on ${\bar{H}}^{{\raisebox{0.05cm}{$\scriptscriptstyle \bullet$}}}_{{{\mathcal F}}{{\mathcal L}}} (X)$ such that in ${{\mathcal D}}' ({{\mathbb{R}}}^*)$ we have: $$\begin{aligned} \label{eq:26} \hspace*{0.5cm} \lefteqn{\sum^{\dim {{\mathcal F}}}_{n=0} (-1)^n {\mathrm{Tr}}(\phi^* {\, | \,}{\bar{H}}^n_{{{\mathcal F}}{{\mathcal L}}} (X)) = }\\ & & \sum_{\gamma} l (\gamma) \left( \sum_{k \ge 1} \varepsilon_{\gamma} (k) \delta_{k l (\gamma)} + \sum_{k \le -1} \varepsilon_{\gamma} (|k|) \det (-T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}}) \delta_{kl (\gamma)} \right) \nonumber \\ & & + \sum_x W_x \; . \nonumber\end{aligned}$$ Here $\gamma$ runs over the closed orbits not contained in a leaf and in the sums over $k$’s any point $x \in \gamma$ can be chosen. The second sum runs over the fixed points $x$ of the flow. The distributions $W_x$ on ${{\mathbb{R}}}^*$ are given by: $$W_x \, |_{{{\mathbb{R}}}^{> 0}} = \varepsilon_x \, |1 - e^{\kappa_x t}|^{-1}$$ and $$W_x \, |_{{{\mathbb{R}}}^{< 0}} = \varepsilon_x \det (-T_x \phi^t {\, | \,}T_x {{\mathcal F}}) \, |1 - e^{\kappa_x |t|}|^{-1} \; .$$ \[t57\] [**0)**]{} It may actually be better to use a version of foliation cohomology where transversally forms are only supposed to be locally $L^2$ instead of being continuous.\ [**1)**]{} In the situation described in \[t58\] below the working hypotheses can be proved if ${\bar{H}}^n_{{{\mathcal F}}{{\mathcal L}}} (X)$ is replaced by $H^n_{{{\mathcal F}}} (X)$, Theorem \[t59\]. In those cases there are no fixed points, only closed orbits. Thus Theorem \[t59\] dictated only the coefficients of $\delta_{kl (\gamma)}$ for $k \in {{\mathbb{Z}}}{\setminus}0$, but not the contributions $W_x$ from the fixed points.\ [**2)**]{} The coefficients of $\delta_{kl (\gamma)}$ for $k \in {{\mathbb{Z}}}{\setminus}0$ can be written in a uniform way as follows. They are equal to: $$\label{eq:27} \frac{\det (1 - T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}})}{|\det (1 - T_x \phi^{|k| l (\gamma)} {\, | \,}T_x X / {{\mathbb{R}}}Y_{\phi , x})|} = \frac{\det (1 - T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}})}{| \det (1 - T_x \phi^{|k| l (\gamma)} {\, | \,}T_x {{\mathcal F}})|} \; .$$ Here $x$ is any point on $\gamma$. Namely, for $k \ge 1$ this equals $\varepsilon_{\gamma} (k)$ whereas for $k \le -1$ we obtain $$\label{eq:28} \varepsilon_{\gamma} (|k|) \det (- T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}}) = \varepsilon_{\gamma} (k) \, |\det (T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}})| \; .$$ The expression on the left hand side of (\[eq:27\]) motivated our conjecture about the contributions on ${{\mathbb{R}}}^*$ from the fixed points $x$. Since $Y_{\phi , x} = 0$, they should be given by: $$\frac{\det (1 - T_x \phi^t {\, | \,}T_x {{\mathcal F}})}{| \det (1 - T_x \phi^{|t|} {\, | \,}T_x X)|} \overset{!}{=} W_x \; .$$ [**3)**]{} One can prove that in the manifold setting of theorem \[t33\] we have $$|\det (T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}})| = 1 \; .$$ By (\[eq:26\]), our working hypotheses \[t56\] is therefore compatible with formula (\[eq:12\]). Compatibility with conjecture \[t31\] is clear.\ [**4)**]{} We will see below that in our new context metrics $g$ on $T{{\mathcal F}}$ can exist for which the flow has the conformal behaviour (\[eq:22\]). Assuming we are in such a situation and that ${{\mathcal F}}$ is $2$-dimensional, we have: $$|\det (T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}})| = e^{kl (\gamma)} \quad \mbox{for} \; x \in \gamma , k \in {{\mathbb{Z}}}$$ and $$|\det (T_x \phi^t {\, | \,}T_x {{\mathcal F}})| = e^t \quad \mbox{for a fixed point} \; x \; .$$ In the latter case, we even have by continuity: $$\det (T_x \phi^t {\, | \,}T_x {{\mathcal F}}) = e^t \; ,$$ the determinant being positive for $t = 0$. Hence by (\[eq:28\]) the conjectured formula (\[eq:26\]) reads as follows in this case: $$\begin{aligned} \label{eq:29} \lefteqn{\sum^2_{n=0} (-1)^n {\mathrm{Tr}}(\phi^* {\, | \,}{\bar{H}}^n_{{{\mathcal F}}{{\mathcal L}}} (X))} \\ & = & \sum_{\gamma} l (\gamma) \left( \sum_{k \ge 1} \varepsilon_{\gamma} (k) \delta_{kl (\gamma)} + \sum_{k \le -1} \varepsilon_{\gamma} (k) e^{kl (\gamma)} \delta_{kl (\gamma)} \right) \nonumber \\ & & + \sum_x W_x \; . \nonumber\end{aligned}$$ Here: $$W_x \, |_{{{\mathbb{R}}}^{> 0}} = \varepsilon_x \, |1 - e^{\kappa_x t}|^{-1}$$ and $$W_x \, |_{{{\mathbb{R}}}^{< 0}} = \varepsilon_x e^t \, |1 - e^{\kappa_x |t|}|^{-1} \; .$$ This fits perfectly with the explicit formula (\[eq:19\]) if all $\varepsilon_{\gamma_{{\mathfrak{p}}}} (k) = 1$ and $\varepsilon_{x_{{\mathfrak{p}}}} = 1$. Namely if $l (\gamma_{{\mathfrak{p}}}) = \log N {\mathfrak{p}}$ for ${\mathfrak{p}}\nmid \infty$ and $\kappa_{x_{{\mathfrak{p}}}} = \kappa_{{\mathfrak{p}}}$ for ${\mathfrak{p}}{\, | \,}\infty$, then we have: $$e^{kl (\gamma_{{\mathfrak{p}}})} = e^{k \log N{\mathfrak{p}}} = N {\mathfrak{p}}^k \quad \mbox{for finite places} \; {\mathfrak{p}}$$ and $$W_{x_{{\mathfrak{p}}}} = W_{{\mathfrak{p}}} \quad \mbox{on} \; {{\mathbb{R}}}^* \; \mbox{for the infinite places} \; {\mathfrak{p}}\; .$$ [**5)**]{} In the setting of the preceeding remark the automorphisms $$e^{-\frac{k}{2} l (\gamma)} T_x \phi^{kl (\gamma)} \quad \mbox{of} \; T_x {{\mathcal F}}\quad \mbox{for} \; x \in \gamma$$ respectively $$e^{-\frac{t}{2}} T_x \phi^t \quad \mbox{of} \; T_x {{\mathcal F}}\quad \mbox{for a fixed point} \; x$$ are orthogonal automorphisms. For a real $2 \times 2$ orthogonal determinant $O$ with $\det O = -1$ we have: $$\det (1 - uO) = 1 - u^2 \; .$$ The condition $\varepsilon_{\gamma} (k) = +1$ therefore implies that $\det (T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}})$ is positive for $k \ge 1$ and hence for all $k \in {{\mathbb{Z}}}$. The converse is also true. For a fixed point we have already seen directly that $\det (T_x \phi^t {\, | \,}T_x {{\mathcal F}})$ is positive for all $t \in {{\mathbb{R}}}$. Hence we have the following information. [**Fact**]{} In the situation of the preceeding remark, $\varepsilon_k (\gamma) = +1$ for all $k \in {{\mathbb{Z}}}{\setminus}0$ if and only if on $T_x {{\mathcal F}}$ we have: $$T_x \phi^{kl (\gamma)} = e^{\frac{k}{2} l (\gamma)} \cdot O_k \quad \mbox{for} \; O_k \in {\mathrm{SO}}(T_x {{\mathcal F}}) \; .$$ For fixed points, $\varepsilon_x = 1$ is automatic and we have: $$T_x \phi^t = e^{\frac{t}{2}} O_t \quad \mbox{for} \; O_t \in {\mathrm{SO}}(T_x {{\mathcal F}}) \; .$$ In the number theoretical case the eigenvalues of $T_x \phi^{\log N{\mathfrak{p}}}$ on $T_x {{\mathcal F}}$ for $x \in \gamma_{{\mathfrak{p}}}$ would therefore be complex conjugate numbers of absolute value $N{\mathfrak{p}}^{1/2}$. If they are real then $T_x \phi^{\log N{\mathfrak{p}}}$ would simply be mutliplication by $\pm N{\mathfrak{p}}^{1/2}$. If not, the situation would be more interesting. Are the eigenvalues Weil numbers (of weight $1$)? If yes there would be some elliptic curve over ${\mathfrak{o}}_K / {\mathfrak{p}}$ involved by Tate–Honda theory.\ [**6)**]{} It would of course be very desirable to extend the hypotheses \[t56\] to a conjectured equality of distributions on all of ${{\mathbb{R}}}$. By theorem \[t33\] we expect one contribution of the form $$\chi_{{\mathrm{Co}}} ({{\mathcal F}}{{\mathcal L}}, \mu) \cdot \delta_0 \; .$$ The analogy with number theory suggests that there will also be somewhat complicated contributions from the fixed points in terms of principal values which are hard to guess at the moment. After all, even the simpler conjecture \[t31\] has not yet been verified in the presence of fixed points!\ [**7)**]{} If there does exist a foliated dynamical system attached to $\overline{{\mathrm{spec}\,}{\mathfrak{o}}_K}$ with the properties dictated by our considerations we would expect in particular that for a preferred transverse measure $\mu$ we have: $$\chi_{{\mathrm{Co}}} ({{\mathcal F}}{{\mathcal L}}, \mu) = - \log |d_{K / {{\mathbb{Q}}}}| \; .$$ This gives some information on the space $X$ with its ${{\mathcal F}}{{\mathcal L}}$-foliation. If $K / {{\mathbb{Q}}}$ is ramified at some finite place i.e. if $d_{K / {{\mathbb{Q}}}} \neq \pm 1$ then $\chi_{{\mathrm{Co}}} ({{\mathcal F}}{{\mathcal L}}, \mu) < 0$. Now, since ${\bar{H}}^2_{{{\mathcal F}}{{\mathcal L}}}$ must be one-dimensional, it follows that $$\chi_{{\mathrm{Co}}} ({{\mathcal F}}{{\mathcal L}}, \nu) < 0 \quad \mbox{for all non-trivial transverse measures} \; \nu \; .$$ Hence by a result of Candel [@Ca] there is a Riemannian metric on $T {{\mathcal F}}{{\mathcal L}}$, such that every ${{\mathcal F}}{{\mathcal L}}$-leaf has constant curvature $-1$. Moreover $(X , {{\mathcal F}}{{\mathcal L}})$ is isomorphic to $${{\mathcal O}}(H , X) / {\mathrm{PSO}}(2) \; .$$ Here ${{\mathcal O}}(H , X)$ is the space of conformal covering maps $u : H \to N$ as $N$ runs through the leaves of ${{\mathcal F}}{{\mathcal L}}$ with the compact open topology. See [@Ca] for details. In the unramified case, $|d_{K / {{\mathbb{Q}}}}| = 1$ we must have $\chi_{{\mathrm{Co}}} ({{\mathcal F}}{{\mathcal L}}, \nu) = 0$ for all transverse measures by the above argument. Hence there is an ${{\mathcal F}}{{\mathcal L}}$-leaf which is either a plane, a torus or a cylinder c.f. [@Ca]. \[t58\] In this final section we describe a simple case where the working hypothesis \[t56\] can be proved. Consider an unramified covering $f : M \to M$ of a compact connected orientable $d$-dimensional manifold $M$. We set $$\bar{M} = \lim_{\leftarrow} ( \ldots \xrightarrow{f} M \xrightarrow{f} M \to \ldots) \; .$$ Then $\bar{M}$ is a compact topological space equipped with the shift automorphism $\bar{f}$ induced by $f$. It can be given the structure of a $C^{\infty ,\infty}$-laminated space as follows. Let $\tilde{M}$ be the universal covering of $M$. For $i \in {{\mathbb{Z}}}$ there exists a Galois covering $$p_i : \tilde{M} \longrightarrow M$$ with Galois group $\Gamma_i$ such that $p_i = p_{i+1} {\mbox{\scriptsize $\,\circ\,$}}f$ for all $i$. Hence we have inclusions: $$\ldots \subset \Gamma_{i+1} \subset \Gamma_i \subset \ldots \subset \Gamma_0 =: \Gamma \cong \pi_1 (M , x_0) \; .$$ Writing the operation of $\Gamma$ on $\tilde{X}$ from the right, we get commutative diagrams for $i \ge 0$: $$\begin{CD} \tilde{M} \times_{\Gamma} (\Gamma / \Gamma_{i+1}) @= \tilde{M} / \Gamma_{i+1} @>{\overset{p_{i+1}}{\sim}}>> M \\ @VV{{\mathrm{id}}\times {\mathrm{proj}}}V @VV{{\mathrm{proj}}}V @VV{f}V \\ \tilde{M} \times_{\Gamma} (\Gamma / \Gamma_i) @= \tilde{M} / \Gamma_i @>{\overset{p_i}{\sim}}>> M \end{CD}$$ It follows that $$\label{eq:30} \tilde{M} \times_{\Gamma} \bar{\Gamma} {\stackrel{\sim}{\longrightarrow}}\bar{M}$$ where $\bar{\Gamma}$ is the pro-finite set with $\Gamma$-operation: $$\bar{\Gamma} = \lim_{\leftarrow} \Gamma / \Gamma_i \; .$$ The isomorphism (\[eq:30\]) induces on $\bar{M}$ the structure of a $C^{\infty , \infty}$-laminated space with respect to which $\bar{f}§$ becomes leafwise smooth. Fix a positive number $l > 0$ and let $\Lambda = l {{\mathbb{Z}}}\subset {{\mathbb{R}}}$ act on $\bar{M}$ as follows: $\lambda = l \nu$ acts by $\bar{f}^{\nu}$. Define a right action of $\Lambda$ on $\bar{M} \times {{\mathbb{R}}}$ by the formula $$(m,t) \cdot \lambda = (- \lambda \cdot m , t + \lambda) = (\bar{f}^{-\lambda / l} (m) , t + \lambda) \; .$$ The suspension: $$X = \bar{M} \times_{\Lambda} {{\mathbb{R}}}$$ is an $a = d+1$-dimensional $C^{\infty , \infty}$-laminated space with a one-codimensional foliation ${{\mathcal F}}$ as in \[t52\]. The leaves of ${{\mathcal F}}$ are the fibres of the natural fibration of $X$ over the circle ${{\mathbb{R}}}/ \Lambda$: $$X \longrightarrow {{\mathbb{R}}}/ \Lambda \; .$$ The leaves are also the images of $\bar{M} \times \{ t \}$ for $t \in {{\mathbb{R}}}$ under the natural projection. Translation in the ${{\mathbb{R}}}$-variable $$\phi^t [m,t'] = [m,t+t']$$ defines an ${{\mathcal F}}$-compatible flow $\phi$ on $X$ which is everywhere transverse to the leaves of ${{\mathcal F}}$ and in particular has no fixed points. The map $$\gamma \longmapsto \gamma_M = \gamma \cap (\bar{M} \times_{\Lambda} \Lambda)$$ gives a bijection between the closed orbits $\gamma$ of the flow on $X$ and the finite orbits $\gamma_M$ of the $\bar{f}$- or $\Lambda$-action. These in turn are in bijection with the finite orbits of the original $f$-action on $M$. We have: $$l (\gamma) = |\gamma_M| l \; .$$ \[t59\] In the situation of \[t58\] assume that all periodic orbits of $\phi$ are non-degenerate. Let ${\mathrm{Sp}}^n (\Theta)$ denote the set of eigenvalues with their algebraic multiplicities of the infinitesimal generator $\Theta$ of $\phi^{t*}$ on $H^n_{{{\mathcal F}}} (X)$. Then the trace $${\mathrm{Tr}}(\phi^* {\, | \,}H^n_{{{\mathcal F}}} (X)) := \sum_{\lambda \in {\mathrm{Sp}}^n (\Theta)} e^{t\Theta}$$ defines a distribution on ${{\mathbb{R}}}$ and the following formula holds true in ${{\mathcal D}}' ({{\mathbb{R}}})$: $$\begin{aligned} \lefteqn{\sum^{\dim {{\mathcal F}}}_{n=0} (-1)^n {\mathrm{Tr}}(\phi^* {\, | \,}H^n_{{{\mathcal F}}} (X)) = \chi_{{\mathrm{Co}}} ({{\mathcal F}}{{\mathcal L}}, \mu) \cdot \delta_0 \; + }\\ && \sum_{\gamma} l (\gamma) \left( \sum_{k \ge 1} \varepsilon_{\gamma} (k) \delta_{kl (\gamma)} + \sum_{k \le -1} \varepsilon_{\gamma} (|k|) \det (-T_x \phi^{kl (\gamma)} {\, | \,}T_x {{\mathcal F}}) \delta_{kl (\gamma)} \right) \; .\end{aligned}$$ Here $\gamma$ runs over the closed orbits of $\phi$ and in the sum over $k$’s any point $x \in \gamma$ can be chosen. Moreover $\chi_{{\mathrm{Co}}} ({{\mathcal F}}{{\mathcal L}}, \mu)$ is the Connes’ Euler characteristic of ${{\mathcal F}}{{\mathcal L}}$ with respect to a certain canonical transverse measure $\mu$. Finally we have the formula: $$\chi_{{\mathrm{Co}}} ({{\mathcal F}}{{\mathcal L}}, \mu) = \chi (M) \cdot l \; .$$ The definition of $\mu$ and the proof of the theorem will be given elsewhere. Let $E / {{\mathbb{F}}}_p$ be an ordinary elliptic curve over ${{\mathbb{F}}}_p$ and let ${{\mathbb{C}}}/ \Gamma$ be a lift of $E$ to a complex elliptic curve with $CM$ by the ring of integers ${\mathfrak{o}}_K$ in an imaginary quadratic field $K$. Assume that the Frobenius endomorphism of $E$ corresponds to the prime element $\pi$ in ${\mathfrak{o}}_K$. Then $\pi$ is split, $\pi \bar{\pi} = p$ and for any embedding ${{\mathbb{Q}}}_l \subset {{\mathbb{C}}}, l \neq p$ the pairs $$(H^*_{{\mathrm{\acute{e}t}}} (E \otimes \bar{{{\mathbb{F}}}}_p , {{\mathbb{Q}}}_l) \otimes {{\mathbb{C}}}, {\mathrm{Frob}}^*) \quad \mbox{and} \quad (H^* ({{\mathbb{C}}}/ \Gamma , {{\mathbb{C}}}) , \pi^*)$$ are isomorphic. Setting $M = {{\mathbb{C}}}/ \Gamma , f = \pi$ we are in the situation of \[t58\] and we find: $$X = ({{\mathbb{C}}}\times_{\Gamma} T_{\pi} \Gamma) \times_{\Lambda} {{\mathbb{R}}}\; .$$ Here $$T_{\pi} \Gamma = \lim_{\leftarrow} \Gamma / \pi^i \Gamma \cong {{\mathbb{Z}}}_p$$ is the $\pi$-adic Tate module of ${{\mathbb{C}}}/ \Gamma$. It is isomorphic to the $p$-adic Tate module of $E$. Setting $l = \log p$, so that $\Lambda = (\log p) {{\mathbb{Z}}}$ and passing to multiplicative time, $X$ becomes isomorphic to $$X \cong ({{\mathbb{C}}}\times_{\Gamma} T_{\pi} \Gamma) \times_{p^{{{\mathbb{Z}}}}} {{\mathbb{R}}}^*_+$$ which may be a more natural way to write $X$. Note that $p^{\nu}$ acts on ${{\mathbb{C}}}\times_{\Gamma} T_{\pi} \Gamma$ by diagonal multiplication with $\pi^{\nu}$. It turns out that the right hand side of the dynamical Lefschetz trace formula established in theorem \[t59\] equals the right hand side in the explicit formulas for $\zeta_E (s)$. Moreover the metric $g$ on $T {{\mathcal F}}$ given by $$g_{[z,y,t]} (\xi, \eta) = e^t {\mathrm{Re}\,}(\xi \bar{\eta}) \quad \mbox{for} \; [z,y,t] \; \mbox{in}\; ({{\mathbb{C}}}\times_{\Gamma} T_{\pi} \Gamma) \times_{\Lambda} {{\mathbb{R}}}$$ satisfies the conformality condition (\[eq:9\]) for $\alpha = 1$. The proof of Theorem \[t21\] can be easily adapted to $X$ above and shows that $\Theta = {\frac{1}{2}}+ S$ on ${\bar{H}}^1_{{{\mathcal F}}{{\mathcal L}}} (X)$ where $S$ is skew symmetric. This gives a dynamical proof for the Riemann hypotheses for $\zeta_E (s)$ along the lines that we hope for in the case of $\zeta (s)$. The construction of $(X , \phi^t)$ that we made for ordinary elliptic curves is misleading however, since it almost never happens that a variety in characteristic $p$ can be lifted to characteristic zero [*together with its Frobenius endomorphism*]{}. Moreover for ordinary elliptic curves the Riemann hypotheses can already be proved using Hodge cohomology of the lifted curve. This was essentially Hasse’s proof. Our present dream for the general situation is this: To an algebraic sum ${{\mathcal X}}/ {{\mathbb{Z}}}$ one should first attach an infinite dimensional dissipative dynamical system, possibly using ${\mathrm{GL}\,}_{\infty}$ in some way. 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[^1]: This is different from the normalization in [@D2] §3.

--- abstract: 'Dynamical models for 17 early-type galaxies in the Coma cluster are presented. The galaxy sample consists of flattened, rotating as well as non-rotating early-types including cD and S0 galaxies with luminosities between $M_B = -18.79$ and $M_B = -22.56$. Kinematical long-slit observations cover at least the major and minor axis and extend to $1-4 \, \,{r_\mathrm{eff}}$. Axisymmetric Schwarzschild models are used to derive stellar mass-to-light ratios and dark halo parameters. In every galaxy the best fit with dark matter matches the data better than the best fit without. The statistical significance is over 95 percent for 8 galaxies, around 90 percent for 5 galaxies and for four galaxies it is not significant. For the highly significant cases systematic deviations between observed and modelled kinematics are clearly seen; for the remaining galaxies differences are more statistical in nature. Best-fit models contain 10-50 percent dark matter inside the half-light radius. The central dark matter density is at least one order of magnitude lower than the luminous mass density, independent of the assumed dark matter density profile. The central phase-space density of dark matter is often orders of magnitude lower than in the luminous component, especially when the halo core radius is large. The orbital system of the stars along the major-axis is slightly dominated by radial motions. Some galaxies show tangential anisotropy along the minor-axis, which is correlated with the minor-axis Gauss-Hermite coefficient $H_4$. Changing the balance between data-fit and regularisation constraints does not change the reconstructed mass structure significantly: model anisotropies tend to strengthen if the weight on regularisation is reduced, but the general property of a galaxy to be radially or tangentially anisotropic, respectively, does not change. This paper is aimed to set the basis for a subsequent detailed analysis of luminous and dark matter scaling relations, orbital dynamics and stellar populations.' author: - | J. Thomas$^{1,2}$[^1], R. P. Saglia$^{2}$, R. Bender$^{1,2}$, D. Thomas$^{3}$, K. Gebhardt$^{4}$, J. Magorrian$^{5}$, E. M. Corsini$^{6}$ and G. Wegner$^{7}$\ $^{1}$Universitätssternwarte München, Scheinerstraße 1, D-81679 München, Germany\ $^{2}$Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstraße, D-85748 Garching, Germany\ $^{3}$Institute of Cosmology and Gravitation, Mercantile House, University of Portsmouth, Portsmouth, PO1 2EG, UK\ $^{4}$Department of Astronomy, University of Texas at Austin, C1400, Austin, TX78712, USA\ $^{5}$Theoretical Physics, Department of Physics, University of Oxford, 1 Keble Road, Oxford U.K., OX1 3NP\ $^{6}$Dipartimento di Astronomia, Università di Padova, vicolo dell’Osservatorio 3, I-35122 Padova, Italy\ $^{7}$Department of Physics and Astronomy, 6127 Wilder Laboratory, Dartmouth College, Hanover, NH 03755-3528, USA date: 'Accepted 1988 December 15. Received 1988 December 14; in original form 1988 October 11' title: 'Dynamical modelling of luminous and dark matter in 17 Coma early-type galaxies' --- \[firstpage\] stellar dynamics – galaxies: elliptical and lenticular, cD – galaxies: kinematics and dynamics — galaxies: structure Introduction {#mass:outline} ============ Elliptical galaxies are numerous among the brightest galaxies and they harbour a significant fraction of the present-day stellar mass in the universe [@Fuk98; @Ren06]. Key parameters for the understanding of elliptical galaxy formation and evolution are, among others, the central dark matter density, the scaling radius of dark matter, the stellar mass-to-light ratio and the distribution of stellar orbits. While the concentration of the dark matter halo puts constraints on the assembly epoch [@nfw96; @J00; @W02], the orbital state contains imprints of the assembly mechanism of ellipticals [e.g. @vanAl82; @Her92; @Her93; @Wei96; @Dub98; @Nab03; @Jes05]. Information about elliptical galaxy masses are in principle offered through various channels. The analysis of X-ray halo temperatures, the kinematics of occasional gas discs and galaxy-galaxy lensing provide evidence for extended dark matter halos around early-type galaxies (e.g. @Ber93 [@Piz97; @Loe99; @Oos02; @Hoe04; @Fuk06; @Hum06; @Kle06; @Man06]). These methods do not constrain the inner halo-profiles strongly, however. At non-local redshifts strong lensing configurations allow a detailed reconstruction of the mass enclosed inside, say, ${r_\mathrm{eff}}$ (e.g. @Kee01 [@Kop06]). None of the above mentioned observational channels is sensitive to dynamical galaxy parameters, such as the distribution of stellar orbits. Dynamical modelling of stellar kinematics has the unique advantage that it allows reconstruction of both the mass structure and the orbital state of a galaxy. High-quality observations of the line-of-sight velocity distributions (LOSVDs) out to several ${r_\mathrm{eff}}$ are needed for this purpose. To overcome the problems of measuring absorption line kinematics in the faint outskirts of ellipticals, discrete kinematical tracers such as planetary nebulae or globular clusters can be used to additionally constrain the mass distribution (e.g. @Sag00 [@R03; @Pie06]). Since stars in galaxies behave collisionlessly to first order, the distribution of stellar orbits is not known a priori and very general dynamical methods are required to probe all the degrees of freedom in the orbital system. So far only one large sample of 21 round, non-rotating giant ellipticals has been probed for dark matter considering at least the full range of [*spherical*]{} models [@Kr00]. These models predict circular velocity curves constant to about 10 per cent and equal luminous and dark matter somewhere inside $1 - 3 \, {r_\mathrm{eff}}$. Reconstructed halos of these models are $\sim 25$ times denser than in comparably bright spirals, which indicates a $\sim 3$ times higher formation redshift [@G01]. Not all apparently round objects need to be intrinsically spherical; some may be face-on flattened systems. Apparently flattened ellipticals have not yet been addressed in much generality. Primarily, because [*axisymmetric*]{} modelling is required to account for intrinsic flattening, inclination effects and rotation. Fully general axisymmetric models involve three integrals of motion, one of which – the non-classical so-called third integral – is not given explicitly in most astrophysically relevant potentials. Only recently, sophisticated numerical methods such as Schwarzschild’s orbit superposition technique [@S79] have provided fully general models involving all relevant integrals of motion. Dynamical studies of samples of elliptical galaxies using this technique are, however, based on kinematical data inside $r \la {r_\mathrm{eff}}$ [@Geb03; @Cap05] and dark matter is not considered. The present paper is part of a project aimed to analyse the luminous and dark matter distributions as well as the orbital structure in a sample of flattened Coma ellipticals. The data for this project has been collected over the last years and consists of ground-based as well as (archival and new) HST imaging and measurements of line-of-sight velocity distributions (LOSVDs) along various position angles out to $1-4 \, {r_\mathrm{eff}}$ (@Meh00; @Weg02; @Cor07). The implementation of our modelling machinery, which is an advanced version of the axisymmetric Schwarzschild code of @Ric88 and @Geb00 has been described in detail in @Tho04 [@Tho05]. In the present paper we survey the models of the whole sample. This sets the basis for subsequent investigations of luminous and dark matter scaling relations and stellar populations in elliptical galaxies (Thomas et al. 2007a, in preparation). In Sec. \[sec:obs\] the observations are summarised and the modelling is outlined in Secs. \[sec:setup\] and \[sec:res\]. The mass structure of our models and the orbital anisotropies are described in Secs. \[sec:mass\] and \[sec:aniso\], respectively. Phase-space distribution functions for luminous and dark matter are the subject of Secs. \[sec:dfstars\] and \[sec:dfhalo\]. We discuss the influence of regularisation on our results in Sec. \[sec:regula\]. The paper closes with a short discussion and summary in Sec. \[sec:sum\]. A detailed comparison of models and data for each galaxy can be found in App. \[sec:fits\]. Summary of observations {#sec:obs} ======================= The Coma sample consists of seventeen early-type galaxies: two cD galaxies, nine ordinary giant ellipticals and six lenticulars or galaxies of intermediate type. They cover the luminosity interval $-20.30<M_B<-22.56$, typical for luminous giant ellipticals/cDs. One single fainter galaxy with $M_B=-18.8$ is also included (cf. Tab. \[dattable\]; magnitudes are from Hyperleda for a distance of $d = 100 \, \mathrm{Mpc}$ to Coma; this corresponds to $H_0 = 69 \, \mathrm{km/s/Mpc}$). Effective radii are mostly between $3\farcs3<{r_\mathrm{eff}}<18\farcs4$. Only the four brightest galaxies have formally very large ${r_\mathrm{eff}}\approx30\arcsec-70\arcsec$ (cf. Tab. \[dattable\]; ${r_\mathrm{eff}}$ are from @Meh00 and based on de-Vaucouleurs fits). All galaxies share the same distance and the spatial resolution in the photometric as well as the kinematical observations is roughly comparable for all galaxies. The photometric input for the modelling is constructed as a composite of ground-based (outer parts) and HST imaging (inner parts). The two surface brightness profiles $\mu_\mathrm{grd}$ and $\mu_\mathrm{HST}$ are joined by shifting the HST profile according to the average $\langle \mu_\mathrm{grd}-\mu_\mathrm{HST} \rangle$ over a region where both data sets overlap and seeing effects are negligible ($\approx 5\arcsec - 16\arcsec$). The shift $\langle \mu_\mathrm{grd}-\mu_\mathrm{HST} \rangle$ is usually well defined (cf. Tab. \[dattable\]). ------ --------- ---------- ------- -------------------- -------------- ------------------------------------------------------------------- --------- --------- ------------------------ ------------------------ ------------------------ ------------------------ type $M_B$ ${r_\mathrm{eff}}$ $\epsilon_e$ $\mathrm{rms}\langle \mu_\mathrm{grd} - \mu_\mathrm{HST} \rangle$ maj min off dia GMP NGC HST grd \[mag\] \[arcsec\] \[mag\] \[${r_\mathrm{eff}}$\] \[${r_\mathrm{eff}}$\] \[${r_\mathrm{eff}}$\] \[${r_\mathrm{eff}}$\] (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) 0144 4957 E L97 M00 $-21.07$ $18.4$ $0.256$ $0.011$ $1.4$ $0.7$ – – 0282 4952 E L97 M00 $-20.69$ $14.1$ $0.315$ $0.009$ $1.7$ $0.5$ – – 0756 4944 S0 W07 M00 $-21.77$ $11.7$ $0.657$ $0.010$ $3.0$ $0.4$ $2.5$ – 1176 4931 S0 W07 M00 $-20.32$ $7.4$ $0.552$ $0.080$ $4.7$ $0.8$ $3.7$ – 1750 4926 E L97 J94 $-21.42$ $11.0$ $0.132$ $0.058$ $0.9$ $0.9$ – – 1990 IC 843 E/S0 W07 M00 $-20.52$ $9.45$ $0.485$ $0.066$ $3.3$ $0.5$ $1.8$ – 2417 4908 E/S0 L97 J94 $-21.06$ $7.1$ $0.322$ $0.042$ $2.2$ $0.9$ $0.9$ – 2440 IC 4045 E W07 J94 $-20.30$ $4.37$ $0.330$ $0.038$ $3.2$ $1.0$ – $1.0$ 2921 4889 D L97 J94 $-22.56$ $33.9$ $0.360$ $0.028$ $0.7$ $0.3$ – – 3329 4874 D H98 J94 $-22.50$ $70.8$ $0.141$ $0.057$ $0.4$ $0.1$ – – 3510 4869 E L97 J94 $-20.40$ $7.6$ $0.112$ $0.033$ $2.0$ $1.1$ – – 3792 4860 E L97 J94 $-20.99$ $8.5$ $0.161$ $0.071$ $1.1$ $1.0$ – – 3958 IC 3947 E L97 J94 $-18.79$ $3.3$ $0.323$ $0.024$ $1.7$ $0.9$ – – 4928 4839 E/S0 (D) L97 J94 $-22.26$ $29.5$ $0.426$ $0.104$ $1.1$ $0.1$ – $0.2$ 5279 4827 E L97 M00 $-21.36$ $13.6$ $0.205$ $0.019$ $1.6$ $0.7$ – – 5568 4816 S0 L97 M00 $-21.53$ $55.7$ $0.284$ $0.075$ $0.5$ $0.1$ $0.1$ – 5975 4807 E L97 M00 $-20.73$ $6.7$ $0.170$ $0.015$ $2.9$ $0.5$ – $1.2$ ------ --------- ---------- ------- -------------------- -------------- ------------------------------------------------------------------- --------- --------- ------------------------ ------------------------ ------------------------ ------------------------ The kinematic data to be fit by the models come from long-slit observations along at least two position angles: the apparent major and minor axis, respectively. Kinematical parameters from different sides of a galaxy are averaged. As error-bars we use the maximum from the two sides or half of the scatter between them, whatever is larger. This assumes that uncertainties in the observations are mostly systematic (see also the discussion in Sec. \[subsec:conf\]). In case of pure statistical errors and an exactly axisymmetric system this would overestimate the errors by a factor $\sqrt{2}$. Thus, we are conservative. The data are described in full detail in @Meh00, @Weg02 and @Cor07. Basic parameters of the photometric and kinematic data set are summarised in Tab. \[dattable\]. Three galaxies deserve further comments: #### GMP5568/NGC4816: {#gmp5568ngc4816 .unnumbered} GMP5568 has been observed along four position angles. In addition to major and minor-axis spectra, two observations were made with the slits parallel to the major-axis: one with an offset of ${r_\mathrm{eff}}/4$, the other with ${r_\mathrm{eff}}/20$. #### GMP0144/NGC4957: {#gmp0144ngc4957 .unnumbered} The velocity dispersion peak of GMP0144 is significantly off the photometric centre. Furthermore, GMP0144 is the only galaxy in our sample that exhibits a significant isophotal twist towards the centre. Thus, GMP0144 is likely triaxial near its centre. To reduce the influence of potentially non-axisymmetric regions on our modelling, kinematic measurements inside $r < 4 \arcsec$ are omitted. #### GMP5975/NGC4807: {#gmp5975ngc4807 .unnumbered} Dynamical models for GMP5975, based on major and minor-axis kinematics, have already been presented in @Tho05. There, a striking depopulation of retrograde orbits was found. To check its significance we also determined kinematics along a diagonal slit. Here we present new models that include this additional kinematical data. Both, the mass structure and the distribution of stellar orbits did not change significantly. Dynamical modelling {#sec:setup} =================== We model the kinematic and photometric observations with Schwarzschild’s orbit superposition method [@S79]. Details about our implementation are given in @Tho04 [@Tho05]. Basic steps of the method are briefly recalled in this section. Deprojection and inclination {#subsec:depro} ---------------------------- The surface photometry is deprojected to obtain the 3d luminosity distribution $\nu$ for each galaxy (using the program of @mag99). We consider radial profiles of surface-brightness, ellipticity and isophotal shape parameters $a_4$ and $a_6$ [@bm87] for the deprojections[^2] (cf. App. \[sec:fits\]). For each galaxy, we probe three different inclinations in the deprojection, and subsequent dynamical modelling, respectively: (1) $i=90\degr$ (edge-on), where the deprojection is intrinsically least flattened; (2) a minimum inclination that is found by requiring the deprojection to be intrinsically as flattened as an E7 galaxy; (3) an intermediate inclination, for which the deprojection looks like an E5 galaxy from the side. This inclination scheme emerges as a compromise between limited computation time on the one side and the strategy to get the most conservative estimate of uncertainties on intrinsic properties on the other. In many galaxies the inclination is only poorly constrained (cf. Sec. \[subsec:inclin\]). In other words, when we quote 68 percent confidence uncertainties on intrinsic properties below this includes in many cases models from all three probed inclinations, including those connected with the rather extreme intrinsic E5 and E7 shapes. In case of GMP0756, GMP1176, GMP1990 and GMP2417 only the edge-on orientation is considered. These galaxies are all highly flattened. In addition, they appear either discy (e.g. GMP1176) or have thin dust features (GMP1990, GMP2417; cf. @Cor07), implying that they are seen close to edge-on. The dynamical modelling of GMP5975 in @Tho05 revealed that only models at $i=90\degr$ were within the one sigma confidence region. It was also found that the deprojection of GMP5975 becomes implausibly boxy in the outer parts, if it is assumed that the galaxy is significantly inclined. We therefore reanalysed the extended data set for GMP5975 only with $i=90\degr$. Mass model {#subsec:mass} ---------- With the luminosity density $\nu$ given, a trial mass density distribution can be defined by $$\label{rhorho} \rho = \Upsilon \, \nu + \rho_\mathrm{DM}.$$ The stellar mass-to-light ratio $\Upsilon$ is assumed constant throughout the galaxy. Concerning the dark matter density $\rho_\mathrm{DM}$ we probe the two following parametric prescriptions. Firstly the NFW-distribution [@nfw96] $$\label{nfw} \rho_\mathrm{NFW}(r,r_s,c) \propto \frac{1}{(r/r_s)(1+r/r_s)^2},$$ where $r_s=r_{200}/c$ is a scaling radius, $r_{200}$ is a measure of the virial radius and $c$ is the concentration of the halo. Simulations predict the two halo parameters $r_s$ and $c$ to be correlated, such that the distribution (\[nfw\]) can be read as a one-parameter family of dark matter halos [@nfw96]. To be explicit, we use $$\label{family} r_s^3 \propto 10^{(A-\log c)/B} \left( 200 \, \frac{4 \pi}{3} \, c^3 \right)^{-1}$$ with $A=1.05$ and $B=0.15$ [@nfw96; @R97]. We consider spherical as well as flattened NFW halos, where the latter are derived from equation (\[nfw\]) by the substitution $r \rightarrow r \sqrt{\cos^2 (\vartheta) + \sin^2 (\vartheta) / q^2}$ ($q$ is the constant flattening of the isodensity contours, $\vartheta$ is the latitude in spherical coordinates). The second halo family used is the logarithmic potential $$\label{nis} \rho_\mathrm{LOG}(r) \propto v_c^2 \frac{3r_c^2+r^2} {(r_c^2+r^2)^2},$$ that gives rise to an asymptotically constant circular velocity $v_c$ and a flat central density core inside $r \la r_c$ (@Bin81). In the gravitational potential generated by the mass distribution (\[rhorho\]) we compute typically about $18000$ orbits as described in @Tho04. Orbit superposition {#subsec:orbitfit} ------------------- The final orbit superposition model is constructed according to the maximum entropy technique of @Ric88. It consists in solving $$\label{maxs} \hat{S} \equiv S - \alpha \, \chi^2_\mathrm{LOSVD} \rightarrow \mathrm{max},$$ with $S$ denoting the Boltzmann entropy $$\label{bentropy} S \equiv - \int f \ln \left( f \right) \, {\mathrm{d}}^3r \, {\mathrm{d}}^3v \, = - \sum_i w_i \ln \left( \frac{w_i}{V_i} \right)$$ and $f$ being the phase-space distribution function (DF) of the model. In Schwarzschild models – by construction – the DF is constant along individual orbits. The corresponding phase-space density $f_i$ along orbit $i$ is the ratio $$\label{eq:fi} f_i \equiv \frac{w_i}{V_i}$$ of the total amount of light $w_i$ on the orbit (the so-called orbital weight) and the orbital phase-space volume $V_i$. The $w_i$ that solve equation (\[maxs\]) are obtained iteratively: starting with a low $\alpha = 10^{-10}$ we solve equation (\[maxs\]) for a fixed set of $\alpha_i$, using the orbital weights obtained at $\alpha_{i-1}$ as initial guess for the solution at $\alpha_i$. ------ -------------------------- ------------------------ --------------------------- ------- ------- ------------------------- --------------------------- ------- ------ ------------------------- ------ ------------------------------- ----------------------------- ----------------- GMP ${\Upsilon_\mathrm{SC}}$ ${\chi^2_\mathrm{SC}}$ ${\Upsilon_\mathrm{LOG}}$ $r_c$ $v_c$ ${\chi^2_\mathrm{LOG}}$ ${\Upsilon_\mathrm{NFW}}$ $c$ $q$ ${\chi^2_\mathrm{NFW}}$ halo $\Delta \chi^2_\mathrm{halo}$ $\Delta \chi^2_\mathrm{DM}$ $i$ (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) 0144 7.0 0.400 5.0 4.4 212 0.383 4.5 17.17 0.7 0.336 NFW -2.45 3.3 $50^{50}_{50}$ 0282 6.5 0.436 5.0 17.0 502 0.244 4.5 11.24 0.7 0.256 LOG 1.01 16.9 $60 ^{70}_{60}$ 0756 3.0 1.253 2.6 12.7 215 0.930 2.2 20.2 0.7 0.942 LOG 1.57 41.3 90 1176 2.5 1.353 2.0 3.4 200 0.724 2.0 18.0 1.0 0.707 NFW -1.8 67.2 90 1750 7.0 0.540 6.0 18.7 500 0.452 6.0 12.5 1.0 0.469 LOG 0.81 4.2 $65^{90}_{65}$ 1990 10.0 0.301 10.0 13.1 105 0.291 9.0 24.0 1.0 0.298 LOG 0.72 1.0 90 2417 8.5 0.244 8.0 23.8 500 0.206 7.0 14.76 0.7 0.216 LOG 0.46 1.8 90 2440 7.0 0.579 6.5 10.9 482 0.453 6.5 16.47 0.7 0.475 LOG 1.69 9.6 $60^{60}_{60}$ 2921 9.0 0.112 6.5 8.2 425 0.073 6.5 9.2 0.7 0.067 NFW -0.47 3.3 $90^{90}_{60}$ 3329 12.0 0.325 7.0 3.6 400 0.307 9.0 10.85 0.7 0.309 LOG 0.22 1.4 $90^{90}_{45}$ 3510 6.0 0.425 5.5 11.6 287 0.398 5.0 16.12 0.7 0.398 LOG 0.67 2.5 $90^{90}_{60}$ 3792 9.0 0.370 8.0 15.3 550 0.339 8.0 10.0 1.0 0.349 LOG 0.54 1.7 $60^{90}_{40}$ 3958 6.0 0.229 5.0 6.8 274 0.162 4.0 14.7 1.0 0.174 LOG 0.42 2.4 $90^{90}_{70}$ 4928 10.0 0.232 8.5 29.1 507 0.109 7.0 12.7 1.0 0.122 LOG 0.66 6.4 $90^{90}_{70}$ 5279 7.0 0.132 6.5 28.4 482 0.099 6.0 15.9 0.7 0.109 LOG 0.71 2.3 $90^{90}_{90}$ 5568 7.0 0.162 6.0 66.7 650 0.103 5.0 17.2 0.7 0.104 LOG 0.12 5.2 $90^{90}_{50}$ 5975 4.0 0.580 3.0 1.7 200 0.333 3.0 15.0 0.7 0.314 NFW -1.37 19.1 90 ------ -------------------------- ------------------------ --------------------------- ------- ------- ------------------------- --------------------------- ------- ------ ------------------------- ------ ------------------------------- ----------------------------- ----------------- The $\chi^2$-term in equation (\[maxs\]) quantifies deviations between model and data. We do not include the photometry in the $\chi^2$, but treat the deprojected luminosity distribution as a boundary condition for the solution of equation (\[maxs\]). To fit the measured LOSVDs, which are parameterised in terms of the Gauss-Hermite parameters $v$, $\sigma$, $H_3$ and $H_4$ [@Ger93; @vdMF93] we proceed as follows: the Gauss-Hermite parameters are used to generate binned data LOSVDs ${\cal L}^{jk}_\mathrm{dat}$. Errors are propagated via Monte-Carlo simulations. These data LOSVDs and the corresponding model quantities ${\cal L}^{jk}_\mathrm{mod}$ are used to get the $\chi^2_\mathrm{LOSVD}$ of equation (\[maxs\]): $$\label{chilosvd} \chi^2_\mathrm{LOSVD} \equiv \sum_{j=1}^{N_{\cal L}} \, \sum_{k=1}^{N_\mathrm{vel}} \left( \frac{{\cal L}^{jk}_\mathrm{mod}-{\cal L}^{jk}_\mathrm{dat}} {\Delta {\cal L}^{jk}_\mathrm{dat}} \right)^2.$$ The above sum includes all $N_{\cal L}$ data points and each LOSVD is represented by $N_\mathrm{vel}$ bins in projected (line-of-sight) velocity. With the orbital weights $w_i$ determined, the dynamical state of the model is completely specified, i.e. the phase-space distribution function is known (cf. equation \[eq:fi\]). In the course of this paper we will not only consider the DF, but also the orbital anisotropy. It can be quantified by the so-called anisotropy parameters $$\label{eq:bt} {\beta_\vartheta}\equiv 1 - \frac{\sigma_\vartheta^2}{\sigma_r^2}$$ (meridional anisotropy) and $$\label{eq:bp} {\beta_\varphi}\equiv 1 - \frac{\sigma_\varphi^2}{\sigma_r^2}$$ (azimuthal anisotropy). Internal velocity dispersions $\sigma$ in the above equations are computed in spherical coordinates $r$, $\vartheta$ and $\varphi$, oriented such that $\varphi$ is the azimuth in the equatorial plane and $\vartheta$ is the latitude. Regularisation {#subsec:regula} -------------- The parameter $\alpha$ in equation (\[maxs\]) controls the relative weight of data-fit and entropy maximisation. The higher $\alpha$ the better the fit and the larger the noise in the derived distribution function, or orbital weights, respectively. Ideally, regularisation has to be optimised case-by-case for each galaxy. This holds in principle for both, the value of the regularisation parameter $\alpha$, as well as for the functional form of $S$. Firstly, because the spatial resolution and/or coverage as well as the signal-to-noise of the observations vary from galaxy to galaxy and regularisation should be adapted to that. This primarily concerns the choice of $\alpha$. Secondly, different galaxies have different intrinsic structures. Specifically, the degree to which the entropy of a stellar system is maximised may vary in phase-space. Consider, for example, a cold disc inside a hot spheroid. The disc has low entropy and to fit its rotation, $\alpha$ needs to be large (models with the maximum entropy according to equation \[bentropy\] have no rotation). On the other hand, the spheroid-dominated region in phase-space can have higher entropy and using a large $\alpha$ in the fit amplifies the noise in the corresponding parts of the phase-space distribution function (DF). The dilemma as to the choice of $\alpha$ in such cases could be solved by adjusting the function $S$ appropriately. For the Coma galaxy modelling we use the same regularisation for all galaxies: $\alpha = 0.02$ and the entropy of equation (\[bentropy\]). The value for $\alpha$ has been obtained by Monte-Carlo simulations of isotropic rotator test galaxies with realistic, noisy mock data [@Tho05]. Applying it to the whole sample is motivated by the similar spatial coverage and resolution of all our Coma observations (cf. Sec. \[sec:obs\]). Furthermore, since it has been obtained from fitting isotropic rotators it has proven to be sufficiently large to fit non-maximum entropy, rotating galaxies. It might be slightly too large for non-rotating galaxies. Thus we expect models of non-rotating galaxies to possibly be noisier than those of rotating galaxies, but we do not expect that our imposed regularisation is too restrictive. In any case, we will explicitly investigate the dependency of model results on the choice of $\alpha$ in Sec. \[sec:regula\]. Best-fit model and uncertainties {#subsec:bfit} -------------------------------- To obtain the best-fit mass model for a given $\alpha$ we calculate orbit models as described in Secs. \[subsec:mass\] and \[subsec:orbitfit\] for various combinations of the relevant parameters: ($r_c,v_c,\Upsilon$) in case of LOG-halos, ($c,\Upsilon,q$) in case of NFW-halos or just $\Upsilon$ for models without dark matter, respectively. Logarithmic halos are probed on a grid with $\Delta r_c \approx {r_\mathrm{eff}}/2$ and $\Delta v_c = 50 \, \mathrm{km/s}$ (in some cases the grid is refined around the location of the lowest $\chi^2$). Typically we explore $N_r \times N_v \approx 100$ halos. The analogous numbers for the NFW halos read $\Delta c = 2.5$, $N_c \approx 12$ and $N_q = 2$ ($q \in \{0.7,1.0\}$). The step-size $\Delta \Upsilon$ for the mass-to-light ratio equals 10-20 percent of the best-fit ${\Upsilon_\mathrm{dyn}}$, independent of the halo type. Around the best-fit model the resolution in $\Upsilon$ is doubled, resulting in $N_\Upsilon \approx 6-10$ models with different mass-to-light ratios for each halo. This sums up to about $600-1000$ models with logarithmic halos and $120-240$ models with NFW halos. The final number of models is up to a factor of three larger, depending on the number $N_i \le 3$ of probed inclinations $i$. The total number of models per galaxy is around $1000-3000$. {width="83mm"} {width="83mm"} Among these models we determine the best-fit according to the lowest $\chi^2_\mathrm{GH}$, defined as $$\begin{aligned} \label{chigheq} \chi^2_\mathrm{GH} \equiv \sum_{j=1}^{N_{\cal L}} \left[ \left( \frac{v^j_\mathrm{mod} - v^j_\mathrm{dat}}{\Delta v^j_\mathrm{dat}} \right)^2 + \left( \frac{\sigma^j_\mathrm{mod} - \sigma^j_\mathrm{dat}}{\Delta \sigma^j_\mathrm{dat}} \right)^2 + \right. \nonumber \\ \left. \left( \frac{H^j_{3,\mathrm{mod}} - H^j_{3,\mathrm{dat}}}{\Delta H^j_{3,\mathrm{dat}}} \right)^2 + \left( \frac{H^j_{4,\mathrm{mod}} - H^j_{4,\mathrm{dat}}}{\Delta H^j_{4,\mathrm{dat}}} \right)^2 \right].\end{aligned}$$ Here, $v^j_\mathrm{dat}$ is the rotation according to the Gauss-Hermite parameterisation of the LOSVDs (other parameters analogously). A detailed discussion about the difference between $\chi^2_\mathrm{GH}$ and $\chi^2_\mathrm{LOSVD}$ can be found in @Tho05. Confidence intervals on model quantities are set by the corresponding minimum and maximum values obtained over all models within $\Delta \chi^2_\mathrm{GH} = 1.1$ from the minimum $\chi^2_\mathrm{GH}$. The value of $\Delta \chi^2_\mathrm{GH} = 1.1$ is slightly more conservative than the classical $\Delta \chi^2_\mathrm{GH} = 1.0$ and has been derived by means of Monte-Carlo simulations [@Tho05]. As a byproduct of the iterative technique to solve equation (\[maxs\]), we get – for each set of ($r_c,v_c,\Upsilon,i$), ($c,q,\Upsilon,i$) and ($\Upsilon,i$), respectively – orbit models for about $N_\alpha \approx 50$ different values of the regularisation parameter. This allows us to derive a best-fit model for each $\alpha_i$ and to explore the dependency of best-fit model parameters on $\alpha$ (cf. Sec. \[sec:regula\]). Modelling results {#sec:res} ================= Modelling results are summarised in Tab. \[modtable\]. In the remainder of this Section we collect some general notes on these results. Goodness-of-fit {#subsec:conf} --------------- The goodness-of-fit obtained under the different assumptions about the overall mass distribution are given in columns (3), (7) and (11) of Tab. \[modtable\]. Thereby $${\chi^2_\mathrm{SC}}\equiv \min \{ {\chi^2_\mathrm{GH}}(\Upsilon,\, i)/N_\mathrm{data}\},$$ $${\chi^2_\mathrm{LOG}}\equiv \min \{ {\chi^2_\mathrm{GH}}(r_c,\, v_c,\, \Upsilon,\, i)/N_\mathrm{data}\}$$ and $${\chi^2_\mathrm{NFW}}\equiv \min \{ {\chi^2_\mathrm{GH}}(c,\, q,\, \Upsilon,\, i)/N_\mathrm{data}\}$$ are minimised over all relevant mass parameters. Differences between models with and without dark matter are further discussed in Sec. \[subsec:ml\]. Here we only refer to the fact that our models are able to reproduce the observations with a ${\chi^2_\mathrm{GH}}$ per data point which is in many cases significantly smaller than unity. The largest deviations between model and data occur for the S0 GMP0756, possibly related to the low $H_4$ along the offset-axis, which are not followed by our models. Fits to some galaxies are as good as ${\chi^2_\mathrm{min}}\la 0.1$, where $$\label{eq:chimm} {\chi^2_\mathrm{min}}\equiv \min \{{\chi^2_\mathrm{LOG}}, \, {\chi^2_\mathrm{NFW}}, {\chi^2_\mathrm{SC}}\}$$ describes the overall minimum of ${\chi^2_\mathrm{GH}}$ for a given galaxy. In many cases where ${\chi^2_\mathrm{min}}$ is particularly low, error bars of the observations are much larger than the point-to-point scatter of the data points. This concerns GMP5279, GMP2921, GMP4928, GMP5568 and GMP3958, where the observational errors are likely overestimated (see also the fits in App. \[sec:fits\]). In some other systems, like for example GMP0144, the error bars used in the modelling are rather large, because they also include side-to-side variations of the kinematics, which are often also larger than the point-to-point scatter on a given side of the galaxy. Thus, both effects might partly explain the low ${\chi^2_\mathrm{min}}$ of these galaxies. Very low ${\chi^2_\mathrm{min}}$ raise the question whether confidence intervals of model properties calculated as described in Sec. \[subsec:bfit\] (and shown in Figs. \[vcirccomparison\], \[denscomparison\], \[anisominor\] and \[anisomaj\] below) are overestimated. In cases where the observational errors are obviously too large it is reasonable to rescale them until ${\chi^2_\mathrm{min}}\approx 1$. In fact, this has been done for GMP5975 in @Tho05, where error bars were scaled such that ${\chi^2_\mathrm{min}}\approx 0.7$. This value was determined from Monte-Carlo simulations of isotropic rotator models. To quantify the effect of rescaling, Fig. \[4928:rescaled\] exemplifies confidence intervals for one galaxy of the sample (GMP4928) once without rescaling the $\chi^2_\mathrm{GH}$ and once with rescaling all observational error bars to ${\chi^2_\mathrm{min}}= 0.7$. As it can be seen, uncertainty regions shrink a lot after rescaling. {width="164mm"} Globally rescaling the error-bars is not appropriate for all galaxies, however. Errors along the major-axis of GMP3792, for example, might be overestimated, but those along the minor-axis seem not. Moreover, in a case like GMP1750 slight minor-axis rotation – which cannot be fit with axisymmetric models – adds a constant to ${\chi^2_\mathrm{min}}$. Just rescaling to ${\chi^2_\mathrm{min}}\approx 0.7$ in all galaxies would introduce an artificial dependency of uncertainty regions on minor-axis rotation, which is not appropriate. In order to treat all galaxies of the sample homogeneously we do not rescale the $\chi^2_\mathrm{GH}$, but give the most conservative error-bars for our models. The corresponding confidence intervals may be interpreted as the maximal uncertainty on derived model quantities, while the shaded regions of Fig. \[4928:rescaled\] may be interpreted as lower limits for these uncertainties. Fig. \[chicomp\] shows the dependency of $$\Delta \chi^2_\mathrm{GH}(\Upsilon) \equiv \chi^2_\mathrm{GH}(\Upsilon) - \mathrm{min} \, (\chi^2_\mathrm{GH}),$$ where $\mathrm{min} \, {\chi^2_\mathrm{GH}}\equiv \chi^2_\mathrm{min} \times N_\mathrm{data}$ (cf. equation \[eq:chimm\]) and $\chi^2_\mathrm{GH}(\Upsilon)$ is minimised over all NFW-fits, logarithmic-halo fits and self-consistent fits with the given $\Upsilon$. For all but one galaxy, we find a clear minimum in ${\chi^2_\mathrm{GH}}(\Upsilon)$. The exceptional case, GMP3329, is peculiar in many respects: (1) It is among the brightest galaxies of the sample and has a very large ${r_\mathrm{eff}}$. The data only cover the region inside $r \la {r_\mathrm{eff}}/2$. (2) The surface-brightness profile shows a break near $0.2 - 0.3 \, {r_\mathrm{eff}}$ (cf. upper panel of Fig. \[isoplotgh3329\]). (3) At about the same projected distance from the centre the velocity dispersion dips and rises again at larger radii (cf. lower panel of Fig. \[isoplotgh3329\]). The poor constraints on the mass-to-light ratio in this system could be related to the poor data coverage. It might also be that GMP3329 is actually composed of two subcomponents. If these have different mass-to-light ratios $\Upsilon$, then the ${\chi^2_\mathrm{GH}}$-curve may have two corresponding local minima and the poor data coverage may smooth out these into a flat plateau. Finally, our modelling of GMP3329 may suffer from the Coma core being possibly not in dynamical equilibrium, as indicated by the kinematics of intra-cluster planetary nebulae [@Ger07]. Model inclinations {#subsec:inclin} ------------------ Most of the best-fit models are edge-on according to the last column of Tab. \[modtable\]. This is surprising at first sight because if galaxies are oriented randomly then one would expect only about 6 out of 17 objects to have inclinations larger than $i\ga70\degr$. Omitting the five systems for which only edge-on models were calculated (GMP0756, GMP1176, GMP1990, GMP2417 and GMP5975; cf. Sec. \[subsec:depro\]) and taking into account the uncertainties quoted in Tab. \[modtable\] there are 3 galaxies out of 11 where inclinations $i<70\degr$ are ruled out by our modelling (at the 68 percent confidence level). This is in good agreement with the expectation for random inclinations. Nevertheless, we now discuss a little more whether our modelling might be subject to a slight inclination bias. First, one possible issue is that using the same regularisation for all galaxies might introduce a subtle bias towards edge-on configurations. Consider a rotating system: the lower the assumed inclination of the model the larger its intrinsic rotation needs to be in order to match the observations after projection. Thus, the system will be dynamically colder and its entropy will be lower. Turning the argument around: usage of a constant $\alpha$ enforces the same weight on entropy in the inclined model as in the edge-on model. Since the inclined model has to have lower entropy, however, its fit may be less good. This might drive models of rotating galaxies towards $i=90\degr$. We do not expect this to affect conclusions drawn from our modelling results strongly, because, as it has been argued in Sec. \[subsec:depro\], error bars on intrinsic properties include in many cases models at different inclinations, even extreme cases. But it might drive the [*best-fit*]{} model to occur preferentially around $i=90\degr$. Second, for face-on galaxies noise in the kinematics may be a source of bias towards edge-on models as well [@Tho07]. It can cause rotation measurements $v \ne 0$ even for exactly face-on, axisymmetric galaxies. An edge-on model can in principle fit these $v\ne0$, whereas face-on models necessarily obey $v \equiv 0$. Thus, everything else fitting equally well, the contribution of noise in $v$ to the $\chi^2$ would be smaller in edge-on than in face-on models. Since we have no clear candidate face-on galaxy in our sample we do not expect this issue to be relevant to the Coma sample, however. ![Top: histogram of apparent short-to-long axis ratios at ${r_\mathrm{eff}}$. Bottom: intrinsic best-fit short-to-long axis ratio $\langle b/a \rangle$ (averaged over $r/{r_\mathrm{eff}}\in [0.5,2.5]$). Black/solid: whole sample; red/dashed: without S0s.[]{data-label="fig:shape"}](figure3.eps){width="84mm"} The third thing to note is that galaxies for our sample may be not at random inclinations. The sample is designed to complement earlier studies on round galaxies and we explicitly selected flattened, rotating ellipticals and S0s to be fitted. The distribution of apparent axis-ratios $(b/a)_e \equiv 1 - \epsilon_e$ (with $\epsilon_e$ from Tab. \[dattable\]) is shown in the upper panel of Fig. \[fig:shape\]. The sample exhibits a tail of highly flattened systems, which lacks, for example, in the distribution of bright galaxies with de-Vaucouleurs profiles in the Sloan Digital Sky Survey [@Vin05]. This tail is produced by the S0 galaxies in our sample and clearly shows that the sample as a whole is biased towards flattened systems. The ellipticity distribution of those galaxies that are classified as ellipticals in Tab. \[dattable\] (dashed line in Fig. \[fig:shape\]) is still shifted slightly towards higher ellipticities with respect to the bright ellipticals of @Vin05. In combination with the lack of round objects in our sample, this indicates that even our 11 ordinary ellipticals are slightly biased, but the sample is too small for a definite appraisal. Fourth, even assuming our sample galaxies are at random inclinations and that the first two just discussed points are irrelevant (regularisation and noise) and that inclinations can be reconstructed uniquely from ideal data with ideal models then we still could be faced with a slight bias in our models: as it has been described in Sec. \[subsec:depro\] we do not probe a fine grid in inclinations but look for extreme cases. Our models provide for each galaxy only the choice between edge-on or intrinsically E5/E7, respectively. Since an intrinsic E5/E7 shape is a rather extreme assumption this might drive the modelling towards the edge-on option as well. {width="164mm"} The distribution of intrinsic axis ratios of the Coma models is shown in the lower panel of Fig. \[fig:shape\]. It peaks at $b/a = 0.8$, consistent with deprojections of the frequency function of elliptical galaxy apparent flattenings [@Tre96; @Vin05]. Compared with these studies, the distribution in the lower panel of Fig. \[fig:shape\] has relatively more galaxies on the flatter side of the peak and relatively fewer galaxies on the rounder side. Now, if the modelling would be subject to a strong bias towards $i\to90\degr$ then we would expect the opposite: an overestimation of intrinsically roundish galaxies. Thus, the lower panel of Fig. \[fig:shape\] argues against a strong modelling bias towards high inclinations. However, the argument is not conclusive, because the sample itself maybe biased against apparently round galaxies. This could partly compensate for an inclination bias in the modelling. In conclusion, although there might be a slight inclination bias in the modelling and/or the sample galaxies, Fig. \[fig:shape\] reveals that either this bias is not very strong, or that modelling and sample biases roughly counterbalance each other. Luminous and dark matter distribution {#sec:mass} ===================================== Now we discuss the distribution of luminous and dark matter in the Coma models. {width="164mm"} {width="164mm"} Does mass follow light? {#subsec:ml} ----------------------- According to Tab. \[modtable\], the best-fit model of each galaxy contains a dark matter halo. Column (14) of the table states that eight galaxies have at least a two sigma detection of a dark matter halo (GMP0282, GMP0756, GMP1176, GMP1750, GMP2440, GMP4928, GMP5568 and GMP5975). The best fitting models with and without dark matter, respectively, are compared to the kinematic data in App. \[sec:fits\]. From this comparison it follows that models without a halo obviously fail to reproduce the kinematic data for the above mentioned galaxies. The evidence for dark matter thereby comes mostly from the innermost and outermost kinematic data points: without dark matter, the energy of the models is too low, when compared to the data at large radii and too high, when compared to the central data (for example illustrated by the dispersion profile of GMP5975). The reason for the differences at small radii is likely that part of the missing outer mass in models without a halo is compensated for by a larger mass-to-light ratio (cf. Sec. \[subsec:mtols\]). This, in turn, causes an increase of the central mass and central velocity dispersion, respectively. In GMP1750 the dispersion profile without dark matter fits systematically worse than the one with dark matter at all radii. Concerning GMP5568, the dispersion along one of the offset-slits is particularly large, larger than in all other slits. It is not entirely clear if these large dispersions are real. If not, then they erroneously increase the evidence for dark matter. However, because the error-bars of the corresponding data points are very large, these dispersions are not the dominant driver for the dark halo detection in GMP5568. In the rest of the sample the detection of dark matter – if at all – is more of statistical nature. Models with and without dark matter for GMP0144 and GMP2921 differ in a similar fashion as those of GMP1750. For GMP3510, GMP3958 and GMP5279 small differences between models with and without dark matter can be seen at the last kinematic data points, but the formal significance for dark matter is less than 90 percent. We believe that the statistical significance for dark matter in these five cases is underestimated due to our very conservative error estimates. In the four remaining objects GMP1990, GMP2417, GMP3329 and GMP3792 the evidence for dark matter is generally low. Poor evidence for dark matter in GMP3329 maybe related to the overall poor constraints that the measured kinematics put on its mass-to-light ratio (cf. Sec. \[subsec:conf\]). GMP1990 is consistent with the assumption that mass follows light. Our sample thus roughly divides into three categories: (1) galaxies that are clearly inconsistent with a constant mass-to-light ratio (8 galaxies out of 17). (2) Cases where models with and without a dark halo differ systematically, but where the formal evidence for a dark halo is less than two sigma (5 galaxies). In these cases we expect that our very conservative error bars lead to an underestimation of the dark matter detection. (3) Systems in which the evidence for dark matter is generally weak (4 galaxies). Models and data of some galaxies with a clear dark halo detection still differ systematically in the outer parts (e.g. GMP0756, GMP1176 and GMP5975). Decreasing the weight on regularisation reduces these differences. However, according to the discussion of Sec. \[subsec:regula\] we do not expect that we have significantly over-regularised our models. Even in case we would have, the derived halos of these systems do not depend much on the choice of the regularisation parameter, such that conclusions upon the masses of these galaxies are robust (cf. Sec. \[sec:regula\]). It might be possible that differences between models and data are related to changes in the stellar population, that other our adopted halo profiles are not appropriate for these systems, or that the corresponding outer regions are not in equilibrium or not axisymmetric. We plan further investigations of these topics for future publications. Circular velocity curves {#rotkurven} ------------------------ Fig. \[vcirccomparison\] shows the best-fit circular velocity curves for the Coma galaxies. The shapes of the curves vary from cases with two local extrema (e.g. GMP5279) to a case of monotonic increase (GMP3958). The sample provides examples of rising as well as falling outer circular-velocity curves. Flattened, rotating galaxies have fairly flat circular velocity curves beyond the central rise (less than 10 percent variation up to the last kinematic data point in GMP0282, GMP0756, GMP2417, GMP3510, GMP1176 and GMP5975). Mass densities and dark matter fractions {#subsec:rho} ---------------------------------------- Spherically averaged density profiles of all Coma galaxies are surveyed in the top panel of Fig. \[denscomparison\]. The luminosity distribution of most galaxies shows a power-law core that smoothly joins with the outer light-distribution. Towards the most luminous galaxies the central slope of the luminosity distribution flattens out (GMP3329 and GMP2921). The inner breaks in the light profiles of GMP1990 ($r \approx 2 \, {\mathrm{kpc}}$), GMP2417 ($r \approx 1 \, {\mathrm{kpc}}$) and GMP2440 ($r \approx 0.3 \, {\mathrm{kpc}}$) originate from prominent dust features. The central regions are dominated by luminous matter. Halo densities in the centre are at least one order of magnitude lower than stellar mass densities – independently of the halo profile being either of the logarithmic or of the NFW type. The radius where dark and luminous mass densities equalise is inside the kinematically sampled region of each galaxy. In some galaxies the transition from the luminous inner parts to the dark matter dominated outskirts is very smooth (for example GMP0756, GMP1176, GMP5975). The corresponding dark halo components are relatively concentrated (NFW halos) and the circular velocity curves are fairly flat. In other galaxies the transition is marked by a break in the total mass profile and a dip in ${v_\mathrm{circ}}$ (for example GMP0282). We will come back to these different circular velocity curve shapes in Thomas et al. (2007a, in preparation). Dark matter fractions of the best-fit Coma models are shown in the bottom panel of Fig. \[denscomparison\]. In most galaxies 10 to 50 percent of the mass inside ${r_\mathrm{eff}}$ is dark. NFW or logarithmic halos? {#subsec:haloprof} ------------------------- The evidence for or against logarithmic and NFW halos is summarised in column (13) of Tab. \[modtable\]. The majority of best-fit models (13 out of 17) is obtained with logarithmic halos. However, the significance for one or the other halo profile providing the better fit is in most cases low. As already implied by the approximate flatness of the circular velocity curves, the overall effect of the halo component is to keep the outer logarithmic slope of the total mass density around $-2$ (i.e. the case of an exactly constant ${v_\mathrm{circ}}$). This can be achieved either with logarithmic halos (asymptotically) or with suitably scaled NFW halos (over a finite radial range around the scaling radius). Differences in the profiles’ inner slopes seem to play a minor role, perhaps because the inner mass profile turns out to closely follow the light profile. If elliptical galaxy circular velocity curves are roughly flat over a very extended radial range, then NFW fits will break down at some point. With the data at hand no clear decision in favour of one of the two profiles can be made. Most of the best-fit NFW-models are obtained with a flattened halo. Since we cannot significantly discriminate between NFW and LOG-halos, firm statements about the flattening of the halos are not possible. {width="164mm"} {width="164mm"} The fraction of mass that follows the light {#subsec:mtols} ------------------------------------------- From Tab. \[modtable\] it can be taken that mass-to-light ratios ${\Upsilon_\mathrm{SC}}$ of self-consistent models are on average ($17 \pm 10$) percent larger than those of models with a dark matter halo. Concerning the difference between logarithmic and NFW halos, the best-fit ${\Upsilon_\mathrm{NFW}}$ is generally equal or lower than the corresponding ${\Upsilon_\mathrm{LOG}}$. Velocity Anisotropy {#sec:aniso} =================== Having explored the mass structure of the models we next focus on their dynamics. In this section we will consider the velocity anisotropies defined in equations (\[eq:bt\]) and (\[eq:bp\]), respectively. The discussion will be restricted to the minor-axis and major-axis bins of the Schwarzschild models, respectively. Each galaxy is covered by kinematical observations along at least these axes and the internal orbital structure is best constrained there. The polar region {#subsec:kinpolar} ---------------- Fig. \[anisominor\] surveys velocity anisotropy profiles along the intrinsic minor-axis of the Coma galaxy sample. According to axial symmetry ${\beta_\vartheta}\equiv {\beta_\varphi}$ holds directly on the symmetry axis. Hence, the upper and lower panel of Fig. \[anisominor\] are overall very similar. The minor-axis bins of the models, however, form a cone with opening angle $\Delta \vartheta=25 \degr$ around the $z$-axis. Thus, they include regions off the symmetry axis, where the equivalence between azimuthal and meridional dispersions does not hold, such that ${\beta_\vartheta}$ and ${\beta_\varphi}$ in Fig. \[anisominor\] are not identical. In Fig. \[anisominor\] the very central anisotropies should not be regarded as reliable. Firstly, because the central bins are affected from incomplete orbit sampling, resulting in artificially large azimuthal dispersions [@Tho04]. Secondly, for numerical reasons the innermost bin is not resolved in $\vartheta$, but averaged over all $\vartheta \in [0\degr, \, 90\degr]$. In the spatial region with kinematical data the Coma galaxies offer different degrees of minor-axis anisotropy, from strongly tangential (GMP5279) to moderately radial (GMP3792). Towards the centre ${\beta_\vartheta}\rightarrow 0$, while ${\beta_\varphi}$ becomes negative (most likely due to the incomplete orbit sampling). Going outward, many but not all galaxies exhibit a gradual change in dynamical structure, often in form of a minimum or maximum in $\beta$. Around the last data point most models are isotropic. The most radial system is GMP3792. {width="164mm"} {width="164mm"} The equatorial plane -------------------- Velocity anisotropy profiles in the equatorial plane are shown in Fig. \[anisomaj\]. Note that unlike along the minor-axis axial symmetry does not imply any relationship between ${\beta_\varphi}$ and ${\beta_\vartheta}$ at low latitudes. ##### **Meridional anisotropy.** {#meridional-anisotropy. .unnumbered} Contrasting the situation around the poles, almost no galaxy exhibits tangential anisotropy ${\beta_\vartheta}<0$. Apart from the peculiar object GMP3329 (cf. Sec. \[subsec:conf\]) all galaxies have ${\beta_\vartheta}>0$ over the kinematically sampled radial range. The average ${\beta_\vartheta}$ turns out to be related to the intrinsic flattening of the galaxies (Thomas et al. 2007a, in preparation). Uncertainties on the intrinsic shape therefore propagate into uncertainties on ${\beta_\vartheta}$. As it has been stated in Sec. \[subsec:depro\], in many cases it is not possible to distinguish between different inclinations with high significance. Hence, intrinsic shapes are likewise poorly constrained and the uncertainties on ${\beta_\vartheta}$ become large. A typical example is GMP3792: the best-fit inclination is $i=60\degr$ and requires a relatively flattened intrinsic configuration with large ${\beta_\vartheta}$. However, models at higher as well as lower $i$ are within the 68 percent confidence region. Consequently, the shaded area includes also models with different flattening and ${\beta_\vartheta}$. ![Short-axis anisotropy against minor-axis $H_4$. The line in the upper panel shows a linear fit (quoted in the panel).[]{data-label="betah4min"}](figure8.eps){width="84mm"} ##### **Azimuthal anisotropy.** {#azimuthal-anisotropy. .unnumbered} More diversity than in ${\beta_\vartheta}$ is offered by azimuthal velocities. In GMP3792, for example, ${\beta_\varphi}<0$ suggests that the system may be composed of two flattened subsystems with low net angular momentum, causing large $\varphi$-motions. GMP3510, GMP3958 and GMP0144 are relatively isotropic ($\sigma_r \approx \sigma_\varphi$) over the kinematically sampled spatial region. GMP5279, instead, offers ${\beta_\vartheta}\approx {\beta_\varphi}>0$, implying $\sigma_r>\sigma_\varphi$ and $\sigma_r > \sigma_\vartheta$ over the region with data. Relations between anisotropy and observed kinematics ---------------------------------------------------- The intrinsic short-axis velocity anisotropies are closely related to the observed (local) $H_4$. This can be taken from Fig. \[betah4min\], where for each (projected) radius $R$ with a measurement of $H_4$ the local $H_4(R)$ is plotted against the internal anisotropy $\beta(r=R)$ at the same radius. Internal radii $r$ have not been corrected for inclination since most models are edge-on (cf. Sec. \[subsec:inclin\]). From the figure a tight correlation of ${\beta_\vartheta}$ with $H_4$ follows (quoted in the plot): the smaller $H_4$, the more tangentially anisotropic the model. A similar trend occurs between ${\beta_\varphi}$ and $H_4$ (lower panel). This reflects that ${\beta_\vartheta}\approx {\beta_\varphi}$ around the symmetry axis (see above). Comparable trends between $\beta$ and $H_4$ have also been found in spherical models (e.g. @Ger93 [@Mag01]). The similarity between spherical models on the one hand and the polar region of axisymmetric models on the other might be connected to the fact that in both cases ${\sigma_\varphi}= {\sigma_\vartheta}$. In other words, effectively there is only one degree of freedom in the stellar anisotropy (${\beta_\varphi}={\beta_\vartheta}$) and, if the potential is fixed, there must be a close relationship between $\beta$ and the shape of the LOSVD (as measured by $H_4$). Experiments with spherical models indicate that the dependency of $H_4$ on the potential is weaker than its variation with $\beta$ (e.g. @Ger93 [@Mag01]). If the same holds for axisymmetric potentials, then this would explain why along the polar axis of axisymmetric models $\beta$ depends in about the same way on $H_4$ as in spherical models. Note, however, that our Schwarzschild models provide many more internal degrees of freedom than the smooth spherical models considered by @Ger93 and @Mag01. This becomes apparent when the influence of regularisation on the fit is lowered and the scatter around the relation shown in Fig. \[betah4min\] increases (cf. Sec. \[kin:reg\]). In contrast to the polar region, no tight correlation between $H_4$ and velocity anisotropy is found around the equatorial plane (cf. Fig. \[betah4maj\]). This holds especially for ${\beta_\vartheta}$, whereas there is a slight trend of ${\beta_\varphi}$ to increase with $H_4$. For comparison, a linear fit is shown in the lower panel. A detailed investigation of the orbital structure will be presented in another paper (Thomas et al. 2007a, in preparation). ![As Fig. \[betah4min\], but for the major-axis.[]{data-label="betah4maj"}](figure9.eps){width="84mm"} {width="164mm"} {width="164mm"} Phase-space distribution function of the stars {#sec:dfstars} ============================================== A more fundamental quantity related to a stellar dynamical system than its anisotropy is its phase-space distribution function $f$. It describes the density of stars in phase-space and offers the most detailed and comprehensive view on its dynamical state. For stationary systems the DF is a function of the (isolating) integrals of motion and, thus, constant along individual orbits (Jeans theorem; e.g. @Bin87). To be considered in axisymmetric potentials are the energy $E$, angular momentum $L_z$ along the axis of symmetry and, in most astrophysically relevant potentials, the so-called third integral $I_3$. To be physically meaningful the DF has to obey the further condition that it is positive everywhere. In Schwarzschild models the constancy of the DF along orbits is explicitly taken into account during the orbit integration. Its positive definiteness is guaranteed as long as the orbital weights $w_i$ are positive (cf. equation \[eq:fi\]). In other words, the very existence of our Schwarzschild models ensures that the luminous component of the model is stationary and physically meaningful (positive density). A detailed investigation of the full dependency of the DF on all integrals of motion and its connection to stellar population properties will be the subject of another publication (Thomas et al. 2007a, in preparation). Here we only consider some general properties of the DF. For this purpose it is convenient to define a mean orbital radius $${\langle r_\mathrm{orb} \rangle}_i \equiv \sum_k \frac{\Delta t^k_i}{T_i} r^k_i,$$ where $T_i$ is the total integration time of orbit $i$ and $r^k_i$ is its radius at time-step $k$ (lasting $\Delta t^k_i$). In rough terms ${\langle r_\mathrm{orb} \rangle}$ can be interpreted as a measure of the orbital binding energy. Fig. \[dflum\] surveys the DFs of all 17 Coma galaxies. Each dot represents the phase-space density of a single orbit. To roughly trace the angular-momentum dependency of the DF prograde orbits with $L_z>0$ are highlighted in the top panel and retrograde orbits ($L_z<0$) are highlighted in the bottom one. The figure shows that with decreasing galaxy mass differences between prograde and retrograde orbits in phase-space become more significant. This partly reflects an increasing importance of rotation in lower mass galaxies of our sample. Often, the highest phase-space densities of prograde orbits are nearly constant over some radial region (e.g. GMP3958 around ${r_\mathrm{eff}}$, GMP2440 inside $r \la {r_\mathrm{eff}}$). In many, but not all, rotating galaxies the dominance of prograde orbits comes along with a strong depression of retrograde orbits. In the outer parts of GMP5975, for example, retrograde orbits have phase-space densities up to 10 orders of magnitude smaller than prograde orbits. Such low-density orbits can actually be regarded as being absent in the models [@Tho05]. Concerning the significance of this depopulation it is interesting to note that in case of GMP5975 it was originally found in models based on major and minor-axis data only [@Tho05], but remains almost unchanged in our new models including additional kinematical information along a diagonal axis (cf. Sec. \[sec:obs\]). The depopulation of retrograde orbits cannot be a general modelling artifact since it does not appear in all rotating galaxies. A counter-example is the least-massive object, GMP3958: it rotates but does not show a strong depression of retrograde orbits in its outer parts. One galaxy of the sample, GMP5568, hosts a counter-rotating central disk [@Meh98], which shows up by a dominance of retrograde orbits around ${\langle r_\mathrm{orb} \rangle}\approx 1\,{\mathrm{kpc}}$ and $w/V \approx 10 \, M_\odot/\mathrm{pc}^3/(\mathrm{km}/\mathrm{s})^3$ in Fig. \[dflum\]. In most rotating galaxies the majority of retrograde orbits follows approximately a power-law like straight line (e.g. GMP3958, GMP2417). Retrograde orbits in GMP1990 follow such a power-law like distribution only outside ${r_\mathrm{eff}}$. Near the centre of the galaxy retrograde orbits exhibit some excess density, compared to a power-law extrapolation of the behaviour outside ${r_\mathrm{eff}}$. This may reflect a faint inner counter-rotating sub-component, although unlike in GMP5568, prograde orbits always dominate in GMP1990 (and the observed sense of rotation is the same at all radii). In some galaxy models orbital phase-space densities are more spread than in others: for example the DFs of GMP3329 and GMP5568 look particularly noisy. In case of GMP3329 its peculiar kinematics, already discussed in Sec. \[subsec:ml\], may be responsible for the distorted phase-space distribution of orbits. {width="164mm"} Phase-space distribution function of dark matter {#sec:dfhalo} ================================================ So far we have only considered the phase-space distribution function of the luminous component of our models. To ensure that these models are physically meaningful we also need the dark halos to be supported by an everywhere positive phase-space distribution function. Without the baryons present, the existence of DFs for our halo profiles is known. In case of NFW-halos it follows trivially from the fact that they arise in $N$-body simulations and DFs for LOG-halos have been constructed explicitly by @Eva93. With a significant contribution of baryons (or any other component) to the overall gravitational potential, the existence of these DFs is no longer guaranteed, however. For example, if a cored halo (central logarithmic density slope $\gamma =0$) is embedded in a cuspy baryonic component ($\gamma = -1$) and if the core radius exceeds a critical limit around $r_C \ga 3 \, {r_\mathrm{eff}}$, then central phase-space densities become negative in isotropic or radially anisotropic systems [@Cio92; @Cio99]. In contrast, a cuspy halo can always be supported [@Cio96]. Thus, the existence of a plausible halo DF for our LOG-halos, which often have core radii near or beyond the critical limit (cf. Tabs. \[dattable\] and \[modtable\]) is not obvious. The main goal of this section is to investigate whether we can find a positive definite DF for all our best-fit models, or whether the phase-space analysis rules out some of our halo profiles. Construction and existence of the halo distribution function ------------------------------------------------------------ Our modelling machinery allows to construct a DF for dark matter in an analogous way as for luminous matter: by solving equation (\[maxs\]) with an orbit superposition. The only difference to the calculation of the luminous matter orbit superposition is that now the dark matter density profile is used as the boundary condition and not the deprojected light-profile $\nu$. In addition, since we lack of any kinematic information about the hypothetical halo constituents we set $\alpha = 0$ in equation (\[maxs\]) and maximise the entropy of the orbits. For our goal of finding at least one positive definite DF this does not imply any loss of generality. Within the numerical resolution of our orbit models we find indeed orbit superpositions with positive orbital weights $w_i>0$ that allow to reconstruct the halo density profile in each case. The corresponding DFs are stationary by construction and positive everywhere. The fact that we even find positive definite DFs for those LOG-halos that are beyond the above cited critical core-radius can have several reasons: our models are slightly different from the ones used in @Cio99 (different radial run in outer parts, baryonic component flattened in our case). In addition, our orbit superpositions can well produce tangential anisotropy, which helps to maintain a positive DF. Finally, our orbit models have a finite resolution. We cannot exclude that reconstructing the halo density with higher resolution would force some orbital weights to become negative. Differences between NFW and LOG-halos ------------------------------------- Apart from the mere existence, there are significant differences in the derived DFs, however. This can be taken from plots of the halo DFs in Fig. \[dfdark\]. NFW-halo DFs (GMP2921, GMP0144, GMP1176 and GMP5975) are monotonic with respect to ${\langle r_\mathrm{orb} \rangle}$ and regular. The high degree of regularity (compared to the corresponding luminous matter DFs) reflects the maximisation of entropy, whereas noise in the stellar kinematics and sub-structuring of stars in phase-space tend to broaden the stellar DF (cf. Sec. \[sec:dfstars\]) . DFs of LOG-halos exhibit a drop of central phase-space density, as predicted by @Cio99. In GMP3329, where the halo is very concentrated, the drop is rather gentle. With increasing core-radius the drop becomes more substantial. In addition, the noise in the DF increases with increasing core radius. Such disturbances in the DF, even though we maximise the entropy, indicate that a fine-tuning of the orbits is necessary for large core-radii to be supported by a positive-definite DF. This, and the non-monotonic dependency of orbital phase-space densities on ${\langle r_\mathrm{orb} \rangle}$ could imply that the corresponding DFs and, thus, also the spatial density profiles are unstable. If this is indeed the case, then the phase-space analysis would provide a strong argument against large-cored halo profiles, independent from the kinematic fits. Of course, the halo DFs shown in Fig. \[dfdark\] are not unique, as stated above: the models have no access to the orbit distribution in the halo, apart from those constraints coming from the shape of the density-profile alone. Details of the DFs in Fig. \[dfdark\] are therefore physically meaningless. However, that the entropy maximisation does not yield smooth DFs for LOG-halos with large cores suggests that – independent from our ignorance about the details of the orbit distribution – smooth dark matter DFs in the corresponding baryonic potential wells are unlikely. In any case, a systematic stability analysis is out of the scope of this paper. What we can conclude here is, that based on the kinematic fits and based on the mere existence of a positive definite halo DF, we cannot rule out one or the other halo profile with our orbit models. Shape and structure of LOG-halo DFs make them the less likely option, however. Central dark matter density --------------------------- According to Sec. \[subsec:rho\] the central density of dark matter is often orders of magnitudes lower than the luminous mass density. This suggests that it is in many cases weakly constrained. Could it be even lower than in LOG-halos? According to the above phase-space analysis this seems unlikely, because lower central densities would most likely augment the disturbances of the halo DF. Thus, the central spatial dark matter densities of our LOG-halos are likely lower limits to the true central dark matter densities. Central dark matter phase-space density --------------------------------------- From the dark matter DFs of Fig. \[dfdark\] we have also calculated a mean central phase-space density $$\label{favdef} {f_h}\equiv \left( \frac{\sum w_i}{\sum V_i} \right)_{0.1},$$ where the sums on the right hand side are intended to comprise all orbits with ${\langle r_\mathrm{orb} \rangle}<0.1 \, {r_\mathrm{eff}}$. These central phase-space densities are flagged in Fig. \[dfdark\] by the large symbols. Comparison with Fig. \[dflum\] shows that central dark matter phase-space densities are particularly low in systems with strong rotation. Exceptions are GMP1176 and GMP5975 with their NFW halos. ![Best-fit ${\Upsilon_\mathrm{dyn}}$ versus regularisation parameter $\alpha$. Dotted line: $\alpha=0.02$, the regularisation adopted for the best-fit models; shaded: one sigma confidence region for $\alpha=0.02$.[]{data-label="reg:star"}](figure12.eps){width="84mm"} The uncertainty in the dark-halo DF related to our ignorance about dark matter kinematics of course affects ${f_h}$. According to our above discussion the drop in LOG-halo DFs seems a feature connected to the density profile, though, and we do not expect that reasonably isotropic or radially anisotropic LOG-halo DFs exist for which ${f_h}$ increases by orders of magnitude. Thus, although ${f_h}$ is subject to many uncertainties, it is likely good as an order of magnitude estimation for the central dark matter phase-space density connected with the mass decomposition made in Sec. \[subsec:mass\]. Regularisation {#sec:regula} ============== As it has been discussed in Sec. \[subsec:regula\] the same regularisation $\alpha = 0.02$ is adopted for all Coma galaxies. In the following we will discuss the dependency of our modelling results on the choice of $\alpha$. ![Dark matter fractions at $0.1 \, {r_\mathrm{eff}}$ (black/solid), $0.5 \, {r_\mathrm{eff}}$ (red/short-dashed) and $1.0 \, {r_\mathrm{eff}}$ (blue/long-dashed) versus regularisation parameter $\alpha$. Vertical dotted lines: $\alpha=0.02$.[]{data-label="reg:dmfrac"}](figure13.eps){width="84mm"} ![Best-fit circular velocity curves for different values of the regularisation parameter: $\alpha = 0.001$ (blue, solid), $\alpha = 2.7$ (blue, dotted) and $\alpha=0.02$ (black, solid).[]{data-label="reg:vcirc"}](figure14.eps){width="84mm"} {width="81mm"} {width="81mm"} {width="81mm"} {width="81mm"} The influence of regularisation on model masses {#mass:reg} ----------------------------------------------- Fig. \[reg:star\] surveys the best-fit stellar mass-to-light ratios ${\Upsilon_\mathrm{dyn}}$ over the regularisation interval $\alpha \in [10^{-5},3]$. Two conclusions can be drawn from the figure. First, no systematic trend of ${\Upsilon_\mathrm{dyn}}$ with $\alpha$ is noticeable. In GMP1990, for example, ${\Upsilon_\mathrm{dyn}}$ increases with $\alpha$, while in GMP1750 it decreases. Second, in most of the sample galaxies the weight on regularisation has barely any effect on ${\Upsilon_\mathrm{dyn}}$ (e.g. GMP5568, GMP0282, GMP0144, GMP5279, GMP2417, GMP3510, GMP1176, GMP5975, GMP2440). The best-fit dark matter fractions at three representative radii are shown in Fig. \[reg:dmfrac\] as a function of $\alpha$. As could have been expected, the dark matter fraction and ${\Upsilon_\mathrm{dyn}}$ are correlated: in most cases where ${\Upsilon_\mathrm{dyn}}$, say, increases, the dark matter fraction decreases (and vice versa). Since there is no systematic trend of ${\Upsilon_\mathrm{dyn}}$ with $\alpha$ it follows that there is also no systematic trend of the dark matter fraction with $\alpha$. Moreover, the variation of dark matter fractions with $\alpha$ is within the quoted error budget of Fig. \[denscomparison\]. Finally, Fig. \[reg:vcirc\] shows circular velocity curves for three different values of $\alpha$. The influence of $\alpha$ on the shape of the circular velocity curve is weak. Only in a few systems the general shape of the circular velocity curve changes with $\alpha$ (for example GMP3510). These changes occur mostly outside the region covered by kinematic data, however. The influence of regularisation on model kinematics {#kin:reg} --------------------------------------------------- Now to the influence of $\alpha$ on the derived velocity anisotropies: the left panels of Fig. \[reg:kin\] show best-fit meridional and azimuthal velocity anisotropies at three representative radii as a function of $\alpha$. The figures indicate that maximum entropy fits ($\alpha \rightarrow 0$) yield isotropy along the minor-axis. Lowering the weight on regularisation generally increases the anisotropy – the absolute value of $\beta$ – in the models, as could have been expected. There is no specific trend of $\beta$ with $\alpha$: some systems gain more tangential anisotropy with increasing $\alpha$ (for example GMP5279), while others become more radial (for example GMP0756). In most cases the dependency of $\beta$ on $\alpha$ is monotonic and $\beta(\alpha)$ does not change sign. In other words, whether or not a galaxy model is tangentially or radially anisotropic does not depend on $\alpha$. Only the exact degree of anisotropy changes with $\alpha$. Major-axis velocity anisotropies are plotted on the right hand side of Fig. \[reg:kin\]. In contrast to the minor-axis case there is no trend of ${\beta_\vartheta}\to 0$ for $\alpha \to 0$. Variations of intrinsic velocity anisotropies with $\alpha$ are slightly weaker along the equator than they are along the minor-axis. Since the trend of $\beta$ with $\alpha$ is again monotonic and sign preserving in most cases, the general property of a galaxy to be radially or tangentially anisotropic is insensitive to the particular choice of $\alpha$. From the top-left panel of Fig. \[reg:kin\] it is clear that the relation between (the $\alpha$-dependent) anisotropy and (the $\alpha$-independent) $H_4$ described in Sec. \[sec:aniso\] must change with different amounts of regularisation in the models. This is illustrated in Fig. \[betah4minNR\], which repeats the upper panel of Fig. \[betah4min\] for two different values of $\alpha$. The consequences of stronger regularisation are displayed in the top panel, while weaker regularisation leads to the distribution shown in the bottom panel. For comparison the linear fit from Fig. \[betah4min\] is shown by the dashed line. Increasing the weight on regularisation makes the correlation tighter, but does not alter the slope. With less regularisation the scatter increases, because models start to fit the noise in the data. The mean relation is in any case robust against different choices of $\alpha$. Note, that a correlation between an [*intrinsic*]{} property (like ${\beta_\vartheta}$) and an [*observed*]{} one (like $H_4$) cannot be the result of the entropy maximisation. In fact, for $\alpha = 0$ the meridional anisotropy along the minor-axis vanishes (${\beta_\vartheta}\approx 0$) and the relation breaks down. Concluding, the dynamical structure of the fits depends more strongly on the choice of $\alpha$ than the mass distribution does (cf. Sec. \[mass:reg\]). Thereby no clear trend of velocity anisotropies with $\alpha$ is noticeable. The monotonic behaviour of $\beta$ with respect to $\alpha$ in most cases ensures that the general property of a model to be radially or tangentially anisotropic is independent of the choice of $\alpha$. To give an example of how $\alpha$ influences the orbit distribution Fig. \[histo\] shows the histogram of orbital weights in the best-fit mass-model of GMP5975 for three different $\alpha$. As can be seen the model at large $\alpha$ (weak regularisation) is dominated by a few orbits that carry almost the entire light. All other orbits are essentially depopulated in the model (only orbits with weights $\log w>-15$ are included in the plot). The model at $\alpha=0.02$ is still relatively close to the maximum entropy distribution. ![As upper panel of Fig. \[betah4min\], but for different values of the regularisation parameter $\alpha$ (indicated in the panels). Dashed: linear fit for $\alpha=0.02$ (cf. Fig. \[betah4min\]).[]{data-label="betah4minNR"}](figure16.eps){width="84mm"} ![Distribution of orbital weights for three differently regularised models (as indicated in the plot). All models are calculated in the best-fitting mass distribution of GMP5975.[]{data-label="histo"}](figure17.eps){width="84mm"} Discussion and summary {#sec:sum} ====================== We have surveyed axisymmetric Schwarzschild models for a sample of 17 Coma early-type galaxies. The models are fitted to measurements of line-of-sight velocity distributions out to $1-4\,{r_\mathrm{eff}}$. Stellar mass-to-light ratios and dark halo parameters are determined for two parameterised halo families with different inner and outer density slopes. The models are regularised towards maximum entropy. Luminous and dark matter ------------------------ In each galaxy, models with dark matter fit better than models without. A constant mass-to-light ratio is significantly ruled out in about half of the sample (eight galaxies where dark matter is detected on at least the 95 percent confidence level). In four galaxies the case for dark matter is weak. The mass distribution in one of these systems (GMP1990) is in fact consistent to follow the light. Five galaxies are intermediate cases where the formal evidence for dark matter is low, although fits with and without dark matter differ systematically in either their radial dispersion profiles or in their outermost LOSVDs. We believe that the low signal for dark matter in these systems is partly due to our very conservative treatment of the error bars. Our inferences about dark matter are based on the mass decomposition provided by equation (\[rhorho\]). In particular, we have assumed that the stellar mass-to-light ratio is constant throughout the galaxy. In this context, GMP1990 is not surrounded by significant amounts of dark matter. However, before finally concluding upon dark matter in our galaxies a detailed comparison to independent estimates of the stellar mass is required (Thomas et al. 2007a, in preparation). For example, GMP1990 has a large mass-to-light ratio $\Upsilon = 10.0$ ($R_C$-band). If the actual stellar mass can only account for a fraction of it, then our result does not argue against dark matter in this galaxy, but merely implies that dark matter follows closely the light of the system. Constant mass-to-light ratios have been reported to be consistent with planetary nebulae kinematics in the outskirts of three roundish objects (spherical modelling; @R03). Also some of the round and non-rotating ellipticals of @Kr00 are consistent with the mass distribution following the light distribution. Many of these latter systems lack of kinematic data beyond ${r_\mathrm{eff}}$, however. In addition to the related uncertainties for the outer dark halo, spherical modelling of round galaxies generally suffers from the ambiguity related to the flattening along the line-of-sight. In this sense, GMP1990 is an interesting case, because its apparent flattening implies a viewing-angle close to $i=90\degr$ and the kinematic data extend relatively far out ($3 \, {r_\mathrm{eff}}$). Best-fit dark matter halos are in 4 out of 17 cases of the NFW-type and in all other cases logarithmic. Differences in the goodness-of-fit based on one or the other halo family are marginal in most cases. Central dark matter densities are at least one to two orders of magnitude lower than the corresponding mass densities in stars. Between 10 and 50 percent of the mass inside the half-light radius ${r_\mathrm{eff}}$ is formally dark in most Coma galaxies. These dark matter fractions are in general agreement with earlier results of dark matter modelling in round, non-rotating ellipticals [@G01] as well as with the analysis of cold gas kinematics [@Ber93; @Oos02], hot halo gas [@Loe99; @Fuk06; @Hum06] and strong lensing studies [@Kee01; @Tre04]. @Cap05 concluded for similar dark matter fractions in the SAURON-ellipticals (although in their models it is assumed that mass follows light). The combination of luminous and dark matter results in circular velocity curves of various shapes: some galaxies have outer decreasing ${v_\mathrm{circ}}$ while others show an indication for a dip in ${v_\mathrm{circ}}$ around $\approx 10 \, {\mathrm{kpc}}$ and a subsequent increase of ${v_\mathrm{circ}}$ towards larger radii. We cannot easily quantify the significance of this dip. Its appearance close to the outermost kinematic radius is suspicious to reflect a modelling artifact. However, in contrast to the orbital structure, whose reconstruction becomes uncertain around (and beyond) the last kinematic data point [@Kra05; @Tho05], the mass reconstruction in these regions is more robust [@Tho05]. In addition, the dip does not appear in all galaxies, suggesting that it is related to some observable property of the corresponding objects. In any case, it is interesting to note that similar dips are also indicated in temperature profiles of elliptical galaxy X-ray halos [@Fuk06] and can also been seen in spherical models of some round galaxies [@G01]. More extended kinematic data sets may in the future allow to better constrain the outer shape of the circular velocity curve. In rotating systems the circular velocity is fairly constant over the observationally sampled radial region (10 percent fractional variation). Similarly flat circular velocity curves have also been inferred from stellar kinematics of round systems (spherical modelling; @G01 [@Mag01]) and from strong gravitational lensing [@Kop06]. To the resolution of our orbit models all dark halos are supported by at least one positive definite phase-space distribution function. In case of NFW-halos smooth DFs can be constructed, but for LOG-halos with large core radii even the maximisation of orbital entropy does not yield smooth DFs. It is not obvious whether the corresponding spatial density profiles are stable or not. Further modelling is required to investigate whether phase-space arguments can be used to rule out logarithmic halos in elliptical galaxies. Kinematics ---------- With decreasing total mass the influence of rotation on the stellar phase-space distribution increases. In many galaxies rotation arises by an overpopulation of prograde orbits and a simultaneous underpopulation of retrograde orbits. At least one system lacks of the depopulation of retrograde orbits, proving that it is not a general artifact of our modelling approach. Some galaxies show strong tangential anisotropy along the minor-axis. This derives from low minor-axis $H_4$-measurements, because observed $H_4$ and modelled orbital anisotropy along the minor-axis turn out to be correlated. Slope and zero-point of this correlation are largely independent of regularisation, but – to some degree – stronger regularisation tightens the relation. Such a relation between an intrinsic quantity on the one hand and an observed one on the other cannot originate from the entropy maximisation alone. Along the major-axis, a slight tendency of increasing ${\beta_\varphi}$ with increasing $H_4$ is noticeable, but with much larger scatter than along the minor-axis. Only one system shows indication of tangential anisotropy (${\beta_\vartheta}<0$), all other galaxies are isotropic or mildly radially anisotropic (${\beta_\vartheta}> 0$, ${\beta_\varphi}\ga 0$). Radial anisotropy also appears characteristic for spherical models of round galaxies [@Kr00; @G01]. A suppression of vertical energy (corresponding to ${\beta_\vartheta}> 0$) has recently been reported for SAURON ellipticals [@Cap07]. We plan a detailed investigation of the orbital structure of the Coma galaxies for the future. Regularisation {#regularisation} -------------- Stellar mass-to-light ratios, dark matter fractions and the shape of circular velocity curves turn out to be robust against different choices of the regularisation parameter $\alpha$. The strongest effect $\alpha$ has is on the reconstructed anisotropies: their absolute values tend to increase if the weight on regularisation constraints is lowered. At a fixed radius the dependency of anisotropy on $\alpha$ is mostly monotonic, such that the general quality of a galaxy to be radially or tangentially anisotropic, respectively, is independent of $\alpha$. What changes instead is the actual amount of anisotropy. Outlook ------- Detailed investigations of luminous and dark matter scaling relations, of stellar population properties and their connection to the phase-space distribution of orbits are in preparation. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Eric Emsellem for his constructive referee report that helped to improve the presentation. JT acknowledges financial support by the Sonderforschungsbereich 375 “Astro-Teilchenphysik” of the Deutsche Forschungsgemeinschaft. EMC receives support from the grant PRIN2005/32 by Istituto Nazionale di Astrofisica (INAF) and from the grant CPDA068415/06 by the Padua University. Support for Program number HST-GO-10884.0-A was provided by NASA through a grant from the Space Telescope Science Institute which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. 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Lines: best-fit deprojection (red) and its edge-on reprojection (blue). Lower panel: stellar kinematics along major (left/red) and minor axis (right/blue); filled and open circles refer to the two sides of the galaxy; dotted: best-fit model without dark matter.[]{data-label="isoplotgh3329"}](figureA1a.eps "fig:"){width="90mm"} ![Upper panel: Joint ground-based and HST photometry of GMP3329/NGC 4874. Lines: best-fit deprojection (red) and its edge-on reprojection (blue). Lower panel: stellar kinematics along major (left/red) and minor axis (right/blue); filled and open circles refer to the two sides of the galaxy; dotted: best-fit model without dark matter.[]{data-label="isoplotgh3329"}](figureA1b.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP2921/NGC 4889.[]{data-label="isoplotgh2921"}](figureA2a.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP2921/NGC 4889.[]{data-label="isoplotgh2921"}](figureA2b.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP5568/NGC 4816. Green/third column: offset to major-axis ${r_\mathrm{eff}}/4$; magenta/fourth column: offset to major-axis ${r_\mathrm{eff}}/20$.[]{data-label="isoplotgh5568"}](figureA3a.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP5568/NGC 4816. Green/third column: offset to major-axis ${r_\mathrm{eff}}/4$; magenta/fourth column: offset to major-axis ${r_\mathrm{eff}}/20$.[]{data-label="isoplotgh5568"}](figureA3b.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP4928/NGC 4839; green/third column: diagonal axis.[]{data-label="isoplotgh4928"}](figureA4a.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP4928/NGC 4839; green/third column: diagonal axis.[]{data-label="isoplotgh4928"}](figureA4b.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP0282/NGC 4952.[]{data-label="isoplotgh0282"}](figureA5a.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP0282/NGC 4952.[]{data-label="isoplotgh0282"}](figureA5b.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP0144/NGC 4957.[]{data-label="isoplotgh0144"}](figureA6a.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP0144/NGC 4957.[]{data-label="isoplotgh0144"}](figureA6b.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP5279/NGC 4827.[]{data-label="isoplotgh5279"}](figureA7a.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP5279/NGC 4827.[]{data-label="isoplotgh5279"}](figureA7b.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP1990/IC 843; green/third column: offset to major-axis ${r_\mathrm{eff}}/3$.[]{data-label="isoplotgh1990"}](figureA8a.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP1990/IC 843; green/third column: offset to major-axis ${r_\mathrm{eff}}/3$.[]{data-label="isoplotgh1990"}](figureA8b.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP1750/NGC 4926.[]{data-label="isoplotgh1750"}](figureA9a.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP1750/NGC 4926.[]{data-label="isoplotgh1750"}](figureA9b.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP3792/NGC 4860.[]{data-label="isoplotgh3792"}](figureA10a.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP3792/NGC 4860.[]{data-label="isoplotgh3792"}](figureA10b.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP0756/NGC 4944; green/third column: offset to major-axis ${r_\mathrm{eff}}/2$ (the two outermost $H_4<-0.1$ are omitted in the plot).[]{data-label="isoplotgh0756"}](figureA11a.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP0756/NGC 4944; green/third column: offset to major-axis ${r_\mathrm{eff}}/2$ (the two outermost $H_4<-0.1$ are omitted in the plot).[]{data-label="isoplotgh0756"}](figureA11b.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP2417/NGC 4908; green/third column: offset to major-axis ${r_\mathrm{eff}}/2$.[]{data-label="isoplotgh2417"}](figureA12a.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP2417/NGC 4908; green/third column: offset to major-axis ${r_\mathrm{eff}}/2$.[]{data-label="isoplotgh2417"}](figureA12b.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP3510/NGC 4869.[]{data-label="isoplotgh3510"}](figureA13a.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP3510/NGC 4869.[]{data-label="isoplotgh3510"}](figureA13b.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh0144\], but for GMP1176/NGC 4931; green/third column: offset to major-axis ${r_\mathrm{eff}}/3$.[]{data-label="isoplotgh1176"}](figureA14a.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh0144\], but for GMP1176/NGC 4931; green/third column: offset to major-axis ${r_\mathrm{eff}}/3$.[]{data-label="isoplotgh1176"}](figureA14b.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP5975/NGC 4807; green/third column: diagonal axis.[]{data-label="isoplotgh5975"}](figureA15a.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP5975/NGC 4807; green/third column: diagonal axis.[]{data-label="isoplotgh5975"}](figureA15b.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP2440/IC 4045; green/third column: diagonal axis.[]{data-label="isoplotgh2440"}](figureA16a.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP2440/IC 4045; green/third column: diagonal axis.[]{data-label="isoplotgh2440"}](figureA16b.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP3958/IC 3947.[]{data-label="isoplotgh3958"}](figureA17a.eps "fig:"){width="90mm"} ![As Fig. \[isoplotgh3329\], but for GMP3958/IC 3947.[]{data-label="isoplotgh3958"}](figureA17b.eps "fig:"){width="90mm"} \[lastpage\] [^1]: E-mail: jthomas@mpe.mpg.de [^2]: In case of GMP1176 isophotal shape parameters up to $a_{12}$ are used to represent the isophotes appropriately (see also @Cor07).

--- abstract: 'The new theoretical input to the analysis of the experimental data of the CCFR collaboration for $F_3$ structure function of $\nu N$ deep inelastic scattering is considered. This input comes from the next-to-next-to-leading order corrections to the anomalous dimensions of the Mellin moments of the $F_3$ structure function and N$^3$LO corrections to the related coefficient funtions. The QCD scale parameter $\Lambda_{\overline{MS}}^{(4)}$ is extracted from higher-twist independent fits. The results obtained demonstrate the minimization of the influence of perturbative QCD contributions to the value of $\Lambda_{\overline{MS}}^{(4)}$.' --- [CERN-TH/2000-343]{}\ hep-ph/0012014 [**Application of new multiloop QCD input\ to the analysis of $xF_3$ data**]{}\ $^{(a)}$, [**G. Parente**]{}$^{(b,1)}$ and [**A.V. Sidorov**]{}$ ^{(c,2)}$\ (a) Theoretical Physics Division, CERN CH - 1211 Geneva 23 and\ Institute for Nuclear Research of the Academy of Sciences of Russia, 117312 Moscow, Russia\ (b) Department of Particle Physics, University of Santiago de Compostela,\ 15706 Santiago de Compostela, Spain\ (c) Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia [**ABSTRACT**]{} The new theoretical input to the analysis of the experimental data of the CCFR collaboration for $F_3$ structure function of $\nu N$ deep inelastic scattering is considered. This input comes from the next-to-next-to-leading order corrections to the anomalous dimensions of the Mellin moments of the $F_3$ structure function. The QCD scale $\Lambda_{\overline{MS}}^{(4)}$ is extracted from higher-twist independent fits. The results obtained demonstrate the minimization of the influence of perturbative QCD contributions to the value of $\Lambda_{\overline{MS}}^{(4)}$. [*Based on Contributed to the Proceedings of Quarks-2000 International Seminar, Pushkin, May 2000, Russia and of ACAT’2000 Workshop, Fermilab, October 2000, USA*]{} $^{1}$ Supported by Xunta de Galicia (PGIDT00PX20615PR) and CICYT (AEN99-0589-C02-02)\ $^{2}$ Supported by RFBI (Grants N 99-01-00091, 00-02-17432) and by INTAS call 2000 (project N587) CERN-TH/2000-343\ November 2000 [**Application of new multiloop QCD input\ to the analysis of $xF_3$ data** ]{} [**A.L. Kataev$^{a}$, G. Parente$^{b}$ and A.V. Sidorov$^{c}$**]{} [$^{a}$Theoretical Physics Division, CERN, CH-1211 Geneva, Switzerland and\ Institute for Nuclear Research of the Academy of Sciences of Rusia,\ 117312 Moscow, Russia\ $^{b}$Department of Particle Physics, University of Santiago de Compostela,\ 15706 Santiago de Compostela, Spain\ $^{c}$ Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,\ 141980 Dubna, Russia]{} Introduction ============ One of the most important current problems of symbolic perturbative QCD studies is the analytical evaluation of the next-to-next-to-leading order (NNLO) QCD corrections to the kernels of the DGLAP equations  [@DGLAP] for different structure functions of the deep-inelastic scattering (DIS) process. In this note we will apply the related information for the fixation of definite uncertainties of the NNLO analysis [@KKPS; @KPS1] of experimental data for $F_3$ structure function (SF) data of $\nu N$ DIS, provided by the CCFR collaboration [@CCFR] at the Fermilab Tevatron and present preliminary results of our improved fits which will be described elsewhere [@KPS2]. Methods of analysis of DIS data =============================== There are several methods of analysis of the experimental data of DIS in the high orders of perturbation theory. The traditional method is based on the solution of the DGLAP equation, which in the case of the $F_3$ SF has the following form: $$Q^2\frac{d}{dQ^2}F_3(x,Q^2)=\frac{1}{2}\int_x^{1}\frac{dy}{y} \bigg[V_{F_3}(y,A_s)+\beta(A_s)\frac{\partial{\rm ln}C_{F_3}(y,A_s)} {\partial A_s}\bigg]F_3\bigg(\frac{x}{y},Q^2\bigg)$$ where $A_s=\alpha_s/(4\pi)$, $\mu\partial A_s/\partial\mu=\beta(A_s)$ is the QCD $\beta$-function and $C_{F_3}(y,A_s)$ is the coefficient function, defined as $$C_{F_3}(y,A_s)=\sum_{n\geq 0} C_{F_3,n}(y) \bigg(\frac{\alpha_s}{4\pi}\bigg)^{n}$$ and $V_{F_3}(z)$ is the DGLAP kernel, related to a non-singlet (NS) $F_3$ SF. The solution of Eq.(1) is describing the predicted by perturbative QCD violation of scaling [@Bj] or automedeling [@BVT] behaviour of the DIS SFs by the logarithmically decreasing order $\alpha_s$-corrections. The coefficient function we are interested in has been known at the NNLO for quite a long period. The term $C_{F_3,2}(y)$ was analytically calculated in Ref.[@VZ]. The results of these calculations were confirmed recently [@MV] using a different technique. The kernel $V_{F_3}(z,\alpha_s)$ is analytically known only at the NLO. However, since there exists a method of symbolic evaluation of multiloop corrections to the renormalization group functions in the $\overline{MS}$-scheme [@T] and its realization at the FORM system, it became possible to calculate analytically the NNLO corrections to the $n=2,4,6,8,10$ Mellin moments of the NS kernel of the $F_2$ SF [@Larin]. They have the following expansion: $$-\int_0^{1} z^{n-1}V_{NS,F_2}(z,\alpha_s)dz = \sum_{i\geq 0}\gamma_{NS,F_2}^{(i)}(n) \bigg(\frac{\alpha_s}{4\pi}\bigg)^{i+1}$$ and are related to the anomalous dimension of NS renormalization group (RG) constants of $F_2$ SF[^1] : $$\mu\frac{\partial\ln Z_n^{NS,F_2}}{\partial\mu} =\gamma_{NS,F_2}^{(n)}(\alpha_s)~~~~.$$ These results were used in the process of the fits of Refs.[@KKPS; @KPS1] of the CCFR data for the $F_3$ SF with the help of the Jacobi polynomial method [@Jacobi]. It allows the reconstruction of the SF $F_3$ from the [**finite**]{} number of Mellin moments $M_{j,F_3}(Q^2)$ of the $xF_3$ SF: $$F_3^{N_{max}}(x,Q^2)=w\sum_{n=0}^{N_{max}} \Theta_n^{\alpha,\beta}(x)\sum_{j=0}^{n}c_j^{(n)}(\alpha,\beta) M_{j+2,F_3}^{TMC}(Q^2)$$ where $w=w(\alpha,\beta)=x^{\alpha-1}(1-x)^{\beta}$, $\Theta_n^{\alpha,\beta}$ are the orthogonal Jacobi polynomials and $c_j^{(n)}(\alpha,\beta)$ is the combination of Euler $\Gamma$-functions, which is factorially increasing with increasing of $N_{max}$ and thus $n$. The expressions for $M_{j+2,F_3}^{TMC}(Q^2)$ include the information about Mellin moments of the coefficient function $$C_{n,F_3}(Q^2)=\int_0^{1}x^{n-1}C_{F_3}(x,\alpha_s)dx =\sum_{i\geq 0}C^{(i)}(n)\bigg(\frac{\alpha_s}{4\pi}\bigg)^{i}$$ where $C^{(0)}(n)=1$. The target mass corrections, proportional to $(M_N^2/Q^2)M_{j+4,F_3}(Q^2)$, are also included into the fits. Therefore, the number of the Jacobi polynomials $N_{max}=6$ corresponds to taking into account the information about RG evolution of 10 moments, and $N_{max}=9$ presumes that the evolution of $n=13$ number of Mellin moments is considered. The procedure of reconstruction of $F_3(x,Q^2)$ from the finite number of Mellin moments and the related fits of the experimental data were implemented in the form of FORTRAN programs. The details of the fits of the CCFR data, based on RG evolution of 10 moments, are desribed in Refs.[@KKPS; @KPS1] (for the brief review see Ref.[@KPSB]). In the process of these analyses the following approximations were made: a) it was assumed that for a large enough number of moments, $\gamma_{NS,F_3}^{(n)}(\alpha_s)\approx\gamma_{NS,F_2}^{(n)}(\alpha_s)$; b) since the odd NNLO terms of $\gamma_{NS,F_2}^{(n)}$ are explicitly unknown, they were fixed using the smooth interpolation procedure proposed in Ref.[@PKK]. It was known that the additional contributions, proportional to the $d^{abc}d^{abc}$ structure of the colour gauge group $SU(N_c)$ are starting to contribute to the coefficients of $\gamma_{NS,F_3}^{(n)}(\alpha_s)$ from the NNLO [@KPS1]. In the process of the analysis of Refs.[@KKPS; @KPS1] it was assumed that they were not dominating and therefore were not taken into account. New inputs for the fits ======================= After recent explicit analytical evaluation of the NNLO coefficients of $\gamma_{NS,F_3}^{(n)}(\alpha_s)$ at $n=$3,5,7\ ,9,11,13 (see Ref.[@RV]) it became possible to fix this uncertainty (it is worth noting that the NNLO contribution to $\gamma_{NS,F_2}^{(n)}(\alpha_s)$ for $n=$12 was analytically evaluated in Ref.[@RV] also). To estimate the NNLO terms of $\gamma_{NS,F_3}^{(n)}(\alpha_s)$ at $n=$4,6,8,10,12 we applied the smooth interpolation procedure, identical to the one used to estimate the odd NNLO terms of $\gamma_{NS,F_2}^{(n)}(\alpha_s)$, while the numerical value of $\gamma_{NS,F_3}^{(2)}(2)$ was fixed with the help of an extrapolation procedure, where we have not used the value at $n=1$. The justification and more details of this procedure will be given elsewhere [@KPS2]. The used numerical results of the NNLO contributions $\gamma_{NS,F_3}^{(2)}(n)$ with and without $d^{abc}d^{abc}$-factors are presented in Table 1, where we marked in parenthesis the estimated even terms. The expressions for the NNLO contributions to the NS anomalous dimensions terms $\gamma_{NS,F_2}^{(2)}(n)$ are also given for comparison. They include the numerical results of the explicit analytical calculations of Refs.[@Larin; @RV], normalized to $f=4$ numbers of active flavours, and the results of the smooth interpolation procedure, in parenthesis, applied for estimating explicitly uncalculated odd terms. The satisfactory agreement between the numbers in the second and third columns supports the assumptions a) and b) mentioned above. $n$ $\gamma_{NS,F_3}^{(2)}(n)$ $d^{abc}d^{abc}$ neglected in $\gamma_{NS,F_3}^{(2)}(n)$ $\gamma_{NS,F_2}^{(2)}(n)$ ----- ---------------------------- ---------------------------------------------------------- ---------------------------- 2 (631) (585) 612.06 3 861.65 836.34 (838.93) 4 (1015.37) (1001.42) 1005.82 5 1140.90 1132.73 (1135.28) 6 (1247) (1241.21) 1242.01 7 1338.27 1334.32 (1334.65) 8 (1420) (1416.73) 1417.45 9 1493.47 1491.13 (1492.02) 10 (1561) (1558.85) 1559.01 11 1622.28 1620.73 (1619.83) 12 (1679.81) (1677.70) 1678.40 : The numerical expressions of the NNLO coefficients of anomalous dimensions of the $n$-th NS moments of the $F_3$ and $F_2$ SFs at $f=4$. The numbers in parenthesis are the estimated results. []{data-label="tab:a"} $n$ $C^{(1)}(n)$ $C^{(2)}(n)$ $C^{(3)}(n)$ $C^{(3)}(n)_{[1/1]}$ ----- -------------- -------------- -------------- ---------------------- 1 $-$4 $-$52 $-$644.35 $-$676 2 $-$1.78 $-$47.47 ($-$1127.45) $-$1268 3 1.67 $-$12.72 $-$1013.17 97 4 4.87 37.12 ($-$410.66) 283 5 7.75 95.41 584.94 1175 6 10.35 158.29 (1893.58) 2421 7 12.72 223.90 3450.47 3940 8 14.90 290.88 (5205.39) 5679 9 16.92 358.59 7120.99 7602 10 18.79 426.44 (9170.21) 9677 11 20.55 494.19 11332.82 11884 12 22.20 561.56 (13590.97) 14205 13 22.76 628.45 15923.91 17353 : The numerical expressions for the coefficients of the coefficient functions for $n$-th Mellin moments of the $F_3$ SF up to N$^3$LO and their \[1/1\] Padé estimates. []{data-label="tab:b"} In Table 2 the numerical expressions for the coefficients of Eq.(6) for $f=4$ numbers of active flavours are given. They include the results of explicit calculations of N$^3$LO corrections of odd moments [@RV], supplemented with the information about the coefficients of the Gross–Llewellyn Smith sum rule [@GL; @LV], defined by the $n=1$ Mellin moment of the $xF_3$ SF. The numbers in parenthesis are the results of the interpolation procedure. In the last column we present the values of $C^{(3)}(n)$, obtained with the help of the \[1/1\] Padé estimates approach. One can see that the agreement of Padé estimates with the used N$^3$LO results is good in the case of the Gross–Llewellyn Smith sum rule (this fact was already known from the considerations of Ref.[@Samuel]). In the case of $n=2$ and $n\geq 6$ moments the results are also in satisfactory agreement. Indeed, one should keep in mind that the difference between the results of column 3 and 4 of Table 2 should be devided by the factor $(1/4)^3$, which comes from our definition of expansion parameter $A_s=\alpha_s/(4\pi)$. Note, that starting from $n\geq 6$ the results of application of \[0/2\] Padé approximants, which in accordance with analysis of Ref.[@Gardi] are reducing scale-dependence uncertainties, are even closer to the the results of the interploation procedure (for the comparison of the estimates, given by \[1/1\] and \[0/2\] Padé approximants in the case of moments of $xF_3$ SF see Ref.[@KPS1], while in Ref.[@PP] the similar topic was analysed within the quantum mechanic model). For $n=$3,4 the interpolation method gives completely different results. The failure of the application of the Padé estimates approach in these cases might be related to the irregular sign structure of the perturbative series under consideration. $N_{max}$ $\Lambda_{\overline{MS}}^{(4)}$ (MeV) ---------------------------------- ----------- --------------------------------------- result of Ref.[@KPS1]: NLO 6 339$\pm$36 7 340$\pm$37 8 343$\pm$37 9 345$\pm$37 10 339$\pm$36                        NNLO 6 326$\pm$35 NNLO results with 6 325$\pm$35 inclusion of NNLO 7 326$\pm$31 terms of $\gamma_{NS,F_3}^{(n)}$ 8 329$\pm$36 9 332$\pm$36 N$^3$LO approximate results with 6 324$\pm$33 inclusion of the interpolated 7 322$\pm$33 values of $C^{(3)}$(n)-terms 8 325$\pm$34 9 326$\pm$33 : The results of the fits of the CCFR data for $xF_3$ SF, taking into account the NNLO approximation for $\gamma_{F_3,NS}^{(n)}$. The initial scale of RG evolution is $Q_0^2$=20 GeV$^2$. []{data-label="tab:c"} Some results of the fits ======================== In Table 3 we present the comparison of the results of the determination of the $\Lambda_{\overline{MS}}^{(4)}$ parameter, made in Ref.[@KPS1], with the new ones, obtained by taking into account more definite theoretical information. Since NNLO corrections to the anomalous dimensions and N$^3$LO contributions to the coefficient functions of odd moments of the $xF_3$ SF are now known up to $n=$13, it became possible to study the dependence of the results of the fits from the value of $N_{max}$, which we can now vary from $N_{max}=6$ to $N_{max}=9$. It should be mentioned that for $N_{max}=6$ the new NNLO result and its $Q_0^2$ dependence are in agreement with the results of Ref.[@KPS1]. However, the incorporation of higher number of moments, and thus the increase of $N_{max}$, make the NNLO (and approximate N$^3$LO ) results almost independent from the variation of $Q_0^2$ in the interval 5 GeV$^2$–100 GeV$^2$. This is the welcome feature of including into the fits the results of the new analytical calculations of the NNLO corrections to anomalous dimensions and N$^3$LO corrections to the coefficient functions of odd moments of the $xF_3$ SF [@RV]. Comparing now the central values of the results of the stable NLO fits of Ref.[@KPS1] with the new NNLO and N$^3$LO results, we observe the decrease of the theoretical uncertainties and, probably, the saturation of the predictive power of the corresponding perturbative series at the 4-loop level. More detailed results of our fits, including extraction of $\alpha_s(M_Z)$, its scale dependence and the information about the behaviour of twist-4 corrections at the NNLO and N$^3$LO, in the case of $N_{max}=9$, will be described elsewhere [@KPS2]. [**Acknowledgements**]{} We present here some of the results, reported at Quarks-2000 International Seminar, Pushkin, May 2000, together the first results from Ref.[@KPS2]. 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--- abstract: 'A junction between two boundaries of a topological superconductor (TSC), mediated by localized edge modes of Majorana fermions, is investigated. The tunneling of fermions across the junction depends on the magnetic flux. It breaks the time-reversal symmetry at the boundary of the sample. The persistent current is determined by the emergence of Majorana edge modes. The structure of the edge modes depends on the magnitude of the tunneling amplitude across the junction. It is shown that there are two different regimes, which correspond to strong and weak tunneling of Majorana fermions, distinctive in the persistent current behavior. In a strong tunneling regime, the fermion parity of edge modes is not conserved and the persistent current is a $2\pi$-periodic function of the magnetic flux. When the tunneling is weak the chiral Majorana states, which are propagating along the edges have the same fermion parity. They form a $4\pi$-phase periodic persistent current along the boundaries. The regions in the space of parameters, which correspond to the emergence of $2\pi$- and of $4\pi$-harmonics, are numerically determined. The peculiarities in the persistent current behavior are studied.' author: - 'Igor N. Karnaukhov' title: Persistent current in 2D topological superconductors --- Introduction {#introduction .unnumbered} ============ The phase coherent tunneling across a junction between two superconductors implies the presence of a $2 \pi-$ periodic persistent current, which is defined by the phase difference between the superconducting order parameters. The Josephson effect has been considered in Refs [@I1; @Er] in the framework of the well-known Kitaev chain model [@Kitaev]. The Kitaev’s proposal that, in the case of the fermion parity conservation, zero-energy Majorana fermion states, which are localized at the ends of the superconducting wire, trigger a $4 \pi$-periodic persistent current explains the so-called ’topological (or fractional) Josephson effect. The $2 \pi$- and $4 \pi$-harmonics of a persistent current correspond to the respective ground states of the system with different fermion parity when the magnetic flux is greater than $\pi$. In [@Kitaev] the author stimulates further research of new topological states that are realized at junctions between 1D TSCs, and Luttinger liquids [@I2; @I3]. In the absence of fermion parity conservation (that is, in those superconductors, in which the total number of particles is not conserved), the system under consideration is relaxing to the phase state with the lowest energy, which leads to the emergence of a $2\pi$-periodic persistent current. Below we discuss the persistent current in a 2D $(p+ip)$ TSC that has the spatial form of a hollow cylinder and is penetrated by a magnetic flux Q. We expect a nontrivial behavior of the persistent current depending on the magnitude of the applied magnetic flux. Due to their nontrivial topology [@TSC5; @TSC6; @TSC7; @TSC8], the superconductors with $(d +id)$ and $(p+ip)$ order parameters exhibit exotic phenomena such as Majorana vortex bound states and gapless chiral edge modes. The 2D TSCs with the ($p+ip)$-pairing of spinless fermions, which have chiral Majorana fermion states propagating along the edges, have been considered in [@TSC5a]. The behavior of topological states in the presence of disorder has been studied in Refs [@A1; @D1; @D3; @D4; @D5; @D6]. A nontraditional approach for description of TSCs has been proposed in [@K1] (see also [@K2]). It was shown that spontaneous breaking of time reversal symmetry is realized due to nontrivial stable phases of the superconducting order parameter (new order parameter). At that, the models of the TSC with the $p-$ and $(p+ip)$-wave superconducting pairing of spinless fermions are the simplest and the most straightforward examples of relevant model systems. In a finite system, the gapless chiral edge modes are localized at the boundaries. The tunneling of fermions across a junction leads to gapped edge modes due to the hybridization (through the weak link) of chiral edge modes localized at the different boundaries of the junction. In the case of a 1D superconductor the fermion parity is associated with zero energy Majorana edge states [@Kitaev; @I5; @I6; @I7; @D7], for a 2D TSC a persistent current is determined by the presence of the Majorana gapless edge modes localized at the boundaries of the junction. The ground state fermion parity changes whenever the energy of a pair of Majorana fermions crosses the zero energy. In the superconductor-topological insulator system the fermion parity of the ground state was associated with the Hopf index [@I4]. The fermion parity conservation, as a rule, is the result of the conservation of the total number of particles in the system, while the total number of particles is not conserved in those superconductors, which were studied in the framework of the Bogoliubov-de Gennes formalism. Nevertheless, we show that the fermion parity conservation is realized due to the conservation of the Chern number that determines the chiral current at the ends of the cylinder. The key point of the paper is that the unconventional behavior of the persistent current is determined by a chiral current along the boundaries of the TSC, while the behavior of the persistent current depends on the value of the tunneling amplitude of Majorana fermions across the junction. We should expect that behavior of the persistent current differs in the cases of the strong and weak tunneling of Majorana fermions. Model Hamiltonian, edge modes {#model-hamiltonian-edge-modes .unnumbered} ============================= We consider a junction between two boundaries of the TSC. The lattice Hamiltonian for a $(p+ip)$-wave superconductor of spinless fermions consists of two terms: ${\cal H} = {\cal H}_{TSC} + {\cal H}_{tun}$. At that, the first term describes the TSC per se: $${\cal H}_{TSC}= - \sum_{<ij>}a^\dagger_{i}a_j - 2\mu \sum_{j} n_j+ (i\Delta \sum_{<ij> x-links} a^\dagger_{i}a^\dagger_{j}+\Delta\sum_{<ij> y-links} a^\dagger_{i}a^\dagger_{j}+h.c.) , \label{eq-H}$$ and the second term describes the tunneling of fermions between two boundaries of a TSC with a junction along the x-direction $${\cal H}_{tun}= - 2\tau e^{i\frac{Q}{2}} \sum_{x-links} a^\dagger_ {x,1} a_{x,L} +h.c., \label{eq-Htun}$$ where $a^\dagger_{j} $ and $a_{j}$ are the spinless fermion operators on a site $j = {x,y}$ obeying usual anticommutation relations, and $n_j$ denotes the density operator. The first term in (1) describes hoppings of spinless fermions between nearest-neighbor lattice sites with equal to the unity magnitude, $\mu$ is the chemical potential (by choosing $ 0 < \mu < 1$ we do not restrict the generality of the study). Remaining terms describe pairing with superconducting order parameter $\Delta > 0$, which is defined along the link. Links are divided into two types depending on their direction: real $\Delta$ along y-links and complex $i\Delta$ along x-links. In practice, values of $\Delta,|\mu| << 1$. Therefore, we consider low energy excitations for $\Delta,|\mu| < 1$. The term ${\cal H}_{tun}$ contains the tunneling amplitude $0 < \tau < 1$ and takes into account the applied flux $Q$. The value of $ Q$ is measured in units of the quantum of flux $hc/(2e)$. Energies of spinless fermions E in the TSC that is described by the Hamiltonian (\[eq-H\]) are arranged symmetrically with respect to the zero energy and are given by the following dispersion relation $$E=\pm[(\mu+\cos k_x + \cos k_y)^2 +\Delta^2 (\sin^2 k_x +\sin^2 k_y)]^{1/2}, \label{eq-3}$$ where the wave vector $\textbf{k}=\{k_x,k_y\}$. In a finite system, the one-particle spectrum of the Hamiltonian $ {\cal H}$ (\[eq-H\]), (\[eq-Htun\]), is also symmetric edge states including. The corresponding edge states are determined by the particle-hole states of Majorana fermions. ![(Color online) Low energy spectra with edge modes of the one-dimensional strip along the *x*-direction as a function of the momentum directed along the edge. The energies are calculated at the Kitaev point $\Delta =1$ for $\mu=\frac{1}{5}$, $Q=\pi$ left), $Q=1\frac{1}{4}\pi$ right) and for different $\tau$. []{data-label="fig:3"}](3.eps){width="1\linewidth"} We analyze the formation of Majorana modes at the edges of the TSC. The gapped spectrum of excitations (\[eq-3\]) is realized in the topological nontrivial phase at $0<| \mu |<2$ (see the excitation spectra in Figs \[fig:1\]a),c)). The topological properties of a system are manifested in the existence of a nontrivial Chern number $C$ and chiral gapless edge modes (see in Figs \[fig:1\]), which are robust to effects of disorder and interactions. The excitation spectrum of the TSC includes chiral edge modes that connect the lower and upper fermion subbands. They are localized near the boundaries of the sample, and, therefore, amplitudes of the corresponding wave functions decrease exponentially with receding from the boundaries. The chiral gapless edge modes do exist in the gap if the Chern number of isolated bands located below the gap is nonzero. The gap of the superconductor collapses at $\mu=0$ and $\mu=\pm 2$. The TSC state with $C=\texttt{sgn} (\mu) $ is realized at $|\mu|<2$ , whereas the Chern number is equal to zero in a trivial topological state at $|\mu| > 2$ [@D7]. ### The Kitaev point $\Delta=1$ {#the-kitaev-point-delta1 .unnumbered} In order to describe in detail edge states of TSCs, we consider a superconductor in the form of a right square prism. Its base is taken to be LxL in size, while its height is assumed to be smaller than the superconducting coherence length. The superconductor can effectively be described by the 2D model (see Fig.\[fig:2\]a)). In the Appendix, we rigorously prove that chiral edge modes exist in the TSC for open boundary conditions. At that, their energy is determined by the wave vector component $k$ that is parallel to the boundary. These edge modes intersect each other at the Dirac point according to the dispersion relation $E_{edge}=\pm \Delta \sin k $. This dispersion relation is valid up to the points, at which edge modes are entering the domain of bulk states (see Figs \[fig:1\]). In the topological state with the Chern number equal to 1, gapless edge modes with wavevectors directed along the x-,y-boundaries have the corresponding Dirac points at $k_x = \pi , k_y =0$ and $k_x=0 , k_y=\pi$, respectively. In the topological state with $C=-1$ gapless edge modes have a different chirality, and their Dirac points are shifted to $\pi$ at $k_x=0$, $k_y =\pi$ and $k_x=\pi , k_y=0$, the TSC state is characterized by a chiral current along the boundary of the 2D system (see in Fig \[fig:2\]a)). This current can be of different chirality depending on the sign of the Chern number. The behavior of chiral edge modes at the junction (see in Fig.\[fig:2\]b)) is examined for the sample in the form of a hollow cylinder with varying the applied flux $Q$. In the case of a contact interaction between fermions at the boundaries, the tunneling Hamiltonian can be expressed in Majorana operators $f_{x,1},g_{x,1}$ and $ f_{x,L},g_{x,L}$ as follows: ${\cal H}_{tun}=- i \tau \cos(Q/2)\sum_{x-links} \delta_{1,L} (f_{x,1} g_{x,L} - g_{x,1} f_{x,L})$. Gapless edge modes are associated with Majorana operators $g_1(k_x)$, $f_1(k_x)$ and $g_L(k_x)$, $f_L(k_x)$ that belong to the boundaries. Cases of $Q=\pm \pi$ are particular because the contact interaction between particles vanishes at the boundaries for arbitrary $\tau$. Thus, the system is reduced to the TSC with open boundary conditions, in which (as noted above) the chiral gapless modes are realized in the topological phase (see Figs \[fig:1\]a),b)).At $\tau \neq 0$ and $Q\neq \pm \pi$ the edge modes at the junction are gapped, as a result of their hybridization at the Dirac point. In addition, we will demonstrate that their behavior depends on the magnitude of $\tau$. Majorana edge states are gapless at the points $Q = \pm \pi$ with the linear dispersion in $k$: $E_{edge}(\delta k)\sim \delta k$, $k=\pi+\delta k$ which is given by Eq (\[A7\]) (see in Figs \[fig:1\],\[fig:3\], the appendix contains some calculation details). Numerical calculations show that in a weak tunneling regime $\tau<\tau_c$ two chiral gapless edge modes are realized in the spectrum of the TSC for $Q=\pm \pi$ (see in Figs \[fig:3\]). These edge modes merge with the bulk states for any other $\tau$ and $Q$ in a weak tunneling regime. The edge modes with different chirality are localized at the different boundaries (at y = 1 and y = L) of the junction. The chiral edge modes yield chiral currents along the boundaries of the junction and form a chiral boundary current. The numerical calculation of $\tau_c$ at $Q=\pi$ as a function of $\mu$ is shown in Fig.\[fig:4\]a) at the Kitaev point. The calculations of $\tau_c$ for arbitrary $Q$ demonstrate that $\tau_c$ has the maximum value at $Q=\pi$. At the point $Q=\pi$ for $\tau >\tau_c$ gapless modes are localized at the junction, but they are non-chiral and do not touch fermion subbands at the arbitrary $Q$ (see Figs \[fig:3\]b),c),d)). For $\tau =1$ the linear dispersion of edge modes vanishes at $k=\pi$. We see that the behavior of the edge modes changes radically at $\tau >1$. In the strong tunneling regime $\tau>\tau_c$ the edge modes are localized at both ($1$ and $L$) boundaries of the junction. They do not connect lower and upper subbands of the superconductor and form localized standing waves. Arbitrary $\Delta$ {#arbitrary-delta .unnumbered} ------------------ The critical value $\tau_c$ depends on $\Delta$ and $\mu$. The minimal value of $\tau_c =\frac{1}{2}$ is reached in the $\Delta\rightarrow 0$ limit. The value of $\tau_c$ calculated for $Q=\pi$ as a function of $\mu$ and $\Delta$ is shown in Fig.\[fig:4\] b). We have plotted the width of the gap in the spectrum of Majorana bound states as a function of $Q$ for different values of $\tau$ (see in Fig\[fig:5\]). It follows from numerical calculations that this gap width is an even functions of $Q$, which can be approximated by $ \pm \tau^* \cos (Q/2)$, where, in the case of a weak tunneling the amplitude $\tau^*\sim \tau$ at $\tau <0.3$. Persistent current {#persistent-current .unnumbered} ================== The current along the boundaries is divided into chiral currents at the ends of the cylinder (red lines in Fig.\[fig:2\]b)) and chiral currents along the junction (blue lines). Chiral currents at the ends of the cylinder are described by the Hamiltonian (\[eq-H\]) with open boundary conditions and do not depend on the tunneling term (\[eq-Htun\]), whereas currents along the junction are described by the total Hamiltonian $\cal H$. The energy of the system $E^P(\tau,Q)=E_{cyl}^{p'} + E_{bulk}^p(\tau,Q)$ is determined by two terms: the energy of chiral edge modes at the ends of the cylinder with fermion parity $p'$ $E_{cyl}^{p'}$ (which does not dependent on $\tau$) and the energy of the superconductor, which takes into account the tunneling of fermions across the junction $E_{bulk}^p(\tau,Q)$, where $p,p'=f,h$ denote the fermion parity of the edge states: fermion (f) or hole (h), the symbol $P$ denotes the fermion parity of the ground state. In the strong tunneling regime, the edge modes, which occur at the junction, are represented by localized standing waves at all Q’s including the points $Q = \pm \pi$. Chiral currents at the ends of the cylinder and the current flowing along the junction, which is equal to zero, are not connected. Their fermion parities $p'$ and $p$ are not conserved. The fermion parities of the edge states are independent. In a contrast, in the weak tunneling regime, chiral currents flowing at the ends of the cylinder and along the junction are connected with each other due to the chiral current along the boundaries. Therefore, the fermion parities of Majorana-bound states located at the ends of the cylinder and at the junction are the same $p=p'$. Let us consider the behavior of the persistent current in the TSC in detail. In the limit $T\to 0$, the magnitude of the persistent current $I(\tau,Q)$ is determined by the ground-state energy of the system $I(\tau, Q)=\partial E^P(\tau, Q)/\partial Q$ (in unities of $2e/\hbar$), where $E^P(\tau,Q)=\int dk \sum_{\epsilon_n (k) <0} \epsilon_n(k)$ is determined by the quasi-particle excitations $\epsilon_n (k)$, and the Fermi energy is equal to zero at half-filling. In the strong coupling regime of tunneling $\tau>\tau_c$, the magnitude $I(\tau, Q)$ is a generic periodic function of the magnetic flux with the period of $2\pi$, so that $I(\tau,Q)=\partial E^f(\tau, Q)/\partial Q$ (see in Fig\[fig:6\] a)). In this case, the persistent current is determined by the energy of the superconductor, which takes into account both bulk excitations renormalized via the tunneling across the junction and the energies of edge modes at the junction. The fermion parity of edge states of Majorana fermions at the ends of the cylinder $p' = f$ and the Chern number, associated with these edge modes, are conserved. The fermion parity of edge Majorana fermions at the junction is not conserved. The system relaxes to the phase state with the minimum energy. In strong coupling tunneling regime $\tau>\tau_c$, $I(\tau,Q)$ is a typically periodic function of a magnetic flux with the period $2\pi$ (see in Fig\[fig:6\] a)). The persistent current is determined by the energy of the superconductor which takes into account the bulk excitations renormalized via the tunneling across the junction and the energies of edge modes at the junction. The fermion parity of edge states of Majorana fermions at the ends of the cylinder $p`=f$ and the Chern number, associated with these edge modes, are conserved. The fermion parity of edge Majorana fermions at the junction is not conserved, the system relaxes to the phase state with the minimum energy. In the weak tunneling regime $\tau<\tau_c$ all edge modes have the same chirality and fermion parity. This leads to a periodic persistent current having the period of $4\pi$.. The fermion parities of edge modes, which form the current along the boundaries, are identical. At $Q<\pi$ and $Q>\pi$ the ground state energy is determined by $E_{cyl}^{p'}+E_{bulk}^p(\tau,Q)$ with $p'=p=f$. At $Q>\pi$ the energy of the edge modes at the ends of the cylinder is negative, while the energy of the edge modes at the junction is positive. The balance of these energies determines the total energy of the system for given values of $\tau$ and $Q>\pi$. According to numerical calculations, a critical value of $Q_c$, at which energy difference of edge modes with different fermion parity changes its sign, is greater than $\pi$. The Chern number of the TSC is conserved, while the phase state of the system may not have the minimum energy at $Q>Q_c$. The resistive current is a periodic function of $Q$ with period $4\pi$, and $I(\tau,Q)$ is a continuous function of $Q$ within the whole interval $[-2\pi,2\pi]$ (see in Fig. \[fig:6\] b)). Conclusions {#conclusions .unnumbered} =========== This work is a step in our understanding of the behavior of a persistent current in topological systems. We have discussed the emergence of a persistent current in 2D TSC, pierced by a magnetic flux. It is proved that, the behavior of a persistent current is different in the case of strong and weak tunneling of Majorana fermions across a junction. The fermion parity of edge modes, forming a current along the boundaries of the sample, is the same, therefore the Chern number conserves a fermion parity of edge modes in the case of a weak interaction. Bulk edge correspondence leads to $4\pi$-periodic tunnel current. In a strong tunneling regime the currents at the ends of the cylinder and along the junction are not connected, therefore the fermion parities of the edge modes at the ends of the cylinder and at the junction are not conserved. At $Q=\pi$ in a strong tunneling regime spontaneous breaking of a fermion parity is realized. In the absence of fermion parity conservation the system relaxes to the minimum energy state, thus triggering a $2\pi$ periodic persistent current in TSC at strong tunneling of Majorana fermions across the junction. The results can be generalized to other topological phases, in particular, to topological insulators. Methods {#methods .unnumbered} ======= Edge modes in the 2D topological superconductor {#edge-modes-in-the-2d-topological-superconductor .unnumbered} ----------------------------------------------- Below we discuss the solution of the Schr$\ddot{o}$dinger equation for the chosen Hamiltonian ${\cal H}$ at the special point $\Delta=\pm 1$ using the formalism proposed for the calculation of Kitaev’s chain in Refs [@A1; @A2]. We focus on a 2D superconductor in the form of a square with the LxL size. Its sketch is shown in Figs \[fig:2\]. The wave function $\psi =\sum_{j=1}^L\sum_{s=1}^L[a^\dagger_{j,s}u_ {j, s}+a_{j,s}v_ {j,s}]$ is determined by amplitudes $u_ {j,s}$ and $v_ {j,s}$,that are solutions of the following equations: for $-L<j,s<L$ $$\begin{aligned} &&(E +\mu) u_ {j, s} = -\frac{1}{2}( u_ {j + 1, s} + u_ {j - 1, s} + u_ {j, s + 1} + u_ {j, s - 1}) + \frac{i}{2} (v_ {j + 1, s} - v_ {j - 1, s}) + \frac{1}{2} (v_ {j, s + 1} - v_ {j, s - 1}),\nonumber\\ && (E -\mu) v_ {j, s} = \frac{1}{2}( v_ {j + 1, s} + v_ {j - 1, s} + v_ {j, s + 1} + v_ {j, s - 1}) + \frac{i}{2} (u_ {j + 1, s} - u_ {j - 1, s}) - \frac{1}{2} (u_ {j, s + 1} - u_ {j, s - 1}), \label{A1}\end{aligned}$$ for $s=1,L$, where $1<j<L$ $$\begin{aligned} &&(E + \mu) u_ {j, 1} = -\frac{1}{2} (u_ {j,2} + u_ {j+1,1}+u_ {j-1,1} - v_ {j,2}) + \frac{i}{2} (v_ {j+1, 1} - v_ {j-1, 1}) - \tau e^{-i\frac{Q}{2}} u_ {j,L},\nonumber\\ &&(E - \mu) v_ {j, 1} =\frac{1}{2}( v_ {j,2} + v_ {j+1,1} +v_ {j-1,1} -u_ {j,2}) + \frac{i}{2} (u_ {j+1,1} - u_ {j-1,1}) + \tau e^{i\frac{Q}{2}} v_ {j,L},\nonumber\\ && (E +\mu) u_ {j,L} = -\frac{1}{2}(u_ {j,L - 1} +u_ {j+1,L}+u_ {j-1,L}+ v_ {j,L - 1}) + \frac{i}{2} (v_ {j+1,L} - v_ {j-1,L})- \tau e^{i\frac{Q}{2}}u_{j,1}, \nonumber\\ &&(E - \mu) v_ {j,L} = \frac{1}{2} (v_ {j,L - 1} + v_ {j+1,L} +v_ {j-1,L}+ u_ {j,L - 1}) +\frac{i}{2} (u_ {j+1,L} - u_ {j-1,L})+ \tau e^{-i\frac{Q}{2}}v_{j,1}, \label{A2}\end{aligned}$$ for $j=1,L$, where $1<s<L$ $$\begin{aligned} (E + \mu) u_ {1, s} = -\frac{1}{2} (u_ {2,s} + u_ {1,s+1}+u_ {1,s-1}- v_ {1,s+1}+v_{1,s-1}) + \frac{i}{2} v_ {2,s},\nonumber\\ (E - \mu) v_ {1,s} =\frac{1}{2}( v_ {2,s} + v_ {1,s+1} +v_ {1,s-1} -u_ {1,s+1}+u_{1,s-1})+\frac{i}{2} u_ {2,s},\nonumber\\ (E +\mu) u_ {L,s} = -\frac{1}{2}(u_ {L-1,s} +u_ {L,s+1}+u_ {L,s-1} -v_ {L,s+1}+v_ {L,s-1})- \frac{i}{2}v_ {L-1,s}\nonumber, \\ (E - \mu) v_ {L,s} = \frac{1}{2} (v_ {L - 1,s} + v_ {L,s+1} +v_ {L,s-1}- u_ {L,s+1}+ u_ {L,s-1}) - \frac{i}{2} u_ {L-1,s}, \label{A3}\end{aligned}$$ and the similar equations for the vertices of the square $\{1,1\};\{1,L\};\{L,1\};\{L,L\}$. The solutions of Eqs (\[A1\]) also satisfy Eqs (\[A2\])-(\[A3\]) at $\tau =0$ and the following boundary conditions $v_ {j,0} + u_ {j,0} = 0$, $v_ {j,L + 1} - u_ {j,L + 1} = 0$ and $u_ {0,s} + i v_ {0,s} = 0$, $u_ {L + 1,s} - iv_ {L + 1,s} = 0$. We determine the amplitudes of the wave function in accordance with the following Ansatz $$\begin{gathered} u_{j,s}=A_u(k_x,k_y)e^{ik_x j+i k_y s}+B_u(k_x,k_y)e^{i k_x j-i k_y s}+C_u(k_x,k_y)e^{-i k_x j +i k_y s}+D_u(k_x,k_y) e^{-i k_x j -i k_y s},\\ v_{j,s}=A_v(k_x,k_y)e^{i k_x j+i k_y s}+B_v(k_x,k_y)e^{i k_x j-i k_y s}+C_v(k_x,k_y)e^{-i k_x j +i k_y s}+D_v(k_x,k_y) e^{-i k_x j -i k_y s}. \label{A5}\end{gathered}$$ Unknown amplitudes in (\[A5\]) are defined as $A_u(k_x,k_y)=G_u(k_x,k_y)$, $B_u(k_x,k_y)=G_u(k_x,k_y)e^{i(-\chi+ \alpha)}$, $C_u=G_u(k_x,k_y)e^{i(- \chi+ \beta)}$, $D_u(k_x,k_y)=G_u(k_x,k_y)e^{i\gamma}$, $A_v(k_x,k_y)=G_v(k_x,k_y)$, $B_v(k_x,k_y)=-G_v(k_x,k_y)e^{i( \chi+ \alpha)}$, $D_v=G_v(k_x,k_y)e^{i(\chi + \beta)}$, $D_v(k_x,k_y)=-G_v(k_x,k_y)e^{i \gamma}$, where $e^{2 i \chi} =\frac{i\sin k_y -\sin k_x}{i\sin k_y +\sin k_x}$, the energies of the eigenstates are determined by Eq (\[eq-3\]) at $\Delta=1$ and the constants $\alpha,\beta,\gamma$ are determined by the boundary conditions. We redefine the unknown $G_u(k_x,k_y)$ and $G_v (k_x,k_y)$ as $G_u(k_x,k_y) =G \cos\varphi/2$ and $G_v(k_x,k_y) = iG \sin\varphi/2$, where $\tan \varphi =\frac{ \sin k_y - i \sin k_x}{\mu +\cos k_y + \cos k_x}$, $G$ is a normalization constant. Let us consider the points $k_y=0$ and $k_y=\pi$ at $\tau =0$ that correspond to zero energy of Majorana modes localized at the boundaries (see Figs \[fig:1\] for the illustration). The solutions for particle-hole excitations localized at the boundary are determined by complex $k_y$-wave vectors $k_y=\pm i \varepsilon $ or $k_y=\pi \pm i \epsilon $ with $$\begin{aligned} \varepsilon =2\sinh^{-1}\left(\frac{1}{2}\sqrt{\frac{ \mu^2+4(1+\mu)\cos^2(k_x/2)-E^2}{-\cos k_x- \mu}}\right),\nonumber\\ - \cos k_x >\mu \nonumber\\ \varepsilon =2\sinh^{-1}\left(\frac{1}{2}\sqrt{\frac{ \mu^2+4(1-\mu)\sin^2(k_x/2)-E^2}{\cos k_x + \mu}}\right),\nonumber\\ \cos k_x >-\mu \label{A6}\end{aligned}$$ $\mu$ defines the bulk gap. Solution (\[A6\]) determines the momentum of an excitation at a given energy, we can invert Eq (\[eq-3\]) yielding the momentum with energy E. At $k_y=0,\pi$ or $k_x=0,\pi$ $\chi=\frac{\pi}{2}$ or $\chi=0$, therefore the boundary conditions are reduced to the following equations $\sin[k_y(L+1)- \varphi] = 0$, $ \sin[k_x(L+1)]=0$. Similar to the 1D model [@A1], the energy of level localized at a boundary is equal to zero at $k_y= i \varepsilon , k_x=\pi$, in the $L\to \infty$ limit $E\sim (-1+\cosh\varepsilon -\mu)\exp(-2\varepsilon L)$. Complex solution for $k_y$ describes the edge modes localized at the $x$-boundary with $k_x$-dispersion. The boundary conditions describe free fermion states with the wave vector directed along the boundary. The solution of Eqs (\[A1\]) are a x-y symmetric. Let us consider the edge modes localized at the $x$-boundary with $k_y=i\varepsilon$ which have zero energy $E\to 0$ at the Dirac point $k_x=\pi$. The solution $E=0$ corresponds to the degenerate solution of Eqs (\[A1\]) for the amplitudes of the wave function $u(k_x,i\varepsilon)=v(k_x,i\varepsilon)$. This solution is valid for arbitrary $k_x$ at $E^\ast\rightarrow E-\sin k_x=0$. We do not use the boundary conditions for calculation of the wave function, as a result, the dispersion of edge modes is determined for an arbitrary value of $\Delta$. We find that the dispersion relation for the energy of the edge modes reads: $E_{edge}(k_x)=\pm \Delta \sin k_x$. The numerical calculations of the spectrum of the edge modes, obtained for arbitrary $\mu$ and $\Delta$, confirm the dispersion (see in Fig.\[fig:1\]b) for example). The energy of the edge modes at the $x-$ and $y-$boundaries have the intebtical dispersion for the wave vector directed along the boundary, that triggers a chiral current along the boundaries of the sample. As we already noted above, points $Q=\pm \pi$ are the special since the gapless edge modes are realized at them for $\tau \neq 0$. The zero-energy solutions for the edge modes at the Dirac point follow from the solutions of Eqs (\[A1\])-(\[A3\]) in the $L\to \infty$ limit at $\tau \neq 0$. Using appropriate boundary conditions we calculate a low energy dispersion of gapless edge modes at $Q=\pm \pi$. The energies of the edge modes propagating along the junction have the following form $$E_{edge}(\delta k_x)=\pm \frac{1}{3}\sin \delta k_x \mp\frac{2}{3} \sqrt{\sin^2 \delta k_x +3 w(0)} \cos (\zeta - 2 \pi/3) \label{A7}$$ where $k_x=\pi+\delta k_x$, $w(Q)=(\mu+1-\cos \delta k_x)^2+\sin^2 \delta k_x + \tau^2 \cos Q$, $\zeta =\frac{1}{3}\arccos\left(\frac{27}{54}\frac{-2 \sin^3 \delta k_x - 9 w(0) \sin \delta k_x + 27 w(Q)}{ (\sin^2 \delta k_x +3 w(0))^{3/2}}\right)$. Equation (\[A7\]) is derived from the low energy solution of the following equation $ E^3 - E^2 \sin \delta k_x- w(0) E +\sin \delta k_x w(Q)=0$. We consider the zero $\delta k_x$ limit of Eq (\[A7\]) and obtain the linear dispersion of edge modes at the Dirac point $E_{edge}(\delta k_x)=\pm v_{edge} \delta k_x$, where $v_{edge}=1-\frac{2\tau^2 }{\mu^2+\tau^2 }$. The linear dispersion of the edge modes vanishes at $\tau=1$. The solutions (\[A5\]) do not satisfy the boundary conditions at $\tau=1$ and, as a result, solution (\[A7\]) does not hold. According to numerical calculations the edge modes have a parabolic dispersion (see in Fig. \[fig:3\]c)). [31]{} Alicea, J., Oreg, Y., Refael, G., von Oppen, F. & Fisher, M.P.A. [Non-Abelian Statistics and Topological Quantum Information Processing in 1D Wire Networks.]{} *Nature Physics*, **7**, 412-417 (2011). Nogueira, F.S. & Eremin, I. [Strong-coupling topological Josephson effect in quantum wires.]{} *J.Phys.:Condens. 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Author contributions statement {#author-contributions-statement .unnumbered} ============================== I.K. is an author of the manuscript Additional information {#additional-information .unnumbered} ====================== The author declares no competing financial interests.

--- abstract: 'We study the clustering of galaxies as function of luminosity and redshift in the range $0.35 < z < 1.25$ using data from the Advanced Large Homogeneous Area Medium Band Redshift Astronomical (ALHAMBRA) survey. The ALHAMBRA data used in this work cover $2.38 \deg^2$ in 7 independent fields, after applying a detailed angular selection mask, with accurate photometric redshifts, $\sigma_z \lesssim 0.014 (1+z)$, down to $I_{\rm AB} < 24$. Given the depth of the survey, we select samples in $B$-band luminosity down to $L^{\rm th} \simeq 0.16 L^{*}$ at $z = 0.9$. We measure the real-space clustering using the projected correlation function, accounting for photometric redshifts uncertainties. We infer the galaxy bias, and study its evolution with luminosity. We study the effect of sample variance, and confirm earlier results that the COSMOS and ELAIS-N1 fields are dominated by the presence of large structures. For the intermediate and bright samples, $L^{\rm med} \gtrsim 0.6L^{*}$, we obtain a strong dependence of bias on luminosity, in agreement with previous results at similar redshift. We are able to extend this study to fainter luminosities, where we obtain an almost flat relation, similar to that observed at low redshift. Regarding the evolution of bias with redshift, our results suggest that the different galaxy populations studied reside in haloes covering a range in mass between $\log_{10}[M_{\rm h}/({\, h^{-1} \, \mathrm{M}_{\sun}})] \gtrsim 11.5$ for samples with $L^{\rm med} \simeq 0.3 L^{*}$ and $\log_{10}[M_{\rm h}/({\, h^{-1} \, \mathrm{M}_{\sun}})] \gtrsim 13.0$ for samples with $L^{\rm med} \simeq 2 L^{*}$, with typical occupation numbers in the range of $\sim 1 - 3$ galaxies per halo.' author: - | P. Arnalte-Mur$^{1,\star}$, V. J. Martínez$^{2,3,4}$, P. Norberg$^1$, A. Fernández-Soto$^{4,5}$, B. Ascaso$^6$, A. I. Merson$^7$, J. A. L. Aguerri$^8$, F. J. Castander$^9$, L. Hurtado-Gil$^{2,5}$, C. López-Sanjuan$^{10}$, A. Molino$^6$, A. D. Montero-Dorta$^{11}$, M. Stefanon$^{12}$, E. Alfaro$^6$, T. Aparicio-Villegas$^{13}$, N. Benítez$^6$, T. Broadhurst$^{14}$, J. Cabrera-Caño$^{15}$, J. Cepa$^{8,16}$, M. Cerviño$^{6,8,16}$, D. Cristóbal-Hornillos$^{10}$, A. del Olmo$^6$, R. M. González Delgado$^6$, C. Husillos$^6$, L. Infante$^{17}$, I. Márquez$^6$, J. Masegosa$^6$, M. Moles$^{10}$, J. Perea$^6$, M. Pović$^6$, F. Prada$^6$, J. M. Quintana$^6$\ $^1$Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK\ $^2$Observatori Astronòmic, Universitat de València, C/ Catedràtic José Beltrán 2, E-46980, Paterna, Spain\ $^3$Departament d’Astronomia i Astrofísica, Universitat de València, E-46100, Burjassot, Spain\ $^4$Unidad Asociada Observatorio Astronómico (IFCA-UV), E-46980, Paterna, Spain\ $^5$Instituto de Física de Cantabria (CSIC-UC), E-39005 Santander, Spain\ $^6$IAA-CSIC, Glorieta de la Astronomía s/n, 18008 Granada, Spain\ $^7$Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK\ $^8$Instituto de Astrofísica de Canarias, Vía Láctea s/n, 38200 La Laguna, Tenerife, Spain\ $^9$Institut de Ciències de l’Espai (IEEC-CSIC), Facultat de Ciències, Campus UAB, 08193 Bellaterra, Spain\ $^{10}$Centro de Estudios de Física del Cosmos de Aragón, Plaza San Juan 1, 44001 Teruel, Spain\ $^{11}$Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA\ $^{12}$Physics and Astronomy Department, University of Missouri, Columbia, MO 65211, USA\ $^{13}$Observatório Nacional-MCT, Rua José Cristino, 77. CEP 20921-400, Rio de Janeiro-RJ, Brazil\ $^{14}$Department of Theoretical Physics, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain\ $^{15}$Departamento de Física Atómica, Molecular y Nuclear, Facultad de Física, Universidad de Sevilla, 41012 Sevilla, Spain\ $^{16}$Departamento de Astrofísica, Facultad de Física, Universidad de La Laguna, 38206 La Laguna, Spain\ $^{17}$Departamento de Astronomía, Pontificia Universidad Católica. 782-0436 Santiago, Chile\ $^{\star}$ E-mail: pablo.arnalte-mur@durham.ac.uk bibliography: - 'ArnalteMur\_ALHclustering\_v2.bib' date: 'Accepted by MNRAS, 2014 April 4.' title: 'The ALHAMBRA survey: evolution of galaxy clustering since $z \sim 1$' --- \[firstpage\] methods: data analysis – methods: statistical – galaxies: distances and redshifts – cosmology: observations – large-scale structure of Universe Introduction {#sec:intro} ============ The large-scale structure (LSS) of the Universe is one of the main observables that we can use to obtain information about the nature of dark matter and cosmic acceleration. The simplest way to study the LSS is to study the spatial distribution of galaxies in surveys covering cosmologically significant volumes. Although the galaxy distribution is closely related to the global matter distribution, they are not equal. The relation between both distributions is known as galaxy bias, and it depends on the processes of galaxy formation and evolution. In the simplest case, one can consider the galaxy contrast to be proportional to the matter contrast. Then, the bias is simply the constant of proportionality, which is independent of scale. Being able to understand and model this bias is crucial for the correct interpretation of the cosmological information that can be obtained from the analysis of galaxy clustering. As the bias encodes information about the galaxy formation and evolution process, it is logical to expect that it will be different for different galaxy populations. In other words, the clustering properties of galaxies should depend on some of their intrinsic properties, such as stellar mass, star formation rate or age, and should evolve with time. This phenomenon, known as galaxy segregation, is observed when studying the dependence of clustering on different observables such as luminosity, colour, or morphology. In general, it is observed that bright, red, elliptical galaxies are more strongly clustered (i.e., they have a larger bias) than faint, blue, spiral ones [see e.g. @dav76a; @ham88a; @mad03a; @ski08a; @mar10a; @zeh10a]. In this work, we focus on the dependence of the galaxy bias on luminosity, and the evolution of this relation with redshift. This dependence has been studied extensively in the local Universe using both the Two-degree Field Galaxy Redshift Survey [2dFGRS, @nor01a; @nor02b] and Sloan Digital Sky Survey [SDSS, @teg04b; @zeh05a; @zeh10a]. @Guo2013a also studied this relationship at $z \sim 0.5$ using data from the Baryon Oscillation Spectroscopic Survey (BOSS). The bias shows a weak dependence on luminosity $L$ for galaxies with $L < L^{*}$, where $L^{*}$ is the characteristic luminosity parameter of the Schechter function. For $L \gtrsim L^{*}$, however, this relation steepens, and the bias clearly increases with luminosity. These studies of galaxy clustering and luminosity segregation have been extended to redshifts in the range $z \sim 0.5 - 1$, using state-of-the-art spectroscopic surveys, such as the VIMOS-VLT Deep Survey [VVDS, @pol06a; @abb10a], the Deep Extragalactic Evolutionary Probe survey [DEEP2, @coi06a; @coi08a], the zCOSMOS survey [@men09a] or the VIMOS Public Extragalactic Redshift Survey [VIPERS, @mar13a], or photometric surveys such as the Canada-France-Hawaii Legacy Survey (CFHTLS) Wide survey [@McCracken2008a; @Coupon2012a]. Recently, @Skibba2013c used an intermediate method, somehow similar to the one presented in this work. They used low-resolution spectroscopy data (with a typical redshift precision of $\sigma_z/(1+z) = 0.005$) from the PRIsm Multi-object Survey (PRIMUS) to study galaxy clustering in the range $0.2 < z < 1$. Overall, these studies show strong evidence for luminosity segregation at these redshifts, with the relation between bias and luminosity being slightly steeper in this case than in local studies. However, the luminosity range covered by these surveys is more limited in these cases, and is restricted typically to $L^{\rm th} \gtrsim 0.3 L^{*}$. The presence of this bias parameter can be understood in a natural way in the context of the halo model [e.g. @sel00a; @pea00a; @coo02a]. In this model, the matter distribution is decomposed into a population of massive virialised dark matter haloes that form at the peaks of the density field, and galaxies form within these haloes. The bias parameter for dark matter haloes can be modelled, and depends on the properties of the halo such as its mass [e.g. @Sheth2001i; @mo02a]. Studying the clustering of a certain population of galaxies gives therefore information on the characteristics of the haloes that host them. In this context, luminosity segregation indicates that more luminous galaxies form preferentially in more massive haloes than fainter ones. In this work, we use data from the Advanced Large Homogeneous Area Medium-Band Redshift Astronomical (ALHAMBRA) survey [@mol08a; @Molino2013a] to study galaxy clustering and luminosity segregation for redshifts in the range $0.35 < z < 1.25$, using the two-point correlation function. ALHAMBRA is a deep photometric survey, which uses a total of 23 optical and near-infrared (NIR) bands in order to obtain accurate and reliable photometric redshifts (photo-$z$) for a large number of objects, in a nominal area of $4~\deg^2$ over 8 independent fields. It is therefore well suited to study the large-scale distribution of galaxies in significant volumes over this redshift range, providing an opportunity to explore the clustering of fainter galaxies than it is possible using spectroscopic surveys. Moreover, the use of several independent fields allows us to use ALHAMBRA to study the effect of sample variance in the clustering measurements, and in particular the effect of large structures present in the samples. @lopez2013a also exploited this independence of the ALHAMBRA fields to study the effect of sample variance on merger fraction studies. In Section \[sec:data\] we present the ALHAMBRA data used in this work (characterised in more detail in @Molino2013a), and our selection of samples. We also present here the mock catalogues created to test our clustering methods. In Section \[sec:selfunc\] we explain how we model the selection function of the survey, and in particular the masks created to reproduce the angular selection. Section \[sec:projcf\] presents our method to estimate the projected correlation function (a real space quantity) taking into account the effect of the photometric redshifts, following @arn09a. We also present our error estimation method, and leave for Appendix \[sec:pimax\] the detailed justification of our line-of-sight integration limit. Our results are presented in Section \[sec:results\]. We show the correlation functions obtained for our different samples, including the modelling in terms of a simple power law model (Sect. \[ssec:powlaw\]), and of a $\Lambda$ cold dark matter ($\Lambda$CDM) model in order to derive the bias parameter (Sect. \[ssec:bias\]). We also make use of the independence of the surveyed fields to study the effect of sample variance on our measurements (Sect. \[sec:cvariance\]), and compare our results with those of previous surveys in a similar redshift range (Sect. \[sec:othersurveys\]). In Appendix \[sec:systematics\] we present the tests done using the mock catalogues to test the reliability of the results, and Appendix \[sec:numer-results\] contains numerical tables of our results. Finally, in Sect. \[sec:conc\] we discuss our results and summarise our conclusions. Unless noted otherwise, we use a fiducial flat $\Lambda$CDM cosmological model with parameters $\Omega_{\rm M} = 0.27$, $\Omega_{\Lambda} = 0.73$, $\Omega_{\rm b} = 0.0458$, and $\sigma_8 = 0.816$ based on the WMAP7 results [@kom11a]. All the distances used are co-moving, and are expressed in terms of the Hubble parameter $h \equiv H_0/100 {\, \mathrm{km} \, \mathrm{s}^{-1} \, \mathrm{Mpc}^{-1}}$. Absolute magnitudes are given as $M - 5 \log_{10}(h)$, even when not explicitly indicated. Data used: the ALHAMBRA Survey {#sec:data} ============================== The ALHAMBRA survey [@mol08a] is a photometric survey which covers a total of $4 \deg^2$ in the sky, using 20 medium-band filters in the optical range, and three standard broad-band filters ($J$, $H$, and $K_s$) in the NIR. The survey was carried out using the 3.5-m telescope at the Centro Astronómico Hispano-Alemán (CAHA)[^1] in Calar Alto (Almería, Spain). The camera used for the optical observations was the Large Area Imager for Calar Alto (LAICA)[^2], and Omega-2000[^3] was used for the NIR observations. The optical filter system for the ALHAMBRA survey was specifically designed to optimise the output of the survey in terms of photo-$z$ accuracy and number of objects with reliable $z$ determination [@ben09a]. It consists of a set of 20 contiguous, equal-width, medium-band filters of width $FWHM~\simeq~310$ Å covering the full optical range, between $3500$ and $9700$ Å [@apa10a]. The survey is complemented by observations in the standard NIR filters $J$, $H$ and $K_s$. The homogeneous spectral coverage of this system minimises the variations in the selection functions of the different objects with redshift. The NIR observations help eliminate some degeneracies in the photo-$z$ determination while at the same time improving the determination of important galaxy properties such as stellar mass. ALHAMBRA imaging data {#ssec:imaging-data} --------------------- ---------------- ------------- --------------- ------------- ------------------------- Field $N_{\rm f}$ $A_{\rm eff}$ $N_{\rm g}$ $N_{\rm g}/A_{\rm eff}$ $(\deg^2)$ $(\deg^{-2})$ ALH-2/DEEP2 8 0.377 26759 70979 ALH-3/SDSS 8 0.404 28331 70126 ALH-4/COSMOS 4 0.203 16877 83138 ALH-5/HDF-N 4 0.216 16629 76986 ALH-6/GROTH 8 0.400 28892 72230 ALH-7/ELAIS-N1 8 0.406 29530 72734 ALH-8/SDSS 8 0.375 27615 73640 Total 48 2.381 174633 73344 ---------------- ------------- --------------- ------------- ------------------------- : Properties of the seven ALHAMBRA fields used in this work. We list the number of frames $N_{\rm f}$ included in the current catalogue in each case (where a completed field corresponds to 8 frames), the area $A_{\rm eff}$ covered by the survey according to our angular selection mask, the number of galaxies $N_{\rm g}$ included in the catalogue used (at $I < 24$), and the resulting surface number density $N_{\rm g}/A_{\rm eff}$. We also list other surveys which have overlap with each of the fields, see @mol08a for details.[]{data-label="tab:fields"} The data used in this work correspond to the photometric catalogue described in @Molino2013a[^4]. It contains data for a nominal area of $3 \deg^2$ distributed over 7 fields in the sky, in order to minimise the effects of sample variance (see Table \[tab:fields\]). The minimum distance between fields is $17^{\circ}$, so we can safely consider them as independent. The fields were primarily chosen because of their low extinction, and trying to have significant overlap with other surveys [@mol08a]. Each field is typically composed of 8 frames forming two strips of $\sim 15' \times 1^{\circ}$, separated by a $\sim 15'$ gap. We discuss in detail the geometry of the different fields in Section \[ssec:angmask\]. We developed our own pipelines for the reduction of the imaging data, including bias, flatfield and fringing corrections. The details of the data reduction can be found in @cri09a [in prep.] for both the optical and NIR data. The detection of objects for inclusion in the catalogue is performed in synthetic images built using a combination of the ALHAMBRA filters in the range $7000 < \lambda < 9700$ Å to match the Hubble Space Telescope $F814W$ filter (hereafter denoted as our $I$ band). Matched photometry is then obtained for these detected objects in the 23 ALHAMBRA filters. We restrict our analysis to the magnitude range $I < 24$, where the catalogue is photometrically complete and we do not expect any significant field-to-field variation in the depth [see section 3.8 in @Molino2013a]. We eliminate stars from the catalogue using the star-galaxy separation method described in @Molino2013a, which uses information on both the geometry and colours of the sources. In particular, we use the stellar flag given in the catalogue, and select only objects with <span style="font-variant:small-caps;">Stellar Flag</span> $< 0.7$. This method is only reliable for $I < 22.5$. However, at $I=22.5$ the fraction of stars in the sample is $\sim 1 \%$, and we expect it to decrease at fainter magnitudes. Therefore the possible effect of stellar contamination at $I > 22.5$ is negligible. The final catalogue used contains a total of $N_{\rm T} = 174,633$ galaxies. Photometric redshifts {#ssec:photo-z} --------------------- ![Redshift distribution of the $174,633$ galaxies in the ALHAMBRA catalogue used in this work. The distribution shown corresponds directly to a histogram of the ‘best’ photo-$z$ for each galaxy, in bins of width 0.08.[]{data-label="fig:zhist"}](ArnalteMur_fig_1){width="\columnwidth"} Photometric redshifts were estimated for this catalogue using an updated version of the Bayesian Photometric Redshift (<span style="font-variant:small-caps;">BPZ</span>) code [@ben00a], including a new prior and spectral template library (Benítez et al., in prep.), and a new technique for the re-calibration of the photometric zero points. @Molino2013a discussed in detail the methods used for the redshift estimation and the characteristics of the photo-$z$ obtained. They performed a comparison for the $\sim 7000$ galaxies with measured spectroscopic redshift (see their figure 25) and showed that the global accuracy in the photo-$z$ is $\sigma_z \lesssim 0.014 (1+z)$ for $I < 24.5$. We show the distribution of photo-$z$ for this catalogue in Fig. \[fig:zhist\]. The median redshift of the catalogue is $z_{\rm med} = 0.75$, with the bulk of the redshift distribution in the range $0.35 < z < 1.25$ that we study in this work. ![Distribution of the relative differences between the ALHAMBRA photometric redshift ($z_{\rm p}$) and the COSMOS photometric redshift ($z_{\rm c}$) for the objects matched between the two catalogues. We show this distribution for different $I$-band magnitude selections, as shown in the label. We quote in each case the dispersion $\sigma$ estimated using the NMAD method.[]{data-label="fig:hist_cosmos"}](ArnalteMur_fig_2){width="\columnwidth"} As an additional test of the reliability of the photometric redshifts used, we made a comparison with the Cosmic Evolution Survey (COSMOS) photo-$z$ catalogue described in @ilb08a. This catalogue contains photometric redshift determinations with comparable accuracy and depth to those in the ALHAMBRA catalogue, and overlaps with the field ALH-4 (see Table \[tab:fields\]). We matched both catalogues using a separation radius of $1''$ in the angular position, and obtained a sample of 12832 objects common to both catalogues. We show the distribution of the relative redshift differences for this sample in Fig. \[fig:hist\_cosmos\], where we also quote the dispersion in the results measured using the normalised median absolute deviation (NMAD) method [see e.g. @bra08a]. We compare the dispersion obtained in this way to a simple estimate based on the redshift errors quoted in both catalogues. In each case, we estimate the typical redshift uncertainty (in both ALHAMBRA and COSMOS) as the mean of the $1\sigma$ errors quoted for each object in the sample. Our estimate for the dispersion in the difference shown in Fig. \[fig:hist\_cosmos\] is then $\sigma_{\rm diff} = \sqrt{\sigma_{\rm ALH}^2 + \sigma_{\rm COS}^2}$. We obtain that this value of the dispersion obtained from the quoted errors is a good estimate of that observed. However, for our faintest samples ($I > 23$), we need to increase this estimate by a factor of $\sim1.3$, suggesting that the photo-$z$ uncertainty could be slightly underestimated for these galaxies in both ALHAMBRA and COSMOS. Hereafter, we quote the error estimates for our samples (e.g. in Table \[tab:samples\]) as the mean of the quoted <span style="font-variant:small-caps;">BPZ</span> error for the objects in the sample. For consistency, we correct this value by the factor of $1.3$ for all samples, although we only see an indication for the underestimation of the errors at the faintest ones. Selection of samples in redshift and luminosity {#ssec:samples} ----------------------------------------------- To study the dependence of clustering properties on both luminosity and cosmic time, we build a series of subsamples splitting the catalogue in redshift and absolute magnitude. The size of the redshift bins has to be larger than the distance we will integrate over the radial direction, $\pi_{\rm max}$. As shown in @arn09a, using smaller bins may introduce systematic effects in the correlation functions we want to measure. Taking this fact into account, and the limitations in volume covered and galaxy density, we decided to use the four redshift bins $0.35 < z_{\rm p} < 0.65$, $0.55 < z_{\rm p} < 0.85$, $0.75 < z_{\rm p} < 1.05$, $0.95 < z_{\rm p} < 1.25$. We allow for overlap between consecutive bins in order to better trace the redshift evolution in our analysis, but one should bear in mind that results for different bins will be therefore correlated. Our low redshift limit $z_{\rm p} = 0.35$ was set in order for the scales of interest to be well sampled given the angular size of the fields. At this redshift, the typical size of a field, $1^{\circ}$, corresponds to a projected comoving separation of $17 {\, h^{-1} \, \mathrm{Mpc}}$. We fixed our high redshift limit at $z_{\rm p} = 1.25$ as, for higher redshifts, the quality of the photo-$z$ and the number density of objects are significantly reduced. In addition to the redshift selection, we also apply a set of cuts in the rest-frame $B$-band absolute magnitude $M_B$. We use this band for the selection as the region of the spectrum corresponding to it is well sampled by the ALHAMBRA filters (including the NIR filters) for the whole redshift range studied. Moreover, as this same band (or similar ones as $g$) is used for luminosity selection by other surveys at these redshifts, this will allow for more direct comparisons. The $M_B$ for each object is obtained as a by-product of the photo-$z$ estimation, and includes the appropriate $K$-correction at the best value of $z_{\rm p}$. We use ‘threshold samples’, meaning that we will impose a faint luminosity threshold, but not a bright limit. In this way, we obtain approximately volume-limited samples, but also we can study the luminosity dependence of clustering, and its evolution. Following @men09a and @abb10a, we apply an absolute magnitude threshold depending linearly on redshift as $$\label{eq:p1alh:Mthevol} M_B^{\rm th}(z) = M_B^{\rm th}(0) + A z_{\rm p} \,,$$ in order to follow the evolution of samples corresponding approximately to the same galaxy population. The value of the constant $A$ characterises the typical luminosity evolution of the galaxies in the catalogue. We use here a value of $A = -0.6$, which we selected to produce samples with similar number density across the whole redshift range. This value is also similar to the observed evolution of the typical luminosity parameter $M^{*}$ derived from luminosity function studies at similar redshifts [@ilb05a; @zuc09a]. ![Selection of samples in the absolute $B$-band magnitude $M_B$ vs. photometric redshift diagram. The different coloured dots show the eight magnitude cuts, while the lines mark the boundaries of our redshift bins. See the main text and Table \[tab:samples\] for the details of the sample selection.[]{data-label="fig:zMB"}](ArnalteMur_fig_3){width="\columnwidth"} We show in Fig. \[fig:zMB\] the actual cuts made in the redshift – absolute magnitude plane to define our samples, and list the properties of all the samples used in Table \[tab:samples\]. We estimate the error in the mean number density $\bar{n}$ of each sample using a block bootstrap method based on the 7 independent fields. For each sample, we compute the typical $z_{\rm p}$ error $\sigma_z/(1+z)$ as described in Sect. \[ssec:photo-z\], and the line-of-sight distance that corresponds to this uncertainty, $r(\sigma_z)$, measured at the median redshift $z_{\rm med}$ of the sample. We also measure the median absolute luminosity, $M_B^{\rm med}$, and express it in terms of the typical luminosity parameter $L^{*}$ at $z_{\rm med}$. We compute $L^{*}(z)$ from a linear fit to the results of @ilb05a. Mock catalogues {#ssec:mock-catalogues} --------------- To test our methods for clustering and error estimation, and to provide a test bench for future ALHAMBRA studies, we use a set of mock catalogues, based on the Millennium dark matter simulation [@spr05a]. We populate the dark matter haloes with galaxies using the @Lagos2011a version of the semi-analytic galaxy formation model <span style="font-variant:small-caps;">Galform</span> [@Cole2000a]. In addition to other physical parameters, we compute the photometry for each of the galaxies in the model using the 24 ALHAMBRA filters, including the synthetic $I$ band and, for completeness, also using the five SDSS broad-band filters $ugriz$. A light-cone is built from the simulation’s snapshots up to $z = 2$, reproducing the photometric depth of the survey. In order to properly model the evolution of structures along the line of sight, the galaxy positions are interpolated between snapshots. The procedure used to generate the light-cone mocks is presented in detail in @mer13a. The cosmological model used for the mocks is set by that of the Millennium simulation, which uses the parameters $\Omega_{\rm M} = 0.25$, $\Omega_{\Lambda} = 0.75$, $\sigma_8 = 0.9$. We will use these parameters when doing tests with the mocks in Appendices \[sec:analyt-model-determ\] and \[sec:systematics\]. We generate a $200 \deg^2$ light-cone, which is divided in 50 non-overlapping mock ALHAMBRA realisations. Each of these realisations reproduces the ideal geometry of the full survey, containing 8 fields covering $0.5 \deg^2$ each, for a total of $4\deg^2$ per realisation. The fields in each realisation are as separated as possible within our light-cone geometry. Each field is formed by two strips of $15' \times 1^{\circ}$, separated by a $15'$ gap, approximately reproducing the geometry of the ALHAMBRA fields, as described in Sect. \[ssec:angmask\]. To simulate the photometric redshifts for the galaxies in the mock we proceeded as follows. We first use the original rest-frame photometry and spectroscopic redshifts in the mock to assign to each galaxy a spectral type from the same <span style="font-variant:small-caps;">BPZ</span> template library used to estimate photo-$z$ in the real data[^5]. Then, we measure consistent ALHAMBRA photometry for these spectral types by using the ALHAMBRA filter curve response. Finally, we estimate the photometric redshifts, together with the spectral types and absolute magnitudes associated with the previous photometry, by running <span style="font-variant:small-caps;">BPZ</span> in normal mode. These photometric redshifts are found to be very realistic as their performance is very similar to those obtained for real data, although with a somewhat larger uncertainty ($\sim 30\%$). All the details can be found in Ascaso et. al (in prep.). Modelling the selection function {#sec:selfunc} ================================ To study the clustering of the galaxies in a survey, it is crucial to understand and to model its selection function. In this work, we separate the angular and radial parts of the selection function, with our angular selection function (or ‘mask’) defining the geometry of the survey on the sky. We assume a uniform depth inside the mask, as the catalogue considered does not reach the photometric limit of the survey. Angular selection mask {#ssec:angmask} ---------------------- The angular selection mask is defined in the first instance by the coverage of the ALHAMBRA survey. It consists of independent fields of $\sim 0.5 \deg^2$ each, with a specific geometry set by the configuration of the detectors in the optical camera used, LAICA. The camera has four $15.5' \times 15.5'$ detectors, distributed in a square leaving a space of $13.6'$ between them. Each of the ALHAMBRA fields consists of two pointings made with this configuration, resulting in two strips of $15.5' \times 58.5'$ with a gap of $13.6'$ between them (see bottom panel of Fig. \[fig:masks\]). For this work, fields ALH-4 and ALH-5 correspond to only one pointing each, and thus are formed by four disjoint $15.5' \times 15.5'$ frames. Based on that geometry, we define a set of masks describing the sky area which has been reliably observed. We start with the *flag images* described in @Molino2013a, that give information on the areas in which the detection of the objects in the synthetic $I$-band images was performed. They exclude areas with low exposure time (less than $60\%$ of the maximum in each frame), which mainly correspond to regions next to the borders of each frame, or corresponding to large saturated stars. To avoid possible variations in depth, which could potentially introduce a spurious clustering pattern, we remove some additional regions from the survey area, taking a conservative approach. We mask out regions around bright stars, using the Tycho-2 catalogue [@Hog2000d]. The masked regions are circles of radius $33\, \mathrm{arcsec}$ centred on each star. For the brightest stars ($V < 11$), we extend this radius to $111\, \mathrm{arcsec}$. We define these radii by observing the typical maximum extension of the stellar haloes in the $I$-band detection images. Furthermore, we select objects showing saturated detections in the ALHAMBRA catalogues (using the <span style="font-variant:small-caps;">Satur\_Flag</span> parameter, see @Molino2013a for details), and mask a region around each of them with a radius twice that of the object itself. Finally, we mask by hand some obvious defects in the image (typically extended stellar spikes), and some small overlap between contiguous frames. The latter is needed to avoid double-counting objects from the overlap regions when computing the clustering for the combined field. To avoid position-dependent differences in the photo-$z$ quality we mask by hand regions which present bad photometric quality in at least 3 of the ALHAMBRA bands (but not necessarily in the $I$ band used for detection). This uses the <span style="font-variant:small-caps;">irms\_opt\_flag</span> and <span style="font-variant:small-caps;">irms\_nir\_flag</span> parameters in the catalogue [see @Molino2013a for details]. ![Illustration of the ALHAMBRA angular mask for field ALH-7. Top: synthetic $I$-band image for one of the 8 frames in the field, showing an area of $\sim 16' \times 16'$. Green dots mark the position of the objects included in the catalogue, and the blue lines show the limits of the angular selection mask. Bottom: angular mask for the ALH-7 field. The shaded area corresponds to the regions of the survey that are included in the calculations. The red rectangle marks the area shown in the top image.[]{data-label="fig:masks"}](ArnalteMur_fig_4a "fig:"){width="\columnwidth"} ![Illustration of the ALHAMBRA angular mask for field ALH-7. Top: synthetic $I$-band image for one of the 8 frames in the field, showing an area of $\sim 16' \times 16'$. Green dots mark the position of the objects included in the catalogue, and the blue lines show the limits of the angular selection mask. Bottom: angular mask for the ALH-7 field. The shaded area corresponds to the regions of the survey that are included in the calculations. The red rectangle marks the area shown in the top image.[]{data-label="fig:masks"}](ArnalteMur_fig_4b "fig:"){width="\columnwidth"} We defined and combined the different masks using the <span style="font-variant:small-caps;">Mangle</span>[^6] software [@ham04a; @swa08a], which allows for an easy manipulation of angular masks, and for some additional routines like generating random catalogues. These angular masks will be publicly available from http://www.alhambrasurvey.com/. Fig. \[fig:masks\] illustrates the resulting mask for ALH-7. The total effective area after applying this mask is $A_{\rm eff} = 2.381 \deg^2$, distributed over the different fields as shown in Table \[tab:fields\]. Overall, this procedure masks an additional $\sim 15\%$ of the area not yet masked by the original *flag images*. This explains the difference in area between this work and @Molino2013a. Radial selection function {#ssec:radsfunc} ------------------------- ![Number density as function of comoving distance (or, equivalently, redshift) for our different cuts in absolute magnitude. We show the function directly measured from the data with a smoothing length of $200 {\, h^{-1} \, \mathrm{Mpc}}$ (continuous lines) and our third-order polynomial fit (dashed lines) in each case. Lines from top to bottom correspond to samples with fainter to brighter luminosity cuts.[]{data-label="fig:radialdens"}](ArnalteMur_fig_5){width="\columnwidth"} We model the radial selection function for our different samples directly using the observed number density of galaxies as function of comoving distance (or, equivalently, redshift), $n(d)$. We show in Fig. \[fig:radialdens\] the number density of our different samples selected in luminosity (solid lines), measured using a smoothing length of $200 {\, h^{-1} \, \mathrm{Mpc}}$. Given our redshift-dependent luminosity cut, the number density for each of the samples is approximately constant over the redshift range considered, as expected for nearly volume-limited samples. However, apart from the small-scale variations due to the presence of structures, we observe some long-range variations in $n(d)$. We assume the latter are part of our selection function, and model them by fitting a third-order polynomial to $n(d)$ over the full range spanned by each of the samples. This model is smooth enough not to include possible variations in $n(d)$ due to large-scale structures, to prevent a systematic underestimation of the clustering signal. We use this smooth model for our clustering measurements as described below. However, we performed some tests assuming either a model with constant $n(d)$, or using directly the measured $n(d)$ as our radial selection. Our results do not change significantly in either case. One particularity of the radial density of ALHAMBRA as measured here is the presence of a series of regularly spaced ‘peaks’. They can be seen more clearly in Fig. \[fig:radialdens\] for the fainter samples (higher $n$), or as a series of vertical ‘strips’ in the distribution of galaxies in Fig. \[fig:zMB\]. The presence of these peaks is the consequence of using only the best value $z_{\rm p}$ of the photometric redshift estimate for each galaxy, instead of the full probability density function $p(z)$ [@ben00a]. We tested whether this issue could introduce any systematic bias in our measurements by creating a new ‘realisation’ of the photometric redshifts: we assigned to each galaxy a new value of $z_{\rm p}$ drawn from a Gaussian distribution centred at the original value, and with a width given by the quoted error. Additionally, we randomly selected $5\%$ of galaxies to be ‘outliers’, and assigned them a random value of $z_{\rm p}$ within the studied range. We computed the projected correlation function for our samples using this new ‘realisation’, and obtained only small changes contained within the quoted errors. We therefore conclude that the presence of these peaks in $n(d)$ does not significantly bias our results. The projected correlation function calculation in photometric redshift catalogues {#sec:projcf} ================================================================================= The two-point correlation function $\xi(\mathbf{r})$ measures the excess probability of finding two points separated by a vector $\mathbf{r}$ compared to that probability in a homogeneous Poisson sample [@pee80a; @mar02a]. If the point process considered is homogeneous and isotropic, the correlation function can be expressed simply in terms of the distance between the points, i.e. $r \equiv | \mathbf{r} |$. However, this is not the case when studying a sample from a redshift galaxy survey. Although the galaxy distribution is intrinsically isotropic, the way in which it is measured is not, as the line-of-sight component of each position is derived from the observed redshift. A way around this issue is the use of the projected correlation function $w_{\rm p}(r_{\rm p})$, first introduced by @dav83a to deal with the redshift-space effects present in spectroscopic samples [@kai87a; @ham98a]. As shown in @arn09a, this same approach can be used to deal with samples of photometric redshifts, and we use it in this paper. In this approach, we first separate the redshift-space distance between any pair of galaxies in two components: parallel ($\pi$) and perpendicular ($r_{\rm p}$) to the line of sight.[^7] We compute the correlation function as function of these components, $\xi(r_{\rm p},\pi)$, and define the projected correlation function $w_{\rm p}(r_{\rm p})$ as $$\label{eq:1} w_{\rm p}(r_{\rm p}) \equiv 2 \int_0^{+\infty} \xi_{\rm s}(r_{\rm p}, \pi) \mathrm{d}\pi \, .$$ We estimate $\xi(r_{\rm p},\pi)$ following @lan93a. We first generate an auxiliary random Poisson process following the same selection function as our sample, as defined in Section \[sec:selfunc\]. We compute, for a given bin in the distance components $(r_{\rm p},\pi)$, the number of pairs in our galaxy catalogue ($DD$), in our random catalogue ($RR$), and the number of crossed pairs between both catalogues ($DR$). The correlation function is estimated as $$\label{eq:2} \xi(r_{\rm p},\pi) = 1 + \left(\frac{N_{\rm R}}{N_{\rm D}}\right)^2 \frac{DD(r_{\rm p},\pi)}{RR(r_{\rm p},\pi)} - 2 \frac{N_{\rm R}}{N_{\rm D}} \frac{DR(r_{\rm p},\pi)}{RR(r_{\rm p},\pi)} \, ,$$ where $N_{\rm D}$ is the number of galaxies in our sample, and $N_{\rm R}$ is the number of points in the auxiliary random catalogue. In this work, we always fix $N_{\rm R} = 20 N_{\rm D}$. We tested that our results do not change if we increase the number of random points used to $N_{\rm R} = 50 N_{\rm D}$. The projected correlation function defined in equation (\[eq:1\]) does not depend on the line-of-sight component of the separation $\pi$ and thus, to first order, is not affected by the uncertainty on the photometric redshift determination. However, in a real survey, we can not use this definition, as we can not calculate the integral in equation (\[eq:1\]) up to infinity. We calculate instead $$\label{eq:4} w_{\rm p}(r_{\rm p}, \pi_{\rm max}) \equiv 2 \int_0^{\pi_{\rm max}} \xi_{\rm s}(r_{\rm p}, \pi) \mathrm{d}\pi \, ,$$ which introduces a bias in the result, which is now dependent on the redshift-space effects. The upper limit $\pi_{\rm max}$ has to be chosen in each case with the aim of minimising this bias, but also of avoiding the introduction of too much additional noise in the calculation. In Appendix \[sec:pimax\] we explore this issue in detail for the case of photometric redshift surveys like ALHAMBRA, using both an analytical model including Gaussian photo-$z$ errors and the full mock catalogues described in Sect. \[ssec:mock-catalogues\]. We study the bias introduced by the finite integration limit, and calculate the minimum value of $\pi_{\rm max}$ needed given the statistical uncertainty in our measurements. Accounting for this study, we use throughout $\pi_{\rm max} = 200 {\, h^{-1} \, \mathrm{Mpc}}$, which is appropriate for the ALHAMBRA samples considered here. As a further test, we study the change of our results with $\pi_{\rm max}$ in Appendix \[sec:test-robustness-our\]. Hereafter, we omit the explicit dependence of $w_{\rm p}$ on the value of $\pi_{\rm max}$, and just write $w_{\rm p}(r_{\rm p}) \equiv w_{\rm p}(r_{\rm p}, \pi_{\rm max} = 200 {\, h^{-1} \, \mathrm{Mpc}})$. Integral constraint {#ssec:integral-constraint} ------------------- The integral constraint [@pee80a] is a bias in the estimation of the correlation function due to the use of a finite volume. It is related to the fact that the correlations are measured with respect to the mean density of the sample considered (the particular survey) instead of with respect to the global mean (that of the parent population). We can derive the effect of this constraint on $w_{\rm p}$ based on that of the three-dimensional correlation function $\xi$. When $\xi$ is measured using an estimator such as that of equation (\[eq:2\]), it can be shown that the bias introduced by the integral constraint is given, at first order, by [@ber02a; @lab10a] $$\label{eq:3} \xi(\mathbf{r}) = \xi^{\rm true}(\mathbf{r}) - K \, ,$$ where $$\label{eq:5} K \equiv \frac{1}{V^2} \int_V \int_V \mathrm{d}^3 \mathbf{r_1} \mathrm{d}^3 \mathbf{r_2} \xi^{\rm true}(\mathbf{r_2} - \mathbf{r_1}) \, ,$$ and $V$ is the volume of the survey. Using equation (\[eq:4\]) this translates into a bias on the estimated projected correlation function $w_{\rm p}(r_{\rm p}, \pi_{\rm max})$ which depends also on $\pi_{\rm max}$, $$\label{eq:6} w_{\rm p}(r_{\rm p}, \pi_{\rm max}) = w_{\rm p}^{\rm true}(r_{\rm p}, \pi_{\rm max}) - 2K \pi_{\rm max} \, .$$ To correct the measured values of $w_{\rm p}$ for the integral constraint using equation (\[eq:6\]), one needs to know the true underlying correlation function. Here we choose an alternative approach, by including the integral constraint correction in the models we fit to the data. In practice, we follow @Roche1999o and make use of the auxiliary Poisson catalogue to compute numerically the double integral in equation (\[eq:5\]) as $$\label{eq:7} K \simeq \frac{\sum_i RR(r_i) \xi^{\rm model}(r_i)}{\sum_i RR(r_i)} = \frac{\sum_i RR(r_i) \xi^{\rm model}(r_i)}{N_R(N_R - 1)} \, ,$$ where we use the same notation as in equation (\[eq:2\]), and where the sum is over bins in distance extending up to the largest separations in the survey. In all cases, however, we check that the value of the integral constraint correction is small compared with the errors on $w_{\rm p}$ (as can be seen in Fig. \[fig:wpresults\]), so our results are not sensitive to the details of the estimation of $K$. Error estimation {#ssec:error-estimation} ---------------- To estimate the statistical error on our $w_{\rm p}(r_{\rm p})$ measurements, we use the standard block bootstrap method [see e.g. @nor08a], making use of the fact that the survey consists of 7 totally independent fields. We generate $N_{\rm b} = 1000$ bootstrap realisations for each calculation, using the fields as bootstrap regions. Each of these realisations is created by selecting 7 fields at random, allowing for repetition. We then compute the projected correlation function for each bootstrap realisation using equations (\[eq:2\]) and (\[eq:4\]). We obtain the error of $w_{\rm p}$ at each bin in $r_{\rm p}$ as the standard deviation of the measurements from the $N_{\rm b}$ bootstrap realisations. To account for the covariance between bins in $r_{\rm p}$ when fitting a model to our data, we repeat the $\chi^2$ fitting for the $N_{\rm b}$ realisations, using only the derived diagonal errors. Our estimate of the error on each model parameter is then the standard deviation of the best values obtained for the $N_{\rm b}$ realisations. We test in Appendix \[sec:systematics\] this error estimation and model fitting procedure for the case of ALHAMBRA using the mock galaxy catalogues described in Sect. \[ssec:mock-catalogues\]. We show that it produces an unbiased estimate of the galaxy bias and of its uncertainty. We also compared our bootstrap error estimate with the standard jackknife method [see e.g. @nor08a]. We obtained that the error on $w_{\rm p}(r_{\rm p})$ estimated using both methods is consistent for $r_p \gtrsim 1 {\, h^{-1} \, \mathrm{Mpc}}$. For $r_{\rm p} \lesssim 1 {\, h^{-1} \, \mathrm{Mpc}}$ the jackknife method slightly underestimates the error with respect to the bootstrap estimate. Correlation functions for ALHAMBRA samples {#sec:results} ========================================== {width="\textwidth"} We show the resulting projected correlation functions $w_{\rm p}(r_{\rm p})$ for the different samples selected in redshift and luminosity in Fig. \[fig:wpresults\]. When comparing the results for samples at a given redshift bin we see clearly the effect of segregation by luminosity: bright galaxies are systematically more clustered than faint ones. This effect can be readily seen in all four redshift bins. Moreover, we see that all results show approximately a power-law behaviour for scales $r_{\rm p} \gtrsim 0.2 {\, h^{-1} \, \mathrm{Mpc}}$. We focus here on these scales, and leave the study of smaller scales for a later work. Power-law modelling of the correlation functions {#ssec:powlaw} ------------------------------------------------ In order to study the change of the clustering properties with luminosity and redshift, we fit the obtained projected correlation function $w_{\rm p}(r_{\rm p})$ of each sample using a power law model. Following the standard practice, we assume the real-space correlation function $\xi(r)$, is given by $$\label{eq:8} \xi^{\rm pl}(r) = \left( \frac{r}{r_0} \right)^{-\gamma} \, .$$ When transforming this model, using equation (\[eq:1\]), to a model for $w_{\rm p}(r_{\rm p})$, we also obtain a power law which, expressed in terms of the parameters $r_0$ and $\gamma$ above is given by $$\label{eq:wppowlaw} w_{\rm p}^{\rm pl}(r_{\rm p}) = r_{\rm p} \left( \frac{r_0}{r_{\rm p}} \right)^{\gamma} \frac{ \Gamma(1/2) \Gamma\left[ (\gamma - 1)/2 \right]}{\Gamma(\gamma/2)} \, ,$$ where $\Gamma(\cdot)$ is Euler’s Gamma function. Fitting the power-law model of equation (\[eq:wppowlaw\]) to our observed data, we can study the change of both the slope $\gamma$ and the correlation length $r_0$ with the properties of each sample. In practice, we modify this power-law model by adding the effect of the integral constraint described in Sect. \[ssec:integral-constraint\]. Following equation (\[eq:6\]) and leaving explicit the dependence on the model parameters $(r_0, \gamma)$, the model projected correlation function is $$\label{eq:9} w_{\rm p}^{\rm model}(r_{\rm p}|r_0, \gamma) = w_{\rm p}^{\rm pl}(r_{\rm p} | r_0, \gamma) - 2 K(r_0, \gamma) \pi_{\rm max} \, ,$$ where $w_{\rm p}^{\rm pl}(r_{\rm p} | r_0, \gamma)$ is given by equation (\[eq:wppowlaw\]), and the integral constraint term $K(r_0, \gamma)$ is obtained from equation (\[eq:7\]) using the power-law model for the three-dimensional correlation function of equation (\[eq:8\]). We fit the model of equation (\[eq:9\]) to the projected correlation function measured for our different samples in the range $0.2 < r_{\rm p} < 17 {\, h^{-1} \, \mathrm{Mpc}}$. We obtain the best-fit parameters $\gamma$, $r_0$ in each case using a standard $\chi^2$ minimisation method, and their error using the method described in Sect. \[ssec:error-estimation\]. ![Parameters $r_0$ and $\gamma$ obtained from the power-law fits for the different samples, as a function of the rest-frame $B$-band median luminosity, for each of the redshift bins.[]{data-label="fig:powlawparams"}](ArnalteMur_fig_7){width="\columnwidth"} The best-fit models obtained are shown as solid lines in Fig. \[fig:wpresults\]. The effect of the integral constrain produces a slight deviation from a straight line (in the log-log plot) at larger scales, very small compared with the errors. We plot in Fig. \[fig:powlawparams\] the resulting parameters $\gamma$, $r_0$ for each of our redshift bins, as function of the median $B$-band luminosity expressed as function of $L^{*}(z)$. From the bottom panel of Fig. \[fig:powlawparams\] we conclude that the slope $\gamma$ is approximately constant, with a value $\gamma \sim 1.75$. This is in agreement with previous studies at similar redshifts [@coi06a; @mar13a], although @pol06a found significantly steeper slopes for the brightest samples. The results for $r_0$ shown in the top panel of Fig. \[fig:powlawparams\], however, show clear evidence of luminosity segregation, as already observed qualitatively in Fig. \[fig:wpresults\]. In all cases, luminous galaxies are more clustered than faint ones. However, the change of $r_0$ with redshift is not monotonic. While the results at $z = 0.5$ and $z = 0.9$ are very similar, the bin at $z = 0.7$ shows a stronger clustering. The bin at $z = 1.1$ shows a behaviour clearly different to the other three redshift bins. On one side, the $r_0$ values for this bin are consistently smaller than those of the lower redshift bins. On the other side, its dependence on luminosity is much weaker. However, it is difficult to interpret the results for this last bin, as there is a possible selection bias affecting it. The reason for this bias is that, for this redshift range, the rest-frame $4000$ Å break is crossing the observer-frame $I$ band used for the selection of our catalogue. This means that the selection function is changing inside the redshift bin, and in particular this will affect the selection of red passive galaxies (which we expect to show a stronger clustering). We do not study further this redshift bin in this work, but will study it in more detail in Hurtado-Gil et al. (in prep.), where we focus on the clustering as function of spectral type. For the three bins at $z \leq 1$, we analyse the clustering properties in detail in the next sections. First, we separate the evolution of the clustering of the underlying matter density field from that of the bias of our different samples in Sect. \[ssec:bias\]. Then, we study the effect sample variance has on our results, and develop a more robust clustering measurement in Sect. \[sec:cvariance\]. Dependence of bias on luminosity and redshift {#ssec:bias} --------------------------------------------- We study the bias $b$ of our samples by comparing the observed projected correlation function $w_{\rm p}$ for each sample to that of the matter distribution at the corresponding median redshift $w_{\rm p}^{\rm m}$. We assume a simple linear model, in which bias is constant and independent of scale, $$\label{eq:13} w_{\rm p}(r_{\rm p}) = b^2 w_{\rm p}^{\rm m}(r_{\rm p}) \, .$$ We restrict our study to the bias in the range $1 < r_{\rm p} < 10 {\, h^{-1} \, \mathrm{Mpc}}$, corresponding mainly to the two-halo term of the correlation function. We leave a more detailed study using the full halo occupation distribution (HOD) formalism [@sco01a; @ber02b] for a future work. We note however, that previous works have shown that the value of the bias obtained using our method is consistent to that using the HOD modelling [@zeh10a]. We use a model for $w_{\rm p}^{\rm m}$ based on $\Lambda$CDM and using values of the cosmological parameters consistent with the WMAP7 results [@kom11a]. In particular, we use a normalisation of the power spectrum $\sigma_8 = 0.816$. We obtain the matter power spectrum at each redshift using the <span style="font-variant:small-caps;">Camb</span> software [@lew00a], including the non-linear corrections of <span style="font-variant:small-caps;">Halofit</span> [@smi03a]. We then Fourier-transform the power spectrum to obtain the real-space correlation function $\xi(r)$ of matter, and finally obtain the projected correlation function using equation (\[eq:1\]). We perform a $\chi^2$ fit to the model in equation (\[eq:13\]) as described in Sect. \[ssec:error-estimation\], keeping our model $w_{\rm p}^{\rm m}(r_{\rm p})$ fixed and with $b$ as the only free parameter. In this case, we use for each sample the value of the integral constraint $K$ obtained from the best-fit power-law model, and correct the observed $w_{\rm p}(r_{\rm p})$ according to equation (\[eq:6\]). ![Top: galaxy bias for the different samples from the fit to equation (\[eq:13\]), as a function of the median luminosity. Bottom: galaxy bias as function of median redshift for the different luminosity cuts. We omit some of the samples for clarity. The horizontal error bars indicate the full extent of each redshift bin. The solid lines correspond to the bias of haloes above a given mass according to the model of @mo02a. The label for each of these lines indicates the minimum halo mass in terms of $\log_{10}[M_{\rm h}/({\, h^{-1} \, \mathrm{M}_{\sun}})]$.[]{data-label="fig:bias"}](ArnalteMur_fig_8a "fig:"){width="\columnwidth"} ![Top: galaxy bias for the different samples from the fit to equation (\[eq:13\]), as a function of the median luminosity. Bottom: galaxy bias as function of median redshift for the different luminosity cuts. We omit some of the samples for clarity. The horizontal error bars indicate the full extent of each redshift bin. The solid lines correspond to the bias of haloes above a given mass according to the model of @mo02a. The label for each of these lines indicates the minimum halo mass in terms of $\log_{10}[M_{\rm h}/({\, h^{-1} \, \mathrm{M}_{\sun}})]$.[]{data-label="fig:bias"}](ArnalteMur_fig_8b "fig:"){width="\columnwidth"} The top panel of Fig. \[fig:bias\] shows the value of the bias obtained as function of the median luminosity of the sample for each of the three redshift bins considered. Not surprisingly we see again the effect of luminosity segregation for all redshift bins, like for $r_0$ (Fig. \[fig:powlawparams\]). In the bottom panel of Fig. \[fig:bias\], we show the bias as function of redshift for a few of our luminosity-selected samples. For comparison, we show the bias of haloes of different masses according to the model of @mo02a. For the samples with faintest luminosities, the evolution of bias with redshift is not significant. For the brightest samples, however, the bias does change with redshift. This evolution is not monotonic, as it seems to have at maximum at $z \sim 0.7$. Given our uncertainties, this result is not very significant. However, we study in the next section whether this behaviour is due to the effects of sample variance, and in particular to the contribution of any particular ALHAMBRA field. Analysis of the impact of sample variance on the clustering results {#sec:cvariance} ------------------------------------------------------------------- The use of 7 independent fields in the ALHAMBRA survey is an opportunity to study the effect of sample variance. Regarding our clustering measurements, we have already used the fact that we have data in several independent fields to estimate the errors in our results, as explained in Sect. \[ssec:error-estimation\]. However, this is based only on a global measure of the variance of the measurements (through the use of the bootstrap technique). We can go one step further and study the impact of individual fields on our final measurements. Given the relatively small volume of the survey and, especially, the typical size of the fields, the presence of a large structure in one of the fields could significantly affect our clustering measurement in a given redshift bin. Similar studies have been performed with other surveys. For example, when using data from the SDSS, @zeh10a studied the effect on their results of including or avoiding the SDSS ‘Great Wall’ [@Gott2005a]. @Wolk2013i performed a similar study for the case of higher-order statistics. To study the impact of these large structures in our measurements we use the jackknife ensemble fluctuation statistic introduced by @Norberg2011g. This statistic is designed as an objective way of identifying ‘outlier regions’: those that, due to the presence of a superstructure, dominate the clustering signal of the whole survey. In the case of ALHAMBRA, it seems natural to take as jackknife regions our 7 independent fields. We present here a basic description of this statistic as used in our case for the projected correlation function, but a more detailed description can be found in @Norberg2011g. For a given sample, we start by computing the projected correlation function removing from the survey a given field $i$, $w_{\rm p}^i(r_{\rm p})$, and the corresponding rescaled quantity $$\label{eq:10} \Delta_i(r_{\rm p}) = \frac{w_{\rm p}^i(r_{\rm p}) - w_{\rm p}^{\rm full}(r_{\rm p})}{w_{\rm p}^{\rm full}(r_{\rm p})} \, ,$$ where $w_{\rm p}^{\rm full}(r_{\rm p})$ refers to the projected correlation function measured from the full sample. This *jackknife re-sampling fluctuation* $\Delta_i(r_{\rm p})$ therefore quantifies the relative change in $w_{\rm p}$ due to the exclusion of a given field. To assess the significance of this change, we define the quantity $\sigma_{{\rm tot} - i}^2(r_{\rm p})$ as the rms error of this resampling fluctuations, omitting field $i$, $$\label{eq:11} \sigma_{{\rm tot} - i}^2(r_{\rm p}) = \frac{1}{N_{\rm fields} - 1} \sum_{j \neq i}^{N_{\rm fields} - 1} \Delta_j^2(r_{\rm p}) \, .$$ In the case of the ALHAMBRA catalogue used here, $N_{\rm fields} = 7$. We finally define the *jackknife ensemble fluctuation* $\delta_i$ as the re-sampling fluctuation normalised to its error $$\label{eq:12} \delta_i(r_{\rm p}) = \frac{\Delta_i(r_{\rm p})}{\sigma_{{\rm tot} - i}(r_{\rm p})} \, .$$ This is a direct measure of how significant the change in the clustering result for a given sample is when a given field $i$ is either included or excluded. @Norberg2011g define an ‘outlier region’ as that for which $\left| \delta_i \right| > 2.5$, where $\delta_i$ is averaged over the range of scales of interest. We adopt this same limit to define ‘outlier fields’ in the case of ALHAMBRA. This choice is somehow arbitrary, as full $N$-body simulations would be needed to test the needed value in this case, as done in @Norberg2011g. ![Ensemble fluctuation $\delta_i$ averaged over the range $r_{\rm p} \in [1,10] {\, h^{-1} \, \mathrm{Mpc}}$ for the different redshift bins, as function of the excluded field. These results correspond to the samples selected with $M_B(z=0) < -19.6$, for which $L^{\rm med} \sim L^{*}$. The dashed lines denote our limits $\left|\delta_i\right| = 2.5$ to identify a field as an ‘outlier’.[]{data-label="fig:delta_aver"}](ArnalteMur_fig_9){width="\columnwidth"} We computed the jackknife ensemble fluctuation $\delta_i$, averaged over the range $1 < r_{\rm p} < 10 {\, h^{-1} \, \mathrm{Mpc}}$ (the same range used to estimate the bias) for the samples selected by $M_B(z=0) < -19.6$ in our three redshift bins, corresponding to $L^{\rm med} \simeq L^*(z)$. However, as the effects we measure here are due to sample variance, we obtain consistent results when using a different luminosity cut. We show the results, for the different ALHAMBRA fields, in Fig. \[fig:delta\_aver\]. As expected, in most cases we obtain values $\left| \delta_i \right| \lesssim 1$ corresponding to the expected variance. However, we can use the criterion explained above to identify outliers in an iterative way. The first outlier we identify is the ALH-4 field, for which we obtain the largest value of $\left| \delta_i \right|$, $\delta_i = -5.01$ for the redshift bin centred at $z = 0.9$. Once this outlier field is identified, we exclude it from the calculation, and repeat the measurement of $\delta_i$. Using these new values, we identify an additional outlier: the ALH-7 field, for which we now obtain $\delta_i = -3.45$ for the redshift bin centred at $z = 0.7$. The original value for this field and redshift bin, when we included also ALH-4 in the calculation, was $\delta_i = -2.74$. We repeat the process again, excluding both the ALH-4 and ALH-7 fields from the calculation, and find now in all cases values of $\left|\delta_i \right| \leq 1.73$, which we interpret as all fields being equally consistent with each other. The most obvious outlier is the ALH-4/COSMOS field. The large negative value of $\delta_i$ obtained means that the inclusion of this field in the survey produces a very significant increase in the measured clustering for this bin. This is consistent with the fact that previous studies of clustering in the COSMOS survey at similar redshifts have obtained values significantly larger than other similar surveys [@McCracken2007a; @men09a; @tor10a; @Skibba2013c]. The excess clustering can be explained by the presence of large over-dense structures in this field [@Guzzo2007a; @Scoville2007d; @Kovac2010a]. In fact, taking into account the particular area covered by the ALH-4 field, we obtain that the four largest structures found by @Scoville2007d [see their table 3] are partially included in our sample. The central redshifts estimated for these structures are $z = 0.73, 0.88, 0.93, 0.71$, so all of them have substantial overlap with the redshift bin $0.75 < z < 1.05$ where we identify this field as an outlier. The particularly large over-density of this field is also observed in ALHAMBRA. The surface density of galaxies is significantly larger in this field than in the rest, as shown in Table \[tab:fields\]. Moreover, the redshift distribution $N(z)$ of this field shows a broad peak centred at $z \sim 0.8$ when compared to the global ALHAMBRA $N(z)$ [see figure 32 in @Molino2013a]. The second ‘outlier’ is the ALH-7/European Large Area ISO Survey North 1 (ELAIS-N1) field. Unfortunately, this field is not as well studied as COSMOS and, to the best of our knowledge, there are no previous studies of clustering or identification of large structures at these redshifts. However, we also find a peak in the density of clusters and groups in this field at $z\sim0.7$ using the same ALHAMBRA data set (see Ascaso et al., in prep., for details), indicating the presence of a large structure at this particular redshift, which could explain the particularly large clustering observed here. ![Both panels are identical to these in Fig. \[fig:bias\], for the case in which we totally omit from the calculation the ‘outlier’ fields ALH-4/COSMOS and ALH-7/ELAIS-N1.[]{data-label="fig:bias_noalh47"}](ArnalteMur_fig_10a "fig:"){width="\columnwidth"} ![Both panels are identical to these in Fig. \[fig:bias\], for the case in which we totally omit from the calculation the ‘outlier’ fields ALH-4/COSMOS and ALH-7/ELAIS-N1.[]{data-label="fig:bias_noalh47"}](ArnalteMur_fig_10b "fig:"){width="\columnwidth"} Figure \[fig:bias\_noalh47\] shows the bias of our samples (measured as described in Sect. \[ssec:bias\]) as function of their median luminosity and redshift, when we completely omit from the calculation the ‘outlier fields’ ALH-4 and ALH-7. We can compare this figure directly to Fig. \[fig:bias\], where we considered the whole survey. We obtain results very similar to the whole survey for the bin centred at $z = 0.5$. This was expected from the results in Fig. \[fig:delta\_aver\]: the low values of $\left| \delta_i \right|$ for the fields ALH-4 and ALH-7 in this case indicated that removing them would not significantly change the result. However, we see significant differences for the bins where the removed fields were ‘outliers’, at $z = 0.7$ and $z = 0.9$. In this case, the bias obtained is smaller now. The dependence of the bias on luminosity, however, does not change significantly except for the overall normalisation. This is due to the fact that, for a given redshift bin, we expect sample variance to affect in the same way all the samples regardless of the luminosity selection. The error on the bias computed using the bootstrap method has also been greatly reduced. This was also expected: as we eliminated the greatest outliers, the variance of the remaining measurements is reduced. However, we note that the original error estimate for the full survey was also affected by the presence of the ‘outlier’ fields, as these imply a very non-Gaussian error distribution. From the bottom panel of Fig. \[fig:bias\_noalh47\] we can analyse the evolution of the bias in this case. For the faintest samples we obtain now an even weaker evolution of the bias. For the brightest ones we see again a clear variation of bias with redshift, but the observed trend is somewhat different to that seen in Fig. \[fig:bias\]. Now, for our three bins at $z<1$, we see a roughly monotonic trend, with bias increasing with increasing redshift. Overall, the evolution observed in Fig. \[fig:bias\_noalh47\] is similar to the bias evolution for haloes above a given mass, according to the model of [@mo02a]. According to that model, the bias we obtain for our different samples correspond to populations of haloes with minimum masses in the range $11.5 \lesssim \log_{10}[M_{\rm h}/({\, h^{-1} \, \mathrm{M}_{\sun}})] \lesssim 13.0$. The bias of galaxies with $L^{\rm med} \simeq L^{*}$ roughly corresponds to that of a halo population with $\log_{10}[M_{\rm h}/({\, h^{-1} \, \mathrm{M}_{\sun}})] \gtrsim 12.2$. ![Galaxy bias as a function of the number density of galaxies for our different samples (points). Galaxy bias is obtained from the fit to equation (\[eq:13\]), for the case in which we omit the ‘outlier’ fields ALH-4/COSMOS and ALH-7/ELAIS-N1. The lines show the prediction of the model of @mo02a for haloes above a given mass. Continuous lines show the prediction for fixed values of the redshift (indicated by the labels in the left). Dashed lines correspond to the prediction for fixed values of the minimum halo mass (indicated by the labels in the bottom, in terms of $\log_{10}[M_{\rm h}/({\, h^{-1} \, \mathrm{M}_{\sun}})]$). Comparing these predictions for haloes to the observed values, we obtain that the typical mean occupation numbers for the ALHAMBRA galaxies are in the range $\sim 1-3$.[]{data-label="fig:ndens_bias"}](ArnalteMur_fig_11){width="\columnwidth"} To further investigate the relationship between our galaxy samples and the halo populations, we show in Fig. \[fig:ndens\_bias\] the bias of our samples as a function of their number density. We compare our results to the prediction for populations of haloes above a given minimum mass from the model of @mo02a, shown as the continuous (for fixed redshift) and dashed (for fixed minimum mass) lines in the plot. We can estimate roughly the halo occupation number (i.e. the mean number of galaxies per halo) for a given sample by comparing its number density to that of the halo population at the same redshift and with similar bias. For the different ALHAMBRA samples, we obtain that the occupation numbers are typically in the range $\sim 1 - 3$, although with a large uncertainty due to our uncertainty in the bias measurement. Comparison to previous results from other surveys {#sec:othersurveys} ------------------------------------------------- We compared our results with previous studies using the largest galaxy surveys to date covering similar redshifts. @Coupon2012a studied galaxy clustering in the range $0.2 < z < 1.2$ using data from CFHTLS-Wide[^8], a broad-band photometric survey covering $\sim 155 \deg^2$. The bias was derived in each case by fitting a HOD model to the angular correlation function of each sample. @mar13a measured the clustering using spectroscopic data from VIPERS[^9] covering $\sim 15 \deg^2$, in the range $0.5 < z < 1.1$. They measured the bias from the measured projected correlation function in the same way as we do here (equation \[eq:13\]), and showed that their results were in rough agreement with other (smaller) spectroscopic surveys at similar redshifts such as DEEP2 [@coi06a] and VVDS [@pol06a]. In both cases, the depth of the data used was $i < 22.5$. We note that the area covered by VIPERS is a subset of that covered by CFHTLS Wide. The ALH-6 field also overlaps with CFHTLS-Wide. We also included in this comparison the results in the range $0.5 < z < 1.0$ of @Skibba2013c using data from PRIMUS. PRIMUS [@Coil2011a] is a survey which uses a low-resolution spectrograph resulting in a typical redshift precision of $\sigma_z/(1+z) = 0.005$ to a depth of $i < 23$. The data covers five independent fields (including the COSMOS field) covering a total[^10] of $7.80 \deg^2$. @Skibba2013c measured the bias of different samples selected in redshift, luminosity and colour using the projected correlation function in the same way as we describe above (equation \[eq:13\]). ![Galaxy bias comparison between ALHAMBRA (this work), VIPERS [@mar13a], CFHTLS-Wide [@Coupon2012a] and PRIMUS [@Skibba2013c]. The solid line in each panel corresponds to the low redshift SDSS results of @zeh10a. The bias measurements have been re-normalised to the fiducial value $\sigma_8^{\rm fid} = 0.816$ used in this work.[]{data-label="fig:bias_comp"}](ArnalteMur_fig_12) In Fig. \[fig:bias\_comp\] we plot the bias obtained in our different redshift bins as a function of the threshold luminosity $L^{\rm th}$ used to select the different samples in ALHAMBRA, CFHTLS Wide, VIPERS and PRIMUS. $L^{\rm th}/L^{*}$ is measured at the median redshift of the sample, taking into account the use of different selection parameters $A$ in equation (\[eq:p1alh:Mthevol\]). We note that $L^{\rm th}$ refers to the $B$ band in the case of ALHAMBRA and VIPERS, and to the $g$ band in the case of CFHTLS-Wide and PRIMUS. In each case, we compare the ALHAMBRA results with the CFHTLS Wide results for the bin centred at the same redshift. As @mar13a used bins centred at different redshifts, we plot in each case the one or two closest bins to the ALHAMBRA one. In the case of PRIMUS, the actual redshift range of each sample is slightly different with mean redshifts in the range $0.60 - 0.74$, so we plot their results in the central panel. In each case, we re-normalise the bias by the value of $\sigma_8$ considered. Changes in bias due to other differences in the cosmology used are much smaller than our errors. For reference, we also plot as a continuous line the relation derived for low redshifts by @zeh10a from the SDSS data, which is very similar to that obtained by @nor01a from the 2dFGRS. We plot the ALHAMBRA results both for the full survey (dashed lines) and for the case in which we have removed the two ‘outlier fields’ ALH-4 and ALH-7 (solid lines). We obtain a good agreement between our results and both the CFHTLS-Wide and VIPERS ones, especially considering the significantly smaller area surveyed by ALHAMBRA. When looking at the $z=0.7$ and $z=0.9$ bins, we see how the result obtained after omitting the outlier fields is in better agreement with the other data than the original results. This confirms the idea that using the jackknife ensemble fluctuation to identify outlier regions results in a good measurement of the typical clustering properties (bias in this case) of the samples. We point out that the comparison presented here was performed only after the full analysis of ALHAMBRA data was finished, so it did not influence the design of the method described in Sect. \[sec:cvariance\]. Our results are also in very good agreement with the PRIMUS results. We note that PRIMUS obtained slightly larger values of the bias than CFHTLS-Wide or VIPERS, and they attributed this fact to the presence of the COSMOS field in their sample. This is compatible with their results lying between our results with and without the outliers fields included. We note, however, that the dependence of bias on luminosity appears to be slightly steeper in ALHAMBRA than in previous works. This is noticeable at the bright end of the $z=0.9$ bin. It is difficult to assess the significance of this discrepancy, as our bias error estimate is affected by the removal of the ‘outlier’ fields, and the different measurements are highly correlated. With these caveats in mind, we estimate that the discrepancy for the most extreme case is at the $\lesssim 2 \sigma$ level. Given its small area, the ALHAMBRA survey is not designed to provide an accurate measurement of low number density samples, nor is the error analysis necessarily adequate for them either. The lowest number density samples (i.e. bright galaxies) require large survey areas to be properly estimated. Fig. \[fig:bias\_comp\] shows the complementarity between the different surveys covering this redshift range to study the dependence of galaxy bias on luminosity and redshift. Large area surveys such as CFHTLS-Wide and VIPERS can measure very accurately the bias of relatively bright samples, $L \gtrsim 0.3 L^{*}(z)$, thus setting the overall normalisation of the $b(L)$ relation at each redshift. Despite its smaller volume, ALHAMBRA can extend this relation to luminosities $\simeq 1.5 {\rm mag}$ fainter, with our study of the outliers showing that the result is robust to sample variance, except for the overall normalisation. This larger luminosity range in ALHAMBRA allows us to see clearly the transition from a nearly flat relation at the faint end to a steep one at the bright end. Discussion and conclusions {#sec:conc} ========================== In this work, we have studied the clustering of galaxies in the ALHAMBRA survey and its dependence on luminosity and redshift, in the range $0.35 < z < 1.25$. To this end, we have used the projected correlation function $w_{\rm p}(r_{\rm p})$, taking into account the uncertainties associated with the use of photometric redshifts, following the method described in @arn09a. We have compared the measured $w_{\rm p}(r_{\rm p})$ to the prediction from our fiducial $\Lambda$CDM model to estimate the bias for the different samples selected in redshift and luminosity. We also used the method introduced in @Norberg2011g to study the effect on the clustering measurements of superstructures located in particular ALHAMBRA fields. The use of the projected correlation function for the case of high-quality photometric redshifts was tested in @arn09a using a simulated halo catalogue. Here, we have tested the method using more realistic galaxy mock catalogues (Appendix \[sec:systematics\]), and have applied it to real data from the ALHAMBRA survey. We obtain results that are consistent with larger-area surveys (Sect. \[sec:othersurveys\]), and in particular the VIPERS spectroscopic survey, while reaching $1.5$ mag deeper. This confirms the reliability of the method, and shows that surveys using a large number of medium-band filters can provide very useful data sets for the study of galaxy clustering. In addition to further results from ALHAMBRA, this indicates good prospects for the planned Javalambre-Physics of the Accelerating Universe Astrophysical Survey[^11] [J-PAS, @ben08a] and Physics of the Accelerating Universe[^12] [PAU, @Castander2012] surveys, which will use a similar technique covering larger cosmological volumes. One of the main characteristics of the ALHAMBRA survey is the mapping of 8 independent fields in the sky (although only 7 are available in the current data set), which provide a useful tool to study the effect of sample variance. We have studied this issue in two complementary ways. On one side, we have used the independence of the fields to obtain a global measure of the clustering uncertainty using the block bootstrap technique described in Sect. \[ssec:error-estimation\]. On the other side, we used the jackknife ensemble fluctuation statistic $\delta_i$ [@Norberg2011g] to assess the impact of particular superstructures in the clustering measurements. This method is based on measuring the clustering omitting one region (field in our case) at a time and comparing it to the global result. In this way, we have identified the fields ALH-4/COSMOS (at $z \sim 0.9$) and ALH-7/ELAIS-N1 (at $z \sim 0.7$) as ‘outliers’, as the inclusion or omission of each of them changes our results significantly. We therefore provide also the results for the bias of our samples when we omit these two fields from the calculation, which give a better description of the ‘typical’ clustering properties of the samples, as evidenced by the comparison with the VIPERS and CFHTLS-Wide surveys. One may want to discuss which is the ‘correct’ result for the bias from this work: that obtained using the full sample (Fig. \[fig:bias\]) or that obtained omitting the outlier fields (Fig. \[fig:bias\_noalh47\]). However, it is the combination of both approaches what gives a more complete view of the information about clustering contained in the survey. On one side, the results obtained after removing the outliers provide information about the typical dependence of galaxy bias on redshift and luminosity. This is confirmed by the comparison to surveys covering larger volumes, discussed in Section \[sec:othersurveys\]. On the other side, the results for the global sample show how this typical behaviour can be affected by the inclusion or omission of particular fields containing extreme super-structures. However, the relatively small number of fields covered by ALHAMBRA, and the fact that we only identify either none or one field as an outlier in each of the redshift bins, does not allow us to assess how rare these super-structures are. Our clustering results give a detailed picture of the dependence of galaxy bias on both luminosity and redshift, summarised in Figs. \[fig:bias\_noalh47\] and \[fig:bias\_comp\]. The depth and photometric redshift reliability of the ALHAMBRA survey allow us to extend the study of the bias to fainter luminosities than previous surveys at similar redshifts. In this way, the full dependence of bias with luminosity is more clearly seen. Moreover, our results in Sect. \[sec:cvariance\] show that this dependence is reliable, and not significantly affected by sample variance. At the faint end this relation is nearly flat, up to $L^{\rm med} \simeq L^{*}$ for $z = 0.5$, and up to $L^{\rm med} \simeq 0.5 L^{*}$ for higher redshifts. At brighter luminosities, the bias increases, following a dependence on $L$ which, for $z = 0.7$ and $z = 0.9$, is significantly steeper than the relation found at low redshift by the SDSS and 2dFGRS surveys. Regarding the evolution of bias, we see very little dependence of bias with redshift for the faint samples ($L^{\rm med} \lesssim 0.8 L^{*}$), while the evolution is strong for the brighter samples. In the latter case, for samples with a approximately fixed number density, bias decreases with cosmic time. This behaviour is consistent with that expected from the halo model, where the bias of the more massive haloes shows much stronger evolution than that of the less massive ones, as illustrated in Figs. \[fig:bias\] and \[fig:bias\_noalh47\]. The comparison of our results with the predicted bias of haloes according to the model of @mo02a suggests that the galaxies studied reside in haloes covering a range in mass between $\log_{10}[M_{\rm h}/({\, h^{-1} \, \mathrm{M}_{\sun}})] \gtrsim 11.5$ (for the samples selected with $M_B(z=0) < -17.6$) and $\log_{10}[M_{\rm h}/({\, h^{-1} \, \mathrm{M}_{\sun}})] \gtrsim 13.0$ (for the samples selected with $M_B(z=0) < -20.6$). The samples with $L^{\rm med} \simeq L^{*}$ ($M_B(z=0) < -19.6$) are found to correspond to haloes with mass $\log_{10}[M_{\rm h}/({\, h^{-1} \, \mathrm{M}_{\sun}})] \gtrsim 12.2$. From the joint comparison of the bias and number density of our samples to the theoretical prediction for haloes, we obtain that the mean number of galaxies per halo is in the range $\sim 1-3$. We excluded from this detailed study of the luminosity dependence of the galaxy bias the redshift bin centred at $z = 1.1$. As explained in Sect. \[ssec:powlaw\], this is due to the fact that our $I$-band selection could be biasing the sample in that redshift range, affecting in a different way active and passive galaxies. In this paper, we have focused the study of galaxy clustering in ALHAMBRA on the effect of luminosity segregation and evolution up to $z \sim 1$. In a companion paper (Hurtado-Gil et al., in prep.) we use this same data set to study the segregation by spectral type in a similar redshift range. We also plan to extend this work to further redshifts by the use of a NIR-selected catalogue, which will allow us to study the clustering of extremely red objects (EROs, Nieves-Seoane et al., in prep.). Acknowledgements {#acknowledgements .unnumbered} ================ This work is based on observations collected at the German-Spanish Astronomical Center, Calar Alto, jointly operated by the Max-Planck-Institut für Astronomie (MPIA) and the Instituto de Astrofísica de Andalucía (CSIC). PAM was supported by an ERC StG Grant (DEGAS-259586). PN acknowledges the support of the Royal Society through the award of a University Research Fellowship and the European Research Council, through receipt of a Starting Grant (DEGAS-259586). This work was supported by the Science and Technology Facilities Council (grant number ST/F001166/1), by the Generalitat Valenciana (project of excellence Prometeo 2009/064), by the Junta de Andalucía (Excellence Project P08-TIC-3531) and by the Spanish Ministry for Science and Innovation (grants AYA2010-22111-C03-01 and CSD2007-00060). This work used the DiRAC Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grant ST/H008519/1, and STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure. Optimal value of $\pi_{\rm max}$ for the estimation of the projected correlation function {#sec:pimax} ========================================================================================= Theoretical determination of the minimum $\pi_{\rm max}$ needed {#sec:analyt-model-determ} --------------------------------------------------------------- As explained in Sect. \[sec:projcf\], it is not possible to estimate the integral of equation (\[eq:1\]) without choosing a finite upper limit $\pi_{\rm max}$, and computing instead $w_{\rm p}(r_{\rm p},\pi_{\rm max})$, as defined in equation (\[eq:4\]). This introduces a bias that has to be accounted for in the modelling. At the same time, if we extend the measurement to large values of $\pi$ where the signal-to-noise ratio of $\xi(r_{\rm p}, \pi)$ is small, we would be introducing additional noise in the measurement. In this appendix, we study the relation of this bias with the photo-$z$ errors of the catalogue used, and what is the minimum value of $\pi_{\rm max}$ needed in the case of ALHAMBRA. To this end, we use the mock catalogues described in Sect. \[ssec:mock-catalogues\], which include photo-$z$ for the galaxies with similar properties to the real data, and a simple analytic model. In this appendix, we use the cosmological parameters used in the creation of the mocks, $\Omega_{\rm M} = 0.25$, $\Omega_{\Lambda} = 0.75$, $\sigma_8 = 0.9$. From equations (\[eq:1\]) and (\[eq:4\]), the bias introduced by the finite integration is given by $$\label{eq:14} \Delta w(r_{\rm p}, \pi_{\rm max}) \equiv w_{\rm p}(r_{\rm p}) - w_{\rm p}(r_{\rm p}, \pi_{\rm max}) = 2 \int_{\pi_{\rm max}}^{+\infty} \xi(r_{\rm p}, \pi) \mathrm{d}\pi \, .$$ In principle, given a model for $\xi(r_{\rm p}, \pi)$, one could do the same finite integration in the model, and obtain a prediction directly for $w_{\rm p}(r_{\rm p}, \pi_{\rm max})$. However, in this case $w_{\rm p}(r_{\rm p}, \pi_{\rm max})$ is not a real space quantity any longer, and it depends on the way in which redshift space distortions (due to peculiar velocities and photo-$z$) are included in the model. If we want to avoid this and keep the statistic used as a real space quantity, we should choose a value of $\pi_{\rm max}$ such that the bias $\Delta w(r_{\rm p}, \pi_{\rm max})$ is negligible. Given the difficulties to model in detail the effect of the photo-$z$ distribution in $\xi(r_{\rm p}, \pi)$, we follow here the latter approach. We use a galaxy sample selected in redshift and absolute magnitude from the mock catalogues with the limits $0.5 < z_{\rm p} < 0.8$, $M_B -5 \log_{10} h < -17.95$, which is similar to our sample ‘Z07M1’. Following the same method described in Sect. \[sec:projcf\], we obtain the correlation function $\xi(r_{\rm p}, \pi)$ for the 50 realisations, and for the combined $200 \deg^2$ mock. We compare the mock results with a simple analytic model obtained using the following steps. First, we obtain the matter power spectrum $P_{\rm m}(k)$ at the median redshift of the sample using <span style="font-variant:small-caps;">Camb</span> [@lew00a]. We then obtain the real-space galaxy power spectrum $P_{\rm g}(k)$ using a simple HOD model, as described in @abb10a. We include the large-scale redshift-space effects following @kai87a to obtain the redshift-space correlation function $\xi_{\rm s}(r_{\rm p}, \pi)$. Finally, we include the effect of the photometric redshifts assuming a simple model in which the redshift errors follow a Gaussian distribution. In this model, the observed correlation function is given by the convolution $$\label{eq:15} \xi_{\rm phot}(r_{\rm p}, \pi) = \int_{-\infty}^{+\infty} \xi_{\rm s}(r_{\rm p}, \pi')f_{\sigma_{\rm pw}}(\pi - \pi') \mathrm{d}\pi' \, ,$$ where $f_{\sigma}(x)$ is the Gaussian distribution of width $\sigma$. In this case, the width of the distribution is given by the pair-wise photometric redshift uncertainty $\sigma_{\rm pw} = \sqrt{2}r(\sigma_z)$. As explained in Sect. \[ssec:samples\], $r(\sigma_z)$ is the comoving separation corresponding to a given photometric redshift uncertainty $\sigma_z$ at the median redshift of the sample. We choose the HOD parameters of this model (including the bias) to reproduce the observed $w_{\rm p}(r_{\rm p})$ of the mock. We use both the results from the mock and the analytical model to compute the finite integration bias $\Delta w(r_{\rm p}, \pi_{\rm max})$ defined in equation (\[eq:14\]), and find the minimum value of $\pi_{\rm max}$ for which this bias is sufficiently small. We express this requirement in terms of the statistical error on $w_{\rm p}(r_{\rm p})$, by requiring the bias to be smaller than 20% of the estimated statistical uncertainty, $$\label{eq:17} \frac{\Delta w(r_{\rm p}, \pi_{\rm max})}{w_{\rm p}(r_{\rm p})} \leq 0.2 \sigma_{w_{\rm p}}^{\rm r}(r_{\rm p}) \, ,$$ where $\sigma_{w_{\rm p}}^{\rm r}(r_{\rm p})$ is the relative uncertainty in the measurement of $w_{\rm p}(r_{\rm p})$ in a single mock ALHAMBRA realisation obtained from the dispersion of the measurements in the 50 mocks. ![Minimum value of $\pi_{\rm max}$ needed to fulfil our condition (\[eq:17\]), as function of transverse separation $r_{\rm p}$, for our different models. The $\pi_{\rm max}$ in each case is expressed in terms of $r(\sigma_z)$, the comoving separation corresponding to the photometric redshift error $\sigma_z$ of the model, at the median redshift of the bin considered (in this case, $z_{\rm med} \simeq 0.65$). The dashed lines correspond to the theoretical model described in the text, which include the effect of the photometric redshifts using a Gaussian distribution as shown in equation (\[eq:15\]). The solid line and points correspond to the measurement in the mock catalogue (combining the 50 ALHAMBRA realisations). For the selected mock sample, we estimate $\sigma_z = 0.025 (1 + z)$, using the method described in Sect. \[ssec:photo-z\].[]{data-label="fig:pimax"}](ArnalteMur_fig_A1){width="\columnwidth"} In Fig. \[fig:pimax\] we plot the minimum value of $\pi_{\rm max}$ needed to fulfil condition (\[eq:17\]), as function of $r_{\rm p}$, for our different models: the computation from the combined $200 \deg^2$ mock, and the analytic Gaussian model with different values of the photometric redshift error between $\sigma_z / (1+z) = 0.010$ and $\sigma_z / (1+z) = 0.025$. In the case of the mock catalogue we estimate $\Delta w(r_{\rm p}, \pi_{\rm max})$ using as the reference $w_{\rm p}(r_{\rm p})$ in equation (\[eq:14\]) the projected correlation function obtained in real space. We estimate the typical redshift uncertainty in the mock sample using the BPZ confidence limits in the same way as explained in Sect. \[ssec:photo-z\], in particular including the correction factor of $1.3$, and obtain $\sigma_z / (1+z) = 0.025$. In all cases, we plot the value of $\pi_{\rm max}$ in terms of $r(\sigma_z)$ for each particular model. From Fig. \[fig:pimax\] we see that overall the required value of $\pi_{\rm max}$ decreases with $r_{\rm p}$. This is a consequence of our condition (equation \[eq:17\]), given by the fact that the relative error of $w_{\rm p}$ increases with $r_{\rm p}$. Comparing the different analytical models we see that the different lines are almost coincident for $r_{\rm p} \lesssim 1 {\, h^{-1} \, \mathrm{Mpc}}$, meaning that the required value of $\pi_{\rm max}$ scales linearly with $r(\sigma_z)$. At larger scales, there is a slight deviation from this proportionality. Regarding the result obtained from the mock, we see how the non-Gaussianity of the photo-$z$ error distribution has an impact on the observed correlation function $\xi(r_{\rm p}, \pi)$. This is clearly seen at the smaller scales, $r_{\rm p} \lesssim 2 {\, h^{-1} \, \mathrm{Mpc}}$, where the needed value of $\pi_{\rm max}$ is significantly larger than that predicted by the analytical Gaussian models. Here, the value of the relative error $\sigma^{\rm r}_{w_{\rm p}}$ is small, so our condition (equation \[eq:17\]) is more stringent and the extended wings of the photo-$z$ distribution have the effect of slowing down the convergence of the integral in equation (\[eq:14\]). At larger scales, $r_{\rm p} \gtrsim 2 {\, h^{-1} \, \mathrm{Mpc}}$, our condition (equation \[eq:17\]) is much weaker (because $\sigma^{\rm r}_{w_{\rm p}}$ is large), so the details of the wings of the photo-$z$ distribution are less relevant. Actually, as the mock photo-$z$ distribution is slightly more peaked at the centre than the equivalent Gaussian distribution, we obtain values of $\pi_{\rm max}$ slightly lower than in the analytic case. Overall, we see that, to fulfil our condition (equation \[eq:17\]) over the full range of scales of interest $0.2 < r_{\rm p} < 20 {\, h^{-1} \, \mathrm{Mpc}}$, the minimum value of $\pi_{\rm max}$ needed is $\pi_{\rm max} \simeq 3.5 - 4 r(\sigma_z)$. This result is in agreement with previous, less detailed estimates [@arn09a]. The particular value of $\pi_{\rm max}$ needed for each ALHAMBRA sample will depend on the details of the correlation function and its error, with the most significant effect being the change in the correlation function error $\sigma^{\rm r}_{w_{\rm p}}(r_{\rm p})$ appearing in equation (\[eq:17\]). As this error depends, at first order, on the sample volume, we repeated the calculation re-scaling it according to the volumes of the actual ALHAMBRA samples used. We obtained only minor changes in the required value of $\pi_{\rm max}$ in all cases. We can therefore estimate the minimum value of $\pi_{\rm max}$ needed for each ALHAMBRA sample from the value of $r(\sigma_z)$ in each case (see Table \[tab:samples\]). Taking $\pi_{\rm max} = 4 r(\sigma_z)$, we obtain values in the range $\pi_{\rm max} \sim 100 - 280 {\, h^{-1} \, \mathrm{Mpc}}$. As increasing the value of $\pi_{\rm max}$ also introduces additional noise in the measurement, a compromise should be made in deciding the actual value of $\pi_{\rm max}$ to use. For simplicity, we decided to use a constant value of $\pi_{\rm max}$ for all our samples, fixing it at $\pi_{\rm max} = 200 {\, h^{-1} \, \mathrm{Mpc}}$. According to the criterion described here, this value of $\pi_{\rm max}$ is adequate for all our samples, except for the faintest samples of each redshift bin. However, as shown below in Appendix \[sec:test-robustness-our\], we find the bias introduced in these cases to be still acceptable. Test of the robustness of our results with respect to changes in $\pi_{\rm max}$ {#sec:test-robustness-our} -------------------------------------------------------------------------------- ![Bias parameter obtained for the different samples from the fit to equation (\[eq:13\]), using different values of $\pi_{\rm max}$ in the estimation of the projected correlation function $w_{\rm p}(r_{\rm p})$ (equation \[eq:4\]). The solid lines and filled symbols with errorbars correspond to the calculation using $\pi_{\rm max} = 200 {\, h^{-1} \, \mathrm{Mpc}}$ and are the same as shown in the top panel of Fig. \[fig:bias\]. The dashed lines and empty symbols (slightly displaced in the horizontal direction for clarity) correspond to the calculation using $\pi_{\rm max} = 350 {\, h^{-1} \, \mathrm{Mpc}}$. We do not show the errorbars in this case, but they are consistently larger than those shown for $\pi_{\rm max} = 200 {\, h^{-1} \, \mathrm{Mpc}}$.[]{data-label="fig:bias-pi300"}](ArnalteMur_fig_A2){width="\columnwidth"} We performed an additional test of the effect of our choice of $\pi_{\rm max}$ in our results. We repeated the calculation of $w_{\rm p}(r_{\rm p})$ for all our samples using a substantially larger value, $\pi_{\rm max} = 350 {\, h^{-1} \, \mathrm{Mpc}}$, in the integration of equation (\[eq:4\]). We did not observe any significant difference in our results but obtained, as expected, larger uncertainties due to the additional noise included in the integration. As an example of the results obtained in this case, we plot in Fig. \[fig:bias-pi300\] the bias as function of luminosity for our samples obtained using different values of $\pi_{\rm max}$. The solid lines and filled symbols correspond to the results when using $\pi_{\rm max} = 200 {\, h^{-1} \, \mathrm{Mpc}}$, and match the results presented in the top panel of Fig. \[fig:bias\]. The dashed lines and open symbols are our results when we use $\pi_{\rm max} = 350 {\, h^{-1} \, \mathrm{Mpc}}$. The results in both cases are consistent, especially noting that the errors in the case of using $\pi_{\rm max} = 350 {\, h^{-1} \, \mathrm{Mpc}}$ (not shown in the figure) are consistently larger than those shown for our main results. Analysis of the reliability of the recovered bias {#sec:systematics} ================================================= We used the mock catalogues to test the full process used to estimate the bias of a given galaxy sample and its uncertainty. We perform this test using the same mock galaxy sample used in Appendix \[sec:analyt-model-determ\], selected in redshift and absolute magnitude as $0.5 < z_{\rm p} < 0.8$, $M_B - 5 \log_{10} h < -17.95$. We estimated the projected correlation function $w_{\rm p}(r_{\rm p})$ and its error for this sample in each of the 50 mock ALHAMBRA realisations available, following the method described in Sect. \[sec:projcf\]. In the calculation, we used a value of $\pi_{\rm max} = 3 r(\sigma_z)$, as discussed in Appendix \[sec:pimax\]. Given that the redshift uncertainty in this mock sample, $\sigma_z/(1+z) = 0.025$, is somewhat larger than that in the data (see Table \[tab:samples\]), this results in a value of $\pi_{\rm max} = 270 {\, h^{-1} \, \mathrm{Mpc}}$, larger than the value used for the data. However, as we have shown in Appendix \[sec:analyt-model-determ\], the optimal value of $\pi_{\rm max}$ scales with $\sigma_z$, so this value provides an adequate comparison with the calculation done with the data. ![Bias parameter obtained for the galaxy sample defined by $0.5 < z_p < 0.8$, $M_B - 5 \log h < -17.95$ in each of the 50 mock ALHAMBRA realisations. The bias and its uncertainty in each case are estimated using the projected correlation function in the same way as is done for the real data in Sects. \[sec:projcf\] and \[ssec:bias\]. The dashed line shows the mean value obtained from the 50 realisations, $\left\langle b \right\rangle = 1.230$, and the shaded area corresponds to a region of $\pm 1 \sigma_{\rm real}$ around it, where $\sigma_{\rm real} = 0.162$ is the standard deviation of the 50 values. The blue solid line corresponds to the mean bias parameter obtained from the spherically averaged correlation function $\xi(r)$ of the 50 realisations.[]{data-label="fig:bias-mockreals"}](ArnalteMur_fig_B1){width="\columnwidth"} We then estimated the bias $b$ and its uncertainty for each realisation using the method described in Sect. \[ssec:bias\], using a model for the matter correlation function matching the Millennium cosmology used in the mocks. The obtained values are shown in Fig. \[fig:bias-mockreals\]. The mean value obtained for the bias is $\left\langle b \right\rangle = 1.230$, and the standard deviation of the values from the 50 realisations is $\sigma_{\rm real}(b) = 0.162$. These values are shown as the dashed line and shaded area in Fig. \[fig:bias-mockreals\]. The values of the uncertainty estimated show a broad distribution, with a mean value of $\left\langle \sigma (b) \right\rangle = 0.159 \pm 0.007$. This shows that the block bootstrap method used provides an unbiased estimation of the galaxy bias uncertainty. Finally, we compare these values obtained for the mock catalogues to the real-space result. To this end, we compute the spherically-averaged real-space correlation function $\xi(r)$ for each of the 50 realisations, using the real position of each galaxy, instead of that estimated from its photometric redshift. We then obtain the real-space bias using a method analogous to that described in Sect. \[ssec:bias\], using $\xi(r)$ instead of $w_{\rm p}(r_{\rm p})$. The mean value of the bias obtained in this way is $\left\langle b_{\rm r} \right\rangle = 1.244 \pm 0.012$ (shown as the continuous blue line in Fig. \[fig:bias-mockreals\]). Therefore, we can estimate the bias in our measurement as $\Delta b =\left\langle b \right\rangle - \left\langle b_{\rm r} \right\rangle = -0.014$. This corresponds to $0.09 \sigma_{\rm real}(b)$, and is therefore consistent with the condition imposed in Appendix \[sec:analyt-model-determ\] to determine the value of $\pi_{\rm max}$ used (equation \[eq:17\]). We therefore conclude that the method used to recover the real-space clustering (and in particular the galaxy bias) from the ALHAMBRA photometric redshift catalogues using the projected correlation function is reliable, as we recover the direct real-space result within the expected accuracy. Tables of numerical results {#sec:numer-results} =========================== We present in Table \[tab:numresults\] the parameters obtained from the fits of different models to the projected correlation function of our different samples. The parameters listed are the correlation length $r_0$ and exponent $\gamma$ obtained from the fit to the power-law model in equation (\[eq:9\]), and the bias $b$ obtained from the fit to the model in equation (\[eq:13\]). We list both the results obtained using the full survey (with 7 fields) and those obtained when we exclude the ‘outlier fields’ ALH-4/COSMOS and ALH-7/ELAIS-N1 (see Sect. \[sec:cvariance\] for details). For completeness, we show in Fig. \[fig:powlawparams\_noout\] the parameters $r_0$ and $\gamma$ obtained from the power-law fits when we exclude the ‘outlier’ fields’ from the calculation. This figure can be directly compared to Fig. \[fig:powlawparams\]. ![Parameters $r_0$ and $\gamma$ obtained from the power law fits (Sect. \[ssec:powlaw\]) for the different samples, as a function of the rest-frame $B$-band median luminosity, for the case in which we omit from the calculation the ‘outlier’ fields ALH-4/COSMOS and ALH-7/ELAIS-N1 (see Sect. \[sec:cvariance\]).[]{data-label="fig:powlawparams_noout"}](ArnalteMur_fig_C1){width="\columnwidth"} \[lastpage\] [^1]: http://www.caha.es [^2]: http://www.caha.es/CAHA/Instruments/LAICA [^3]: http://www.caha.es/CAHA/Instruments/O2000 [^4]: This ALHAMBRA catalogue is publicly available at http://www.alhambrasurvey.com/ [^5]: We do this assignment running <span style="font-variant:small-caps;">BPZ</span> with the <span style="font-variant:small-caps;">Only\_Type</span> option [^6]: http://space.mit.edu/$\sim$molly/mangle/ [^7]: Taking $\mathbf{s}_1$ and $\mathbf{s}_2$ to be the position vectors of the two galaxies, these components are defined as $\pi \equiv \left| \mathbf{s} \cdot \mathbf{l} \right| / \left| \mathbf{l} \right|$ and $r_{\rm p} \equiv \sqrt{\mathbf{s} \cdot \mathbf{s} - \pi^2}$, where $\mathbf{s} \equiv \mathbf{s}_2 - \mathbf{s}_1$, and $\mathbf{l} \equiv \mathbf{s}_2 + \mathbf{s}_1$. [^8]: http://www.cfht.hawaii.edu/Science/CFHLS/ [^9]: http://vipers.inaf.it/ [^10]: This corresponds to the study at $z > 0.5$, where they excluded two additional fields from their analysis. The total area covered by the survey is $9.05 \deg^2$. [^11]: http://j-pas.org/ [^12]: http://www.pausurvey.org/

--- abstract: 'We have studied crystal structure, magnetism and electric transport properties of a europium fulleride Eu$_6$C$_{60}$ and its Sr-substituted compounds, Eu$_{6-x}$Sr$_x$C$_{60}$. They have a $bcc$ structure, which is an isostructure of other $M_6$C$_{60}$ ($M$ represents an alkali atom or an alkaline earth atom). Magnetic measurements revealed that magnetic moment is ascribed to the divalent europium atom with $S$ = 7/2 spin, and a ferromagnetic transition was observed at $T_C$ = 10 - 14 K. In Eu$_6$C$_{60}$, we also confirm the ferromagnetic transition by heat capacity measurement. The striking feature in Eu$_{6-x}$Sr$_x$C$_{60}$ is very large negative magnetoresistance at low temperature; the resistivity ratio $\rho$($H$ = 9 T)/$\rho$($H$ = 0 T) reaches almost 10$^{-3}$ at 1 K in Eu$_6$C$_{60}$. Such large magnetoresistance is the manifestation of a strong $\pi$-$f$ interaction between conduction carriers on C$_{60}$ and 4$f$ electrons of Eu.' author: - Kenji Ishii - Akihiko Fujiwara - Hiroyoshi Suematsu - Yoshihiro Kubozono bibliography: - 'eu.bib' title: 'Ferromagnetism and giant magnetoresistance in the rare earth fullerides Eu$_{6-x}$Sr$_x$C$_{60}$' --- INTRODUCTION ============ Since the discovery of fullerenes, C$_{60}$ compounds have given us various opportunities for the research in condensed matter physics and materials science. Much attention was attracted to the superconductivity in $A_3$C$_{60}$ ($A$ is an alkali atom) [@Hebard1]. As for the magnetism, TDAE-C$_{60}$ (TDAE is tetrakisdimethylaminoethylene) shows a ferromagnetic transition [@Allemand1], while antiferromagnetic (or spin density wave) ground state was observed in polymeric $A_1$C$_{60}$ [@Chauvet1], Na$_2$Rb$_{0.3}$Cs$_{0.7}$C$_{60}$ [@Arcon1], and three-dimensional (NH$_3$)$A_3$C$_{60}$ [@Takenobu1]. In these compounds a magnetic moment is considered to be carried by an electron on C$_{60}$ molecule. Because various atoms and molecules can be intercalated into C$_{60}$ crystal, we also expect the magnetic C$_{60}$ compounds in which magnetic moment is carried by intercalants. In this viewpoint, rare earth metal is a good candidate. The research of rare earth fullerides was reported for Yb [@Oezdas1] and Sm [@Chen1] in relation to the superconductivity, but little effort has been made to study the magnetic properties. The only case of magnetic study in rare earth fullerides is for europium. Europium has a magnetic moment of 7$\mu_B$ ($S$ = 7/2, $L$ = 0, and $J$ = 7/2) in the divalent state, while it is non-magnetic ($S$ = 3, $L$ = 3, and $J$ = 0) in the trivalent state. A photoemission study of C$_{60}$ overlayered on Eu metal revealed the charge transfer from Eu to C$_{60}$ and the formation of fulleride [@Yoshikawa1]. Ksari-Habiles [*et al*]{}. [@Ksari1; @Claves1] investigated the crystal structure and magnetic properties of Eu$_{\sim3}$C$_{60}$ and Eu$_6$C$_{60}$; they observed some magnetic anomalies in Eu$_6$C$_{60}$. In this paper, we report the ferromagnetic transition of Eu$_6$C$_{60}$, which was observed at $T_C \sim$ 12 K in magnetic and heat capacity measurements. We also investigated the substitution effect from Eu to non-magnetic Sr, and the ferromagnetic transition temperature was found to change little with the Sr concentration. In the resistivity measurement we found a huge negative magnetoresistance below around $T_C$; the reduction ratio of resistivity $\rho(H)/\rho(0)$ is almost 10$^{-3}$ at 1 K in Eu$_6$C$_{60}$. This ratio is comparable to those in perovskite manganese oxides, which is known as colossal magnetoresistance (CMR). However Eu$_6$C$_{60}$ should be categorized as a new class of giant magnetoresistive compounds in the sense that (1) the magnitude of magnetoresistance increases very steeply with decreasing temperature rather than the vicinity of $T_C$, (2) the compound consists of a molecule with novel structure. These features can open the further possibility to find a new magnetic and magnetoresistive material. EXPERIMENTAL PROCEDURES ======================= Polycrystalline samples of Eu$_{6-x}$Sr$_x$C$_{60}$ were synthesized by solid-state reaction. A stoichiometric amount of mixture of Eu, Sr and C$_{60}$ powders, which was pressed into a pellet and sealed in a quartz tube in vacuum, was heat-treated at 600 $^{\circ}$C for about 10 days. In the course of the heat treatment the sample was ground for ensuring the complete reaction. Because the sample is very unstable in air, we treated it in a glove box with inert atmosphere. Powder x-ray diffraction experiments were carried out by using synchrotron radiation x-rays at BL-1B in Photon Factory, KEK, Tsukuba. The samples was put into a glass capillary in 0.3 mm diameter and an imaging plate was used for the detection [@Fujiwara1]. Magnetic measurements were performed using a SQUID magnetometer. In the heat capacity measurement by relaxation method, the sample was pressed into a pellet and sealed by grease to keep from exposure to air. Eu $L_{III}$-edge XANES (x-ray absorption near edge structure) was measured in the fluorescence method at BL01B1 of SPring-8, Harima. The resistivity measurements were carried out by the 4-probe method. Four gold wires were attached to a pressed pellet of polycrystalline sample with sliver paste. The sample was put into a capsule and sealed in He atmosphere. RESULTS ======= X-ray diffraction spectra of Eu$_{6-x}$Sr$_x$C$_{60}$ are shown in Fig. \[fig:structure\](a). The spectrum of Sr$_6$C$_{60}$ is also presented as a reference. The wavelength of x-ray is 0.8057 Å for $x$ = 0, 3, 5, and 0.8011 Å for $x$ = 6. They all can be understood by a $bcc$ structure which is an isostructure of other $M_6$C$_{60}$ in alkali [@Zhou1], alkaline earth[@Kortan1] and rare earth (Sm) [@Chen2] fullerides. The Rietveld refinements based on the space group $Im\overline{\it 3}$ were performed with use of the RIETAN program [@Izumi1; @Kim1]. In the refinements, only two atomic coordinates ($x$ for C1 and C3) are refined in C$_{60}$ molecule, which corresponds to the refinement of the length of 6:6 bond (the bond between two hexagons) and 5:6 bond (the bond between hexagon and pentagon). In the compounds of $x$ = 3 and 5, the sum of the metal concentration is fixed to unity. The results of refinement are presented in Table \[tab:structure\] and obtained structure is shown in Fig. \[fig:structure\](b). This crystal structure of Eu$_6$C$_{60}$ is consistent with the previous works [@Claves1; @Ootoshi1], but we observed little trace of the secondary phase in the present sample. In the Sr-substituted compounds, the values of Eu concentration are in good agreement with the nominal ones. As seen in Fig. \[fig:structure\](c), the obtained lattice constants change linearly with the nominal Eu concentration, which means they follow the Vegard’s law, and confirms the formation of solid solution at $x$ = 3 and 5. This result is attributed to the fact that ionic radius of Eu$^{2+}$ and Sr$^{2+}$ is quite similar, while the substitution of Ba for Eu results in the phase separation. Figures \[fig:magnetism\] show the result of magnetic measurements of Eu$_{6-x}$Sr$_x$C$_{60}$. Above 30 K, magnetic susceptibility ($\chi$) follows the Curie-Weiss law, as shown in Figs.\[fig:magnetism\](a)-(c). The effective Bohr magneton estimated from Curie constant and the Weiss temperature are summarized in Table \[tab:magnetism\] [@eumag]. The former agrees with the Eu$^{2+}$ state ($S$ = 7/2, $L$ = 0, and $J$ = 0). The field dependence of magnetization at 2 K gives the saturation moment close to 7$\mu_{B}$, which is consistent with the magnetic moment of Eu$^{2+}$. Moreover the Eu$^{2+}$ state has been also confirmed by Eu $L_{III}$-edge XANES experiments, as seen in Fig. \[fig:xanes\]. The spectra of EuS and Eu$_2$O$_3$ was also presented as a reference of divalent and trivalent of Eu, and absorption edges of Eu$_{6-x}$Sr$_x$C$_{60}$ are very close to that of EuS. The divalent state of Eu also observed in the case that Eu atom exists inside the C$_{60}$ cage, namely, metallofullerene Eu@C$_{60}$ [@Inoue1]. Temperature dependence of magnetization at a weak field of 3 mT (Figs. \[fig:magnetism\](g)-(i)) shows a steep increase of magnetization below 10-14 K, indicating a ferromagnetic transition. To confirm the presence of the ferromagnetic phase transition, we measured heat capacity for Eu$_6$C$_{60}$. In Fig.\[fig:magnetism\](g) we show the temperature dependence of heat capacity including that of grease. An obvious peak can be seen near the transition temperature, which is an evidence of the ferromagnetic phase transition. The $T_C$ is determined to be 11.6 K from the peak position. We ascertained that there was no anomaly in specific heat in this temperature region for grease, which was used to keep the sample from exposure to air. We can also see a smaller peak near 16 K, whose origin has not been clarified yet, but we consider that it does not come from a magnetic origin because of no anomaly in the temperature dependence of magnetization. The transition temperatures for Eu$_3$Sr$_3$C$_{60}$ and Eu$_1$Sr$_5$C$_{60}$ are estimated from the Arrott plot [@Arrott1] at 12.8 K and 10.4 K, respectively. These values are very close to the Weiss temperature mentioned above. In Eu$_6$C$_{60}$, the transition temperature estimated from the Arrott plot is a little larger value (13.7 K) than that from the heat capacity measurement, but it is not so important in the following discussion. From these evidences we conclude that Eu$_{6-x}$Sr$_x$C$_{60}$ shows a ferromagnetic transition at $T_C$ = 10-14 K, and the magnetic moment is ascribed to Eu$^{2+}$. In the previous work of Eu$_6$C$_{60}$, Ksari-Habiles [*et al*]{}. [@Ksari1] observed a mixed valence state of Eu (Eu$^{2+}$ and Eu$^{3+}$) and three successive magnetic anomalies, which is different from the present work; a possible reason is that their sample might contain a secondary phase other than Eu$_6$C$_{60}$. Figure \[fig:resistivity\] (a) show the temperature dependence of electric resistivity of Eu$_6$C$_{60}$ measured at some magnetic fields. A most striking feature in resistivity is the huge negative magnetoresistance below around $T_C$. The negative magnetoresistance becomes much more significant at lower temperature. In the case of Eu$_6$C$_{60}$, magnetoresistivity $\rho$($H$ = 9 T) is three orders magnitude smaller than $\rho$($H$ = 0 T) at 1 K, as seen in Fig.\[fig:resistivity\] (b). This large negative magnetoresistance is comparable to those of the colossal magnetoresistance (CMR) materials such as perovskite manganese oxides, where CMR effect is seen only near the ferromagnetic transition temperature. We also observed a relatively large magnetoresistance in the Sr-substituted compounds, as shown in \[fig:resistivity\] (c). There is no difference between magnetoresistances in the transverse ($H \perp I$) and longitudinal ($H \parallel I$) configurations ($I$ represents electric current), suggesting the magnetoresistance in Eu$_6$C$_{60}$ is not ascribed to the orbital motion of free carriers. DISCUSSION ========== Such giant magnetoresistance is a manifestation of the strong interaction between conduction carriers and localized magnetic moments; namely, the strong $\pi$-$f$ interaction exists in Eu$_6$C$_{60}$. When we consider formal valence state of (Eu$^{2+}$)$_6$C$_{60}^{12-}$, $t_{1g}$ band of C$_{60}$ is completely filled and the compound should become an insulator. In this case, the interaction between conduction carrier and localized moment is considered to be week, assuming that the conduction carrier mainly passes on C$_{60}$ molecules. If Eu orbitals hybridize with C$_{60}$ orbitals and form a part of conduction band, much enhancement of the interaction must occur. In the band calculation for Sr$_6$C$_{60}$ and Ba$_6$C$_{60}$ [@Saito1] which have the same $bcc$ crystal structure and the same valence state as Eu$_6$C$_{60}$, the hybridization of the $d$ orbital of metal atom and the $t_{1g}$ orbital of C$_{60}$ exists and make the compounds metallic. This fact is confirmed experimentally [@Gogia1]. The hybridization is more significant in Sr$_6$C$_{60}$ than Ba$_6$C$_{60}$ due to the smaller lattice constant of Sr$_6$C$_{60}$. The band structure of Eu$_6$C$_{60}$ has not been studied yet, but such hybridization of the 5$d$ and/or 6$s$ orbitals of Eu and the $t_{1g}$ orbital of C$_{60}$ is plausible in Eu$_6$C$_{60}$ because Eu$_6$C$_{60}$ has a further smaller lattice constant than Sr$_6$C$_{60}$. The $\pi$-$f$ interaction is likely to affect to magnetic interaction of 4$f$ electrons and the origin of ferromagnetism in Eu$_6$C$_{60}$ may be ascribed to the indirect exchange interaction. In the $bcc$ structure, an Eu atom has 4 nearest neighbor Eu atoms (the distance between two Eu atoms is 3.89 Å). Therefore, in the case of Eu$_1$Sr$_5$C$_{60}$, 5 of 6 Eu atoms are replaced by non-magnetic Sr atoms, Eu atoms can no longer have the three-dimensional Eu network, so that the direct interaction fails completely. Nevertheless $T_C$ does not show a drastic change. This is a quite contrast with the case of magnetic semiconductor EuO, where the direct exchange interaction between Eu atoms is important [@Kasuya2] and the substitution of Ca for Eu significantly reduces the ferromagnetic transition temperature [@Samokhvalov1]. This fact indicates that the ferromagnetism in Eu$_6$C$_{60}$ comes from the indirect exchange interaction via C$_{60}$ molecules, and the $\pi$-$f$ interaction has an important role in the present system. Now we discuss the origin of the giant magnetoresistance. The features of magnetoresistance in Eu$_6$C$_{60}$ are (1) negative magnetoresistance occurs below around $T_C$, (2) saturation field of magnetizaion is close to that of magnetoresistance ratio ($\rho(H)/\rho(0)$), as seen in the top curve of Fig.\[fig:magnetism\](d) and the bottom curve of Fig.\[fig:resistivity\](c), (3) magnetoresistance is much enhanced at lower temperatures; The magnetoresistance ratio ($\rho(H)/\rho(0)$) does not seem to saturate with decreasing temperature, while the magnetization almost saturates at 2 K and 5.5 T. The feature (1) indicates that the present MR is closely related to the ferromagnetic transition. In usual ferromagnetic metal, spin fluctuation scatters conduction electrons and causes negative magnetoresistance [@Gennes1; @Fisher1]. This effect may be an origin of the magnetoresistance near $T_C$, but this is not the case for the giant magnetoresistance in Eu$_6$C$_{60}$ at lower temperature, because such effect is remarkable in the vicinity of $T_C$ inconsistent with the feature (3). The feature (2) suggests that magnetoresistance is related to the magnetization. Furthermore, when we see $\rho(H)/\rho(0)$ in log scale, the difference of $\rho(H)/\rho(0)$ between 2 K and 1 K is almost one order (Fig.\[fig:resistivity\](b), while that of magnetization must be small. This means there is another factor, in addtion to the magnetization, to determine the magnetoresistance. It is probably temperature, that is, an activation process needs to be considered in the origin of the magnetoresistance. One possible interpretation of magnetoresistance in Eu$_6$C$_{60}$ is the spin-dependent tunneling at the grain boundary [@Helman1]. In the case of ferromagnetic granular metal, the conductivity is dependent on the tunneling probability of carriers through insulating barrier between grains, and the probability crucially depends on the spin polarization of carriers. In this case, each grain is assumed to be conductive and surrounded by less conductive surface. Because we measured the resistivity in a pellet of polycrystalline sample and Eu$_6$C$_{60}$ is very unstable in air, the surface may react to be insulative barrier, even if the sample is treated in high purity inert atmosphere. However we should note that the insulating region is considered to be limited only on the thin surface because unidentified peaks in XRD spectrum are very weak and they are considered not to affect to the magnetic and heat capacity measurements. As shown in the inset of Fig. \[fig:resistivity\](a), the temperature dependence of resistivity is represented as $\rho(T) \propto \exp(T_0/T)^{1/\alpha} (\alpha \sim 2)$, rather than the activation type which is expected in a usual semiconductor. The value of $T_0$ is about 180 K at 0T. This fact suggests that the resistivity in our sample might be governed by the tunneling at the boundaries [@Sheng1]. Our preliminary Hall effect measurement gives $R_H$ = +5$\times$10$^{-2}$ cm$^3$/C at 250 K, corresponding to the hole density of 1$\times$10$^{20}$ cm$^{-3}$ (0.1 hole per C$_{60}$); this means that intrinsic Eu$_6$C$_{60}$ can have relatively high conductivity and is possibly metallic by hybridization of the C$_{60}$ and metal orbitals mentioned above. Note that Hall voltage is less sensitive to the grain boundary effect. Helman and Abeles [@Helman1] considered the magnetic exchange energy $E_M$ and gave the mangetoconductivity as $$\label{eqn:pnot0} \sigma(H,T) = \sigma_0\left[\cosh(E_M/2k_BT)-P\sinh(E_M/2k_BT)\right],$$ where $P$ is the spin polarization of carrier and $E_M$=(1/2)$J$\[1-$m^2$\]. $J$ is the exchange coupling constant between a conduction carrier and a ferromagnetic metal grain and $m$ is the magnetization normalized by the saturation value. The equation (\[eqn:pnot0\]) gives a negative magnetoresistance of orders of magnitude only when $P$ is very close to unity. If $P$ = 1, we obtain $$\label{eqn:pequal0} \rho(H)/\rho(0) = \exp(-Jm^2/4k_BT).$$ Magnetoresistance in equation (\[eqn:pequal0\]) becomes large with decreasing temperature, which agree qualitatively with the feature (2) mentioned above. The assumption of $P$ = 1 might be unrealistic in usual ferromagnetic metals. However, if the exchange interaction between Eu atoms is accomplished via $\pi$-bands of C$_{60}$ as discussed earlier, we can expect a large spin polarization of $\pi$-electrons. We can also consider the effect of magnetic polaron. In magnetic semiconductors such as Eu chalcogenides, a carrier makes surrounding magnetic moments be polarized via exchange interaction and forms a magnetic polaron [@Kasuya1]. At zero field, magnetic polarons have to move with flipping some magnetic moments which are more or less randomly oriented, and their conduction is suppressed. Application of magnetic field aligns spin directions and carriers become mobile. As a result, negative magnetoresistance occurs. The negative magnetoresistance above $T_C$ can be attributed to this picture. Even in the ferromagnetic phase, magnetic moments have to be flipped at a magnetic domain boundary for the motion of carrier. Because remnant magnetic moment is little, as seen in Figs. \[fig:magnetism\] (d), many magnetic domains exist in our sample of Eu$_6$C$_{60}$. The crucial point of above two interpretations (spin dependent tunneling and magnetic polaron) are that carriers must overcome large exchange interaction with localized spins when they go into the region of different orientation of magnetic moments. SUMMARY ======= We have measured crystal structure, magnetic properties and magnetoresistance in polycrystalline Eu$_6$C$_{60}$ and its Sr-substituted compounds, Eu$_{6-x}$Sr$_x$C$_{60}$. They all have a $bcc$ structure and the compounds of $x$ = 3 and 5 form a solid solution concerning the occupation of metal atom. A ferromagnetic transition is observed at $T_C \sim$ 12 K in Eu$_6$C$_{60}$ and all Eu atoms are in divalent state with a magnetic moment of 7$\mu_B$ ($S$ = 7/2). The fact that the substitution of non-magnetic Sr for Eu affects little to $T_C$ indicates the ferromagnetic interaction is caused through the conduction carriers. In the resistivity measurement, we have found that Eu$_6$C$_{60}$ showed a huge negative magnetoresistance and $\rho(H)/\rho(0)$ reduced almost 10$^{-3}$ at $H$ = 9 T and $T$ = 1 K. The precise mechanism of magnetoresistance has not clarified yet, but it manifests a strong interaction between $\pi$-conduction electrons of C$_{60}$ and 4$f$ electrons on Eu. We acknowledge to Prof. Y. Iwasa, Dr. T. Takenobu and S. Moriyama for the suggestions for synthesis and heat capacity measurements. We also thank to Prof. A. Asamitsu for the advice of the resistivity measurements at low temperature. This work was supported by “Research for the Future” of Japan Society for the Promotion of Science (JSPS), Japan.     [ccccccc]{}\ &Site&Occupancy&$x$&$y$&$z$&$B$(Å$^{2}$)\ C1&$24g$&1&0.0672(5)&0&0.3200&0.5(2)\ C2&$48h$&1&0.1325&0.1056&0.2797&0.5\ C3&$48h$&1&0.0653(3)&0.2144&0.2381&0.5\ Eu$^{2+}$&$12e$&1&0&0.5&0.2768(2)&2.29(4)\ [ccccccc]{}\ &Site&Occupancy&$x$&$y$&$z$&$B$(Å$^{2}$)\ C1&$24g$&1&0.0686(6)&0&0.3186&1.1(3)\ C2&$48h$&1&0.1328&0.1038&0.2790&1.1\ C3&$48h$&1&0.0641(4)&0.2148&0.2366&1.1\ Eu$^{2+}$&$12e$&0.51(2)&0&0.5&0.2792(3)&2.99(7)\ Sr$^{2+}$&$12e$&0.49&0&0.5&0.2792&2.99\ [ccccccc]{}\ &Site&Occupancy&$x$&$y$&$z$&$B$(Å$^{2}$)\ C1&$24g$&1&0.0660(5)&0&0.3207&2.1(3)\ C2&$48h$&1&0.1321&0.1069&0.2798&2.1\ C3&$48h$&1&0.0661(3)&0.2138&0.2390&2.1\ Eu$^{2+}$&$12e$&0.18(2)&0&0.5&0.2803(3)&2.73(6)\ Sr$^{2+}$&$12e$&0.82&0&0.5&0.2803(3)&2.73\ [ccccccc]{}\ ---------------------- -------------------------- -------------- -------------------- ----------- $\mu_{eff}$/Eu ($\mu_B$) $\Theta$ (K) $M_S$/Eu ($\mu_B$) $T_C$ (K) Eu$^{2+}$ 7.94 7 Eu$_6$C$_{60}$ 7.77 10.6 6.97 11.6 (13.7) Eu$_3$Sr$_3$C$_{60}$ 8.13 12.6 7.73 12.8 Eu$_1$Sr$_5$C$_{60}$ 8.26 8.0 7.68 10.4 ---------------------- -------------------------- -------------- -------------------- ----------- : \[tab:magnetism\] Summary of the magnetic properties of Eu$_{6-x}$Sr$_x$C$_{60}$. $\mu_{eff}$, $\Theta$, $M_S$, and $T_C$ denote effective Bohr magneton ($g_J\sqrt{J(J+1)}\mu_B$), Weiss temperature, saturation moment, and ferromagnetic transition temperature, respectively.

--- abstract: 'We investigate the supersymmetry (SUSY) structures for inductor-capacitor circuit networks on a simple regular graph and its line graph. We show that their eigenspectra must coincide (except, possibly, for the highest eigenfrequency) due to SUSY, which is derived from the topological nature of the circuits. To observe this spectra correspondence in the high frequency range, we study spoof plasmons on metallic hexagonal and lattices. The band correspondence between them is predicted by a simulation. Using terahertz time-domain spectroscopy, we demonstrate the band correspondence of fabricated metallic hexagonal and lattices.' author: - Yosuke Nakata - Yoshiro Urade - Toshihiro Nakanishi - Fumiaki Miyamaru - Mitsuo Wada Takeda - Masao Kitano nocite: '[@*]' title: ' Supersymmetric correspondence in spectra on a graph and its line graph: From circuit theory to spoof plasmons on metallic lattices ' --- \#1[[~~\#1~~]{}]{} \#1[[\#1]{}]{} \#1[[Ø u ‘=‘Ø=]{}]{} \#1[\#1|]{} \#1[\#1|]{} \#1[|]{} \#1[|]{} \#1 \#1\#2[|]{} \#1\#2\#3[||]{} \#1[[ ]{}]{} \#1 \#1\#2 \#1[\_]{} \#1[\^]{} Introduction ============ Supersymmetry (SUSY) is a conjectured symmetry between fermions and bosons. Although the concept of SUSY was introduced in high-energy physics and remains to be experimentally confirmed, the underlying algebra is also found in quantum mechanics. When the SUSY algebra is applied to the field of quantum mechanics it is called supersymmetric quantum mechanics (SUSYQM) [@Cooper1994]. The algebraic relations of SUSY link two systems that at first glance might seem to be very different. The linkage through SUSY can be utilized to construct exact solutions for various systems in quantum mechanics. Recently, SUSYQM has been applied to construct quantum systems enabling exotic quantum wave propagations: reflectionless or invisible defects in tight-binding models [@Longhi2010] and complex crystals [@Longhi2013a], transparent interface between two isospectral one-dimensional crystals [@Longhi2013], reflectionless bent waveguides for matter-waves [@Campo2014], and disordered systems with Bloch-like eigenstates and band gaps [@Yu2015]. The SUSY structure was also found in other physics fields besides quantum mechanics, e.g., statistical physics through the Fokker-Planck equations [@Bernstein1984]. Through the similarity between quantum-mechanical probability waves and electromagnetic waves, the SUSY structure can be formulated for electromagnetic systems. Electromagnetic SUSY structures have been found in one-dimensional refractive index distributions [@Chumakov1994; @Miri2013], coupled discrete waveguides [@Longhi2010; @Miri2013], weakly guiding optical fibers with cylindrical symmetry [@Miri2013], planar waveguides with varying permittivity and permeability [@Laba2014], and non-uniform grating structures [@Longhi2015]. Even a quantum optical deformed oscillator with $\mathrm{SU}(1,1)$ group symmetry and its SUSY partner were constructed as a classical electromagnetic system [@Zuniga-Segundo2014]. The SUSY transformation generates new optical systems whose spectra coincide with those of the original system (except possibly for the highest eigenvalue of the fundamental mode of original or generated systems). The SUSY transformations have been utilized to synthesize mode filters [@Miri2013] and distributed-feedback filters with any desired number of resonances at the target frequencies [@Longhi2015]. The scattering properties of the optical systems paired by the SUSY transformation are related to each other [@Longhi2010; @Miri2013]. It is possible to design an optical system family with identical reflection and transmission characteristics by using the SUSY transformations [@Miri2014]. A reflectionless potential derived from the trivial system by SUSY transformation was applied to design transparent optical intersections [@Longhi2015a]. Moreover, SUSY has also been intensively investigated in non-Hermitian optical systems. If a system is invariant under the simultaneous operations of the space and time inversions, it is called $\mathcal{PT}$-symmetric. The SUSY transformation for the $\mathcal{PT}$-symmetric system allows for arbitrarily removing bound states from the spectrum [@Miri2013a]. In addition, non-Hermitian optical couplers can be designed [@Principe2015]. By using double SUSY transformations, the bound states in the continuum were also formulated in tight-binding lattices [@Longhi2014; @Longhi2014a] and continuous systems [@Correa2015]. The SUSY transformation in the $\mathcal{PT}$-symmetric system can also reduce the undesired reflection of one-way-invisible optical crystals [@Midya2014]. From an experimental perspective, it is still challenging to extract the full potential of electromagnetic SUSY because of fabrication difficulties. However, using dielectric coupled waveguides, researchers have realized a reflectionless potential [@Szameit2011], interpreted as a transformed potential derived from the trivial one by a SUSY transformation [@Longhi2010], and SUSY mode converters [@Heinrich2014]. The SUSY scattering properties of dielectric coupled waveguides have also been observed [@Heinrich2014a]. As we have described so far, many studies have been done for the electromagnetic SUSY, but their focusing point is mainly limited to dielectric structures. Recent progress of plasmonics [@AlexanderMaier2007] and metamaterials [@Solymar2009] using metals in optics demands further studies of SUSY for metallic systems. To design and analyze the characteristics of metallic structures, intuitive electrical circuit models are very useful, because they extract the nature of the phenomena despite reducing the degree of freedom for the problem [@Nakata2012a]. Actually, a circuit-theoretical design strategy called [*metactronics*]{} has been proposed even in the optical region [@Engheta2007] and the circuit theory for plasmons has also been developed [@Staffaroni2012]. If we could design circuit models enabling exotic phenomena, they open up new possibilities for application to higher frequency ranges due to the scale invariance of Maxwell equations. Thus, in this paper we develop how SUSY appears in inductor-capacitor circuit networks and demonstrate the SUSY correspondence in the high frequency region. In particular, we focus on the SUSY structure for inductor-capacitor circuit networks on a graph and its line graph. This article is organized as follows. In Sec. \[sec:2\], we start by introducing the graph-theoretical concepts and formulate a general class of inductor-capacitor circuit network pairs related through SUSY, derived from the topological nature of the graphs representing the circuits. In Sec. \[sec:3\], we theoretically and experimentally demonstrate the SUSY eigenfrequency correspondence for paired metallic lattices in the terahertz frequency range. In Sec. \[sec:4\], we summarize and conclude the paper. Theory \[sec:2\] ================ Eigenequation for inductor-capacitor circuit networks ------------------------------------------------------ We consider an inductor-capacitor circuit network on a simple directed graph $G=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ and $\mathcal{E}$ are the sets of vertices and directed edges, respectively. The modifier [*simple*]{} means that there are no multiple edges between any vertex pair and no edge (loop) that connects a vertex to itself. The number of the edges connected to a vertex $v$ of a graph is called the degree of $v$. A [*regular*]{} graph is a graph whose every vertex has the same degree. We assume that $G$ is an $m$-[*regular*]{} graph with all vertices having degree $m$. The capacitors, all with the same capacitance $C$, are connected between each vertex $v\in \mathcal{V}$ and the ground. Coils, all with the same inductance $L$, are loaded along all $e \in \mathcal{E}$. An example of $G$ and the inductor-capacitor circuit network on it are shown in Fig. \[fig:lc\_ladder\](a) and (b). ![\[fig:lc\_ladder\] (a) Example of simple $3$-regular directed graph. (b) Inductor-capacitor circuit network on the graph. (c) Line graph of the graph shown in (a). (d) Inductor-capacitor circuit network on the line graph (c). ](lc_ladder.eps){width="86mm"} For $v\in \mathcal{V}$ and $e\in \mathcal{E}$, the incidence matrix $\mathsf{X}=[X_{ve}]$ of a directed graph $G$ is defined as follows: $X_{ve}=-1$ ($e$ enters $v$), $X_{ve}=1$ ($e$ leaves $v$), otherwise $X_{ve}=0$. Using vector notation, we represent the current distribution $J_e$ flowing along $e\in \mathcal{E}$ as a column vector $\vct{J}=[J_e]^\mathrm{T}$. The charge distribution is denoted by $\vct{q}=[q_v]^\mathrm{T}$ with a stored charge $q_v$ at $v\in \mathcal{V}$. The charge conservation law is given by $$\dot{\vct{q}}=-\mathsf{X} \vct{J}, \label{eq:1}$$ where the time derivative is represented by the dot. The scalar potential $\Phi_v$ at $v \in {\mathcal V}$ must satisfy Faraday’s law of induction, so we have $$\dot{\vct{J}}=\frac{1}{L} \mathsf{X}^\mathrm{T}\vct{\Phi} , \label{eq:2}$$ with $\vct{\Phi}=[\Phi_v]^\mathrm{T}$. The scalar potential $\vct{\Phi}$ is written as $$\vct{\Phi}=\mathsf{P} \vct{q}, \label{eq:3}$$ with a potential matrix $\mathsf{P}$. In our case, $\mathsf{P}$ is given by $$\mathsf{P}=C^{-1}\mathsf{I}, \label{eq:4}$$ where we use the identity matrix $\mathsf{I}$. From Eqs. (\[eq:1\])–(\[eq:4\]), we obtain $$\ddot{\vct{q}}=-{\omega_0}^2\mathsf{X}\mathsf{X}^\mathrm{T} \vct{q}, \nonumber$$ with $\omega_0=1/\sqrt{LC}$. Assuming $\vct{q}=\tilde{\vct{q}}\exp(-\ii \omega t)+\cc$, we have an eigenequation $$\mathsf{L}\tilde{\vct{q}}= \left(\frac{\omega}{\omega_0}\right)^2\tilde{\vct{q}} \label{eq:5}$$ with the Laplacian $\mathsf{L}=\mathsf{X}\mathsf{X}^\mathrm{T}$. We introduce an adjacency matrix $\mathsf{A}=[A_{vw}]$, where $A_{vw}$ is 1 if $v, w\in \mathcal{V}$ are connected by an edge, otherwise 0. From a direct calculation, we can write $\mathsf{L}$ by $\mathsf{A}$ as $$\mathsf{L}=\mathsf{X}\mathsf{X}^\mathrm{T}=-\mathsf{A}+m\mathsf{I}, \label{eq:6}$$ where $m$ is the degree of the vertex of $G$. Note that $\mathsf{L}$ is independent of the direction of the edges in $G$ because $\mathsf{L}$ is expressed in terms of $\mathsf{A}$ and $\mathsf{I}$. The directed graph $G$ can be also regarded as an undirected graph. For $e\in \mathcal{E}$, we can make an undirected edge $\bar{e}$, where the bar operator ignores the direction of the edge. Then, we have $\bar{G}=(\mathcal{V},\bar{\mathcal{E}})$ with $\bar{\mathcal{E}}=\{\bar{e}|e\in \mathcal{E}\}$. We can also define the undirected incidence matrix $\bar{\mathsf{X}}=[\bar{X}_{ve}]$ as $\bar{X}_{ve}=1$ ($e$ and $v$ are connected), otherwise $\bar{X}_{ve}=0$. Using $\bar{\mathsf{X}}$, $\mathsf{A}$ is written as follows [@Biggs1994]: $$\bar{\mathsf{X}}\bar{\mathsf{X}}^\mathrm{T}=\mathsf{A}+m \mathsf{I}. \label{eq:7}$$ From Eqs. (\[eq:6\]) and (\[eq:7\]), we obtain $$\mathsf{L}=- \bar{\mathsf{X}}\bar{\mathsf{X}}^\mathrm{T}+2m\mathsf{I}. \label{eq:8}$$ SUSY correspondence in spectra on a simple regular graph and its line graph --------------------------------------------------------------------------- Next, we introduce the line graph concept [@Biggs1994]. The line graph $L(G)=(\mathcal{V}\sub{L}, \mathcal{E}\sub{L})$ of a directed graph $G$ is constructed as follows. Each edge in $G$ is considered to be a vertex of $L(G)$. Two vertices of $L(G)$ are connected if the corresponding edges in $G$ have a vertex in common. There are two possible choices for the direction of each edge in $L(G)$ and we adapt one of them. From here on, we only consider $L(G)$ of a simple $m$-regular graph $G$. In this case, the line graph $L(G)$ is a simple $m\sub{L}$-regular graph. The degree $m\sub{L}$ can be represented by $m$. For a vertex $v \in \mathcal{V}$ included in $e\in \mathcal{E}$, there are $m-1$ edges $e'\in \mathcal{E}\setminus\{e \}$ connected to $v$. Then, we obtain $$m\sub{L}=2(m-1). \label{eq:9}$$ Note that $(m/2)\#\mathcal{V}= \#\mathcal{E}=\#\mathcal{V}\sub{L}=(2/m\sub{L})\#\mathcal{E}\sub{L}$ is satisfied for a finite graph $G$, where $\#\mathcal{S}$ represents the numbers of the elements of the set $\mathcal{S}$. Figure \[fig:lc\_ladder\](c) is an example of the line graph of the graph $G$ shown in Fig. \[fig:lc\_ladder\](a). Figure \[fig:lc\_ladder\](d) is the inductor-capacitor circuit network on $L(G)$. In the context of mathematics, it is known that the spectra of the Laplacians for the graph and its line graph are related to each other [@Shirai2000]. For the convenience of the readers, we rederive this property in a simple manner and apply it to the inductor-capacitor circuit networks. The Laplacian of $L(G)$ is written as $$\mathsf{L}\sub{L}=\mathsf{X}\sub{L}{\mathsf{X}\sub{L}}^\mathrm{T}=-\mathsf{A}\sub{L}+m\sub{L}\mathsf{I}\sub{L}, \label{eq:10}$$ with the identity matrix $\mathsf{I}\sub{L}$, the incidence matrix $\mathsf{X}\sub{L}$, and the adjacency matrix $\mathsf{A}\sub{L}$ for $L(G)$. The adjacency matrix of $L(G)$ is represented as follows [@Biggs1994]: $$\mathsf{A}\sub{L}=\bar{\mathsf{X}}^\mathrm{T} \bar{\mathsf{X}}-2\mathsf{I}\sub{L}. \label{eq:11}$$ From Eqs. (\[eq:10\]) and (\[eq:11\]), we have $$\mathsf{L}\sub{L}=-\bar{\mathsf{X}}^\mathrm{T} \bar{\mathsf{X}}+(m\sub{L}+2)\mathsf{I}\sub{L}. \label{eq:12}$$ Now, we consider the composite system of $L(G)$ and $G$. Then the composite Laplacian $\mathcal{L}\sub{c}$ is given by $$\mathcal{L}\sub{c}=-\mathcal{K}\sub{c} +2m\mathcal{I}\sub{c}, \label{eq:13}$$ with $$\mathcal{K}\sub{c}= \begin{bmatrix} \bar{\mathsf{X}}^\mathrm{T}\bar{\mathsf{X}}&0\\ 0&\bar{\mathsf{X}}\bar{\mathsf{X}}^\mathrm{T} \end{bmatrix},\ \mathcal{I}\sub{c}= \begin{bmatrix} \mathsf{I}\sub{L}&0\\ 0&\mathsf{I} \end{bmatrix}, \nonumber$$ where we have used Eqs. (\[eq:8\]), (\[eq:9\]), and (\[eq:12\]). The composite operator $\mathcal{K}\sub{c}$ is written as $$\mathcal{K}\sub{c}=\mathcal{Q}\mathcal{Q}^\dagger+\mathcal{Q}^\dagger\mathcal{Q}, \label{eq:14}$$ where the symbol $\dagger$ represents the Hermitian conjugate, and we define the supercharge as $$\mathcal{Q} = \begin{bmatrix} 0 & 0\\ \bar{\mathsf{X}} & 0 \end{bmatrix}.$$ These operators satisfy the superalgebra [@Cooper1994]: $$[\mathcal{K}\sub{c},\mathcal{Q}]=[\mathcal{K}\sub{c},\mathcal{Q}^\dagger]=0,$$ $$\{\mathcal{Q},\mathcal{Q}^\dagger\}=\mathcal{K}\sub{c},\ \{\mathcal{Q},\mathcal{Q}\}=\{\mathcal{Q}^\dagger,\mathcal{Q}^\dagger\}=0,$$ where $\{\mathcal{A},\mathcal{B}\}$ and $[\mathcal{A},\mathcal{B}]$ are the anticommutator and the commutator, respectively. Therefore, the eigenspectra of the inductor-capacitor circuit networks on the simple regular graph and its line graph must coincide except, possibly, for the highest eigenfrequency. Actually, if we have eigenvector $\vct{x}$ satisfying $\mathsf{L}\vct{x}=E\vct{x}$ with the eigenvalue $E$, we obtain $\bar{\mathsf{X}}^\mathrm{T}\vct{x}$, satisfying $\mathsf{L}\sub{L}(\bar{\mathsf{X}}^\mathrm{T}\vct{x})=E(\bar{\mathsf{X}}^\mathrm{T}\vct{x})$. Then, we have an eigenvector $\bar{\mathsf{X}}^\mathrm{T}\vct{x}$ for $\mathsf{L}\sub{L}$ when $\bar{\mathsf{X}}^\mathrm{T}\vct{x}\ne \vct{0}$ ($E\ne 2m$). The eigenvalue $E$ of $\mathsf{L}$ and $\mathsf{L}\sub{L}$ must satisfy $E\leq 2m$, because $\bar{\mathsf{X}}\bar{\mathsf{X}}^\mathrm{T}$ and $\bar{\mathsf{X}}^\mathrm{T}\bar{\mathsf{X}}$ are positive-semidefinite. For an eigenvector $\vct{y}_{2m}$ of $\mathsf{L}$ ($\mathsf{L}\sub{L}$) with eigenvalue $E=2m$, the partner mode cannot be obtained by multiplying $\bar{\mathsf{X}}^\mathrm{T}$ ($\bar{\mathsf{X}}$), because of $\bar{\mathsf{X}}^\mathrm{T}\vct{y}_{2m}=0$ ($\bar{\mathsf{X}}\vct{y}_{2m}=0$). The condition for complete spectral coincidence of the eigenvalues of $\mathsf{L}$ and $\mathsf{L}\sub{L}$ is discussed in Appendix \[sec:appA\]. Note that quantum tight-binding models represented by Eqs. (\[eq:8\]) and (\[eq:12\]) are isospectral except, possibly, for the highest eigenenergy, but an accurate tuning of the on-site potential satisfying Eqs. (\[eq:8\]) and (\[eq:12\]) is usually difficult to achieve. The significant point of SUSY for the inductor-capacitor circuit networks is that the on-site potential tuning is accomplished naturally. If $G$ is a periodic graph with lattice vectors $\{\thrvct{a}_i\}$, we can show that the spectral coincidence except possibly for the highest eigenfrequency holds for [*each wave vector.* ]{} We define parallel translation with $\thrvct{a}_i$ as $\mathsf{P}^\mathcal{E}_i$ and $\mathsf{P}^\mathcal{V}_i$ for edges and vertices, respectively. From the translational symmetry, we have $\bar{\mathsf{X}}\mathsf{P}^\mathcal{E}_i=\mathsf{P}^\mathcal{V}_i\bar{\mathsf{X}}$. For a Bloch vector $\vct{x}_{\thrvct{k}}$ satisfying $\mathsf{P}^\mathcal{E}_i \vct{x}_{\thrvct{k}}=\exp({\ii\thrvct{k}\cdot\thrvct{a}_i}) \vct{x}_{\thrvct{k}}$, we have $\mathsf{P}^\mathcal{V}_i(\bar{\mathsf{X}}\vct{x}_{\thrvct{k}})=\exp(\ii\thrvct{k}\cdot\thrvct{a}_i)(\bar{\mathsf{X}}\vct{x}_{\thrvct{k}})$. This means that $\bar{\mathsf{X}}\vct{x}_{\thrvct{k}}$ is also a Bloch vector. Similar discussion can be applied to $\bar{\mathsf{X}}^\mathrm{T}$. Then, $\bar{\mathsf{X}}$ and $\bar{\mathsf{X}}^\mathrm{T}$ map Bloch vectors to Bloch vectors without changing $\thrvct{k}$. Therefore, the decomposition shown in Eq. (\[eq:13\]) is valid in the subspace of Bloch vectors with a wave vector $\thrvct{k}$. This means that the spectral coincidence except, possibly, for the highest eigenfrequency holds for each wave vector. Examples \[subsec:c\] --------------------- ### finite case\[subsub:1\] For the graph shown in Fig. \[fig:lc\_ladder\](a), we have $$\bar{\mathsf{X}}=\begin{bmatrix} 0 &1 &1 &1 &0 &0\\ 1 &0 &1 &0 &1 &0\\ 1 &1 &0 &0 &0 &1\\ 0 &0 &0 &1 &1 &1 \end{bmatrix}\nonumber.$$ Then, we get $\omega=2\omega_0,\ 2\omega_0,\ 2\omega_0,\ 0$ for the inductor-capacitor circuit network on $G$. On the other hand, $\omega=\sqrt{6}\omega_0,\ \sqrt{6}\omega_0,\ 2\omega_0,\ 2\omega_0,\ 2\omega_0,\ 0$ are obtained for the inductor-capacitor circuit network on $L(G)$. We can see that all angular eigenfrequencies for $G$ are included in those for $L(G)$. ### Infinite case ![\[fig:kagome\_hex\] (color online) (a) Hexagonal lattice. (b) lattice. (c) Dispersion relation for a hexagonal inductor-capacitor circuit network. (d) Dispersion relation for a inductor-capacitor circuit network. (e) Eigenmodes of the higher band at the $\Gamma$ and $\mathrm{M}$ points for the hexagonal lattice. (f) Eigenmodes of the middle band at the $\Gamma$ and $\mathrm{M}$ points for the lattice. ](kagome_hex.eps){width="86mm"} Here, we consider a hexagonal lattice as $G$ \[Fig. \[fig:kagome\_hex\](a)\]. The line graph $L(G)$ is a lattice \[Fig. \[fig:kagome\_hex\](b)\]. To see the spectra coincidence directly, we calculate the angular eigenfrequencies $\omega$. At first, we calculate the eigenvalue $\alpha$ and $\alpha\sub{L}$ for $\mathsf{A}$ and $\mathsf{A}\sub{L}$, respectively. Due to the Bloch theorem, it is enough to calculate them in the restricted space $\mathcal{W}_{\thrvct{k}_\parallel}$ of waves with wave vector $\thrvct{k}_\parallel$. For the hexagonal lattice, we have two vertices $v_p\in \mathcal{V}$ $ (p=1,2)$ in a unit cell. The vertex displaced from $v_p$, with $i\thrvct{a}_1+j\thrvct{a}_2$, is denoted by $v_p^{(i,j)}$ for $(i,j)\in \mathbb{Z}^2$, where $\thrvct{a}_1$ and $\thrvct{a}_2$ are lattice vectors for $G$. Now, we define $\vct{\Psi}_1 (\thrvct{k}_\parallel)=\sum_{(i,j)\in\mathbb{Z}^2} \exp(\ii \thrvct{k}_\parallel\cdot(i\thrvct{a}_1+j\thrvct{a}_2) ){\vct{v}_1^{(i,j)}}$ and $\vct{\Psi}_2 (\thrvct{k}_\parallel)=\sum_{(i,j)\in\mathbb{Z}^2} \exp(\ii \thrvct{k}_\parallel\cdot(i\thrvct{a}_1+j\thrvct{a}_2) ){\vct{v}_2^{(i,j)}}$, where $\{\vct{v}_{p}^{(i,j)}|(i,j)\in\mathbb{Z}^2, p=1,2\}\subset \mathcal{H}$ is a complete orthogonal basis of the Hilbert space $\mathcal{H}$. The vector subspace $\mathcal{W}_{\thrvct{k}_\parallel}$ is spanned by $\vct{\Psi}_1 (\thrvct{k}_\parallel)$ and $\vct{\Psi}_2 (\thrvct{k}_\parallel)$. The action of $\mathsf{A}$ in the restricted space $\mathcal{W}_{\thrvct{k}_\parallel}$ is represented by a $2\times 2$ matrix $\mathsf{A}({\thrvct{k}_\parallel})=[A_{ij}({\thrvct{k}_\parallel})]$, satisfying $\mathsf{A}\vct{\Psi}_i({\thrvct{k}_\parallel})= \sum_{j=1}^{2}\vct{\Psi}_j({\thrvct{k}_\parallel}) A_{ji}({\thrvct{k}_\parallel})$. Diagonalizing $\mathsf{A}({\thrvct{k}_\parallel})$ we have $$\alpha(\thrvct{k}_\parallel)=\pm\sqrt{3+2F(\thrvct{a}_1, \thrvct{a}_2;\thrvct{k}_\parallel)}, \label{eq:15}$$ with $F(\thrvct{u}_1, \thrvct{u}_2; \thrvct{k}_\parallel)=\cos (\thrvct{k}_\parallel\cdot\thrvct{u}_1)+\cos( \thrvct{k}_\parallel\cdot\thrvct{u}_2)+\cos \thrvct{k}_\parallel\cdot(\thrvct{u}_1-\thrvct{u}_2)$. By applying a similar calculation to the lattice, we obtain $$\alpha\sub{L}(\thrvct{k}_\parallel) =-2, 1\pm\sqrt{3+2F(\thrvct{a}_1\sur{L}, \thrvct{a}_2\sur{L};\thrvct{k}_\parallel)}, \label{eq:16}$$ with lattice vectors $\thrvct{a}_1\sur{L}$ and $\thrvct{a}_2\sur{L}$ for $L(G)$. Using Eqs. (\[eq:5\]), (\[eq:6\]), and (\[eq:15\]), we obtain $$\frac{\omega}{\omega_0}=\sqrt{3\pm\sqrt{3+2F(\thrvct{a}_1, \thrvct{a}_2;\thrvct{k}_\parallel)}} \label{eq:17}$$ for the hexagonal lattice. From Eqs. (\[eq:5\]), (\[eq:10\]), and (\[eq:16\]), the lattice also has the dispersion relation $$\frac{\omega}{\omega_0}=\sqrt{6}, \sqrt{3\pm\sqrt{3+2F(\thrvct{a}_1\sur{L}, \thrvct{a}_2\sur{L};\thrvct{k}_\parallel)}}. \label{eq:18}$$ The obtained dispersion relations are shown in Figs. \[fig:kagome\_hex\](c) and (d). The lower two bands are identical as we expected. Note that these bands are determined only by the product $LC$ and are independent of the ratio $L/C$. We also show examples of the eigenmodes for the hexagonal and lattices in Figs. \[fig:kagome\_hex\](e) and (f). Band correspondence between metallic hexagonal and lattices \[sec:3\] ===================================================================== In the previous section, we developed inductor-capacitor circuit networks that are related through SUSY. As an example, we saw that the bands of hexagonal and inductor-capacitor circuit networks are isospectral by SUSY, except for the highest band of the lattice. In this section, we examine this correspondence for realistic system. It is known that bar-disk resonators composed of metallic disks connected by metallic bars can be qualitatively modeled by the inductor-capacitor circuit networks discussed in Sec. \[sec:2\], because charges on the disks are coupled dominantly by the current flowing along the bars [@Nakata2012; @Kajiwara2016]. The modes on the bar-disk resonators are called spoof plasmons. Here, we study the spoof plasmons of metallic hexagonal and lattices whose designs are shown in Fig. \[fig:kagome\_hex\_lattice\_design\]. ![\[fig:kagome\_hex\_lattice\_design\] Designs for (a) metallic hexagonal lattice and (b) metallic lattice. The following parameters are used: $d=10\,\U{\mu m}$, $r=150\,\U{\mu m}$, $b=800/\sqrt{3}\,\U{\mu m}$, and thickness $h=30\,\U{\mu m}$.](kagome_hex_lattice_design.eps){width="86mm"} Simulation\[sec:3-1\] --------------------- ![\[fig:hex\_kagome\_bands\] (color online) Dispersion relations obtained by simulation for a (a) metallic hexagonal lattice and (b) metallic lattice. The fitting parameters for the theoretical models discussed in Appendix \[sec:appB\] are given by $\omega_0\sur{hex}=2\pi \times 0.107\,\U{THz}$, $\eta\sur{hex}_0=0.0916$ for the hexagonal lattice and $\omega_0\sur{kag}=2\pi\times 0.101\,\U{THz}$ and $\eta\sur{kag}_0=0.142$ for the lattice. ](hex_kagome_band.eps){width="86mm"} {width="172mm"} We perform an eigenfrequency analysis for the metallic hexagonal and lattices by the finite element method solver (<span style="font-variant:small-caps;">Comsol Multiphysics</span>). The parameters of the structures for Fig. \[fig:kagome\_hex\_lattice\_design\] are as follows: bar width $d=10\,\U{\mu m}$, radius of disks $r=150\,\U{\mu m}$, distance between nearest disks $b=800/\sqrt{3}\,\U{\mu m}$, and thickness $h=30\,\U{\mu m}$. In each simulation, the finite thickness metallic lattice parallel to $z=0$ is located in $z\in [-h/2,h/2]$. The unit cell in the $xy$ plane is the rhombus spanned by the lattice vectors and denoted by $U$. To reduce the degrees of freedom, we use the mirror symmetry with respect to $z=0$. A simulation domain with the material parameters of a vacuum is set in $U\times [0,6l]$ with $l=\sqrt{3} b=800\,\U{\mu m}$, and a perfect magnetic conductor condition imposed on the surface $z=0$. Half of the structure in $z\in [0,h/2]$ is engraved in the simulation domain and a perfect electric conductor (PEC) boundary condition is imposed on the structure surface. A perfect matched layer (PML) in $U\times [5l, 6l]$ with a PEC boundary at $z=6l$ is used to truncate the infinite effect. The periodic boundary condition with a phase shift (Floquet boundary conditions with a wave vector $\thrvct{k}_\parallel$) is applied to $\partial U \times [0,6l]$. Changing $\thrvct{k}_\parallel$ along the Brillouin zone boundary, we calculate the eigenfrequencies. To remove the modes which are not localized near the metallic surface [@Parisi2012], we select the modes with $$\xi=\frac{\int_{U\times [9l/2, 6l]} |\tilde{\thrvct{E}}|^2 \dd V}{\int_{U\times [h/2, l/2]} |\tilde{\thrvct{E}}|^2 \dd V}<1,\nonumber$$ where the complex amplitude of the electric field of the mode is denoted by $\tilde{\thrvct{E}}$. The calculated eigenfrequencies for the metallic hexagonal and lattices are shown in Fig. \[fig:hex\_kagome\_bands\] as circles. Note that some points are missing because unphysical modes located near PML accidentally exist or couple with the modes. As we explained earlier, we eliminated such modes with $\xi\geq 1$. In Fig. \[fig:hex\_kagome\_bands\], we observe the lower two band correspondence between the metallic hexagonal and lattices. The bands of the metallic hexagonal lattice are about 5% higher than those of the lattice. However, we can say that the band correspondence is qualitatively established. The detailed theoretical models for fitting curves are discussed later in Appendix. \[sec:appB\]. Figure \[fig:eigenmodes\] shows the electric flux density $D_z$ on $z=h/2$ of the specific modes, where $D_z$ corresponds to the surface charge on the metal. These mode profiles agree with the theoretically calculated eigenmodes shown in Figs. \[fig:kagome\_hex\](e) and (f). ![\[fig:hdbr\_kdbr\] (color online) Microphotographs of the (a) metallic hexagonal and (b) metallic lattices.](hdbr_kdbr.eps){width="86mm"} Experiment ---------- {width="172mm"} To investigate the dispersion relation experimentally, we fabricated the metallic hexagonal and lattices by etching and performed transmission measurement on them using the terahertz time-domain spectroscopy technique. The samples made of stainless steel (SUS304) are shown in Fig. \[fig:hdbr\_kdbr\]. The geometrical parameters of these samples are the same as those for the simulation model in Sec. \[sec:3-1\]. The area where structures are patterned is $4\,\U{cm}\times 4\,\U{cm}$. The terahertz beam is generated by a spiral antenna and collimated by a combination of a hyper-hemispherical silicon lens and a Tsurupica$^\text{\textregistered}$ lens. The beam diameter is set to $13\, \U{mm}$ by an aperture. Wire-grid polarizers are located near the emitter and detector, which are adjusted so that the emitted and detected fields have the same linear polarization. The transmission spectrum $T(\omega)$ in the frequency domain is obtained from $T(\omega)=|\tilde{E}(\omega)/\tilde{E}\sub{ref}(\omega)|^2$, where $\tilde{E}(\omega)$ and $\tilde{E}\sub{ref}(\omega)$ are Fourier transformed electric fields with and without the sample. To scan the Brillouin zone, power transmission spectra are measured with changing incident angle $\theta$ from $\theta=0^\circ$ to $60^\circ$ with step $2.5^\circ$. Here, the magnitude of the in-plane wave vector $\thrvct{k}_\parallel$ is given by $k_\parallel = (\omega/c) \sin \theta$, where $c$ is the speed of light. To observe the higher band of the hexagonal lattice and the middle band of the lattice, the incident waves are set as follows: (i) transverse electric (TE) modes in $\Gamma$–$\mathrm{K}$ scan and (ii) transverse magnetic (TM) modes in $\Gamma$–$\mathrm{M}$ scan. Figures \[fig:transmission\](a) and (b-1) show the power transmission spectra for these incident waves entering into the metallic hexagonal and lattices, respectively. The calculated eigenfrequencies are shown simultaneously as circles in Fig. \[fig:transmission\]. We can see that the transmission dips form a band from $0.15$ to $0.3\,\U{THz}$. The calculated eigenfrequencies are located around the experimental transmission dips. Thus, the SUSY band correspondence for the second band is experimentally demonstrated. The highest band for the metallic lattice can be observed for differently polarized incident waves. Figure \[fig:transmission\](b-2) shows the transmission spectra for the metallic lattice, where the incident waves are set as (i) TM modes in the $\Gamma$–$\mathrm{K}$ scan and (ii) TE modes in the $\Gamma$–$\mathrm{M}$ scan. In Fig. \[fig:transmission\](b-2), we can see the flat band reported in Ref. . Note that the frequencies of the lowest band modes are under the light line, so it is impossible to excite them by free-space plane waves. To excite the lowest band modes, another method, e.g., attenuated total reflection measurement, is needed [@Kajiwara2016]. Discussion \[sec:3.3\] ---------------------- ![\[fig:tuned\_kagome\] (color online) Comparison between eigenfrequencies for a tuned metallic lattice with $d=28\,\U{\mu m}$ and metallic hexagonal lattice with $d=10\,\U{\mu m}$. The other parameters are $r=150\,\U{\mu m}$, $b=800/\sqrt{3}\,\U{\mu m}$, and thickness $h=30\,\U{\mu m}$. ](tuned_kagome.eps){width="86mm"} In the previous subsections, the band correspondence between the metallic hexagonal and lattices was demonstrated, but a $5\%$ discrepancy between the bands was also observed. Here, we investigate the possibility to compensate empirically for the discrepancy. We calculated the eigenfrequencies for the metallic lattice with the bar width $d=28\,\U{\mu m}$. The other parameters are the same as the previous one. The calculated results are shown in Fig. \[fig:tuned\_kagome\] as circles. To compare with the previous result, the eigenfrequencies for the metallic hexagonal lattice with $d=10\,\U{\mu m}$ are also plotted in Fig. \[fig:tuned\_kagome\]. We can see the improvement of band correspondence between the metallic hexagonal lattice and tuned metallic lattice. Conclusion \[sec:4\] ==================== In this paper, we showed that the inductor-capacitor circuit networks on a simple regular graph and its line graph are related through SUSY, and their spectra must coincide (except possibly for the highest eigenfrequency). The SUSY structure for the circuits was derived from the topological nature of the graphs. To observe SUSY correspondence of the bands in the high frequency range, we investigated the metallic hexagonal and lattices. The band correspondence between them was predicted by a simulation. We performed terahertz time-domain spectroscopy for these metallic lattices and observed the band correspondence. Finally, we proposed an empirical tuning method to reduce the discrepancy of the corresponding bands of the metallic hexagonal and lattices. The theoretical results are formulated for the inductor-capacitor circuit networks and independent of the implementations. Therefore, our results is also applicable to transmission-line systems such as microstrip. The SUSY correspondence in the spectra of inductor-capacitor circuit networks has the potential to extend mode filters [@Miri2013] and mode converters [@Heinrich2014] in two-dimensional (spoof) plasmonic systems. The present research was supported by the JSPS KAKENHI Grant No. 25790065 and Grant-in-Aid for JSPS Fellows No. 13J04927. Two of the authors (Y. N. and Y. U.) were supported by JSPS Research Fellowships for Young Scientists. Condition for complete spectral coincidence of eigenvalues of $\mathsf{L}$ and $\mathsf{L}\sub{L}$\[sec:appA\] ============================================================================================================== In this Appendix, we consider condition for complete spectral coincidence of the eigenvalues of $\mathsf{L}$ and $\mathsf{L}\sub{L}$. Here, we mainly consider the finite graph cases. At first, we introduce a bipartite graph. A bipartite graph is a graph whose vertex set can be separated into two disjoint sets $\mathcal{V}_1$ and $\mathcal{V}_2$ such that every edge is connected between a vertex in $\mathcal{V}_1$ and that in $\mathcal{V}_2$. Using the concept of a bipartite graph, we obtain the following lemma: Consider a connected graph $G=(\mathcal{V}, \mathcal{E})$ satisfying $\#\mathcal{V}>0$ and $\#\mathcal{E}>0$. Let $\bar{\mathsf X}$ be an undirected incidence matrix of $\bar{G}$. In this case, $\rank \bar{\mathsf X}< \#\mathcal{V}$ is satisfied if and only if $G$ is bipartite. $\Leftarrow$: If the graph is bipartite, $\mathcal{V}$ is represented by the disjoint union of $\mathcal{V}_1$ and $\mathcal{V}_2$ as $\mathcal{V}=\mathcal{V}_1 \sqcup \mathcal{V}_2$. The $v_i$-component row vector of $\bar{\mathsf X}$ is denoted by $\bar{\vct{X}}_{i}$, where $v_i \in \mathcal{V}$ $(i=1,2,\cdots, n)$, and $n=\#\mathcal{V}$. Without loss of generality, we can assume $v_i \in \mathcal{V}_1$ $(i=1,2,\cdots, l)$ and $v_i\in \mathcal{V}_2$ $(i=l+1, l+2,\cdots, n)$. Because the graph is bipartite, $\sum_{i=1}^{l}\bar{\vct{X}}_{i}=\sum_{i=l+1}^{n}\bar{\vct{X}}_{i}$ is satisfied.\ $\Rightarrow$: If we assume $\rank \bar{\mathsf X}\ne \#\mathcal{V}$, $$\sum_{i=1}^{n}c_i \bar{\vct{X}}_{i}=0 \label{eq:19}$$ is satisfied for $c_i \in \mathbb{R}$ and not every $c_i=0$. At first we show $c_i\ne 0$ for all $i\in\{1,2,\cdots,n\}$. We assume $c_{q}=0$ for $q \in \{1,2,\cdots,n\}$. For arbitrary $r \in \{1,2,\cdots,n\}$, we consider a path $(v_{f(1)}, e_{1}, v_{f(2)}, e_2, \cdots, e_{p}, v_{f(p+1)})$ from $v_{q}$ to $v_{r}$, where $p$ is the path length, $f$ is a function from $\{1,2,\cdots, p+1\}$ to $\{1,2,\cdots, n\}$ satisfying $f(1)=q$ and $f(p+1)=r$, and the edge $e_{s}$ connects vertices $v_{f(s)}$ and $v_{f(s+1)}$ $(s=1,2,\cdots, p)$. Considering $e_{s}$-column of Eq. (\[eq:19\]), we have $c_{f(s)}=-c_{f(s+1)}$. Then, we obtain $c_i=0$ for all $i$. This leads a contradiction because we assumed not every $c_i=0$. Therefore, we have $c_i\ne0$ for all $i$. We assume $c_i>0$ $(i=1,2,\cdots, l)$, and $c_i<0$ $(i=l+1,l+2,\cdots, n)$, without loss of generality. For a given arbitrary $e\in \mathcal{E}$, we consider the column component about $e$ of Eq. (\[eq:19\]). Then, we find $e$ is connected between a vertex in $\mathcal{V}_1=\{v_1,v_2,\cdots,v_l\}$ and that in $\mathcal{V}_2=\{v_{l+1},v_{l+2},\cdots,v_{n}\}$. This shows $G$ is bipartite. This proof is based on Ref. . From this lemma, we can prove the following theorem: Consider a simple $m$-regular connected graph $G=(\mathcal{V}, \mathcal{E})$ with $\#\mathcal{E}>0$. There exists at least one mode with $\omega/\omega_0=\sqrt{2m}$ for the inductor-capacitor circuit network on $G$ if and only if $G$ is bipartite. Let $\bar{\mathsf X}$ be an undirected incidence matrix of $\bar{G}$. $G$ is bipartite $\Leftrightarrow$ $\rank{\bar{\mathsf X}}<\#\mathcal{V}$ $\Leftrightarrow$ $\dim \ker{\bar{\mathsf{X}}^\mathrm{T}}=\#\mathcal{V}-\rank{\bar{\mathsf X}}>0$ $\Leftrightarrow$ $^\exists \vct{x}$ satisfying $\bar{\mathsf{X}}^\mathrm{T}\vct{x}=0$ $\Leftrightarrow$ $^\exists \vct{x}$ satisfying $\mathsf{L}\vct{x}=2m\vct{x}$. On the other hand, we can formulate the condition for presence of eigenvalue $E=2m$ of $\mathsf{L}\sub{L}$ as follows: Consider a simple $m$-regular graph $G=(\mathcal{V}, \mathcal{E})$ with $\#\mathcal{V}>0$. There exists at least one mode with $\omega/\omega_0=\sqrt{2m}$ for the inductor-capacitor circuit network on $L(G)$ if $m> 2$. Let $\bar{\mathsf X}$ be an undirected incidence matrix of $\bar{G}$. We have $\dim \ker{\bar{\mathsf X}}=\#\mathcal{E}-\rank{\bar{\mathsf{X}}}\geq \#\mathcal{E}-\#\mathcal{V}=\frac{m-2}{2}\#\mathcal{V}>0$ for $m>2$. Then, $^\exists \vct{x}$ satisfying $\bar{\mathsf{X}}\vct{x}=0$. Finally, $^\exists \vct{x}$ satisfying $\mathsf{L}\sub{L}\vct{x}=2m\vct{x}$. From these theorems, all spectra of the inductor-capacitor circuit networks on a simple $m$-regular ($m>2$) connected bipartite graph and its line graph completely coincide. On the other hand, we find that $\mathcal{L}\sub{c}$ have a non-degenerated eigenvector with eigenvalue $E=2m$ for a simple $m$-regular ($m>2$) connected non-bipartite graph $G$ (e.g. Sec. \[subsub:1\]). For $m=2$, all spectra of the inductor-capacitor circuit networks on a simple connected $m$-regular graph $G$ and its line graph $L(G)$ coincide because of $G=L(G)$. In this case, there are $\omega/\omega_0=\sqrt{2m}$ modes if and only if $\#\mathcal{V}$ is even. To analyze an infinite periodic graph with lattice vectors $\{\thrvct{a}_i\}$, we take a supercell spanned by $\{N_i \thrvct{a}_i\}$, $N_i>1$. We impose a periodic boundary condition (called Born–von Karman boundary condition) on the sides of the supercell. Note that this boundary condition just leads to discretization of wave vectors in the Brillouin zone. Now, the infinite graph is reduced to a finite graph and we can use the theorems. For example, the hexagonal lattice is a simple 3-regular connected bipartite graph. Then, the spectra of the inductor-capacitor circuit networks on hexagonal and lattices include $\omega/\omega_0=\sqrt{6}$, simultaneously. Note that the spectra of the inductor-capacitor circuit networks on hexagonal and lattices completely coincide, but their dispersion relations do not. Detailed theoretical model for the metallic hexagonal and lattices \[sec:appB\] =============================================================================== In this appendix, we derive detailed theoretical models for fitting the eigenfrequencies of the metallic hexagonal and lattices. For the metallic hexagonal and lattices, the circuit models treated in Sec. \[sec:2\] are approximately valid. To improve the model accuracy, we have to take into account capacitive couplings between disks. Considering only nearest neighboring couplings, we modify Eq. (\[eq:4\]) as $\mathsf{P}=C^{-1}( \mathsf{I} + \eta \mathsf{A})$. Here, $C^{-1} \eta \mathsf{A}$ represents the capacitive coupling between the adjacent disks [@Kajiwara2016]. Then, we obtain $$\frac{\omega}{\omega_0}=\sqrt{ \big[m-\alpha(\thrvct{k}_\parallel)\big]\big[1+\eta \alpha(\thrvct{k}_\parallel)\big] }$$ for the metallic hexagonal lattice and $$\frac{\omega}{\omega_0}=\sqrt{ \big[m\sub{L}-\alpha\sub{L}(\thrvct{k}_\parallel)\big]\big[1+\eta \alpha\sub{L}(\thrvct{k}_\parallel)\big] }$$ for the metallic lattice. Generally, the coupling constant $\eta$ depends on $\omega$ as $\eta=\eta_0 \exp[\ii (\omega/c) b]$, where $b$ is the distance between the nearest disks [@Yeung2011]. The imaginary part of $\eta$ represents the resistive component. If we only focus on the real part of the eigenfrequencies, we may ignore the resistive term (note that we have already ignored the imaginary part of $L$ and $C$). Then, we assume $\eta=\eta_0 \cos[(\omega/c) b]$. Using the real part of the eigenvalues calculated by simulation, we numerically minimize the error $$\mathrm{err}\sur{hex}(\omega_0, \eta_0)=\sum_{(\omega,\thrvct{k}_\parallel) \in \text{\{data points\}} }g\big(\omega_0, \eta_0;\omega, m, \alpha(\thrvct{k}_\parallel)\big)^2$$ for the hexagonal lattice and $$\mathrm{err}\sur{kag}(\omega_0, \eta_0)=\sum_{(\omega,\thrvct{k}_\parallel) \in \text{\{data points\}} }g\big(\omega_0, \eta_0;\omega, m\sub{L}, \alpha\sub{L}(\thrvct{k}_\parallel)\big)^2$$ for the lattice, where we define $$g(\omega_0, \eta_0;\omega, m, \alpha)=\omega- \omega_0\sqrt{\left[m-\alpha\right]\left[1+\alpha \eta_0 \cos\left(\frac{\omega b}{c}\right)\right]}.$$ The obtained fitting parameters are as follows: $\omega_0=\omega_0\sur{hex}=2\pi \times 0.107\,\U{THz}$, $\eta_0=\eta\sur{hex}_0=0.0916$ for the hexagonal lattice, and $\omega_0=\omega_0\sur{kag}=2\pi\times 0.101\,\U{THz}$ and $\eta_0=\eta\sur{kag}_0=0.142$ for the lattice. Because the magnetic coupling and higher order effects (beyond the nearest capacitive coupling) are included in these parameters, the parameters for the hexagonal and lattice can be different. The dispersion curves with these fitting parameters are shown in Fig. \[fig:hex\_kagome\_bands\]. These curves agree with the simulated data despite the simplicity of the model. 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--- abstract: 'The performance of Gallager’s error-correcting code is investigated via methods of statistical physics. In this approach, the transmitted codeword comprises products of the original message bits selected by two randomly-constructed sparse matrices; the number of non-zero row/column elements in these matrices constitutes a family of codes. We show that Shannon’s channel capacity is saturated for many of the codes while slightly lower performance is obtained for others which may be of higher practical relevance. Decoding aspects are considered by employing the TAP approach which is identical to the commonly used belief-propagation-based decoding.' address: | $^{1}$ Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, Yokohama 2268502, Japan.\ $^{2}$The Neural Computing Research Group, Aston University, Birmingham B4 7ET, UK. author: - 'Yoshiyuki Kabashima$^{1}$, Tatsuto Murayama$^{1}$ and David Saad$^{2}$' title: 'Typical Performance of Gallager-type Error-Correcting Codes' --- The ever increasing information transmission in the modern world is based on communicating messages reliably through noisy transmission channels; these can be telephone lines, deep space, magnetic storing media etc. Error-correcting codes play an important role in correcting errors incurred during transmission; this is carried out by encoding the message prior to transmission, and decoding the corrupted received codeword for retrieving the original message. In his ground breaking papers, Shannon[@Shannon] analyzed the capacity of communication channels, setting an upper bound to the achievable noise-correction capability of codes, given their code (or symbol) rate. The latter represents the ratio between the number of bits in the original message and the transmitted codeword. Shannon’s bound is non-constructive and does not provide explicit rules for devising optimal codes. The quest for more efficient codes, in the hope of saturating the bound set by Shannon, has been going on ever since, providing many useful but sub-optimal codes. One family of codes, presented originally by Gallager[@Gallager], attracted significant interest recently as it has been shown to outperform most currently used techniques[@MacKay]. In fact, irregular versions of Gallager-type codes have recently been shown to get very close to saturating Shannon’s bound in the case of infinitely long messages[@Richardson]. Gallager-type codes are characterized by several parameters, the choice of which defines a particular member of this family of codes. Most studies of Gallager-type codes conducted so far have been carried out via numerical simulations. Some analytical results have been obtained via methods of information theory [@MacKay], setting bounds on the performance of certain code types, and by combinatorical/statistical methods [@Richardson]; no quantitative results have been obtained for their [*typical*]{} performance. In this Letter we analyze the typical performance of Gallager-type codes for several parameter choices via methods of statistical mechanics. We then validate the analytical solution by comparing the results to those obtained by the TAP approach to diluted systems and via numerical methods. In a general scenario, a message represented by an $N$ dimensional Boolean/binary vector ${\mbox{\boldmath{$\xi$}}}$ is encoded to the $M$ dimensional vector ${\mbox{\boldmath{$J^{0}$}}}$ which is then transmitted through a noisy channel with some flipping probability $p$ per bit (other noise types may also be considered but will not be examined here). The received message ${\mbox{\boldmath{$J$}}}$ is then decoded to retrieve the original message. One can identify several slightly different versions of Gallager-type codes. The one used in this Letter, termed the MN code[@MacKay] is based on choosing two randomly-selected sparse matrices $A$ and $B$ of dimensionality $M\!\times \!N$ and $M\!\times\! M$ respectively; these are characterized by $K$ and $L$ non-zero unit elements per row and $C$ and $L$ per column respectively. The finite, usually small, numbers $K$, $C$ and $L$ define a particular code; both matrices are known to both sender and receiver. Encoding is carried out by constructing the modulo 2 inverse of $B$ and the matrix $B^{-1}A$ (modulo 2); the vector ${\mbox{\boldmath{$J^{0}$}}}\! =\! B^{-1}A \ {\mbox{\boldmath{$\xi$}}}$ (modulo 2, ${\mbox{\boldmath{$\xi$}}}$ in a Boolean representation) constitutes the codeword. Decoding is carried out by taking the product of the matrix $B$ and the received message ${\mbox{\boldmath{$J$}}}\! = \! {\mbox{\boldmath{$J^{0}$}}}\! +\! {\mbox{\boldmath{$\zeta$}}}$ (modulo 2), corrupted by the Boolean noise vector ${\mbox{\boldmath{$\zeta$}}}$, resulting in $A{\mbox{\boldmath{$\xi$}}}\! + \! B{\mbox{\boldmath{$\zeta$}}}$. The equation $$\label{eq:decoding} A{\mbox{\boldmath{$\xi$}}}+ B{\mbox{\boldmath{$\zeta$}}}= A{\mbox{\boldmath{$S$}}}+ B{\mbox{\boldmath{$\tau$}}}$$ is solved via the iterative methods of Belief Propagation (BP)[@MacKay] to obtain the most probable Boolean vectors ${\mbox{\boldmath{$S$}}}$ and ${\mbox{\boldmath{$\tau$}}}$; BP methods in the context of error-correcting codes have recently been shown to be identical to a TAP[@tap] based solution of a similar physical system[@us_sourlas]. The similarity between error-correcting codes of this type and Ising spin systems was first pointed out by Sourlas[@Sourlas], who formulated the mapping of a simpler code, somewhat similar to the one presented here, onto an Ising spin system Hamiltonian. We recently extended the work of Sourlas, that focused on extensively connected systems, to the finite connectivity case[@us_sourlas]. To facilitate the current investigation we first map the problem to that of an Ising model with finite connectivity. We employ the binary representation $(\pm1)$ of the dynamical variables ${\mbox{\boldmath{$S$}}}$ and ${\mbox{\boldmath{$\tau$}}}$ and of the vectors ${\mbox{\boldmath{$J$}}}$ and ${\mbox{\boldmath{$J^{0}$}}}$ rather than the Boolean $(0,1)$ one; the vector ${\mbox{\boldmath{$J^{0}$}}}$ is generated by taking products of the relevant binary message bits $J^{0}_{\left\langle i_{1}, i_{2} \ldots \right\rangle} \! = \! \xi_{i_{1}} \xi_{i_{2}} \ldots $, where the indices $i_{1},i_{2}\ldots $ correspond to the non-zero elements of $B^{-1}A$, producing a binary version of ${\mbox{\boldmath{$J^{0}$}}}$. As we use statistical mechanics techniques, we consider the message and codeword dimensionality ($N$ and $M$ respectively) to be infinite, keeping the ratio between them $R \!=\! N/M$, which constitutes the code rate, finite. Using the thermodynamic limit is quite natural as Gallager-type codes are usually used for transmitting long ($10^{4}\!-\!10^{5}$) messages, where finite size corrections are likely to be negligible. To explore the system’s capabilities we examine the Hamiltonian $$\begin{aligned} \label{eq:Hamiltonian} {\cal H} &=& \!\!\!\!\! \sum_{<i_1,..,i_K;j_1,..,j_L>} \mbox{\hspace*{-5mm}} {{\cal D}}_{<i_1,..,i_K;j_1,..,j_L>} \ \delta \biggl[-1 \ ; \ {{\cal J}}_{<i_1,..,i_K;j_1,..,j_L>} \nonumber \\ &\cdot & S_{i_1}\ldots S_{i_K} \tau_{j_1}\ldots\tau_{j_L} \biggr] - \frac{F_s}{\beta} \sum_{i=1}^{N} S_i - \frac{F_{\tau}}{\beta} \sum_{j=1}^{M} \tau_j \ .\end{aligned}$$ The tensor product ${{\cal D}}_{<i_1,..,i_K;j_1,..,j_L>} {{\cal J}}_{<i_1,..,i_K;j_1,..,j_L>}$, where ${{\cal J}}_{<i_1,..,j_L>} \! = \! \xi_{i_{1}} \xi_{i_{2}} \ldots \xi_{i_{K}} \zeta_{j_{1}} \zeta_{j_{2}} \ldots \zeta_{j_{L}}$, is the binary equivalent of $A{\mbox{\boldmath{$\xi$}}}\! + \! B{\mbox{\boldmath{$\zeta$}}}$, treating both signal (${\mbox{\boldmath{$S$}}}$ and index $i$) and noise (${\mbox{\boldmath{$\tau$}}}$ and index $j$) simultaneously. Elements of the sparse connectivity tensor ${{\cal D}}_{<i_1,..,j_L>}$ take the value 1 if the corresponding indices of both signal and noise are chosen (i.e., if all corresponding indices of the matrices $A$ and $B$ are 1) and 0 otherwise; it has $C$ unit elements per $i$-index and $L$ per $j$-index representing the system’s degree of connectivity. The $\delta$ function provides $1$ if the selected sites’ product $S_{i_1}\ldots S_{i_K} \tau_{j_1}\ldots\tau_{j_L}$ is in disagreement with the corresponding element ${{\cal J}}_{<i_1,..,j_L>}$, recording an error, and $0$ otherwise. Notice that this term is not frustrated, as there are $M\! +\!N$ degrees of freedom and only $M$ constraints from Eq.(\[eq:decoding\]), and can therefore vanish at sufficiently low temperatures. The last two terms on the right represent our prior knowledge in the case of sparse or biased messages $F_s$ and of the noise level $F_{\tau}$ and require assigning certain values to these additive fields. The choice of $\beta\! \rightarrow \! \infty$ imposes the restriction of Eq.(\[eq:decoding\]), limiting the solutions to those for which the first term of Eq.(\[eq:Hamiltonian\]) vanishes, while the last two terms, scaled with $\beta$, survive. Note that the noise dynamical variables ${\mbox{\boldmath{$\tau$}}}$ are irrelevant to measuring the retrieval success $ m = \frac{1}{N} \ \left\langle \sum_{i=1}^{N} \ \xi_{i} \ \mbox{sign} \left\langle S_{i} \right\rangle_{\beta} \right\rangle_{\xi} \ .$ The latter monitors the normalized mean overlap between the Bayes-optimal retrieved message, shown to correspond to the alignment of $\left\langle S_{i} \right\rangle_{\beta}$ to the nearest binary value[@Sourlas], and the original message; the subscript $\beta$ denotes thermal averaging. Since the first part of Eq.(\[eq:Hamiltonian\]) is invariant under the transformations $S_{i} \!\rightarrow\! S_{i} \xi_{i}$, $\tau_{j}\! \rightarrow\! \tau_{j} \zeta_{j}$ and ${{\cal J}}_{<i_1,..,j_L>} \!\rightarrow\! {{\cal J}}_{<i_1,..,j_L>} \xi_{i_{1}} .. \xi_{i_{K}} \zeta_{j_{1}} \zeta_{j_{2}}.. \zeta_{j_{L}} \! =\! 1$, it would be useful to decouple the correlation between the vectors ${\mbox{\boldmath{$S$}}}$, ${\mbox{\boldmath{$\tau$}}}$ and ${\mbox{\boldmath{$\xi$}}}$, ${\mbox{\boldmath{$\zeta$}}}$. Rewriting Eq.(\[eq:Hamiltonian\]) one obtains a similar expression apart from the last terms on the right which become $ F_s/\beta \sum_{k} S_{k} \ \xi_{k}$ and $ F_{\tau}/\beta \sum_{k} \tau_{k} \ \zeta_{k}$. The random selection of elements in ${{\cal D}}$ introduces disorder to the system which is treated via methods of statistical physics. More specifically, we calculate the partition function ${\cal Z} ({{{\cal D}}},\mbox{\boldmath $J$}) = \mbox{Tr}_{\{{\mbox{\boldmath{$S$}}},{\mbox{\boldmath{$\tau$}}}\}} \exp [-\beta {\cal H}]$ averaged over the disorder and the statistical properties of the message and noise, using the replica method[@us_sourlas; @Wong_Sherrington; @Dedom]. Taking $\beta \!\rightarrow\!\infty$ gives rise to a set of order parameters $$\begin{aligned} \label{eq:order_parameters} q_{\alpha, \beta,.., \gamma} &=& \left\langle \frac{1}{N} \sum_{i=1}^{N} Z_{i} \ S_{i}^{\alpha} \ S_{i}^{\beta},..,S_{i}^{\gamma} \right\rangle_{\beta\rightarrow\infty} \nonumber \\ r_{\alpha, \beta,.., \gamma} &=& \left\langle \frac{1}{M} \sum_{i=1}^{M} Y_{j} \ \tau_{j}^{\alpha} \ \tau_{j}^{\beta},.., \tau_{j}^{\gamma} \right\rangle_{\beta\rightarrow\infty}\end{aligned}$$ where $\alpha$, $\beta,..$ represent replica indices, and the variables $Z_{i}$ and $ Y_{j}$ come from enforcing the restriction of $C$ and $L$ connections per index respectively[@us_sourlas]: $$\label{eq:delta} \delta \left( \sum_{\left\langle i_{2},..,i_{K} \right\rangle} \!\!\!\!\!\! {{\cal D}}_{<i, i_{2},..,j_L>} - C \right) = \oint_{0}^{2 \pi} \frac{d Z}{2 \pi} \ Z^{\sum_{\left\langle i_{2},.., i_{K} \right\rangle} \!\!\! {{\cal D}}_{<i, i_2,..,j_L>} -(C+1)} \ ,$$ and similarly for the restriction on the $j$ indices. To proceed with the calculation one has to make an assumption about the order parameters symmetry. The assumption made here, and validated later on, is that of replica symmetry in the following representation of the order parameters and the related conjugate variables $$\begin{aligned} \label{eq:order_parameters_RS} q_{\alpha, \beta.. \gamma} &=& a_{q} \int d x \ \pi(x) \ x^{l} \ , \ \widehat{q}_{\alpha, \beta.. \gamma} = a_{\widehat{q}} \int d \hat{x} \ \widehat{\pi}(\hat{x}) \ \hat{x}^{l} \\ r_{\alpha, \beta.. \gamma} &=& a_{r} \int d y \ \rho(y) \ y^{l} \ , \ \widehat{r}_{\alpha, \beta.. \gamma} = a_{\widehat{r}} \int d \hat{y} \ \widehat{\rho}(\hat{y}) \ \hat{y}^{l} \ , \nonumber\end{aligned}$$ where $l$ is the number of replica indices, $a_{*}$ are normalization coefficients, and $\pi(x), \widehat{\pi}(\hat{x}), \rho(y)$ and $\widehat{\rho}(\hat{y})$ represent probability distributions. Unspecified integrals are over the range $[-1,+1]$. One then obtains an expression for the free energy per spin expressed in terms of these probability distributions $$\begin{aligned} \label{eq:free_energy} &\frac{1}{N}& \left\langle \ln {\cal Z} \right\rangle_{\xi,\zeta,{{\cal D}}} = \mbox{Extr}_{\{ \pi,\widehat{\pi},\rho,\widehat{\rho} \}} \biggl\{ \frac{C}{K} \int \left[ \prod_{k=1}^{K} dx_{k} \ \pi(x_{k}) \right] \left[ \prod_{l=1}^{L} dy_{l} \ \rho(y_{l}) \right] \ln \left[ 1 + \prod_{k=1}^{K} x_{k} \prod_{l=1}^{L} y_{l} \right] \nonumber \\ &-& C \int dx \ d\hat{x} \ \pi(x) \ \widehat{\pi}(\hat{x}) \ \ln \left[ 1 + x \hat{x} \right] - \frac{CL}{K} \int dy \ d\hat{y} \ \rho(y) \ \widehat{\rho}(\hat{y}) \ \ln \left[ 1 + y \hat{y} \right] \\ &+& \int \left[ \prod_{k=1}^{C} dx_{k} \ \widehat{\pi}(\hat{x}_{k}) \right] \left\langle \ln \left[\prod_{k=1}^{C} \left(1+\hat{x}_{k} \right) \ e^{F_{s}\xi} + \prod_{k=1}^{C} \left(1-\hat{x}_{k} \right) \ e^{-F_{s}\xi} \right] \right\rangle_{\xi} \nonumber \\ &+& \frac{C}{K} \int \left[ \prod_{l=1}^{L} d\hat{y}_{l} \ \widehat{\rho}(\hat{y}_{l}) \right] \left\langle \ln \left[\prod_{l=1}^{L} \left(1+\hat{y}_{l} \right) \ e^{F_{\tau}\zeta} + \prod_{l=1}^{L} \left(1-\hat{y}_{l} \right) \ e^{-F_{\tau}\zeta} \right] \right\rangle_{\zeta} -\frac{C}{K} \ln 2 \biggr\}\nonumber \ ,\end{aligned}$$ where $\langle\cdot\rangle_{\xi}$ and $\langle\cdot\rangle_{\zeta}$ denote averages over the input and noise distributions of the form $$\langle\cdot\rangle_{\xi} = \sum_{\xi=\pm 1} \left\{ \frac{1+\tanh F_{s}}{2} \ \delta_{\xi, -1} + \frac{1-\tanh F_{s}}{2} \ \delta_{\xi, 1} \right\} (\cdot )$$ and similarly for $\langle\cdot\rangle_{\zeta}$ where $F_{s}$ is replaced by $F_{\tau}$. The free energy can then be calculated via the saddle point method. Solving the equations obtained by varying Eq.(\[eq:free\_energy\]) w.r.t the probability distributions $\pi(x), \widehat{\pi}(\hat{x}), \rho(y)$ and $\widehat{\rho}(\hat{y})$, is generally difficult. The solutions obtained in the case of unbiased messages (the most interesting case as most messages are compressed prior to transmission) are for the ferromagnetic phase: $$\begin{aligned} \label{eq:sol_ferro} \pi(x) &=& \delta (x-1) \ , \ \widehat{\pi}(\hat{x}) = \delta (\hat{x}-1) \nonumber \\ \rho(y) &=& \delta (y-1) \ , \ \widehat{\rho}(\hat{y}) = \delta (\hat{y}-1) \ ,\end{aligned}$$ and for the paramagnetic phase (there is no spin-glass solution due to lack of frustration): $$\begin{aligned} \label{eq:sol_para} \pi(x) &=& \delta (x) \ , \ \widehat{\pi}(\hat{x}) = \delta (\hat{x}) \ , \ \widehat{\rho}(\hat{y}) = \delta (\hat{y}) \nonumber \\ \rho(y) &=& \frac{1+\tanh F_{\tau}}{2} \ \delta (y-\tanh F_{\tau}) + \frac{1-\tanh F_{\tau}}{2} \ \delta (y+\tanh F_{\tau}) \ .\end{aligned}$$ It is easy to verify that these solutions obey the saddle point equations. However, it is necessary to validate the stability of the solutions and the replica symmetric ansatz itself. To address these questions we obtained solutions to the system described by the Hamiltonian (\[eq:Hamiltonian\]) via the TAP method of finitely connected systems[@us_sourlas]; we solved the saddle point equations derived from Eq.(\[eq:free\_energy\]) numerically, representing all probability distributions by up to $10^4$ bin models and by carrying out the integrations via Monte-Carlo methods; finally, to show the consistency between theory and practice we carried out large scale simulations for several cases, which will be presented elsewhere. The results obtained by the various methods are in complete agreement. The various methods indicate that the solutions may be divided to two different categories characterized by $K\!=\! L\!=\! 2$ and by either $K\!\ge\! 3$ or $L\!\ge\! 3$, which we therefore treat separately. For unbiased messages and either $K\!\ge\! 3$ or $L\!\ge\! 3$ we obtain the solutions (\[eq:sol\_ferro\]) and (\[eq:sol\_para\]) both by applying the TAP approach and by solving the saddle point equations numerically. The former was carried out at the value of $F_{\tau}$ which corresponds to the true noise and input bias levels (for unbiased messages $F_s\!=\! 0$) and thus to Nishimori’s condition[@Nishimori], where no replica symmetry breaking effect is expected. This is equivalent to having the correct prior within the Bayesian framework[@Sourlas_EPL] and enables one to obtain analytic expressions for some observables as long as some gauge requirements are obeyed[@Nishimori]. Numerical solutions show the emergence of stable dominant delta peaks, consistent with those of (\[eq:sol\_ferro\]) and (\[eq:sol\_para\]). The question of longitudinal mode stability (corresponding to the replica symmetric solution) was addressed by setting initial conditions for the numerical solutions close to the solutions (\[eq:sol\_ferro\]) and (\[eq:sol\_para\]), showing that they converge back to these solutions which are therefore stable. The most interesting quantity to examine is the maximal code rate, for a given corruption process, for which messages can be perfectly retrieved. This is defined in the case of $K,L\!\ge\! 3$ by the value of $R \! = \! K/C \! = \! N/M$ for which the free energy of the ferromagnetic solution becomes smaller than that of the paramagnetic solution, constituting a first order phase transition. A schematic description of the solutions obtained is shown in the inset of Fig.1a. The paramagnetic solution ($m\!=\!0$) has a lower free energy than the ferromagnetic one (low/high free energies are denoted by the thick and thin lines respectively, there are no axis lines at $m\!=\!0,1$) for noise levels $p\! >\! p_c$ and vice versa for $p\! \le \! p_c$; both solutions are stable. The critical code rate is derived by equating the ferromagnetic and paramagnetic free energies to obtain $$R_{c}\!=\! 1\!-\!H_{2}(p)\!=\! 1\!+\!\left(p \log_{2} p \!+\!(1-p) \log_{2} (1-p) \right) \ .$$ This coincides with [*Shannon’s capacity*]{}. To validate these results we obtained TAP solutions for the unbiased message case ($K\!=\! L\!=\! 3$, $C\!=\!6$). Averages over 10 solutions obtained for different initial conditions in the vicinity of the stable solutions are presented in Fig.1a (as $+$) in comparison to Shannon’s capacity (solid line). Analytical solutions for the saddle point equations cannot be obtained for the case of biased patterns and we therefore resort to numerical methods and the TAP approach. The maximal information rate (i.e., code rate $\times H_2(f_{s}=(1+\tanh F_{s})/2)$ - the source redundancy) obtained by the TAP method ($\Diamond$) and numerical solutions of the saddle point equations ($\Box$), averaged for each noise level over solutions obtained for 10 different starting points in the vicinity of the analytical solution, are shown in Fig.1a. Numerical results have been obtained using $10^3 \!-\! 10^4$ bin models for each probability distribution and had been run for $10^{5}$ steps per noise level point. The various results are highly consistent and practically saturate Shannon’s bound for the same noise level. The MN code for $K,L \ge 3$ seems to offer optimal performance. However, the main drawback is rooted in the co-existence of the stable $m=1$ and $m=0$ solutions, shown in Fig.1a (inset), which implies that from some initial conditions the system will converge to the undesired paramagnetic solution. Moreover, studying the ferromagnetic solution numerically shows a highly limited basin of attraction, which becomes smaller as $K$ and $L$ increase, while the paramagnetic solution at $m=0$ [*always*]{} enjoys a wide basin of attraction. As initial conditions for the decoding process are typically of close-to-zero magnetization (almost no prior information about the original message is assumed) it is highly likely that the decoding process will converge to the paramagnetic solution. This performance has been observed via computer simulations by us and by others[@MacKay]. While all codes with $K,L \ge 3$ saturate Shannon’s bound and are characterized by a first order, paramagnetic to ferromagnetic, phase transition, codes with $K\!=\! L\!=\! 2$ show lower performance and different physical characteristics. The analytical solutions (\[eq:sol\_ferro\]) and (\[eq:sol\_para\]) are unstable at some flip rate levels and one resorts to solving the saddle point equations numerically and to TAP based solutions. The picture that emerges is sketched in the inset of Fig.1b: The paramagnetic solution dominates the high flip rate regime (appearing as a dominant delta peak in the numerical solutions) up to the point $p_{1}$ (denoted as 1 in the inset) in which a stable, ferromagnetic solution, of higher free energy, appears (thin lines at $m\!=\!\pm 1$). At a lower flip rate value $p_{2}$ the paramagnetic solution becomes unstable (dashed line) and is replaced by two stable sub-optimal ferromagnetic (broken symmetry) solutions which appear as a couple of peaks in the various probability distributions; typically, these have a lower free energy than the ferromagnetic solution until $p_{3}$, after which the ferromagnetic solution becomes dominant (at some code rate values it is dominant directly following the disappearance of the paramagnetic solution). Still, only once the sub-optimal ferromagnetic solutions disappear, at the spinodal point $p_{s}$, a unique ferromagnetic solution emerges as a single delta peak in the numerical results (plus a mirror solution). The point in which the sub-optimal ferromagnetic solutions disappear constitutes the maximal practical flip rate for the current code rate and was defined numerically ($\Diamond$) and via TAP solutions ($+$) as shown in Fig.1b. Notice that initial conditions for both TAP and the numerical solutions were chosen almost randomly, with a very slight bias of ${\cal O}(10^{-12})$, in the initial magnetization. The TAP dynamical equations are identical to those used for practical BP decoding[@us_sourlas], and therefore provide equivalent results to computer simulations with the same parameterization, supporting the analytical results. The excellent convergence results obtained point out the existence of a unique pair of global solutions to which the system converges (below $p_{s}$) [*from practically all initial conditions*]{}. This observation and the practical implications of using the $K\!=\! L\!=\! 2$ code have not been obtained by information theory methods (e.g.[@MacKay]); these prove the existence of very good codes for $C,L\!\ge\! 3$, and examine decoding properties only via numerical simulations. In this Letter we examined the typical performance of Gallager-type codes. We discovered that for a certain choice of parameters, either $K\!\ge\! 3$ or $L\!\ge\! 3$, one obtains optimal performance, saturating Shannon’s bound. This comes at the expense of a decreasing basin of attraction making the decoding process increasingly impractical. Another code, $K\!=\! L\!=\! 2$, shows close to optimal performance with a very large basin of attraction, making it highly attractive for practical purposes. Studying the typical performance of Gallager-type codes, which complements the methods used in the information theory literature, is the first step towards understanding their exceptional performance and in the search for a principled method for designing optimal Gallager-type codes. Important aspects that are yet to be investigated include other noise types, irregular constructions and the significance of finite size effects. 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--- abstract: 'In this paper we develop Algebraic Morse Theory for the case where a group acts on a free chain complex. Algebraic Morse Theory is an adaption of Discrete Morse Theory to free chain complexes.' address: 'Fachbereich Mathematik, Universität Bremen, Bibliothekstraße 1, 28359 Bremen, Germany' author: - Ralf Donau title: Equivariant Algebraic Morse Theory --- Discrete Morse Theory,Acyclic matching,Algebraic Morse Theory Introduction ============ There exists an equivariant version of the Main Theorem of Discrete Morse Theory, see [@freij]. In this paper I present an equivariant version of Theorem 11.24 in [@buch Chapter 11.3]. We use the same notion of an equivariant acyclic matching as in Equivariant Discrete Morse Theory. An example for working with equivariant acyclic matchings can be found in [@donau3]. Equivariant acyclic matchings ============================= The definition of an acyclic matching on a poset can be found in [@clmap; @buch]. Let $P$ be a poset and let $G$ be a group acting on $P$. Let $M$ be an acyclic matching on $P$. We call $M$ an *$G$-equivariant acyclic matching* if $(a,b)\in M$ implies $(ga,gb)\in M$ for all $g\in G$ and $a,b\in P$. There exists a characterization of acyclic matchings by means of order-preserving maps with small fibers, see Definition 11.3 and Theorem 11.4 in [@buch Chapter 11]. In a similar way we can also characterize $G$-equivariant acyclic matchings by means of order-preserving $G$-maps with small fibers. For an order-preserving map with small fibers $\varphi$ let $M(\varphi)$ denote its associated acyclic matching which consists of all fibers of cardinality $2$, see [@buch Chapter 11]. \[smallfibers\] Let $G$ be a group acting on a finite poset $P$. For any order-preserving $G$-map $\varphi:P\longrightarrow Q$ with small fibers, the acyclic matching $M(\varphi)$ is $G$-equivariant. On the other hand, any $G$-equivariant acyclic matching $M$ on $P$ can be represented as $M=M(\varphi)$, where $\varphi:P\longrightarrow Q$ is an order-preserving $G$-map with small fibers. The proof of Proposition \[smallfibers\] can be found in [@donau3]. The main result =============== We consider chain complexes of modules over some fixed commutative ring $R$ with unit. Furthermore we consider group actions on such chain complexes. Let $G$ be a group and let $C_*=(\dots\overset{\partial_{n+2}}\longrightarrow C_{n+1}\overset{\partial_{n+1}}\longrightarrow C_n\overset{\partial_n}\longrightarrow\dots)$ be a finitely generated free chain complex with an action of the group $G$ and let $\Omega=(\Omega_n)_n$ be a $G$-basis of $C_*$, i.e. each $\Omega_n$ is closed under the action of $G$. For $b\in\Omega_n$ let $k_b:C_n\longrightarrow R$, $\sum_{b'\in\Omega_n}\lambda_{b'}b'\longmapsto\lambda_b$ denote the linear function which maps any $x\in C_n$ to the coefficient of $b$ inside the linear representation of $x$. Let $P(C_*,\Omega):=\bigcup_n\Omega_n$. We define an order relation on $P(C_*,\Omega)$ as follows. For $a\in\Omega_n$ and $b\in\Omega_{n+1}$ we denote the *weight of the covering relation* by $w(b\succ a):=k_a(\partial_n b)$, we set $a\leq b$ if $w(b\succ a)\not=0$. Furthermore for $a\in\Omega_n$ and $b\in\Omega_m$ with $n\leq m$, we set $a\leq b$ if there exists a sequence $(c_i)_{n\leq i\leq m}$ with $c_i\leq c_{i+1}$ for $n\leq i<m$ such that $a=c_n$ and $b=c_m$. This defines a partial order relation on $P(C_*,\Omega)$, which can be easily verified. Notice that $P(C_*,\Omega)$ is a $G$-poset since $\partial_n$ is a $G$-map. Let $M$ be an $G$-equivariant acyclic matching on $P(C_*,\Omega)$ such that $w(b\succ a)$ is invertible for any $(a,b)\in M$. Let $\varphi$ denote the order-preserving $G$-map with small fibers with $M(\varphi)=M$, which exists by Proposition \[smallfibers\]. For $b\in C_n$ we define the $G$-subcomplex ${\cal A}(Gb)$ as follows. $$\dots\longrightarrow 0\longrightarrow\span(Gb)\overset{\partial^{Gb}_n}\longrightarrow\span(G\partial_n b)\longrightarrow 0\longrightarrow\dots$$ $\partial^{Gb}_n$ denotes the restriction of $\partial_n:C_n\longrightarrow C_{n-1}$ to $\span(Gb)$, i.e. $\partial^{Gb}_n(x):=\partial_n(x)$ for $x\in\span(Gb)$. Notice that $\partial^{Gb}_n$ is surjective by construction. \[poset\_orbit\] Let $G$ be a finite group acting on a finite poset $Q$. Let $q\in Q$ and $g\in G$. Then $gq\leq q$ implies $gq=q$. In other words, the elements inside an orbit are not comparable to each other. Let $(a,b)\in M$ be a matching pair. Then for any $g\in G$, $a\prec gb$ implies $\varphi(b)=\varphi(a)\leq\varphi(gb)=g\varphi(b)$ which implies $\varphi(b)=\varphi(gb)$ by Remark \[poset\_orbit\]. Hence $b=gb$ since $\varphi$ has small fibers. In particular $k_a(\partial gb)=0$ for $gb\not=b$. \[iso\] Let $(a,b)\in M$ be a matching pair, assume $b\in C_n$. Then $\partial^{Gb}_n$ is an isomorphism and ${\cal A}(Gb)$ is $G$-homotopy equivalent to the zero complex. We have to show that $\partial^{Gb}_n$ is injective. Assume $\partial_n(\sum_{b'\in Gb}\lambda_{b'}b')=0$. Then $0=k_a(\sum_{b'\in Gb}\lambda_{b'}\partial_nb')=\lambda_{b}k_a(\partial_nb)$, which implies $\lambda_b=0$, since $k_a(\partial_nb)$ is invertible. This implies $\lambda_{b'}=0$ for all $b'\in Gb$, since $\partial_n$ is a $G$-map. The composition ${\cal A}(Gb)\longrightarrow0\longrightarrow{\cal A}(Gb)$ is homotop to $\id:{\cal A}(Gb)\longrightarrow{\cal A}(Gb)$ via the $G$-chain homotopy $P=(P_i)$, where - $P_{n-1}:=(\partial^{Gb}_n)^{-1}$ - $P_i:\equiv 0$ for $i\not=n-1$ On the other hand $0\longrightarrow{\cal A}(Gb)\longrightarrow 0$ equals $\id:0\longrightarrow 0$. Hence ${\cal A}(Gb)$ is $G$-homotopy equivalent to the zero complex. \[eqthm\] Let $M$ be an $G$-equivariant acyclic matching on $P(C_*,\Omega)$ such that $w(b\succ a)$ is invertible. Then there exists a $G$-chain complex $C_*^M$ such that $C_*\cong_G C_*^M\oplus T_*$ where $T_*=\bigoplus_{G(a,b)\in M/G}{\cal A}(Gb)$. This in an $G$-equivariant version of Theorem 11.24 in [@buch Chapter 11.3]. Notice that $C_*^M$ is unique up to $G$-isomorphism. By induction on the number $m=|M/G|$ of orbits in $M$. For $m=0$, we set $C_*^M:=C_*$. Now assume $m>0$. By Proposition \[smallfibers\], there exists a finite poset $Q$ and an order-preserving $G$-map with small fibers $\varphi:P(C_*,\Omega)\longrightarrow Q$ such that $M(\varphi)=M$. We consider the following subposet of $Q$: $$Q':=\{q\in Q\mid\varphi^{-1}(q)=\{a,b\}\text{ with }a\not=b\}$$ Let $q\in Q'$ be a minimal element. Assume $a\in\Omega_{n-1}$, $b\in\Omega_n$ and $\varphi^{-1}(q)=\{a,b\}$. We construct a new basis $\widetilde\Omega$ by replacing $\Omega_{n-1}$ and $\Omega_n$ as follows. We construct two $G$-automorphisms $f_{n-1}:C_{n-1}\longrightarrow C_{n-1}$ and $f_n:C_n\longrightarrow C_n$. $$\begin{aligned} f_{n-1}:C_{n-1}\supset\Omega_{n-1}&\longrightarrow&C_{n-1}\\ ga&\longmapsto&g\partial_n b\text{ for all $g\in G$}\\ x&\longmapsto&x\text{ for $x\not\in Ga$}\end{aligned}$$ We set $\Omega^{Gb}_{n-1}:=\Omega_{n-1}\setminus Ga$, then $\widetilde\Omega_{n-1}:=f_{n-1}(\Omega_{n-1})=\Omega^{Gb}_{n-1}\cup Gf_{n-1}(a)$ generates $C_{n-1}$ since the linear representation of $f_{n-1}(a)$ is $$f_{n-1}(a)=\partial_n b=w(b\succ a)\cdot a+\sum_{x\in\Omega^{Gb}_{n-1}}w(b\succ x)\cdot x$$ which implies $$a=\frac 1{w(b\succ a)}\cdot f_{n-1}(a)-\sum_{x\in\Omega^{Gb}_{n-1}}\frac{w(b\succ x)}{w(b\succ a)}\cdot x$$ hence $a\in\span(f_{n-1}(\Omega_{n-1}))$. This implies $ga\in\span(f_{n-1}(\Omega_{n-1}))$ for all $g\in G$ since $f_{n-1}$ is a $G$-map by construction. Assume $$\sum_{x\in\Omega^{Gb}_{n-1}}\lambda_xx+\sum_{a'\in Ga}\lambda_{a'}f_{n-1}(a')=0$$ Then in particular $$\begin{aligned} &k_a\left(\sum_{x\in\Omega^{Gb}_{n-1}}\lambda_xx+\sum_{a'\in Ga}\lambda_{a'}f_{n-1}(a')\right)&=0\\ \Longrightarrow&\lambda_a\cdot k_a(\partial_n b)&=0\\ \Longrightarrow&\lambda_a\cdot w(b\succ a)&=0\\ \Longrightarrow&\lambda_a&=0\end{aligned}$$ For any $g\in G$ this implies $\lambda_{ga}=0$, because $f_{n-1}$ is a $G$-map. Since $\Omega^{Gb}_{n-1}$ is linear independent, we have $\lambda_x=0$ for all $x\in\Omega^{Gb}_{n-1}$. $$\begin{aligned} f_n:C_n\supset\Omega_n&\longrightarrow&C_n\\ gb&\longmapsto&gb\text{ for all $g\in G$}\\ x&\longmapsto&x-\sum_{b'\in Gb}w(x\succ\partial_n b')\cdot b'\text{ for $x\not\in Gb$}\end{aligned}$$ We set $\Omega_n^{Gb}:=f_n(\Omega_n\setminus Gb)$, then $\widetilde\Omega_n:=f_n(\Omega_n)=\Omega_n^{Gb}\cup Gb$ generates $C_n$, since for all $x\in\Omega_n\setminus Gb$ we have $$x=f_n(x)+\sum_{b'\in Gb}w(x\succ\partial b')\cdot b'\in\span(\widetilde\Omega_n)$$ where $f_n(x)\in\span(\Omega_n^{Gb})$ and $\sum_{b'\in Gb}w(x\succ\partial b')\cdot b'\in\span(Gb)$. Clearly we have $k_x(f_n(x))=1$ and $k_x(f_n(x'))=0$ for all $x'\in\Omega_n\setminus\{x\}$. Assume $$\sum_{x'\in\Omega_n\setminus Gb}\lambda_{x'} f_n(x')+\sum_{b'\in Gb}\lambda_{b'}b'=0$$ Then for all $x\in\Omega_n\setminus Gb$: $$\begin{aligned} &k_x\left(\sum_{x'\in\Omega_n\setminus Gb}\lambda_{x'} f_n(x')+\sum_{b'\in Gb}\lambda_{b'}b'\right)&=0\\ \Longrightarrow&\lambda_xk_x(f_n(x))&=0\\ \Longrightarrow&\lambda_x&=0\end{aligned}$$ Since $Gb$ is linear independent, this also implies $\lambda_{gb}=0$ for all $g\in G$. Hence $\widetilde\Omega_{n-1}$ and $\widetilde\Omega_n$ are bases of $C_{n-1}$ and $C_n$ respectively. Let $\widetilde\Omega$ denote the basis of $C_*$ which is obtained from $\Omega$ by replacing $\Omega_{n-1}$ and $\Omega_n$ with these newly constructed bases. Then $\widetilde\Omega$ has the following properties: - $\partial_n(\span(\Omega_n^{Gb}))\subset\span(\Omega_{n-1}^{Gb})$ - $\im\partial_{n+1}\subset\span(\Omega_n^{Gb})$ For $b\in\widetilde\Omega$ let $\widetilde k_b$ denote the linear function which maps an element to the coefficient of $b$ inside its linear representation. For all $x\in\Omega_n\setminus Gb$ we have $$\begin{aligned} \widetilde k_{f_{n-1}(a)}(\partial_n f_n(x))&=&\widetilde k_{f_{n-1}(a)}(\partial_n x)-\sum_{b'\in Gb}w(x\succ\partial_n b')\widetilde k_{f_{n-1}(a)}(\partial_n b')\\ &=&\widetilde k_{f_{n-1}(a)}(\partial_n x)-w(x\succ\partial_n b)\\ &=&\widetilde k_{f_{n-1}(a)}(\partial_n x)-w(x\succ f_{n-1}(a))\\ &=&0\end{aligned}$$ Since $f_n$ is a $G$-map, we also have $\widetilde k_{f_{n-1}(ga)}(\partial_n f_n(x))=0$ for all $g\in G$ and all $x\in\Omega_n\setminus Gb$. Hence (1) is proven. We have $\widetilde k_{f_{n-1}(a)}(\partial_n f_n(b'))=\widetilde k_{f_n(b)}(f_n(b'))$ for all $b'\in Gb$ by construction of $f_{n-1}$, on the other hand we have $0=\widetilde k_{f_{n-1}(a)}(\partial_n x)=\widetilde k_{f_n(b)}(x)$ for all $x\in\Omega_n^{Gb}$ as shown before. Hence $\widetilde k_{f_{n-1}(a)}(\partial_n z)=\widetilde k_{f_n(b)}(z)$ for all $z\in C_n$. In particular $0=\widetilde k_{f_{n-1}(a)}(\partial_n\partial_{n+1}(y))=\widetilde k_{f_n(b)}(\partial_{n+1}(y))$ for all $y\in C_{n+1}$. Since $C_*$ is a $G$-complex, we also have $\widetilde k_{f_n(gb)}(\partial_{n+1}(y))=0$ for all $y\in C_{n+1}$ and all $g\in G$. Hence (2) is proven. We obtain the following two $G$-subcomplexes ${\cal A}(Gb)$ and $C_*^{Gb}$: $$\dots\longrightarrow 0\longrightarrow\span(Gb)\overset{\partial_n^{Gb}}\longrightarrow\span(Gf_{n-1}(a))\longrightarrow 0\longrightarrow\dots$$ $$\dots\longrightarrow C_{n+1}\overset{\partial_{n+1}}\longrightarrow\span(\Omega_n^{Gb})\overset{\partial_n}\longrightarrow\span(\Omega_{n-1}^{Gb})\overset{\partial_{n-1}}\longrightarrow C_{n-2}\longrightarrow\dots$$ Clearly we have $C_*\cong_G C_*^{Gb}\oplus{\cal A}(Gb)$. Furthermore $C_*^{Gb}$ is a free $G$-chain complex with $G$-basis $\Omega^{Gb}$, which is obtained from $\Omega$ by replacing $\Omega_n$ and $\Omega_{n-1}$ by $\Omega_n^{Gb}$ and $\Omega_{n-1}^{Gb}$ respectively. Now we construct an $G$-equivariant acyclic matching $M^{Gb}$ on $P(C_*^{Gb},\Omega^{Gb})$ such that $w(b\succ a)$ is invertible for any $(a,b)\in M^{Gb}$. Let $R\subset P(C_*,\Omega)$ be the set of all elements appearing in any matching pair in $M\setminus G(a,b)$. We define the map $$\begin{aligned} f:R&\longrightarrow&P(C_*^{Gb},\Omega^{Gb})\\ x&\longmapsto&f_n(x)\text{ for $x\in\Omega_n\setminus Gb$}\\ x&\longmapsto&x\text{ else}\end{aligned}$$ which has the following property: Let $x,y\in R$ such that $x\succ y$, then we have $w(f(x)\succ f(y))=w(x\succ y)$. This is clear for $x,y\not\in C_n$. It remains to prove the two other cases: Case $x\in\Omega_n$: $\varphi(b)=q$ is minimal in $Q'$ by choice. We have $b\not\succ y$, since $b>y$ implies $\varphi(b)>\varphi(y)\in Q'$ which contradicts the minimality of $q$. Because $C_*^{Gb}$ is a $G$-complex, this implies $w(gb\succ y)=0$ for all $g\in G$. Hence $w(f(x)\succ f(y))=w(f(x)\succ y)=w(x\succ y)-\sum_{b'\in Gb} w(x\succ\partial_n b')\cdot w(b'\succ y)=w(x\succ y)$. Case $y\in\Omega_n$: As mentioned (how $k_x$ behaves on $\widetilde\Omega$) in the proof of the injectivity of $f_n$, we have $\widetilde k_{f(y)}(\partial_{n+1} x)=k_y(\partial_{n+1} x)$. Hence $w(f(x)\succ f(y))=w(x\succ f(y))=w(x\succ y)$. We set $$M^{Gb}:=\{(f(a'),f(b')\mid(a',b')\in M\setminus G(a,b)\}$$ By the property of $f$, any cycle in $M^{Gb}$ would map to a cycle in $M$ under $f^{-1}$. It is easy to see that $M^{Gb}$ has all desired properties. By induction hypothesis there exists a $G$-chain complex $X_*$ such that $$\begin{aligned} C_*^{Gb}&\cong_G&X_*\oplus\bigoplus_{G(a',b')\in M^{Gb}/G}{\cal A}(Gb')\\ &=&X_*\oplus\bigoplus_{G(a',b')\in M\setminus G(a,b)/G}{\cal A}(Gf_n(b'))\end{aligned}$$ $f_n$ induces a $G$-chain isomorphism $(\varphi_n)_n:{\cal A}(Gb')\longrightarrow{\cal A}(Gf_n(b'))$ for any $b'\in R$, where $$\begin{aligned} \varphi_n:\span(Gb')&\longrightarrow&\span(Gf_n(b'))\\ x&\longmapsto&f_n(x)\end{aligned}$$ $$\begin{aligned} \varphi_{n-1}:\span(G\partial_nb')&\longrightarrow&\span(G\partial_nf_n(b'))\\ x&\longmapsto&(\partial_n\circ f_n)((\partial^{Gb'}_n)^{-1}(x))\end{aligned}$$ By Lemma \[iso\], the restrictions of $\partial_n$ used in the definition of $\varphi_{n-1}$ are isomorphisms. We set $C_*^M:=X_*$. $$\begin{aligned} C_*\cong_G C_*^{Gb}\oplus{\cal A}(Gb)&\cong_G&\left(C_*^M\oplus\bigoplus_{G(a',b')\in M\setminus G(a,b)/G}{\cal A}(Gb')\right)\oplus{\cal A}(Gb)\\ &=&C_*^M\oplus\bigoplus_{G(a',b')\in M/G}{\cal A}(Gb')\end{aligned}$$ In the case $G$ is the trivial group, our result is the same as Theorem 11.24 in [@buch Chapter 11.3], where $C_*^M$ is the Morse complex defined in [@buch Chapter 11.3]. By forgetting the action of $G$ we obtain $$\bigoplus_{G(a,b)\in M/G}{\cal A}(Gb)\cong\bigoplus_{(a,b)\in M}{\cal A}(b)$$ Since the Morse complex for the matching $M$ is also a $G$-complex, it is usable in the $G$-equivariant case too. $C_*$ is $G$-homotopy equivalent to $C_*^M$. $T_*$ is $G$-homotopy equivalent to the zero complex by Lemma \[iso\]. Hence $C_*^M\oplus 0$ is $G$-homotopy equivalent to $C_*^M\oplus T_*$ which is $G$-isomorphic to $C_*$ by Proposition \[eqthm\], in particular $G$-homotopy equivalent. [00]{} R. Donau, *A nice acyclic matching on the nerve of the partition lattice*, `arXiv:1204.2693 [math.AT]` (2015). R. Freij, *Equivariant discrete Morse theory*, Discrete Mathematics **309** (12) (2009), pp. 3821-3829. D.N. Kozlov, *Closure maps on regular trisps*, Topology and its Applications **156** (15) (2009), pp. 2491-2495. D. Kozlov, *Combinatorial Algebraic Topology*, Algorithms and Computation in Mathematics **21**, Springer-Verlag Berlin Heidelberg, 2008.

--- abstract: | We discuss an approach to obtaining black hole quasinormal modes (QNMs) using the asymptotic iteration method (AIM), initially developed to solve second order ordinary differential equations. We introduce the standard version of this method and present an improvement more suitable for numerical implementation. We demonstrate that the AIM can be used to find radial QNMs for Schwarzschild, Reissner-Nordström (RN) and Kerr black holes in a unified way. An advantage of the AIM over the standard continued fraction method (CFM) is that for differential equations with more than three regular singular points Gaussian eliminations are not required. However, the convergence of the AIM depends on the location of the radial or angular position, choosing the best such position in general remains an open problem. This review presents for the first time the spin $0, 1/2$ & $2$ QNMs of a Kerr black hole and the gravitational and electromagnetic QNMs of the RN black hole calculated via the AIM, and confirms results previously obtained using the CFM. We also presents some new results comparing the AIM to the WKB method. Finally we emphasize that the AIM is well suited to higher dimensional generalizations and we give an example of doubly rotating black holes.\ author: - 'H. T. Cho' - 'A. S. Cornell' - Jason Doukas - 'T. -R. Huang' - Wade Naylor date: '18$^{th}$ November, 2011' title: | A New Approach to Black Hole Quasinormal Modes:\ A Review of the Asymptotic Iteration Method --- Introduction {#sec:1} ============ The study of quasinormal modes (QNMs) of black holes is an old and well established subject, where the various frequencies are indicative of both the parameters of the black hole and the type of emissions possible. Initially the calculation of these frequencies was done in a purely numerical way, which requires selecting a value for the complex frequency, integrating the differential equation, and checking whether the boundary conditions are satisfied. Note that in the following we shall use the definition that QNMs are defined as solutions of the perturbed field equations with boundary conditions: $$\psi(x) \to \left\{ \begin{array}{cl} e^{-i \omega x} &\qquad x\to -\infty \\ e^{i \omega x} &\qquad x\to \infty \end{array} \right. \; , \label{QNMdef}$$ for an $e^{-i \omega t}$ time dependence (which corresponds to ingoing waves at the horizon and outgoing waves at infinity). Also note the boundary condition as $x\to\infty$ does not apply to asymptotically anti-de Sitter spacetimes, where instead something like a Dirichlet boundary condition is imposed, for example see Ref. [@Moss:2001ga]. Since those conditions are not satisfied in general, the complex frequency plane must be surveyed for discrete values that lead to QNMs. This technique is time consuming and cumbersome, making it difficult to systematically survey the QNMs for a wide range of parameter values. Following early work by Vishveshwara [@Vishveshwara:1970zz], Chandrasekhar and Detweiler [@Chandrasekhar:1975zza] pioneered this method for studying QNMs. In order to improve on this, a few semi-analytic analyses were also attempted. In one approach, employed by Mashoon [*et al.*]{} [@Ferrari:1984zz], the potential barrier in the effective one-dimensional Schrödinger equation is replaced by a parameterized analytic potential barrier function for which simple exact solutions are known. The overall shape approximates that of the true black hole barrier, and the parameters of the barrier function are adjusted to fit the height and curvature of the true barrier at the peak. The resulting estimates for the QNM frequencies have been applied to the Schwarzschild, Reissner-Nordström and Kerr black holes, with agreement within a few percent with the numerical results of Chandrasekhar and Detweiler in the Schwarzschild case [@Chandrasekhar:1975zza], and with Gunter [@Gunter:1980] in the Reissner-Nordström case. However, as this method relies upon a specialized barrier function, there is no systematic way to estimate the errors or to improve the accuracy. The method by Leaver [@Leaver1985], which is a hybrid of the analytic and the numerical, successfully generates QNM frequencies by making use of an analytic infinite-series representation of the solutions, together with a numerical solution of an equation for the QNM frequencies which involves, typically by applying a Frobenius series solution approach, the use of continued fractions. This technique is known as the continued fraction method (CFM). Historically, another commonly applied technique is the WKB approximation [@Seidel:1989bp]. Even though it is based on an approximation, this approach is powerful as the WKB approximation is known in many cases to be more accurate, and can be carried to higher orders, either as a means to improve accuracy or as a means to estimate the errors explicitly. Also it allows a more systematic study of QNMs than has been possible using outright numerical methods. The WKB approximation has since been extended to sixth-order [@Konoplya:2003ii]. However, all of these approaches have their limitations, where in recent years a new method has been developed which can be more efficient in some cases, called the asymptotic iteration method (AIM). Previously this method was used to solve eigenvalue problems [@Ciftci:2005xn] as a semi-analytic technique for solving second-order homogeneous linear differential equations. It has also been successfully shown by some of the current authors that the AIM is an efficient and accurate technique for calculating QNMs [@Cho:2009cj]. As such, we will review the AIM as applied to a variety of black hole spacetimes, making (where possible) comparisons with the results calculated by the WKB method and the CFM á la Leaver [@Leaver1985]. Therefore, the structure of this paper shall be: In Sec. \[sec:2\] we shall review the AIM and the improved method of Ciftci [*et al.*]{} [@Ciftci:2005xn] (also see Ref. [@Barakat:2006ki]), along with a discussion of how the QNM boundary conditions are ensured. Applications to simple concrete examples, such as the harmonic oscillator and the Poschl-Teller potential are also provided. In Sec. \[sec:3\] the case of Schwarzschild (A)dS black holes shall be discussed, developing the integer and half-spin equations. In Sec. \[sec:4\] a review of how the QNMs of the Reissner-Nordström black holes shall be made, with several frequencies calculated in the AIM and compared with previous results. Sec. \[sec:5\] will review the application of the AIM to Kerr black holes for spin $0, 1/2, 2$ fields. Sec. \[sec:6\] will discuss the spin-zero QNMs for doubly rotating black holes. We then summarize and conclude in Sec. \[sec:7\]. The Asymptotic Iteration Method {#sec:2} =============================== The Method ---------- To begin we shall now review the idea behind the AIM, where we first consider the homogeneous linear second-order differential equation for the function $\chi (x)$, $$\chi'' = \lambda_{0} ( x ) \chi' + s_{0} ( x ) \chi \; , \label{eq:Chapter 4 Equation 14}$$ where $\lambda_{0} ( x )$ and $s_{0} ( x )$ are functions in $C_{\infty} ( a , b )$. In order to find a general solution to this equation, we rely on the symmetric structure of the right-hand of Eq. (\[eq:Chapter 4 Equation 14\]) [@Ciftci:2005xn]. If we differentiate Eq. (\[eq:Chapter 4 Equation 14\]) with respect to $x$, we find that $$\chi''' = \lambda_{1} ( x ) \chi' + s_{1} ( x ) \chi \; ,$$ where $$\lambda_{1} = \lambda'_{0} + s_{0} + ( \lambda_{0})^{2} \; \mathrm{and} \; s_{1} = s'_{0} + s_{0} \lambda_{0} \; .$$ Taking the second derivative of Eq. (\[eq:Chapter 4 Equation 14\]) we get $$\chi'''' = \lambda_{2} ( x ) \chi' + s_{2} ( x ) \chi \; ,$$ where $$\lambda_{2} = \lambda'_{1} + s_{1} + \lambda_{0} \lambda_{1} \hspace{1cm} \mathrm{and} \hspace{1cm} s_{1} = s'_{0} + s_{0} \lambda_{0} \; .$$ Iteratively, for the $( n + 1 )^{ t h }$ and the $( n + 2 )^{ th }$ derivatives, $n = 1 , 2 , . . .$, we have $$\chi^{ ( n + 1 ) } = \lambda_{n - 1} ( x ) \chi' + s_{n - 1} ( x ) \chi \; , \label{eq:Chapter 4 Equation 15}$$ and thus bringing us to the crucial observation in the AIM is that differentiating the above equation $n$ times with respect to $x$, leaves a symmetric form for the right hand side: $$\chi^{(n+2)} = \lambda_n(x) \chi' + s_n(x) \chi \; ,$$ where $$\lambda_n(x) = \lambda'_{n-1} (x)+ s_{n-1}(x) + \lambda_0(x) \lambda_{n-1}(x) \hspace{1cm} \mathrm{and} \hspace{1cm} s_n(x) = s'_{n-1}(x) + s_0(x) \lambda_{n-1} (x) \; . \label{eqn:2-2}$$ For sufficiently large $n$ the asymptotic aspect of the “method" is introduced, that is: $$\frac{s_n (x)}{\lambda_n (x)} = \frac{s_{n-1}(x)}{\lambda_{n-1}(x)} \equiv \beta(x) \; , \label{eqn:2-3}$$ where the QNMs are obtained from the “quantization condition" $$\delta_n = s_n \lambda_{n-1} - s_{n-1} \lambda_n =0 \;, \label{eqn:quantcond}$$ which is equivalent to imposing a termination to the number of iterations [@Barakat:2006ki]. From the ratio of the $(n+1)^{th}$ and the $(n+2)^{th}$ derivatives, we have $$\frac{d}{dx} \ln (\chi^{(n+1)}) = \frac{\chi^{(n+2)}}{\chi^{(n+1)}} = \frac{\lambda_n \left( \chi' + \frac{s_n}{\lambda_n}\chi\right)}{\lambda_{n-1} \left( \chi' + \frac{s_{n-1}}{\lambda_{n-1}}\chi\right)} \; .$$ From our asymptotic limit, this reduces to $$\frac{d}{dx} \ln (\chi^{(n+1)}) = \frac{\lambda_n}{\lambda_{n-1}} \; ,$$ which yields $$\chi^{(n+1)}(x) = C_1 \exp \left( \int^x \frac{\lambda_n (x')}{\lambda_{n-1} (x')} dx' \right) = C_1 \lambda_{n-1} \exp \left( \int^x (\beta + \lambda_0) dx' \right) \; ,$$ where $C_1$ is the integration constant and the right-hand side of Eq. (\[eqn:2-2\]) and the definition of $\beta(x)$ have been used. Substituting this into Eq. (\[eq:Chapter 4 Equation 15\]), we obtain the first-order differential equation $$\chi' + \beta\chi = C_1 \exp \left( \int^x (\beta + \lambda_0) dx' \right) \; ,$$ which leads to the general solution $$\chi(x) = \exp \left[ - \int^x \beta (x') dx' \right] \left( C_2 + C_1 \int^x \exp \left \{ \int^{x'} \left[ \lambda_0 (x'') + 2 \beta ( x'') \right] dx'' \right\} dx' \right) \; .$$ The integration constants, $C_1$ and $C_2$, can be determined by an appropriate choice of normalisation. Note, that for the generation of exact solutions $C_1=0$. The Improved Method ------------------- Ciftci [*et al.*]{} [@Ciftci:2005xn] were among the first to note that an unappealing feature of the recursion relations in Eqs. (\[eqn:2-2\]) is that at each iteration one must take the derivative of the $s$ and $\lambda$ terms of the previous iteration. This can slow the numerical implementation of the AIM down considerably and also lead to problems with numerical precision. To circumvent these issues we developed an improved version of the AIM which bypasses the need to take derivatives at each step [@Cho:2009cj]. This greatly improves both the accuracy and speed of the method. We expand the $\lambda_n$ and $s_n$ in a Taylor series around the point at which the AIM is performed, $\xi$: $$\begin{aligned} \lambda_n(\xi)&=&\sum_{i=0}^{\infty}c_n^i(x-\xi)^i,\\ s_n(\xi)&=&\sum_{i=0}^{\infty}d_n^i(x-\xi)^i,\end{aligned}$$ where the $c_n^i$ and $d_n^i$ are the $i^{th}$ Taylor coefficient’s of $\lambda_n(\xi)$ and $s_n(\xi)$ respectively. Substituting these expressions into Eqs. (\[eqn:2-2\]) leads to a set of recursion relations for the coefficients: $$\begin{aligned} c_n^i&=&(i+1)c_{n-1}^{i+1}+d_{n-1}^{i}+\sum_{k=0}^{i}c_0^kc_{n-1}^{i-k}\; ,\label{eqn:coeffiterlam}\\ d_n^i&=&(i+1)d_{n-1}^{i+1}+\sum_{k=0}^id_0^kc_{n-1}^{i-k}\; . \label{eqn:coeffiters}\end{aligned}$$ In terms of these coefficients the “quantization condition” Eq. (\[eqn:quantcond\]) can be re-expressed as $$d_n^{0} c_{n-1}^{0}-d_{n-1}^{0}c_n^0=0 \; ,$$ and thus we have reduced the AIM into a set of recursion relations which no longer require derivative operators. Observing that the right hand side of Eqs. (\[eqn:coeffiterlam\]) and (\[eqn:coeffiters\]) involve terms of order at most $n-1$, one can recurse these equations until only $c_0^i$ and $d_0^i$ terms remain (that is, the coefficients of $\lambda_0$ and $s_0$ only). However, for large numbers of iterations, due to the large number of terms, such expressions become impractical to compute. We avert this combinatorial problem by beginning at the $n=0$ stage and calculating the $n+1$ coefficients sequentially until the desired number of recursions is reached. Since the quantisation condition only requires the $i=0$ term, at each iteration $n$ we only need to determine coefficients with $i<N-n$, where $N$ is the maximum number of iterations to be performed. The QNMs that we calculate in this paper will be determined using this improved AIM. Two Simple Examples ------------------- ### The Harmonic Oscillator {#sec:2-1} In order to understand the effectiveness of the AIM, it is appropriate to apply this method to a simple concrete problem: The harmonic oscillator potential in one dimension, $$\left( - \frac{d^2}{dx^2} + x^2 \right) \phi = E \phi \; . \label{harm1}$$ When $|x|$ approaches infinity, the wave function $\phi$ must approach zero. Asymptotically the function $\phi$ decays like a Gaussian distribution, in which case we can write $$\phi (x) = e^{-x^2/2}f(x) \; , \label{harm2}$$ where $f(x)$ is the new wave function. Substituting Eq. (\[harm2\]) into Eq. (\[harm1\]) then re-arranging the equation and dividing by a common factor, one can obtain $$\frac{d^2 f}{dx^2} = 2 x \frac{df}{dx} + (1 - E)f \; . \label{harm3}$$ We recognise this as Hermite’s equation. For convenience we let $1 - E = - 2 j$, such that in our case $\lambda_0 = 2 x$ and $s_0 = - 2 j$. We define $$\delta_n = \lambda_n s_{n-1} - \lambda_{n-1} s_n \; , \hspace{1cm} \mathrm{for} \hspace{1cm} n = 1, 2, 3, \ldots \label{harm4}$$ Thus using Eqs. (\[eqn:2-2\]) one can find that $$\delta_n = 2^{n+1} \prod_{i=0}^n (j - i) \; , \label{harm5}$$ and the termination condition Eq. (\[eqn:quantcond\]) can be written as $\delta_n = 0$. Hence $j$ must be a non-negative integer, which means $$E_k = 2 k +1 \; , \hspace{1cm} \mathrm{for}\hspace{1cm} k = 0 , 1, 2, \ldots \label{harm6}$$ and this is the exact spectrum for such a potential. Moreover, the wave function $\phi(x)$ can also be derived in this method. We should point out that in this case the termination condition, $\delta_n = 0$, is dependent only on the eigenvalue $j$ for a given iteration number $n$, and this is the reason why we can obtain an exact eigenvalue. However, for the black hole cases in subsequent sections, the termination condition depends also on $x$, and therefore one can only obtain approximate eigenvalues by terminating the procedure after $n$ iterations. ### The Poschl-Teller Potential {#sec:2-2} To conclude this section we will also demonstrate that the AIM can be applied to the case of QNMs, which have unbounded (scattering) like potentials, by recalling that we can find QNMs for Scarf II (upside-down Poschl-Teller-like) potentials [@OzerRoy]. This is based on observations made by one of the current authors [@ChoLin] relating QNMs from quasi-exactly solvable models. Indeed bound state Poschl-Teller potentials have been used for QNM approximations previously by inverting black hole potentials [@Ferrari:1984zz]. However, the AIM does not require any inversion of the black hole potential as we shall show. Starting with the potential term $$V(x) = \frac{1}{2} \mathrm{sech}^2 x \; ,$$ and the Schrödinger equation, we obtain: $$\frac{d^2 \psi}{dx^2} + \left( \omega^2 - \frac{1}{2} \mathrm{sech}^2 x \right) \psi = 0 \; .$$ As we shall also see in the following sections, it is more convenient to transform our coordinates to a finite domain. Hence, we shall use the transformation $y = \tanh x$, which leads to $$\begin{aligned} (1 - y^2) \frac{d}{dy} \left[ (1-y^2) \frac{d\psi}{dy} \right] + \left[ \omega^2 - \frac{1}{2}\left(1-y^2\right)\right] \psi &=& 0 \; , \nonumber \\ \Rightarrow \frac{d^2\psi}{dy^2} - \left( \frac{2 y}{1-y^2}\right) \frac{d\psi}{dy} + \left[ \frac{\omega^2}{(1-y^2)^2} - \frac{1}{2(1-y^2)} \right] \psi &=& 0 \; ,\end{aligned}$$ where $-1<y<1$. The QNM boundary conditions in Eq. (\[QNMdef\]) can then be implemented as follows. As $y \to 1$ we shall have $\psi \sim e^{\mp i \omega x}\sim ( 1- y)^{\pm i \omega/2} $. Hence our boundary condition $\psi \sim e^{i \omega x} \Rightarrow \psi \sim (1-y)^{- i \omega /2}$. Likewise, as $y \to -1$ we have $\psi \sim e^{\pm i \omega x}\sim (1+y)^{\pm i \omega/2} $ and the boundary condition $\psi \sim e^{- i \omega x} \Rightarrow \psi \sim (1+y)^{-i \omega/2}$. As such we can take the boundary conditions into account by writing $$\psi = (1-y)^{-i\omega/2}(1+y)^{-i \omega/2} \phi \; ,$$ and therefore have $$\frac{d^2 \phi}{dy^2} = \frac{2 y (1 - i \omega)}{1 - y^2} \frac{d\phi}{dy} + \frac{1 - 2 i \omega - 2 \omega^2}{2 ( 1-y^2)} \phi \; ,$$ where $$\begin{aligned} \lambda_0 &=& \frac{2 y (1 - i \omega)}{1 - y^2} \; , \\ s_0 &=& \frac{1 - 2 i \omega - 2 \omega^2}{2 ( 1-y^2)} \; .\end{aligned}$$ Following the AIM procedure, that is, taking $\delta_{n}=0$ successively for $n=1,2,\cdots$, one can obtain exact eigenvalues: $$\omega_{n}=\pm\frac{1}{2}-i\left(n+\frac{1}{2}\right).$$ This exact QNM spectrum is the same as the one in Ref. [@ChoLin] obtained through algebraic means. The reader might wonder about approximate results for cases where Poschl-Teller approximations can be used, such as Schwarzschild and SdS backgrounds, e.g., see Refs. [@Ferrari:1984zz; @Moss:2001ga]. In fact when the black hole potential can be modeled by a Scarf like potential the AIM can be used to find the eigenvalues exactly [@OzerRoy] and hence the QNMs numerically. We demonstrate this in the next section. Schwarzschild (A)dS Black Holes {#sec:3} =============================== We shall now begin the core focus of this review, the study of black hole QNMs using the AIM. Recall that the perturbations of the Schwarzschild black holes are described by the Regge-Wheeler [@Regge:1957td] and Zerilli [@Zerilli:1971wd] equations, and the perturbations of Kerr black holes are described by the Teukolsky equations [@Teukolsky:1972my]. The perturbation equations for Reissner-Nordström black holes were also derived by Zerilli [@Zerilli:1974ai], and by Moncrief [@Moncrief:1974gw; @Moncrief:1974ng; @Moncrief:1975sb]. Their radial perturbation equations all have a one-dimensional Schrödinger-like form with an effective potential. Therefore, we shall commence in the coming subsections by describing the radial perturbation equations of Schwarzschild black holes first, where our perturbed metric shall be $g_{\mu\nu} = g^0_{\mu\nu} + h_{\mu\nu}$, and where $g^0_{\mu\nu}$ is spherically symmetric. As such it is natural to introduce a mode decomposition to $h_{\mu\nu}$. Typically we write $$\Psi_{lm} (t, r, \theta, \phi) = \frac{e^{-i\omega t} u_l(r)}{r} Y_{lm}(\theta,\phi) \; ,$$ where $Y_{lm}(\theta,\phi)$ are the standard spherical harmonics. The function $u_l(r,t)$ then solves the wave equation $$\left( \frac{d^2}{d x^2} - \omega^2 - V_l (r) \right) u_l (r) = 0 \; , \label{eqn:3-3}$$ where $x$, defined by $dx = dr/f(r)$, are the so-called tortoise coordinates and $V(x)$ is a master potential of the form [@Berti:2009kk] $$V(r) = f(r) \left[ \frac{\ell (\ell +1)}{r^2} +(1-s^2)\left( \frac{2M}{r^3} - {(4-s^2) \Lambda\over 6} \right)\right] \; . \label{mastpot}$$ In this section $$f(r)= 1- {2M\over r} -{\Lambda\over 3}r^2 \; ,$$ with cosmological constant $\Lambda$. Here $s=0,1,2$ denotes the spin of the perturbation: scalar, electromagnetic and gravitational (for half-integer spin see Refs. [@Brill:1957; @Medved:2003rga; @Cho:2003qe] and Sec. \[sec:3-2\]). The Schwarschild Asymptotically Flat Case {#sec:3-11} ----------------------------------------- To explain the AIM we shall start with the simplest case of the radial component of a perturbation of the Schwarzschild metric outside the event horizon [@Zerilli:1971wd]. For an asymptotically flat Schwarzschild solution ($\Lambda=0$) $$f(r)= 1- {2M\over r} \; ,$$ where from $dx = dr/f(r)$ we have $$x(r)=r + 2M \ln\left({r\over 2M}-1\right)\; ,$$ for the tortoise coordinate $x$. Note that for the Schwarzschild background the maximum of this potential, in terms of $r$, is given by [@Iyer:1986nq] $$r_0= {3M\over 2} {1\over \ell (\ell+1)} \Big[ \ell(\ell +1) - (1-s^2) + \big(\ell^2(\ell+1)^2 + {14\over 9} \ell(\ell+1)(1-s^2) +(1-s^2)^2\big)^{1/2}\Big]\; .$$ The choice of coordinates is somewhat arbitrary and in the next section (for SdS) we will see how an alternative choice leads to a simpler solution. Firstly, consider the change of variable: $$\xi =1 - \frac{2 M}{r} \;,$$ with $0 \leq \xi < 1$. In terms of $\xi$, our radial equation then becomes $$\frac{d^2 \psi}{d\xi^2} + \frac{1 - 3 \xi}{\xi (1 - \xi)} \frac{d \psi}{d\xi} + \left[ \frac{4 M^2\omega^2}{\xi^2 (1 - \xi)^4} - \frac{\ell (\ell + 1)}{\xi (1 - \xi)^2} -\frac{1-s^2}{\xi (1- \xi)} \right] \psi = 0 \; .$$ To accommodate the out-going wave boundary condition $\psi \to e^{i\omega x}=e^{i \omega(r+2M \ln(r/2M-1))}$ as $(x,r)\to \infty$ in terms of $\xi$ (which is the limit $\xi \to 1$) and the regular singularity at the event horizon ($\xi\to 0$), we define $$\psi (\xi)= \xi^{- 2iM\omega} (1-\xi)^{-2iM\omega} e^{\frac{2 i M \omega}{1 - \xi} }\chi (\xi) \; ,$$ where the Coulomb power law is included in the asymptotic behaviour (cf. Ref. [@Leaver1985] Eq. (5)). The radial equation then takes the form: $$\begin{aligned} \chi''& = & \lambda_0(\xi) \chi' + s_0(\xi) \chi \; , \label{AIMform}\end{aligned}$$ where $$\begin{aligned} \lambda_0(\xi) & = & \frac{4M i \omega (2\xi^2 - 4 \xi + 1) - ( 1 - 3 \xi)(1 - \xi)}{\xi (1 - \xi) ^2} \; , \\ s_0 (\xi)& = & \frac{16M^2 \omega^2(\xi - 2) - 8M i \omega ( 1 - \xi)+\ell (\ell + 1) + (1-s^2)(1 - \xi)}{\xi (1 - \xi)^2} \; .\end{aligned}$$ Note that primes of $\chi$ denote derivatives with respect to $\xi$. Using these expressions we have tabulated several QNM frequencies and compared them to the WKB method of Ref. [@Iyer:1986nq] and the CFM of Ref. [@Leaver1985] in Table \[tab:1\]. For completeness Table \[tab:1\] also includes results from an approximate semi-analytic 3rd order WKB method [@Iyer:1986nq]. More accurate semi-analytic results with better agreement to Leaver’s method can be obtained by extending the WKB method to 6th order [@Konoplya:2003ii] and indeed in Sec. \[sec:5\] we use this to compare with the AIM for results where the CFM has not been tabulated. It might also be worth mentioning that a different semi-analytic perturbative approach has recently been discussed by Dolan and Ottewill [@Dolan:2009nk], which has the added benefit of easily being extended to any order in a perturbative scheme. -------------------------------------------------------------------------------------------------------------- $\ell$ $n$ $\omega_{Leaver}$ $\omega_{AIM} ~{\rm(after~15~iterations)} $ $ \omega_{WKB}$ -------- ----- ------------------------- --------------------------------------------- ----------------------- 2 0 0.3737 - 0.0896 i\[\*\] 0.3737 - 0.0896 i 0.3732 - 0.0892 i ($<$0.01%)($<$0.01%) (-0.13%)(0.44%)\[\*\] 1 0.3467 - 0.2739 i 0.3467 - 0.2739 i 0.3460 - 0.2749 i ($<$0.01%)($<$0.01%) (-0.20%)(-0.36%) 2 0.3011 - 0.4783 i 0.3012 - 0.4785 i 0.3029 - 0.4711 i (0.03%)(-0.04%) (0.60%)(1.5%) 3 0.2515 - 0.7051 i 0.2523 - 0.7023 i 0.2475 - 0.6703 i (0.32%)(0.40%) (-1.6%)(4.6%) 3 0 0.5994 - 0.0927 i 0.5994 - 0.0927 i 0.5993 - 0.0927 i ($<$0.01%)($<$0.01%) (-0.02%)(0.0%) 1 0.5826 - 0.2813 i 0.5826 - 0.2813 i 0.5824 - 0.2814 i ($<$0.01%)($<$0.01%) (-0.03%)(-0.04%) 2 0.5517 - 0.4791 i 0.5517 - 0.4791 i 0.5532 - 0.4767 i ($<$0.01%)($<$0.01%) (0.27%)(0.50%) 3 0.5120 - 0.6903 i 0.5120 - 0.6905 i 0.5157 - 0.6774 i ($<$0.01%)(-0.03%) (0.72%)(1.9%) 4 0.4702 - 0.9156 i 0.4715 - 0.9156 i 0.4711 - 0.8815 i (0.28%)($<$0.01%) (0.19%)(3.7%) 5 0.4314 - 1.152 i 0.4360 - 1.147 i 0.4189 - 1.088 i (1.07%)(0.43%) (-2.9%)(5.6%) 4 0 0.8092 - 0.0942 i 0.8092 - 0.0942 i 0.8091 - 0.0942 i ($<$0.01%)($<$0.01%) (-0.01%)(0.0%) 1 0.7966 - 0.2843 i 0.7966 - 0.2843 i 0.7965 - 0.2844 i ($<$0.01%)($<$0.01%) (-0.01%)(-0.04%) 2 0.7727 - 0.4799 i 0.7727 - 0.4799 i 0.7736 - 0.4790 i ($<$0.01%)($<$0.01%) (0.12%)(0.19%) 3 0.7398 - 0.6839 i 0.7398 - 0.6839 i 0.7433 - 0.6783 i ($<$0.01%)($<$0.01%) (0.47%)(0.82%) 4 0.7015 - 0.8982 i 0.7014 - 0.8985 i 0.7072 - 0.8813 i (-0.01%)(-0.03%) (0.81%)(1.9%) -------------------------------------------------------------------------------------------------------------- : *QNMs to 4 decimal places for gravitational perturbations ($s = 2$) where the fifth column is taken from Ref. [@Iyer:1986nq]. Note that the imaginary part of the $n=0$, $\ell=2$ result in [@Iyer:1986nq] has been corrected to agree with Ref. [@Leaver1985]. \[\*\] Note also that if the number of iterations in the AIM is increased, to say $50$, then we find agreement with Ref. [@Leaver1985] accurate to $6$ significant figures.*[]{data-label="tab:1"} The de-Sitter Case {#sec:3-12} ------------------ We have presented the QNMs for Schwarzchild gravitational perturbations in Table \[tab:1\], however, to further justify the use of this method, it is instructive to consider some more general cases. As such, we shall now consider the Schwarzschild de Sitter (SdS) case, where we have the same WKB-like wave equation and potential as in the radial equation earlier, though now $$f(r) = 1 - {2M\over r} - \Lambda {r^2\over 3}\; ,$$ where $\Lambda >0$ is the cosmological constant. Interestingly the choice of coordinates we use here leads to a simpler AIM solution, because there is no Coulomb power law tail; however, in the limit $\Lambda=0$ we recover the Schwarzschild results. Note that although it is possible to find an expression for the maximum of the potential in the radial equation, for the SdS case, it is the solution of a cubic equation, which for brevity we refrain from presenting here. In our AIM code we use a numerical routine to find the root to make the code more general. In the SdS case it is more convenient to change coordinates to $\xi = 1/r$ [@Moss:2001ga], which leads to the following master equation (cf. Eq. (\[mastpot\])) $$\frac{d^2 \psi}{d\xi^2} + \frac{p'}{p} \frac{d\psi} {d\xi} + \left[ \frac{\omega^2}{p^2} - { \ell (\ell + 1)+ (1 - s^2)\left(2 M \xi - (4-s^2){\Lambda\over 6\xi^2}\right) \over p}\right]\psi = 0\; , \label{masterxi}$$ where we have defined $$p= \xi^2 - 2M \xi^3 -\Lambda /3 \hspace{1cm} \Rightarrow \hspace{1cm} p' = 2\xi(1-3M\xi) \; .$$ It may be worth mentioning that for SdS we can express [@Moss:2001ga]: $$e^{i\omega x} = (\xi-\xi_1)^{{i\omega\over2 \kappa_1}} (\xi-\xi_2)^{{i\omega\over2 \kappa_2}} (\xi-\xi_3)^{{i\omega\over2 \kappa_3}}$$ in terms of the roots of $f(r)$, where $\xi_1$ is the event horizon and $\xi_2$ is the cosmological horizon (and $\kappa_n$ is the surface gravity at each $\xi_n$). This is useful for choosing the appropriate scaling behaviour for QNM boundary conditions. Based on the above equation an appropriate choice for QNMs is to scale out the divergent behaviour at the cosmological horizon:[^1] $$\psi(\xi) = e^{i\omega x} u (\xi) \; , \label{SdScale}$$ which implies $$p u'' + (p'- 2 i\omega)u' - \left[\ell(\ell+1)+ (1 - s^2)\left(2 M \xi - (4-s^2){\Lambda\over 6\xi^2} \right)\right]u=0 \; , \label{youeq}$$ in terms of $\xi$. Furthermore, based on the scaling in Eq. (\[SdScale\]), the correct QNM condition at the horizon $\xi_1$ implies $$u(x)= (\xi-\xi_1)^{-{i\omega\over\kappa_1}}\chi(x) \; ,$$ where $$\kappa_1 = \left.\frac 1 2 {d f\over dr}\right|_{r\to r_1} =M \xi_1^2 - \frac 1 3 {\Lambda\over \xi_1} \; ,$$ with $\xi_1=1/r_1$, and $r_1$ is the smallest real solution of $f(r) =0$, implying $p=0$. The differential equation then takes the standard AIM form: $$\begin{aligned} \chi''& = & \lambda_0(\xi) \chi' + s_0(\xi) \chi \; ,\end{aligned}$$ where $$\begin{aligned} \lambda_0(\xi) &=& -\frac 1 p \left[p'- {2i\omega \over \kappa_1(\xi-\xi_1)} - 2 i\omega\right] \; , \\ s_0 (\xi) &= & \frac 1 p \left[\ell(\ell+1)+ (1-s^2) \left(2 M \xi - (4-s^2){\Lambda\over 6\xi^2} \right)\ +{i \omega\over \kappa_1(\xi-\xi_1)^2}\Big({i\omega\over\kappa_1} +1\Big) +(p'- 2 i\omega) {i\omega\over \kappa_1(\xi-\xi_1)}\right] \; .\end{aligned}$$ Using these equations, we present in Table \[tab:2\] results for SdS with $s=2$. Identical results were generated by the AIM and CFM, both after 50 iterations. Though results are presented for $n = 1, 2, 3$, $\ell = 2, 3$ modes only, the AIM is robust enough to be applied to any other case; where like the $\Lambda = 0$ case, agreement with other methods in more extreme parameter choices would only require further iterations. As far as we are aware only Ref. [@Zhidenko:2003wq] (who used a semi-analytic WKB approach) has presented tables for general spin fields for the SdS case. We have also compared our results to those in Ref. [@Zhidenko:2003wq] for the $s=0,1$ cases and find identical results (to a given accuracy in the WKB method). It may be worth mentioning that a set of three-term recurrence relations was derived in Ref. [@Cho:2009cj] for the CFM, valid for electromagnetic and gravitational perturbations ($s=1,2$), while for $s=0$ this reduces to a five-term recurrence relation. However, for the AIM we can treat the $s=0,1,2$ perturbations on an equal footing, see Ref. [@Cho:2009cj] for more details. Typically $n$ Gaussian Elimination steps are required to reduce an $n+3$ recurrence to a $3$-term continued fraction, e.g., for Reissner-Nördtrom see Ref. [@Leaver:1990zz] and for higher-dimensional Schwarzschild backgrounds see Ref. [@Zhidenko:2006rs] (for an application of the CFM to higher dimensional asymptotic QNMs see [@Cardoso:2003vt]). However, all that is necessary in the AIM is to factor out the correct asymptotic behaviour at the horizon(s) and infinity (we showed this for higher-dimensional scalar spheroids in Ref. [@Cho:2009wf]). $\Lambda ~(\ell=2)$ $n=1$ $n=2$ $n=3$ --------------------- -------------------------- ------------------------- ------------------------- 0 0.373672 - 0.0889623 i 0.346711 - 0.273915 i 0.301050 - 0.478281 i 0.02 0.338391 - 0.0817564 i 0.318759 - 0.249197 i 0.282732 - 0.429484 i 0.04 0.298895 - 0.0732967 i 0.285841 - 0.221724 i 0.259992 - 0.377092 i 0.06 0.253289 - 0.0630425 i 0.245742 - 0.189791 i 0.230076 - 0.319157 i 0.08 0.197482 - 0.0498773 i 0.194115 - 0.149787 i 0.187120 - 0.250257 i 0.09 0.162610 - 0.0413665 i 0.160789 - 0.124152 i 0.157042 - 0.207117 i 0.10 0.117916 - 0.0302105 i 0.117243 - 0.0906409 i 0.115876 - 0.151102 i 0.11 0.0372699 - 0.00961565 i 0.0372493 - 0.0288470 i 0.0372081 - 0.0480784 i : *QNMs to 6 significant figures for Schwarzschild de Sitter gravitational perturbations ($s=2$) for $\ell=2$ and $\ell=3$ modes. We only present results for the AIM method, because the results are identical to those of the CFM after a given number of iterations (in this case $50$ iterations for both methods). The $n=1,2$ modes can be compared with the results in Ref. [@Zhidenko:2003wq] for $s=2$.*[]{data-label="tab:2"} $\Lambda ~(\ell=3)$ $n=1$ $n=2$ $n=3$ --------------------- -------------------------- ------------------------- ------------------------- 0 0.599443 - 0.0927030 i 0.582644 - 0.281298 i 0.551685 - 0.479093 i 0.02 0.543115 - 0.0844957 i 0.530744 - 0.255363 i 0.507015 - 0.432059 i 0.04 0.480058 - 0.0751464 i 0.471658 - 0.226395 i 0.455011 - 0.380773 i 0.06 0.407175 - 0.0641396 i 0.402171 - 0.192807 i 0.392053 - 0.322769 i 0.08 0.317805 - 0.0503821 i 0.315495 - 0.151249 i 0.310803 - 0.252450 i 0.09 0.261843 - 0.0416439 i 0.260572 - 0.124969 i 0.257998 - 0.208412 i 0.10 0.189994 - 0.0303145 i 0.189517 - 0.0909507 i 0.188555 - 0.151609 i 0.11 0.0600915 - 0.00961888 i 0.0600766 - 0.0288567 i 0.0600469 - 0.0480945 i : *QNMs to 6 significant figures for Schwarzschild de Sitter gravitational perturbations ($s=2$) for $\ell=2$ and $\ell=3$ modes. We only present results for the AIM method, because the results are identical to those of the CFM after a given number of iterations (in this case $50$ iterations for both methods). The $n=1,2$ modes can be compared with the results in Ref. [@Zhidenko:2003wq] for $s=2$.*[]{data-label="tab:2"} The Spin-Zero Anti-de-Sitter Case {#sec:3-13} --------------------------------- There are various approaches to finding QNMs for the SAdS case (an eloquent discussion is given in the appendix of Ref. [@Berti:2003ud], see also [@Mann1999]). One approach is that of Horowitz and Hubeny [@Horowitz:1999jd], which uses a series solution chosen to satisfy the SAdS QNM boundary conditions. This method can easily be applied to all perturbations ($s=0,1,2$). The other approach is to use the Frobenius method of Leaver [@Leaver1985], but instead of developing a continued fraction the series must satisfy a boundary condition at infinity, such as a Dirichlet boundary condition [@Moss:2001ga]. The AIM does not seem easy to apply to metrics where there is an asymptotically anti-de Sitter background, because for general spin, $s$, the potential at infinity is a constant and hence would include a combination of ingoing and outgoing waves, leading to a sinusoidal dependence [@Cardoso:2001bb]. However, for the scalar spin zero ($s=0$) case, the potential actually blows up at infinity and is effectively a bound state problem. In this case the AIM can easily be applied as we show below. Let us consider the scalar wave equation in SAdS spacetime, where $\Lambda= - 3/R^2$, and $R$ is the AdS radius. The master equation takes the same form as for the graviational case, except that the potential becomes $$V=\left(1-\frac{2}{r}+r^{2}\right) \left(\frac{2}{r^{3}}+2\right)=\frac{2(r-1)(r^{2}+r+2)(r^{3}+1)}{r^{4}} \; .$$ Here for simplicity we have taken the AdS radius $R=1$, the mass of the black hole $M=1$, and the angular momentum number $l=0$. Hence the horizon radius $r_{+}=1$. Thus, with this choice we can compare with the data in Table 3.2 on page 37 of Ref. [@Cardoso:2003pj] (see Table \[tab:3\] below). To implement the AIM we first look at the asymptotic behavior of $\psi$. As $r\rightarrow r_{+}=1$, the potential $V$ goes to zero. In addition, $$\begin{aligned} \psi&\sim&e^{\pm i\left[\frac{\omega}{4}{\rm ln}(r-1)\right]}\sim (r-1)^{\pm i\omega/4}\sim \left(1-\frac{1}{r}\right)^{\pm i\omega/4} \; .\end{aligned}$$ For QNMs we choose the “out-going" (into the black hole) boundary condition. That is, $$\psi\sim e^{-i\omega x}\sim \left(1-\frac{1}{r}\right)^{-i\omega/4} \; .$$ On the other extreme of our space, $r\rightarrow\infty$, the potential goes to infinity. This is a crucial difference from the case of gravitational perturbations. In that case, the potential goes to a constant. However, in the scalar case, as $r\rightarrow\infty$, $\psi\sim\left(1/r\right)^{\pm\sqrt{2}+1/2}$ and to implement the Dirichlet boundary condition, we take $$\psi\sim\left(\frac{1}{r}\right)^{\frac{1}{2}+\sqrt{2}} \; .$$ For the AIM one possible choice of variables is $$\xi=1-\frac{1}{r} \; ,$$ and we see that to accommodate the asymptotic behaviour of the wavefunction we should take $$\psi=\xi^{-i\omega/4}(1-\xi)^{\sqrt{2}+\frac{1}{2}}\chi \; .$$ Finally, after some work we find the scalar perturbation equation is $$\begin{aligned} \chi''& = & \lambda_0(\xi) \chi' + s_0(\xi) \chi \; ,\end{aligned}$$ where $$\begin{aligned} \lambda_{0}&=&-\frac{-i\omega q+2[-4+2(9+4\sqrt{2})\xi-(21+10\sqrt{2})\xi^{2}+4(2+\sqrt{2})\xi^{3}]} {2\xi q} \; , \nonumber\\ \\ s_{0}&=&\frac{1}{16\xi q^{2}}\bigg\{4i\omega[9+8\sqrt{2}-2(7+5\sqrt{2})\xi+(6+4\sqrt{2})\xi^{2}]q +\omega^{2}(-1+\xi)^{2}(-40+41\xi-20\xi^{2}+4\xi^{3})\nonumber\\ &&\ \ -4[4-5\xi+2\xi^{2}][-8(3+2\sqrt{2})+8(10+7\sqrt{2})\xi-(91+64\sqrt{2})\xi^{2}+(34+24\sqrt{2})\xi^{3}]\bigg\} \; ,\nonumber\\\end{aligned}$$ and $q=(-4+9\xi-7\xi^{2}+2\xi^{3})$. Using the AIM we find the results presented in Table \[tab:3\] below. $n$ HH method AIM ----- --------------------------- -------------------- 0 2.7982 - 2.6712 i 2.79823 -2.67121 i 1 4.75849 - 5.03757 i 4.75850 -5.03757 i 2 6.71927 - 7.39449 i 6.71931 -7.39450 i 3 8.68223\[\*\] - 9.74852 i 8.68233 -9.74854 i 4 10.6467 - 12.1012 i 10.6469 -12.1013 i 5 12.6121 - 14.4533 i 12.6125 -14.4533 i 6 14.5782 - 16.8049 i 14.5788 -16.8050 i 7 16.5449 - 19.1562 i 16.5457 -19.1563 i : *Comparison of the first few QNMs to 6 significant figures for Schwarzschild anti de Sitter scalar perturbations ($s=0$) for $\ell=0$ modes with $r_+=1$. The second column corresponds to data [@Cardoso:2003pj] using the Horowitz and Hubeny (HH) method [@Horowitz:1999jd], while the third column is for the AIM using 70 iterations. \[\*\] Note the mismatch for the real part of the $n=3$ mode in Ref. [@Cardoso:2003pj]; we have confirmed this using the [*Mathematica*]{} notebook provided in Ref. [@Berti:2009kk].*[]{data-label="tab:3"} Reissner-Nordström Black Holes {#sec:4} ============================== The procedure for obtaining the quasinormal frequencies of Reissner-Nordström black holes in four-dimensional spacetime is similar to that of our earlier cases. Starting with the Reissner-Nordström metric $$ds^2 = - f(r) dt^2 + \frac{dr^2}{f(r)} + r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2 \; , \label{eqn:4-1}$$ where $f(r) = \left( 1 - \frac{1}{r} + \frac{Q^2}{r^2} \right)$ and $|Q| \leq \frac{1}{2}$ is the charge of the black hole. If we consider perturbations exterior to the event horizon, the perturbation equations of the Reissner-Nordström (charged and non-rotating) geometry can be separated into two pairs of Schrödinger-like equations, which describe the even- and odd-parity oscillations respectively [@Zerilli:1974ai; @Moncrief:1974gw; @Moncrief:1974ng; @Moncrief:1975sb]. They are given by $$\left( \frac{d^2}{d x^2} - \rho^2 - V_i^{(\pm)} \right) Z_i^{(\pm)}=0 \; , \label{eqn:4-2}$$ where $(+)$ corresponds to even- and $(-)$ to odd-parity modes: $$\begin{aligned} V_i^{(-)}(r) & = & \frac{\Delta}{r^5} \left( A r - q_j + \frac{4 Q^2}{r} \right) \; , \label{eqn:4-3} \\ V_i^{(+)}(r) & = & V_i^{(-)} (r) + 2 q_j \frac{d}{d x} \left( \frac{\Delta}{r^2 [ (l-1)(l+2) r + q_j]} \right) \; , \label{eqn:4-4}\end{aligned}$$ for $i=j=1,2$ ($i\neq j$) where $$\begin{aligned} \frac{dr}{dx} & = & \frac{\Delta}{r} \; , \label{eqn:4-5} \\ \Delta & = & r^2 - r + Q^2 \equiv (r- r_+)(r-r_-) \; , \label{eqn:4-6} \\ A & = & l(l+1) \; , \label{eqn:3-7} \\ q_1 & = & \frac{1}{2} \left[ 3 + \sqrt{9 + 16 Q^2 (l - 1)(l+2)} \right] \; , \label{eqn:4-8} \\ q_2 & = & \frac{1}{2} \left[ 3 - \sqrt{9 + 16 Q^2 (l - 1)(l+2)} \right] \; , \label{eqn:4-9}\end{aligned}$$ and $\rho = - i \omega$. Here $\omega$ is the frequency, $l$ the angular momentum parameter and $r_-$ and $r_+$ the radii of the inner and outer (event) horizons of the black hole respectively. Note that $r_+ = 1$ and $r_- = 0$ at the Schwarzschild limit $(Q=0)$; $r_+ = r_- = \frac{1}{2}$ at the extremal limit $(Q=\frac{1}{2})$. Here the tortoise coordinate is given by $$x= \int \frac{r^2}{\Delta} dr = r+ \frac{r_+^2}{r_+ - r_-} \ln (r-r_+) - \frac{r^2}{r_+ - r_-} \ln (r- r_-) \; , \label{eqn:4-10}$$ which ranges from $-\infty$ at the event horizon to $+\infty$ at spatial infinity. The QNMs of the Reissner-Nordström black holes are ordinarily accompanied by the emission of both electromagnetic and gravitational radiation, except at the Schwarzschild limit [@Kokkotas:1988fm; @Leaver:1990zz]. Eq. (\[eqn:4-2\]) corresponds to purely gravitational perturbations for the radial wave functions $Z_2^{(\pm)}$ and purely electromagnetic perturbations for $Z_1^{(\pm)}$ at the Schwarzschild limit. Chandrasekhar [@Chandrasekhar:1985kt] has shown that the solution $Z_i^{(+)}$ for the even-parity oscillations and $Z_i^{(-)}$ for the odd-parity oscillations have the relationship $$\begin{aligned} \left[ A(A-2) - 2 \rho q_j\right]Z_i^{(+)} & = & \left\{ A(A-2) + \frac{2 q_j^2 \Delta}{r^3\left[ (A-2)r + q_j\right]} \right\} Z_i^{(-)} + 2 q_j \frac{d Z_i^{(-)}}{dx} \; , \label{eqn:4-10b}\end{aligned}$$ so one can just consider solutions for a specific parity, as in the Schwarzschild case, to understand the property of the black hole. Since the formalism of the effective potential $V_i^{(-)}$ in the odd-parity equation is much simpler than $V_i^{(+)}$ in the even-parity equation, it is customary to compute the QNMs for the odd-parity modes. Note that the mass $M$ of the Reissner-Nordström black hole has been scaled to $2 M = 1$, so its quasinormal frequencies are uniquely determined by the charge $Q$, the angular momentum $l$, and the overtone number $n$ of the mode. The following procedure is similar to that in Sec. \[sec:3\]. At first we change $r$ to the variable $x$ in Eq. (\[eqn:4-2\]) for the odd-parity mode. From Eq. (\[eqn:4-5\]) we have $$\frac{d}{dx} = \frac{\Delta}{r^2} \frac{d}{dr} \; , \label{eqn:4-11}$$ and $$\frac{d^2}{dx^2} = \left(\frac{\Delta}{r^2}\right)\left( \frac{r - 2 Q^2}{r^3} \right) \frac{d}{dr} + \left(\frac{\Delta}{r^2}\right)^2 \frac{d^2}{dr^2} \; . \label{eqn:4-12}$$ Substituting Eq. (\[eqn:4-12\]) into Eq. (\[eqn:4-2\]) for the odd-parity mode, we get $$\left(\frac{\Delta}{r^2}\right)^2 \frac{d^2}{dr^2} Z_i^{(-)}+ \left(\frac{\Delta}{r^2}\right)\left( \frac{r - 2 Q^2}{r^3} \right) \frac{d}{dr} Z_i^{(-)} - \left[ \rho^2 + V_i^{(-)} \right] Z_i^{(-)}= 0 \; . \label{eqn:4-13}$$ Considering the QNM boundary conditions in the Reissner-Nordström case $$Z_i^{(-)} \to \left\{ \begin{array}{cl} e^{-i \omega x} & ; x \to - \infty \\ e^{i \omega x} & ; x \to + \infty \end{array} \right. \; , \label{eqn:4-14}$$ and incorporating this into the radial wave function $Z_i^{(-)}$, we have a form involving the asymptotic behaviour [@Leaver:1990zz] $$Z_i^{(-)} = e^{-\rho r} r^{-1} (r-r_-)^{1 - \rho - \frac{\rho r_+^2}{r_+ - r_-}} (r - r_+)^{\frac{\rho r_+^2}{r_+ - r_-}} \chi_{Z_i} (r) \; . \label{eqn:4-15}$$ Differentiating Eq. (\[eqn:4-15\]) one and two times with respect to $r$, we have $$Z_{i,r}^{(-)} \equiv \frac{d}{dr} Z_i^{(-)} = e^{-\rho r} r^{-1} (r-r_-)^{1 - \rho - \frac{\rho r_+^2}{r_+ - r_-}} (r - r_+)^{\frac{\rho r_+^2}{r_+ - r_-}} \left( \chi_{Z_i , r} + \Gamma_Z \chi_{Z_i} \right) \; , \label{eqn:4-16}$$ and $$Z_{i,rr}^{(-)} = e^{-\rho r} r^{-1} (r-r_-)^{1 - \rho - \frac{\rho r_+^2}{r_+ - r_-}} (r - r_+)^{\frac{\rho r_+^2}{r_+ - r_-}} \left[ \chi_{Z_i , rr} + 2 \Gamma_Z \chi_{Z_i, r} + \left( \Gamma_Z^2 + \Gamma_{Z,r} \right) \chi_{Z_i} \right] \; , \label{eqn:4-17}$$ where $\Gamma_Z$ is defined by $$\Gamma_Z = - \rho - \frac{1}{r} + \frac{(1-\rho)( r_+ - r_-)-\rho r_+^2}{(r_+ - r_-)(r - r_-)} + \frac{\rho r_+^2}{ (r_+ - r_-)(r - r_+)} \; . \label{eqn:4-18}$$ Substituting Eqs. (\[eqn:4-16\]) and (\[eqn:4-17\]) into Eq. (\[eqn:4-13\]), we obtain $$\begin{aligned} &&\left(\frac{\Delta}{r^2}\right)^2 \chi_{Z_i , rr} + \left[ 2 \Gamma_Z \left(\frac{\Delta}{r^2}\right)^2 + \left(\frac{\Delta}{r^2}\right) \left( \frac{r - 2 Q^2}{r^3} \right) \right] \chi_{Z_i , r} \nonumber\\ &&\ \ \ + \left\{ \left(\frac{\Delta}{r^2}\right)^2 \left( \Gamma_Z^2 + \Gamma_{Z,r} \right) + \left(\frac{\Delta}{r^2}\right) \left( \frac{r - 2 Q^2}{r^3} \right)\Gamma_Z - \left[ \rho^2 + V_i^{(-)} \right] \right\} \chi_{Z_i}=0 \; . \label{eqn:4-19}\end{aligned}$$ For the same reason as in Sec. \[sec:3\], here we change the variable $r$ to $\xi$ by the definition $\xi = 1 - \frac{r_+}{r}$, which ranges from 0 at the event horizon to 1 at spatial infinity. Thus we have $$\frac{d}{dr} = \frac{(1-\xi)^2}{r_+} \frac{d}{d\xi} \; , \label{eqn:4-20}$$ and $$\frac{d^2}{dr^2} = \frac{(1-\xi)^4}{r_+^2} \frac{d^2}{d\xi^2} - 2\frac{(1-\xi)^3}{r_+^2} \frac{d}{d\xi} \; . \label{eqn:4-21}$$ Substituting Eqs. (\[eqn:4-20\]) and (\[eqn:4-21\]) into Eq. (\[eqn:4-19\]), and rewriting the equation in the AIM form, we obtain $$\chi_{Z_i, \xi\xi} = \lambda_{Z_i} (\xi) \chi_{Z_i , \xi} + s_{Z_i} (\xi) \chi_{Z_i} \; , \label{eqn:4-23}$$ where $$\begin{aligned} \lambda_{Z_i} (\xi) & = & \frac{2}{1-\xi} - \frac{ 2 r_+ \Gamma_Z}{(1-\xi)^2} - \frac{r_+ - 2 Q^2 (1-\xi)}{\Delta (1-\xi)^2} \; , \label{eqn:4-24} \\ s_{Z_i} (\xi) & = & \frac{r_+^6 \left[ \rho^2 + V_i^{(-)}\right]}{\Delta^2 (1-\xi)^8} - \frac{r_+ \Gamma_Z}{\Delta (1- \xi)^4} \left[ r_+ - 2 Q^2 (1- \xi)\right] - \frac{r_+^2 \Gamma_Z^2}{\Delta (1-\xi)^4} - \frac{r_+ \Gamma_{Z , \xi}}{(1-\xi)^2} \; , \label{eqn:4-25} \\ \Gamma_Z & = & -\rho - \frac{1-\xi}{r_+} + \frac{\left[ (1-\rho)(r_+ - r_-)- \rho r_+^2 \right] (1-\xi)}{(r_+ - r_-)\left[ r_+ - r_-(1-\xi)\right]} + \frac{\rho r_+(1-\xi)}{(r_+ - r_-)\xi} \; , \label{eqn:4-26} \\ V_i^{(-)} & = & \Delta \frac{(1-\xi)^5}{r_+^5} \left[ \frac{A r_+}{1-\xi} - q_j + \frac{4 Q^2 (1-\xi)}{r_+} \right] \; , \label{eqn:4-27} \\ \Delta & = & \frac{r_+ \xi \left[ r_+ - r_- (1 - \xi) \right]}{(1-\xi)^2} \; . \label{eqn:4-28}\end{aligned}$$ The numerical results to four decimal places are presented in Table \[tab:RS3\], \[tab:RS4\] and \[tab:RS5\]. They are compared with $\rho_{Leaver}$ and $\rho_{WKB}$ from Refs. [@Leaver:1990zz] and [@Kokkotas:1988fm] respectively. The quasinormal frequencies appear as complex conjugate pairs in $\rho$; we list only the ones with $Im(\rho)>0$. Note that we arrange $\rho$ as $(Im(\rho),Re(\rho))$. In Table \[tab:RS5\] the quasinormal frequencies obtained by the WKB method are not available. It is apparent that the quasinormal frequencies obtained by the AIM are very accurate except for $n = 2$ in the extremal case $Q = \frac{1}{2}$ in Tables \[tab:RS3\] and \[tab:RS5\]. The QNMs of $l = 2$ and $i = 2$ in Table \[tab:RS3\] reduce to the purely gravitational QNMs in the Schwarzschild case at $Q = 0$, while the QNMs of $l = 2$ and $i = 1$ in Table \[tab:RS4\] reduce to the purely electromagnetic QNMs at $Q = 0$. Some comments on the higher ($n=2$) overtones for the Reissner-Nordström black hole for the extremal limit ($Q=1/2$) are perhaps necessary. In general, much like the CFM the AIM begins to break down for larger overtones, requiring more iterations. However, near the extremal limit ($Q=1/2$) the horizons become degenerate and the singularity structure of the corresponding differential (radial) equation changes [@Andersson:1996xw] (the number of singular points are different in the non-extremal and the extremal cases), and causes the current implementation of the AIM (cf. \[eqn:4-15\]) to break down. Thus, we see in Tables \[tab:RS3\]-\[tab:RS5\] that for $Q=0.495$ some of the values have large errors when compared to the CFM. $n$ $Q$ $\rho_{Leaver}$ $\rho_{AIM}$ $\rho_{WKB}$ ----- ------- ------------------ ---------------------- ------------------ 0 0 (0.7473,-0.1779) (0.7473,-0.1779) (0.7463,-0.1784) ($<$0.01%)($<$0.01%) (-0.13%)(-0.28%) 0 0.2 (0.7569,-0.1788) (0.7569,-0.1788) (0.7558,-0.1793) ($<$0.01%)($<$0.01%) (-0.15%)(-0.28%) 0 0.4 (0.8024,-0.1793) (0.8024,-0.1793) (0.8011,-0.1797) ($<$0.01%)($<$0.01%) (-0.16%)(-0.22%) 0 0.495 (0.8586,-0.1685) (0.8586,-0.1685) (0.8566,-0.1706) ($<$0.01%)($<$0.01%) (-0.23%)(-1.25%) 1 0 (0.6934,-0.5478) (0.6934,-0.5478) (0.6920,-0.5478) ($<$0.01%)($<$0.01%) (-0.20%)(-0.37%) 1 0.2 (0.7035,-0.5503) (0.7035,-0.5502) (0.7020,-0.5522) ($<$0.01%)(0.02%) (-0.21%)(-0.36%) 1 0.4 (0.7538,-0.5499) (0.7538,-0.5499) (0.7510,-0.5525) ($<$0.01%)($<$0.01%) (-0.37%)(-0.47%) 1 0.495 (0.8070,-0.5140) (0.8067,-0.5164) (0.8068,-0.5287) (-0.04%)(0.47%) (0.01%)(-2.21%) 2 0 (0.6021,-0.9566) (0.6021,-0.9566) (0.6059,-0.9421) ($<$0.01%)($<$0.01%) (0.63%)(1.52%) 2 0.2 (0.6129,-0.9599) (0.6128,-0.9599) (0.6164,-0.9458) (0.02%)($<$0.01%) (0.57%)(1.47%) 2 0.4 (0.6703,-0.9531) (0.6703,-0.9531) (0.6717,-0.9455) ($<$0.01%)($<$0.01%) (0.21%)(0.80%) 2 0.495 (0.7078,-0.8872) (0.8350,-0.8347) (0.7344,-0.9135) (17.97%)(5.92%) (2.66%)(-2.96%) : *Reissner-Nordström quasinormal frequency parameter values ($\rho = -i \omega$) for the fundamental ($n = 0$) and two lowest overtones for $l = 2$ and $i = 2$.*[]{data-label="tab:RS3"} $n$ $Q$ $\rho_{Leaver}$ $\rho_{AIM}$ $\rho_{WKB}$ ----- ------- ------------------ ---------------------- ------------------ 0 0 (0.9152,-0.1900) (0.9152,-0.1900) (0.9143,-0.1901) ($<$0.01%)($<$0.01%) (-0.10%)(-0.05%) 0 0.2 (0.9599,-0.1929) (0.9599,-0.1929) (0.9590,-0.1930) ($<$0.01%)($<$0.01%) (-0.09%)(-0.05%) 0 0.4 (1.1403,-0.1984) (1.1403,-0.1981) (1.1395,-0.1980) ($<$0.01%)(0.15%) (-0.07%)(-0.20%) 0 0.495 (1.3855,-0.1773) (1.3855,-0.1773) (1.3850,-0.1783) ($<$0.01%)($<$0.01%) (-0.04%)(-0.56%) 1 0 (0.8731,-0.5814) (0.8731,-0.5814) (0.8717,-0.5819) ($<$0.01%)($<$0.01%) (-0.16%)(-0.09%) 1 0.2 (0.9200,-0.5894) (0.9200,-0.5894) (0.9186,-0.5897) ($<$0.01%)($<$0.01%) (-0.15%)(-0.05%) 1 0.4 (1.1100,-0.6021) (1.1100,-0.6021) (1.1081,-0.6014) ($<$0.01%)($<$0.01%) (-0.17%)(0.12%) 1 0.495 (1.3573,-0.5350) (1.3573,-0.5350) (1.3579,-0.5423) ($<$0.01%)($<$0.01%) (0.04%)(-1.36%) 2 0 (0.8024,-1.0032) (0.8024,-1.0032) (0.8046,-0.9917) ($<$0.01%)($<$0.01%) (0.27%)(1.15%) 2 0.2 (0.8530,-1.0143) (0.8530,-1.0143) (0.8548,-1.0037) ($<$0.01%)($<$0.01%) (0.21%)(1.05%) 2 0.4 (1.0582,-1.0263) (1.0582,-1.0263) (1.0568,-1.0181) ($<$0.01%)($<$0.01%) (-0.13%)(0.80%) 2 0.495 (1.3019,-0.9024) (1.3019,-0.9024) (1.3141,-0.9222) ($<$0.01%)($<$0.01%) (0.94%)(-2.19%) : *Reissner-Nordström quasinormal frequency parameter values ($\rho = -i \omega$) for the fundamental ($n = 1$) and two lowest overtones for $l = 2$ and $i = 1$.*[]{data-label="tab:RS4"} $n$ $Q$ $\rho_{Leaver}$ $\rho_{AIM}$ ----- ------- ------------------ ---------------------- 0 0 (0.4965,-0.1850) (0.4965,-0.1850) ($<$0.01%)($<$0.01%) 0 0.2 (0.5238,-0.1883) (0.5238,-0.1883) ($<$0.01%)($<$0.01%) 0 0.4 (0.6470,-0.1965) (0.6470,-0.1965) ($<$0.01%)($<$0.01%) 0 0.495 (0.8428,-0.1742) (0.8428,-0.1742) ($<$0.01%)($<$0.01%) 1 0 (0.4290,-0.5873) (0.4292,-0.5873) ($<$0.01%)(0.05%) 1 0.2 (0.4598,-0.5953) (0.4598,-0.5953) ($<$0.01%)($<$0.01%) 1 0.4 (0.5980,-0.6107) (0.5980,-0.6107) ($<$0.01%)($<$0.01%) 1 0.495 (0.7979,-0.5293) (0.7978,-0.5280) (0.01%)(0.25%) 2 0 (0.3496,-1.0504) (0.3495,-1.0504) (0.03%)($<$0.01%) 2 0.2 (0.3832,-1.0596) (0.3832,-1.0596) ($<$0.01%)($<$0.01%) 2 0.4 (0.5340,-1.0660) (0.5340,-1.0659) ($<$0.01%)($<$0.01%) 2 0.495 (0.7104,-0.9055) (0.6248,-1.0574) (-12.05%)(-16.78%) : *Reissner-Nordström quasinormal frequency parameter values ($\rho = -i \omega$) for the fundamental ($n = 1$) and two lowest overtones for $l = 1$ and $i = 1$.*[]{data-label="tab:RS5"} Kerr Black Holes {#sec:5} ================ A rotating black hole carrying angular momentum is described by the Kerr metric (in Boyer-Lindquist coordinates) as $$ds^{2}=-\left(1-\frac{r}{\Sigma}\right)dt^{2}-\frac{2ar\sin^{2}\theta}{\Sigma}dtd\phi+\frac{\Sigma}{\Delta}dr^{2} \Sigma d\theta^{2}+\left(r^{2}+a^{2}+\frac{a^{2}r\sin^{2}\theta}{\Sigma}\right)\sin^{2}\theta d\phi \; ,$$ with $$\begin{aligned} \Delta&=&r^{2}+a^2-2Mr\equiv(r-r_{-})(r-r_{+}) \; ,\\ \Sigma&=&r^{2}+a^{2}\cos^{2}\theta \; ,\end{aligned}$$ and where $a$ is the Kerr rotation parameter with $0\leq a \leq M$, $M$ being included as a general black hole mass. The horizons $r_{-}$ and $r_{+}$ are again the inner and the outer (event) horizons respectively. Teukolsky [@Teukolsky:1972my] showed that the perturbation equations in the Kerr geometry are separable, where the separated equations for the angular wave function ${}_sS_{lm}(\theta)$ and the radial wave function $R(r)$ are given by: $$\begin{aligned} [(1-u^{2})S_{,u}]_{,u}+\left[a^{2}\omega^{2}u^{2}-2a\omega su+s+{}_sA_{lm}-\frac{(m+su)^{2}}{1-u^{2}}\right]{}_sS_{lm}&=&0 \; , \label{kerrang}\\ \Delta R_{,rr}+(s+1)(2r-1)R_{,r}+K(r)R&=&0 \; , \label{kerrrad}\end{aligned}$$ where the function $$K(r)=\frac{1}{\Delta}\left\{\left(r^{2}+a^{2}\right)^{2}\omega^{2}-2am\omega r+a^{2}m^{2} +is\left[am\left(2r-1\right)-\omega\left(r^{2}-a^{2}\right)\right]\right\}+2is\omega r-a^{2}\omega^{2}-{}_sA_{lm} \; .$$ In the above $u=\cos\theta$, $s$ is the spin weight, ${}_sA_{lm}$ is the spin-weighted separation constant for the angular equation, and $m$ is another angular momentum parameter. For completeness the evaluation of the separation constant ${}_sA_{lm}$ using the AIM is discussed in Appendix \[sec:A\]. In order to use the AIM we need to solve for the angular solution in the radial equation. However, for nonzero $s$ the effective potential of the radial equation is in general complex. A straight forward application of the AIM does not give the correct answer. In fact a similar problem occurs in both numerical [@Detweiler1977] and WKB [@Seidel:1989bp] methods. For this reason we shall look at each of the spin cases ($0,\frac 1 2,2$) separately in the following subsections. The Spin-Zero Case {#sec:5-2} ------------------ Because the AIM works better on a compact domain, we define a new variable $y = 1 - \frac { r _ { + } } { r }$, which ranges from $0$ at the event horizon $( r = r _ { + } )$ to $1$ at spatial infinity. It is then necessary to incorporate the boundary conditions, which expressed in the new compact domain, where $$R(r)=(r^2+a^2)^{-1/2} \psi(r)$$ is $$\psi ( y ) = \left ( 1 - \frac { r _ { - } } { r _ { + } } ( 1 - y ) \right ) ^ { - i \sigma _ { - } } y ^ { - \sigma _ { + } } ( 1 - y ) ^ { - r _ { + } \omega } e ^ { i \omega \frac { r _ { + } } { 1 - y } } \chi ( y ) \; . \label{eqn:5-6}$$ By making the change of coordinates and change of function, Eq. (\[eqn:5-6\]) takes the form $$\chi ( y ) = \lambda _ { 0 } ( y ) + s _ { 0 } ( y ) \; ,\label{eqn:5-7}$$ where $$\lambda _ { 0 } = - 2 \frac { 1 } { g } \frac { \mathrm { d } g } { \mathrm { d } y } - \frac { 1 } { f } \frac { \mathrm { d } f } { \mathrm { d } y } \; ,$$ and $$s _ { 0 } = - \frac { 1 } { g } \frac { \mathrm { d } ^ { 2 } g } { \mathrm { d } y ^ { 2 } } - \frac { 1 } { f } \frac { \mathrm { d } f } { \mathrm { d } y } \times \frac { 1 } { g } \frac { \mathrm { d } g } { \mathrm { d } y } - \frac { 1 } { f ^ { 2 } } \left ( \omega ^ { 2 } - V | _ { r = r _ { + } ( 1 - y ) ^ { - 1 } } \right ) \; .$$ In the above we have defined $$f = \left. \left( \frac { \Delta } { r ^ { 2 } + a ^ { 2 } } \frac { \mathrm { d } y } { \mathrm { d } r } \right) \right|_{ r = r _ { + } ( 1 - y ) ^ { - 1 } } \; ,$$ and $$g = ( 1 - y ) ^ { - 2 i \omega } \left ( 1 - \frac { r _ { - } } { r _ { + } } ( 1 - y ) \right ) ^ { i \sigma _ { - } } y ^ { - i \sigma _ { + } } e ^ { i \omega r _ { + } ( 1 - y ) ^ { - 1 } } \; ,$$ where $\Delta = r^2 + a^2 - 2 M r$, $$\sigma _ { \pm } = \frac { 1 } { r _ { + } - r _ { - } } [ ( r ^ { 2 } _ { \pm } + a ^ { 2 } ) \omega + a m ] \; , \label{eqn:2-last}$$ and $a$ is again our rotation parameter. Eq. (\[eqn:5-7\]) is now in the correct form to use the AIM for QNM frequency calculations. Note that the potential, $V$ is $$V=-{1\over\Delta} ((K-2(r^2+a^2))^2-\Delta\lambda)~,$$ where the angular separation constant is defined via $\lambda=A_{l,m}+a^2\omega^2 - 2a m \omega$.[^2] As presented in Tables \[tab:Blake51\] and \[tab:Blake52\] are the QNM frequencies for the scalar perturbations of the Kerr black hole with the two “extreme” (minimum and maximum) values of the angular momentum per unit mass, that is, $a = 0.00$ and $a = 0.80$. $m$ was set to $0$, while $l$ was given values of $0$, $1$ and $2$ and $n$ varied accordingly. Included in Table \[tab:Blake51\] are the numerically determined QNM frequencies published by Leaver in 1985 [@Leaver1985]. The percentages bracketed under each QNM frequency via the AIM are the percentage differences between the calculated value and the numerical value published by Leaver. With the exceptions of the QNM frequencies for $l = 0$, $n = 0$ and $l = 2$, $n = 2$, the AIM values correspond to the CFM up to four decimal places and even those “anomalies” differ by less than $0.30\%$. Proving, at least in this case, the AIM is a precise semi-analytical technique. In Table \[tab:Blake52\], all three values were calculated in this work, even though published values are available for the third order WKB(J), at least graphically, where numerical values using the CFM were taken from Ref. [@Berti:2005eb]. Since the WKB(J) is a generally accepted semi-analytical technique for QNM frequency calculations, the percentages below the AIM values are the differences to the sixth order WKB(J) values. Only in the case of $l = 0$, $n = 0$ does the AIM QNM frequency significantly differ from the sixth order WKB(J) value. Note that in an upcoming work, further values will be presented for values of $a = 0.20$, $a = 0.40$ and $a = 0.60$, with the same variations of $l$ and $n$ with $M=1$ [@Blake]. l n Numerical Third Order WKB(J) Sixth Order WKB(J) AIM --- --- ------------------ --------------------- ----------------------- ----------------------- 0 0 0.1105 - 0.1049i 0.1046 - 0.1001i 0.1105 - 0.1008i 0.1103 - 0.1046i (-5.34% , 9.82%) ($<$0.01% , -2.91%) (-0.18% , -0.29%) 1 0 0.2929 - 0.0977i 0.2911 - 0.0989i 0.2929 - 0.0978i 0.2929 - 0.0977i (-0.61% , 1.23%) ($<$0.01% , 0.10%) ($<$0.01% , $<$0.01%) 1 0.2645 - 0.3063i 0.2622 - 0.3074i 0.2645 - 0.3065i 0.2645 - 0.3063i (-0.87% , 0.36%) ($<$0.01% , 0.07%) ($<$0.01% , $<$0.01%) 2 0 0.4836 - 0.0968i 0.4832 - 0.0968i 0.4836 - 0.0968i 0.4836 - 0.0968i (-0.08% , $<$0.01%) ($<$0.01% , $<$0.01%) ($<$0.01% , $<$0.01%) 1 0.4639 - 0.2956i 0.4632 - 0.2958i 0.4638 - 0.2956i 0.4639 - 0.2956i (-0.15% , 0.07%) (-0.02% , $<$0.01%) ($<$0.01% , $<$0.01%) 2 0.4305 - 0.5086i 0.4317 - 0.5034i 0.4304 - 0.5087i 0.4306 - 0.5086i (0.28% , -1.02%) (-0.02% , 0.02%) (0.02% , $<$0.01%) : *The QNM Frequencies for the Scalar Perturbations of the Kerr Black Hole, with $a = 0.00$, that is, the Schwarzschild limit ($M=1,~m=0$). Numerical data via the CFM taken from [@Berti:2005eb], where the AIM was set to run at $15$ iterations.*[]{data-label="tab:Blake51"} l n Numerical Third Order WKB(J) Sixth Order WKB(J) AIM --- --- ----------------- -------------------- -------------------- ------------------- 0 0 0.1145 -0.0957i 0.1005 - 0.1007i 0.1211 - 0.0897i 0.1141 - 0.0939i (-12.2% , 5.22%) (5.76% ,- 6.23%) (-0.35% , -1.88%) 1 0 0.3067 -0.0901i 0.3029 - 0.0891i 0.3053 - 0.0893i 0.3052 - 0.0892i (-1.24% , -1.11%) (-0.46% , -0.89%) (-0.49% , -1.00%) 1 0.2820 -.2783i 0.2758 - 0.2779i 0.2821 - 0.2755i 0.2817 - 0.2756i (-2.20%,-0.14%) (0.04%, -1.01%) (-0.11% , -0.97%) 2 0 0.5071 -0.0897i 0.5035 - 0.0885i 0.5041 - 0.0886i 0.5041 - 0.0886i (-0.71%, -1.34%) (-0.59%, -1.23%) (-0.59% , -1.23%) 1 0.4906 -0.2722i 0.4866 - 0.2693i 0.4885 - 0.2690i 0.4885 - 0.2689i (-0.82%, -1.07%) (-0.43%, -1.18%) (-0.43% , -1.21%) 2 0.4609 -0.4634i 0.4585 - 0.4570i 0.4607 - 0.4581i 0.4606 - 0.4579i (-0.52%, -1.38%) (-0.04%, -1.14%) (-0.06%, -1.19%) : *The QNM Frequencies for the Scalar Perturbations of the Kerr Black Hole, with $a = 0.80$ ($M=1,~m=0$). Numerical data via the CFM taken from [@Berti:2005eb], where the AIM was set to run at $15$ iterations.*[]{data-label="tab:Blake52"} The Spin-Half Case {#sec:3-2} ------------------ For the spin-1/2 case we would like to know how the AIM can be used to derive an appropriate form of the Dirac equation in this spacetime background using the basis set up by four null vectors which are the basis of the Newman-Penrose formalism, for further details see Ref. [@Blake]. That is, in the Kerr background we adopt the following vectors as the null tetrad: $$\begin{aligned} l _ { j } & = & \frac { 1 } { \Delta } ( \Delta , - \rho ^ { 2 } , 0 , - a \Delta \sin ^ { 2 } { \theta } )\; ,\nonumber\\ n _ { j } & = & \frac { 1 } { 2 \rho ^ { 2 } } ( \Delta , \rho ^ { 2 } , 0 , - a \Delta \sin ^ { 2 } { \theta } )\; ,\nonumber\\ m _ { j } & = & \frac { 1 } { \sqrt { 2 } \bar { \rho } } ( i a \sin { \theta } , 0 , - \rho ^ { 2 } , - i ( r ^ { 2 } + a ^ { 2 } ) \sin { \theta } )\; ,\nonumber\\ l ^ { j } & = & \frac { 1 } { \Delta } ( r ^ { 2 } + a ^ { 2 } , \Delta , 0 , a )\; ,\nonumber\\ n ^ { j } & = & \frac { 1 } { \sqrt { 2 } \bar { \rho } } ( r ^ { 2 } + a ^ { 2 } , - \Delta , 0 , a )\; ,\nonumber\\ m ^ { j } & = & \frac { 1 } { \sqrt { 2 } \bar { \rho } } ( i a \sin { \theta } , 0 , 1 , - i \frac { 1 } { \sin { \theta } } )\; ,\nonumber\end{aligned}$$ where $\bar { m } _ { j }$ and $\bar { m } ^ { j }$ are nothing but complex conjugates of $m _ { j }$ and $m ^ { j }$ respectively. It is clear that the basis vectors basically become derivative operators when these are applied as tangent vectors to the function $e ^ { i ( \omega t + m \phi ) }$. Therefore we can write $$\vec { l } = D = \mathcal { D } _ { 0 } \mbox { , } \vec { n } = D ^ { * } = - \frac { \Delta } { 2 \rho ^ { 2 } } \mathcal { D } ^ { \dagger } _ { 0 } \mbox { , } \vec { m } = \delta = \frac { 1 } { \sqrt { 2 } \bar { \rho } } \mathcal { L } ^ { \dagger } _ { 0 } \mbox { , } \vec { \bar { m } } = \delta ^ { * } = \frac { 1 } { \sqrt { 2 } \bar { \rho } ^ { * } } \mathcal { L } _ { 0 } \; ,$$ where, $$\begin{aligned} \mathcal { D } _ { n } & = & \partial _ { r } + i \frac { K } { \Delta } + 2 n \frac { r - M } { \Delta } \; ,\nonumber\\ \mathcal { D } ^ { \dagger } _ { n } & = & \partial _ { r } - i \frac { K } { \Delta } + 2 n \frac { r - M } { \Delta } \; ,\nonumber\\ \mathcal { L } _ { n } & = & \partial _ { \theta } + Q + n \cot { \theta }\; ,\nonumber\\ \mathcal { L } ^ { \dagger } _ { n } & = &\partial _ { \theta } - Q + n \cot { \theta }\; ,\nonumber\end{aligned}$$ and $K = ( r ^ { 2 } + a ^ { 2 } ) \omega + a m$ with $Q = a \omega \sin { \theta } + m \csc { \theta }$. The spin coefficients can be written as a combination of basis vectors in the Newman-Penrose formalism which are now expressed in terms of the elements of different components of the Kerr metric. So by combining these different components of basis vectors in a suitable manner we get the spin coefficients as $$\kappa = \sigma = \lambda = \nu = \varepsilon = 0 \; .$$ $$\begin{aligned} \tilde{ \rho } = - \frac { 1 } { \bar { \rho } ^ { * } } \mbox { , } \beta & = & \frac { \cot { \theta } } { 2 \sqrt { 2 } \bar { \rho } ^ { * } } \mbox { , } \pi = \frac { i a \sin { \theta } } { \sqrt { 2 } ( \bar { \rho } ^ { * } ) ^ { 2 } }\; ,\nonumber\\ \tau = - \frac { i a \sin { \theta } } { \sqrt { 2 } \rho ^ { 2 } } \mbox { , } \mu & = & - \frac { \Delta } { 2 \bar { \rho } ^ { * } \rho ^ { 2 } } \mbox { , } \gamma = \mu + \frac { r - M } { 2 \bar \rho ^ { 2 } }\; ,\\ \alpha & = & \pi - \beta ^ { * }\; .\nonumber\end{aligned}$$ Using the above definitions, and by choosing: $f _ { 1 } = \bar { \rho } ^ { * } F _ { 1 }$, $g _ { 2 } = \bar { \rho } G _ { 2 }$, $f _ { 2 } = F _ { 2 }$ and $g _ { 1 } = G _ { 1 }$ (where $F_{1,2}$ and $G_{1,2}$ are a pair of spinors) the Dirac equation reduces to $$\begin{aligned} \mathcal { D } _ { 0 } f _ { 1 } + \frac { 1 } { \sqrt { 2 } } \mathcal { L } _ { \frac { 1 } { 2 } } f _ { 2 } = 0 \; ,\nonumber\\ \Delta \mathcal { D } ^ { \dagger } _ { \frac { 1 } { 2 } } f _ { 2 } - \sqrt { 2 } \mathcal { L } ^ { \dagger } _ { \frac { 1 } { 2 } } f _ { 1 } = 0 \; ,\label{eq:Blake221}\\ \mathcal { D } _ { 0 } g _ { 2 } - \frac { 1 } { \sqrt { 2 } } \mathcal { L } ^ { \dagger } _ { \frac { 1 } { 2 } } f _ { 1 } = 0 \; ,\nonumber\\ \Delta \mathcal { D } ^ { \dagger } _ { \frac { 1 } { 2 } } g _ { 1 } + \sqrt { 2 } \mathcal { L } _ { \frac { 1 } { 2 } } g _ { 2 } = 0 \; .\nonumber\end{aligned}$$ We separate the Dirac equation into radial and angular parts by choosing, $$\begin{aligned} f _ { 1 } ( r , \theta ) = R _ { - \frac { 1 } { 2 } } ( r ) S _ { - \frac { 1 } { 2 } } ( \theta ) \mbox { , } f _ { 2 } ( r , \theta ) = R _ { \frac { 1 } { 2 } } ( r ) S _ { \frac { 1 } { 2 } } ( \theta )\; ,\nonumber\\ g _ { 1 } ( r , \theta ) = R _ { \frac { 1 } { 2 } } ( r ) S _ { - \frac { 1 } { 2 } } ( \theta ) \mbox { , } g _ { 2 } ( r , \theta ) = R _ { - \frac { 1 } { 2 } } ( r ) S _ { \frac { 1 } { 2 } } ( \theta )\; .\nonumber\end{aligned}$$ Replacing these $f _ { j }$ and $g _ { j }$ $( j = 1 , 2 )$ and using $\lambda$ as the separation constant, we get, $$\begin{aligned} \mathcal { L } _ { \frac { 1 } { 2 } } S _ { \frac { 1 } { 2 } } = - \lambda S _ { - \frac { 1 } { 2 } }\; ,\label{eq:Blake222}\\ \mathcal { L } ^ { \dagger } _ { \frac { 1 } { 2 } } S _ { - \frac { 1 } { 2 } } = \lambda S _ { \frac { 1 } { 2 } }\; ,\nonumber\end{aligned}$$ $$\begin{aligned} \Delta ^ { \frac { 1 } { 2 } } \mathcal { D } _ { 0 } R _ { - \frac { 1 } { 2 } } = \lambda \Delta ^ { \frac { 1 } { 2 } } R _ { \frac { 1 } { 2 } }\; ,\label{eq:Blake223}\\ \Delta ^ { \frac { 1 } { 2 } } \mathcal { D } ^ { \dagger } _ { 0 } \Delta ^ { \frac { 1 } { 2 } } R _ { \frac { 1 } { 2 } } = \lambda R _ { - \frac { 1 } { 2 } }\; ,\nonumber\end{aligned}$$ where $2 ^ { \frac { 1 } { 2 } } R _ { - \frac { 1 } { 2 } }$ is redefined as $R _ { - \frac { 1 } { 2 } }$. Eqs. (\[eq:Blake222\]) and (\[eq:Blake223\]) are the angular and radial Dirac equation respectively, in a coupled form with the separation constant $\lambda$ [@Chandrasekhar:1985kt]. Decoupling Eq. (\[eq:Blake222\]) gives the eigenvalue/angular equation for spin half particles as $$\left [ \mathcal { L } _ { \frac { 1 } { 2 } } \mathcal { L } ^ { \dagger } _ { \frac { 1 } { 2 } } + \lambda ^ { 2 } \right ] S _ { - \frac { 1 } { 2 } } = 0 \; ,$$ and $S _ { \frac { 1 } { 2 } }$ satisfies the ‘adjoint’ equation (obtained by replacing $\theta$ by $\pi - \theta$).[^3] Decoupling Eq. (\[eq:Blake223\]) then gives the radial equation for spin half particles as $$\left [ \Delta \mathcal { D } ^ { \dagger } _ { \frac { 1 } { 2 } } \mathcal { D } _ { 0 } - \lambda ^ { 2 } \right ] R _ { - \frac { 1 } { 2 } } = 0 \; , \label{eq:Blake224}$$ and $\Delta ^ { \frac { 1 } { 2 } } R _ { \frac { 1 } { 2 } }$ satisfies the complex-conjugate equation. Furthermore, unlike the case of a scalar particle, a spin half particle is not capable of extracting energy from a rotating black hole, that is, there is no Penrose Process (superradiance) equivalent scenario [@Blake]. Returning now to the AIM, recall that it shall work better on a compact domain, where we define a new variable $y ^ { 2 } = 1 - \frac { r _ { + } } { r }$, which ranges from $0$ at the event horizon $(r= r_{ + } )$ to $1$ at spatial infinity. It is then necessary to incorporate the boundary conditions, which expressed in the new compact domain is $$\psi ( y ) = \left ( 1 - \frac { r _ { - } } { r _ { + } } ( 1 - y ^ { 2 } ) \right ) ^ { - \frac { 1 } { 2 }- i \sigma _ { - } } ( y ^ { 2 } ) ^ { \frac { 1 } { 2 } - \sigma _ { + } } ( 1 - y ^ { 2 } ) ^ { - r _ { + } \omega } e ^ { i \omega \frac { r _ { + } } { 1 - y ^ { 2 } } } \chi ( y ) \; ,$$ where we have defined $$\psi=\sqrt\Delta R_{1/2}+ R_{-1/2}~$$ and $\psi$ satisfies the WKB(J)-like equation: $${d^2 \psi\over dy^2} + (\omega^2-V)\psi=0$$ with potential $$V=\lambda^2 {\Delta\over \bar K^2}+ \lambda {d\over dx} \Big( {\sqrt\Delta \over \bar K} \Big)$$ and $\bar K =K/\omega= ( r ^ { 2 } + a ^ { 2 } ) + {a } m/\omega$ (for more details see [@Blake]). By making the change of coordinates and change of functions, our equation takes the form $$\chi ( y ) = \lambda _ { 0 } ( y ) + s _ { 0 } ( y ) \; ,\label{eq:Blake59}$$ where as in Sub-Sec.\[sec:5-2\] we have $$\lambda _ { 0 } = - 2 \frac { 1 } { g } \frac { \mathrm { d } g } { \mathrm { d } y } - \frac { 1 } { f } \frac { \mathrm { d } f } { \mathrm { d } y } \; ,$$ and $$s _ { 0 } = - \frac { 1 } { g } \frac { \mathrm { d } ^ { 2 } g } { \mathrm { d } y ^ { 2 } } - \frac { 1 } { f } \frac { \mathrm { d } f } { \mathrm { d } y } \times \frac { 1 } { g } \frac { \mathrm { d } g } { \mathrm { d } y } - \frac { 1 } { f ^ { 2 } } \left ( \omega ^ { 2 } - V | _ { r = r _ { + } ( 1 - y ^ { 2 } ) ^ { - 1 } } \right ) \; .$$ As presented in Table \[tab:Blake53\] and Table \[tab:Blake54\] the QNM frequencies for the spin half perturbations of the Kerr black hole with the two “extreme” values of the angular momentum per unit mass, that is $a = 0.00$ and $a = 0.80$, $m$ was set to $0$, while $l$ was given values of $0$, $1$ and $2$ and $n$ varied accordingly. Included in Table \[tab:Blake53\] are the numerically determined QNM frequencies published by Jing [*et al.*]{} [@Jing:2005pk]. Even though the WKB method has been used to calculate the Schwarzschild limit QNM frequencies before [@Cho:2003qe], the sixth order WKB values and AIM values are novel to this work and shall be explored more fully in Ref. [@Blake]. The percentages bracketed under each QNM frequency, is the percentage difference between the calculated value and the numerical value published by Jing [*et al.*]{} [@Jing:2005pk]. As expected, since there are additional correction terms, the sixth order WKB QNM frequencies are closer to the numerical values than the third order WKB values. While the AIM does not prove as accurate in its calculation of the spin half QNM frequencies as it did with the scalar values (both for $15$ iterations), none of the differences between the AIM values and the numerical values exceed $0.30\%$, except for when $l = 2$, $n = 2$ (better accuracy can be achieved by increasing the number of iterations). Similarly in Table \[tab:Blake54\] are the numerically determined QNM frequencies published by Jing [*et al.*]{} [@Jing:2005pk]. Both the third and sixth order WKB values along with the AIM values are novel to this work and shall also be explored more fully in Ref. [@Blake]. The percentages bracketed under each QNM frequency, are the percentage differences between the calculated value and the numerical value published by Jing [*et al.*]{}, at least for $l = 0$ and $l = 1$. For $l = 2$, the AIM values are compared to the sixth order WKB values. As already noted, since there are additional correction terms, the sixth order WKB QNM frequencies are closer to the numerical values than the third order WKB values. Again the AIM does not appear to be as precise in calculating the QNM frequencies for spin half perturbations of the Kerr black hole as it was for the scalar perturbations (at least for $15$ iterations). As we mentioned, additional tables and plots of these Kerr processes shall constitute a future work [@Blake]. l n Numerical Third Order WKB Sixth Order WKB AIM --- --- ------------------ --------------------- ----------------------- ----------------------- 0 0 0.1830 - 0.0970i 0.1765 - 0.1001i 0.1827 - 0.0949i 0.1830 - 0.0969i (-3.55% , 3.20%) (-0.16% , -2.16%) ($<$0.01% , -0.10%) 1 0 0.3800 - 0.0964i 0.3786 - 0.0965i 0.3801 - 0.0964i 0.3800 - 0.0964i (-0.37% , 0.10%) (0.03% , $<$0.01%) ($<$0.01% , $<$0.01%) 1 0.3558 - 0.2975i 0.3536 - 0.2987i 0.3559 - 0.2973i 0.3568 - 0.2976i (-0.62% , 0.40%) (0.03% , -0.07%) (0.28% , 0.03%) 2 0 0.5741 - 0.0963i 0.5737 - 0.0963i 0.5741 - 0.0963i 0.5741 - 0.0963i (-0.07% , $<$0.01%) ($<$0.01% , $<$0.01%) ($<$0.01% , $<$0.01%) 1 0.5570 - 0.2927i 0.5562 - 0.2930i 0.5570 - 0.2927i 0.5573 - 0.2928i (-0.14% , 0.10%) ($<$0.01% , $<$0.01%) (0.05% , 0.03%) 2 0.5266 - 0.4997i 0.5273 - 0.4972i 0.5265 - 0.4997i 0.5189 - 0.5213i (0.13% , -0.50%) (-0.02% , $<$0.01%) (-1.46% , 4.32%) : *The QNM Frequencies for the Spin Half Perturbations of the Kerr black hole, with $a = 0.00$, that is, the Schwarzschild limit ($M=1,~m=0$). Numerical data via the CFM taken from [@Jing:2005pk], where the AIM was set to run at $15$ iterations.*[]{data-label="tab:Blake53"} l n Numerical Third Order WKB(J) Sixth Order WKB(J) AIM --- --- ------------------ -------------------- -------------------- -------------------- 0 0 0.1932 - 0.0891i 0.1883 - 0.0896i 0.1914 - 0.0865i 0.1920 - 0.0872i (-2.54% , 0.56%) (-0.93% , -2.92%) (-0.62% , -2.13%) 1 0 0.3993 - 0.0893i 0.3956 - 0.0881i 0.3967 - 0.0880i 0.3965 - 0.0880i (-0.93% , -1.34%) (-0.65% , -1.46%) (-0.70% , -1.46%) 1 0.3789 - 0.2728i 0.3751 - 0.2701i 0.3777 - 0.2687i 0.3764 - 0.2517i (-1.00% , -0.99%) (-0.32% , -1.50%) (-0.66%, -7.73%) 2 0 0.5984 - 0.0881i 0.5987 - 0.0881i 0.5987 - 0.0882i ($<$0.01% , 0.11%) 1 0.5844 - 0.2669i 0.5855 - 0.2667i 0.5846 - 0.2644i (-0.15% , -0.86%) 2 0.5600 - 0.4512i 0.5609 - 0.4517i 0.6023 - 0.4260i (7.38%, -5.70%) : *The QNM Frequencies for the Spin Half Perturbations of the Kerr black hole, with $a = 0.80$ ($M=1,~m=0$). Numerical data via the CFM taken from [@Jing:2005pk], where the AIM was set to run at $15$ iterations.*[]{data-label="tab:Blake54"} The Spin-Two Case {#sec:5-3} ----------------- As we have mentioned earlier, the radial equation for nonzero spin $s$ is in general complex. In fact, it does not even reduce to the Regge-Wheeler and Zerilli equations when the rotation parameter $a\rightarrow 0$. Detweiler [@Detweiler1977] has found a way to overcome this problem, where he defined a new function $$X=\Delta^{s/2}\left(r^{2}+a^{2}\right)^{1/2}\left[\alpha(r)R+\beta(r)\Delta^{s+1}\frac{dR}{dr}\right] \; .$$ If the functions $\alpha(r)$ and $\beta(r)$ are required to satisfy $$\alpha^{2}-\alpha'\beta\Delta^{s+1}+\alpha\beta'\Delta^{s+1}-\beta^{2}\Delta^{2s+1}K={\rm constant} \; ,$$ then it can be shown that the radial equation in Eq. (\[kerrrad\]) becomes $$\frac{d^{2}X}{dx^{2}}-VX=0\; ,\label{xeqn}$$ where $$\begin{aligned} V&=&\frac{\Delta U}{\left(r^{2}+a^{2}\right)^{2}}+G^{2}+\frac{dG}{dx}\; ,\\ G&=&\frac{s(2r-1)}{2\left(r^{2}+a^{2}\right)}+\frac{r\Delta}{\left(r^{2}+a^{2}\right)^{2}}\; ,\\ U&=&K+\frac{2\alpha'+\left(\beta'\Delta^{s+1}\right)'}{\beta\Delta^{s}}\; ,\\ x&=&r+\frac{r_{+}}{r_{+}-r_{-}}{\rm ln}\left(r-r_{+}\right)-\frac{r_{-}}{r_{+}-r_{-}}{\rm ln}\left(r-r_{-}\right)\; .\end{aligned}$$ As Detweiler has indicated, it is possible to choose the functions $\alpha(r)$ and $\beta(r)$ so that the resulting effective potential $V(r)$ is real and has the form $$\begin{aligned} V&=&\frac{\rho^{2}\Delta}{\left(r^{2}+a^{2}\right)^{2}}\left\{\frac{f(f+2)}{g+b\Delta}-\frac{b\Delta}{\rho^{4}} +\frac{\left[\kappa\rho\Delta-\left(g'\Delta-g\Delta'\right)\right]\left[\kappa\rho g-b\left(g'\Delta-g\Delta'\right)\right]} {\rho^{2}\left(g+b\Delta\right)\left(g-b\Delta\right)^{2}}\right\}\nonumber\\ &&\ \ \ +\left[\frac{ram\Delta}{\omega\rho\left(r^{2}+a^{2}\right)^{2}}\right]^{2} -\frac{\Delta}{r^{2}+a^{2}}\frac{d}{dr}\left[\frac{ram\Delta}{\omega\rho\left(r^{2}+a^{2}\right)^{2}}\right]-\frac{Y^{2}}{\left(r^{2}+a^{2}\right)^{2}}\; , \label{realpot}\end{aligned}$$ where $$\begin{aligned} g&=&f\rho^{2}+3\rho(r^{2}+a^{2})-3r^{2}\Delta\; ,\\ \rho&=&r^{2}+a^{2}-\frac{am}{\omega}\; ,\\ \kappa&=&\pm\left\{9-2f\left[\left(a^{2}-\frac{am}{\omega}\right)(5f+6)-12a^{2}\right]+2bf(f+2)\right\}^{1/2}\; ,\\ b&=&\pm3\left(a^{2}-\frac{am}{\omega}\right)\; ,\label{beqn}\\ Y&=&am-(r^{2}+a^{2})\omega\; ,\\ f&=&A+a^{2}\omega^{2}-2am\omega\; .\end{aligned}$$ When the Kerr rotation parameter $a$ approaches zero, the potential $V$ in Eq. (\[realpot\]) coincides with the Regge-Wheeler potential for negative $\kappa$, and coincides with the Zerilli potential for positive $\kappa$. Here we choose $\kappa$ to be negative, where the choice of the sign in Eq. (\[beqn\]) is determined by the sign of $m$ [@Detweiler1977; @Seidel:1989bp]. The QNM boundary conditions for $X$ are $$X\rightarrow\left\{ \begin{array}{cl} e^{-ikx} & ; \ x\rightarrow-\infty \\ e^{i\omega x} & ; \ x\rightarrow\infty \end{array}\right.\; ,$$ where $$k=\omega-\frac{am}{r_{+}}\; .$$ Hence, we write $$X=e^{i\omega r}r^{\frac{i\omega}{r_{+}-r_{-}}}\left[\frac{(r-r_{+})^{r_{+}}}{(r-r_{-})^{r_{-}}}\right]^{-\frac{ik}{r_{+}-r_{-}}}\chi_{G}\; .$$ Substituting this into Eq. (\[xeqn\]), we have $$\begin{aligned} &&\chi_{G,rr}+\left[2\Gamma_{G}+\frac{r^{2}-a^{2}}{\Delta\left(r^{2}+a^{2}\right)}\right]\chi_{G,r} +\left[\Gamma_{G}^{2}+\Gamma_{G,r}+\frac{\Gamma_{G}\left(r^{2}-a^{2}\right)}{\Delta\left(r^{2}+a^{2}\right)} -\left(\frac{r^{2}+a^{2}}{\Delta}\right)^{2}V\right]\chi_{G}=0\; ,\label{chieqn}\end{aligned}$$ where $$\Gamma_{G}=i\omega+\frac{i\omega}{r(r_{+}-r_{-})}-\frac{ikr}{\Delta}\; .$$ As we did earlier, we define the variable $\xi=1-r_{+}/r$ which has a compact domain $0<\xi<1$. Eq. (\[chieqn\]) can then be written in the AIM form $$\chi_{G,\xi\xi}=\lambda_{G}(\xi)\chi_{G,\xi}+s_{G}(\xi)\chi_{G}\; ,$$ where $$\begin{aligned} \lambda_{G}&=&\frac{2}{1-\xi}-\frac{r_{+}}{(1-\xi)^{2}}\left[2\Gamma_{G} +\frac{1}{\Delta}\frac{r_{+}^{2}-a^{2}(1-\xi)^{2}}{r_{+}^{2}+a^{2}(1-\xi)^{2}}\right]\; ,\\ s_{G}&=&\frac{r_{+}^{2}}{(1-\xi)^{4}}\left\{\left[\frac{r_{+}^{2}+a^{2}(1-\xi)^{2}}{\Delta(1-\xi)^{2}}\right]^{2}V -\frac{\Gamma_{G}}{\Delta}\frac{r_{+}^{2}-a^{2}(1-\xi)^{2}}{r_{+}^{2}+a^{2}(1-\xi)^{2}}-\Gamma_{G}^{2}\right\} -\frac{r_{+}}{(1-\xi)^{2}}\Gamma_{G,\xi}\; .\end{aligned}$$ The results for the gravitational (spin-two) case are presented in Tables \[tab:Blake55S\] and \[tab:Blake56S\]. In general the error in the separation constant is smaller than that of the quasinormal frequencies. As for the quasinormal frequencies the error in the Kerr case is larger than that of either the Schwarzschild or the Reissner-Nordström cases, where this is due to our consideration of the angular and the radial equations simultaneously. The number of iterations that can be performed in the code is relatively small, much like the number of continued fractions in the CFM is typically smaller due to the coupling between radial and angular equations. $a$ $A_{Leaver}$ $A_{AIM}$ $\omega_{Leaver}$ $\omega_{AIM}$ ------ ------------------ ---------------------- ------------------- ------------------- 0 (4.0000, 0.0000) (4.0000, 0.0000) (0.7473, -0.1779) (0.7413, -0.1780) ($<$0.01%)($<$0.01%) (-0.80%)(-0.06%) 0.1 (3.9972, 0.0014) (3.9973, 0.0014) (0.7502, -0.1774) (0.7444, -0.1775) ($<$0.01%)($<$0.01%) (-0.77% , -0.06%) 0.2 (3.9886, 0.0056) (3.9887, 0.0056) (0.7594, -0.1757) (0.7540, -0.1763) ($<$0.01%)($<$0.01%) (-0.71%)(-0.03%) 0.3 (3.9730, 0.0126) (3.9733, 0.0126) (0.7761, -0.1720) (0.7715, -0.1722) ($<$0.01%)($<$0.01%) (-0.59%)(-0.12%) 0.4 (3.9480, 0.0223) (3.9482, 0.0222) (0.8038, -0.1643) (0.8025, -0.1639) ($<$0.01%)(-0.45%) (-0.16%)(0.24%) 0.45 (3.9304, 0.0276) (3.9303, 0.0280) (0.8240, -0.1570) (0.8250, -0.1591) ($<$0.01%)(1.45%) (0.12%)(-1.34%) : *Spin-2 angular separation constants and Kerr gravitational quasinormal frequencies for the fundamental mode corresponding to $l=2$ and $m=0$ compared with the CFM [@Leaver1985] ($M=1/2$).*[]{data-label="tab:Blake55S"} $a$ $A_{Leaver}$ $A_{AIM}$ $\omega_{Leaver}$ $\omega_{AIM}$ ------ ------------------ ---------------------- ------------------- ------------------- 0 (4.0000, 0.0000) (4.0000, 0.0000) (0.7473, -0.1779) (0.7413, -0.1780) ($<$0.01%)($<$0.01%) (-0.80%)(-0.06%) 0.1 (3.8932, 0.0252) (3.8937, 0.0250) (0.7765, -0.1770) (0.7726, -0.1755) (-0.01%)(-0.89%) (-0.51% , 0.86%) 0.2 (3.7676, 0.0532) (3.7681, 0.0526) (0.8160, -0.1745) (0.8143, -0.1726) (0.01%)(-1.12%) (-0.02%)(1.09%) 0.3 (3.6125, 0.0835) (3.6123, 0.0826) (0.8719, -0.1693) (0.8722, -0.1674) ($<$0.01%)(-0.99%) (0.03%)(1.00%) 0.4 (3.4023, 0.1122) (3.4011, 0.1110) (0.9605, -0.1559) (0.9620, -0.1543) (-0.03%)(-1.00%) (0.16%)(1.05%) 0.45 (3.2535, 0.1195) (3.2491, 0.1173) (1.0326, -0.1396) (1.0376, -0.1369) (-0.03%)(-1.82%) (0.48%)(1.97%) : *Spin-2 angular separation constants and Kerr gravitational quasinormal frequencies for the fundamental mode corresponding to $l=2$ and $m=1$ compared with the CFM [@Leaver1985] ($M=1/2$).*[]{data-label="tab:Blake56S"} Doubly Rotating Kerr (A)dS Black Holes {#sec:6} ====================================== Rotating black holes in higher dimensions were first discussed in the seminal paper by Myers and Perry [@Myers:1986un]. One of the unexpected results to come from this work was that some families of solutions were shown to have event horizons for arbitrarily large values of their rotation parameters. The stability of such black holes is certainly in question [@Emparan:2003sy; @Konoplya:2011qq], with numerical evidence recently provided by Shibata and Yoshino [@Shibata:2009ad]. Another new feature of the Myers-Perry (MP) solutions is that they in general have $\lfloor\frac{D-1}{2}\rfloor$ spin parameters, making them more complex than the four dimensional Kerr solution. The first asymptotically non-flat five-dimensional MP metric was given in Ref. [@Hawking:1998kw]. Subsequent generalizations to arbitrary dimensions was done in Ref. [@Gibbons:2004js], and finally the most general Kerr-(A)dS-NUT metric was found by Chen, Lü and Pope [@Chen:2006xh]. In this section we review how the AIM can be used to solve the $D\geq 6$ two-rotation scalar perturbation equations (for more details on the metric and resulting separation see Ref. [@Cho:2011yp]). The scalar field master equations are found to be [@Cho:2011yp]: $$\begin{aligned} 0&=&\frac{1}{r^{D-6}}\frac{d}{dr}\left(r^{D-6}\Delta_{r} \frac{d R_r}{dr}\right)+\left( \frac{(r^2+a_1^2)^2(r^2+a_2^2)^2}{\Delta_r}\tilde{\omega}_r^2-\frac{a_1^2 a_2^2 j(j+D-7)}{r^2}-b_1 r^2 -b_2\right) R_r\;, \label{eqn:Rr}\\ 0&=&\left(\frac{a_i}{y_i}\right)^{ D-6} \frac{d}{dy_i}\left[\left(\frac{y_i}{a_i}\right)^{D-6}\Delta_{y_i} \frac{d R_{\theta_i}}{dy_i}\right]-\left\{ \frac{(a_1^2-y_i^2)^2(a_2^2-y_i^2)^2}{\Delta_{y_i}}\tilde{\omega}_{y_i}^2+\frac{a_1^2 a_2^2 j(j+D-7)}{y_i^2}+b_1 y_i^2 -b_2\right\} R_{\theta_i} \; , \nonumber \\ && \label{eqn:Ryi}\end{aligned}$$ where $$\begin{aligned} \Delta_{r}&=&(1+g^{2}r^{2})(r^{2}+a_{1}^{2})(r^{2}+a_{2}^{2})-2Mr^{7-D},\\ \Delta_{y_{i}}&=&(1-g^{2}y_{i}^{2})(a_{1}^{2}-y_{i}^{2})(a_{2}^{2}-y_{i}^{2})\;,\end{aligned}$$ the radial and angular frequencies are defined by: $$\begin{aligned} \label{suprad} \tilde{\omega}_r&=&\omega-(1+g^2 r^2) \left(\frac{m_1 a_1}{r^2+a_1^2}+\frac{m_2 a_2}{r^2+a_2^2}\right),\\ \tilde{\omega}_{y_i}&=&\omega-(1-g^2 y_i^2) \left(\frac{m_1 a_1}{a_1^2-y_i^2}+\frac{m_2 a_2}{a_2^2-y_i^2}\right),\end{aligned}$$ and $i=1,2$. In the above $g$ is the curvature of the spacetime satisfying $R_{\mu\nu}=-3g^2 g_{\mu\nu}$ (e.g., see Ref. [@Chen:2006xh]), and $a_1,a_2$ are the two rotation parameters and for later reference we define $\epsilon=a_2/a_1$. Doubly rotating black holes are more complicated than simply rotating black holes (cf. Ref. [@Kodama:2009rq]), because two rotation planes lead to two coupled spheroids which are also needed for the solution of the radial equation. Radial Quasi-Normal Modes {#sec:6-1} ------------------------- For simplicity we will consider the flat case, setting $g=0$, which leads to easier QNM boundary conditions (cf. Schwarzschild to Schwarzschild-dS). These satisfy the boundary condition that there are only waves ingoing at the black hole horizon and outgoing waves at asymptotic infinity. As we have shown with the previous examples, it is easier to work on a compact domain and define the variable $x=1/r$, so that infinity is mapped to zero and the outer horizon stays at $x_h=1/r_h=1$. The domain of $x$ will therefore be $[0,1]$. Thus the QNM boundary condition is translated into the statement that the waves move leftward at $x=0$ and rightward at $x=1$. We again choose the AIM point in the middle of the domain, that is, at $x=1/2$. In terms of $x$ the radial equation (\[eqn:Rr\]) becomes: $$\label{eqn:radialx} 0=-x^{D-4}\frac{d}{dx}\left(-x^{8-D}\Delta_{x} \frac{d R}{dx}\right)+\left( \frac{(x^{-2}+a_1^2)^2(x^{-2}+a_2^2)^2}{\Delta_x}\tilde{\omega}_x^2-a_1^2 a_2^2 j(j+D-7)x^2-\frac{b_1}{x^2} -b_2\right) R\;,$$ where $\Delta_x(x)\equiv\Delta_r(r=1/x)$ and $\omega_x(x)\equiv\omega_r(r=1/x)$. After performing some asymptotic analysis, we find that for the solutions to satisfy the QNM boundary conditions we must have: $$\label{eqn:y} R \sim (1-x)^{i \tilde{\omega}_h\alpha_h} x^{(D-2)/2}e^{i \omega_x/x} y(x)\;,$$ where $$\begin{aligned} \tilde{\omega}_h&\equiv&\omega_x(x=1)\;,\\ \alpha_h&\equiv&\frac{(1+a_1^2)(1+a_2^2)}{\Delta_x'(x=1)}\;.\end{aligned}$$ We then substitute this ansatz into Eq. (\[eqn:radialx\]) and rewrite into the AIM form: $$y''=\lambda_0 y' +s_0 y\;.$$ This final step above can be performed in Mathematica, where the resulting expressions for $\lambda_0$ and $s_0$ are fed into the AIM routine. The method we use to find the QNMs proceeds in a fashion similar to that used in Ref. [@Berti:2005gp; @Berti:2005ys] (see also Sec \[sec:5-3\]) except we use the AIM instead of the CFM. First we set the number of AIM iterations in both the eigenvalue and QNM calculations to sixteen. We start with the Schwarzschild values $(b_1,b_2,\omega)$, that is, at the point $(a_1,a_2)\sim 0$ and then increment $a_1$ and $a_2$ by some small value. We take the initial eigenvalues $(b_1,b_2)$, insert them into the radial equation (\[eqn:y\]) then use the AIM to find the new QNM that is closest to $\omega$ using the Mathematica routine [FindRoot]{}. ![An example of the $D=6$ fundamental $(j,m_1,m_2,n_1,n_2)=(0,0,0,0,0)$ QNM. On the left is a plot of the imaginary part and on the right plot of the real part. \[fig:D6000plots\] ](Results_GradientsIm_0006.pdf "fig:") ![An example of the $D=6$ fundamental $(j,m_1,m_2,n_1,n_2)=(0,0,0,0,0)$ QNM. On the left is a plot of the imaginary part and on the right plot of the real part. \[fig:D6000plots\] ](Results_GradientsRe_0006.pdf "fig:") Taking this new value of omega, $\omega'$, we insert it into the two angular equations (at the same value of $a_1$ and $a_2$) then solve using the AIM, searching closest to the previous $b_1$ and $b_2$ values. Thereby obtaining the new eigenvalues $b_1'$, $b_2'$. We then repeat this process with the new $(\omega',b_1',b_2')$ as the starting point until the results converge and we have achieved four decimal places of accuracy. When this occurs we increment $a_1$ and $a_2$ again and repeat the process. In this way, we are able to find the QNMs and eigenvalues along lines passing approximately through the origin (that is, starting from the near Scwharzschild values) in the $(a_1,a_2)$ parameter space. As an example, we have plotted various values of $\epsilon=a_2/a_1$ ($=0,0.2,0.4,0.6,0.8,1$) against $a_1$ and used an interpolating function to interpolate between these values as shown in Fig. \[fig:D6000plots\] (for further details see Ref. [@Cho:2011yp]). Summary and Outlook {#sec:7} =================== In this review we have shown that the AIM can be used to calculate the radial QNMs of a variety of black hole spacetimes. In particular, we have used it to calculate perturbations of Schwarzschild (in asymptotically flat, de Sitter and anti-DeSitter), RN and Kerr (for spin $0, 1$ and $2$ perturbations) black holes in four dimensions. We argued that the method will be of use in studies of extra dimensional black holes and gave an explicit example of this in the case of the doubly rotating Myers Perry black hole. We have hopefully demonstrated how the AIM can also be applied to radial QNMs and not just to spheroidal eigenvalue problems [@Barakat:2006ki; @Cho:2009wf]. Given the fact that the AIM can be used in both the radial and angular wave equations [@Cho:2009wf] we expect no problems in obtaining QNMs for Kerr-dS black holes in four and higher dimensions. Note that this was only recently accomplished via the CFM in Ref. [@Yoshida:2010zzb] using Heun’s equation [@Suzuki:1998vy] to reduce the problem to a 3-term recurrence relation. In higher dimensions a similar method was used for simply rotating Kerr-AdS black holes [@Kodama:2009rq]. It remains to be seen if the AIM can be tailored to handle asymptotic QNMs (see Ref. [@Nollert:1993zz] for an adapted version of the CFM). However, given the close relationship between the AIM and the exact WKB approach [@Matamala:2007], it might be possible to adapt the AIM to find asymptotic QNMs [@Motl:2003cd; @Andersson:2003fh; @Das:2004db; @Ghosh:2005aq] numerically or even semi-analytically. The AIM might be of some topical use, for example, in the angular spheroids/QNMs needed in the phenomenology of Hawking radiation from spinning higher-dimensional black holes, for a recent review see Ref. [@Frost:2009cf]. We recently used a combination of all the techniques discussed in this work to evaluate the angular eigenvalues, ${}_0A_{kjm}$, for real $c=a\omega$, which are needed for the tensor graviton emission rates on a [*simply*]{} rotating Kerr-de Sitter black hole background in $(n+4)$-dimensions [@Doukas:2009cx] (also see Ref. [@Kanti:2009sn]) and it might also be interesting to find QNMs of doubly rotating Kerr-(A)dS black holes (for asymptotically flat Kerr see Ref. [@Cho:2011yp]). Finally, attempting to solve the QNMs for all spins on the Schwarzschild-AdS background via the AIM also seems an interesting problem. As such we hope to have provided the reader with enough technical details, and to have addressed some of the possible questions to allow them to pursue the study of QNMs with the AIM.[^4] HTC was supported in part by the National Science Council of the Republic of China under the Grant NSC 99-2112-M-032-003-MY3, and the National Science Centre for Theoretical Sciences. The work of JD was supported by the Japan Society for the Promotion of Science (JSPS), under fellowship no. P09749. WN would like to thank the Particle Physics Theory Group, Osaka University for computing resources. Angular Eigenvalues for Spin-Weighted Spheroidal Harmonics {#sec:A} ========================================================== $l$ $c=0.1$ (n${}_A$=n${}_C$=15, ) $c=0.8$ (n${}_A$=35, n${}_C$=70) $c=-10 i$ (n${}_A$=80, n${}_C$=130 CFM) $c=10 $ (n${}_A$=75, n${}_C$=145) ----- -------------------------------- ---------------------------------- ----------------------------------------- ----------------------------------- 2 -0.1391483511 -1.462479552 (12.44128209, 0.8956143162) -101.8949078 3 5.929826236 5.247141863 (32.31138608, 1.302608040) -63.74900642 4 13.95640426 13.45636668 (51.27922784, 1.946041848) -30.35607486 5 23.96944247 23.54163307 (69.25750923, 3.012877154) -4.557015739 6 35.97681567 35.58524928 (85.86796852, 4.990079008) -2.555206382 7 9.98139515 49.61077286 (99.20081385, 6.801617108) 12.32203552 : *Selected spin two eigenvalues, ${}_2A_{lm}$, obtained from the AIM for a Kerr black hole with varying values of $c=a\omega$ and $m=1$. The same number of iterations in the AIM, $n_A$ and the number of recursions in the CFM, $n_C$ for $c=0.1, 0.8$ at a working precision of $15$ digit precision, where results are presented to 10 s.f.*[]{data-label="tab:spinTwo"} As mentioned in Sec. \[sec:5-3\], aside from radial QNMs the AIM can also be applied to various kinds of spin-weighted spheroidal harmonics, ${}_sS_{lm}(\theta)$, e.g. see Ref. [@Berti:2005ys]. Therefore, in this appendix, we briefly compare the AIM with the CFM for the four-dimensional spin-weighted spheroids. With the regular boundary conditions, the angular wave function can be written as [@Leaver1985] $$S=e^{a\omega u}(1+u)^{\frac{1}{2}|m-s|}(1-u)^{\frac{1}{2}|m+s|}\chi_{A}(u)\; .$$ Putting this back into Eq. (\[kerrang\]) and rewriting the equation in AIM form, we have $$\chi_{A,uu}=\lambda_{A}(u)\chi_{A,u}+s_{A}(u)\chi_{A}\; ,$$ where $$\begin{aligned} \lambda_{A}(u)&=&\frac{2u}{1-u^{2}}-2N\; ,\\ s_{A}(u)&=&\frac{1}{1-u^{2}}\left[\frac{(m+su)^{2}}{1-u^{2}}+2a\omega su+2uN-\left(a^{2}\omega^{2}u^{2}+s+{}_sA_{lm}\right)\right]-N^{2}-N_{,u}\; ,\\ N&=&a\omega+\frac{|m-s|}{2(1+u)}-\frac{|m+s|}{2(1-u)}\; .\end{aligned}$$ These are the relevant equations for calculating the eigenvalues of the spheroidal harmonics in the four-dimensional case. It was noticed in Ref. [@Barakat:2006ki] that the AIM converges fastest at the maximum of the potential, when the AIM is written in WKB form. In four dimensions this occurs at $x=0$ and is true for general spin-$s$ as we have verified. Note that for higher dimensional generalizations it is not easy to explicitly find a maximum [@Cho:2009wf]. It may be worth mentioning that for the case where $c=0$ an exact analytic solution of the above equations leads to [@Berti:2005ys]: $${}_sA_{lm}= l(l+1)-s(s+1)\; . \label{spherhar}$$ In the exact limit $c=0$ the AIM recovers the result for spherical harmonics, Eq. (\[spherhar\]) above, while for the CFM taking $c=0$ leads to singularities [@Berti:2005ys]; however for $c\ll 1$ we find agreement with the CFM and Eq. (\[spherhar\]). For the purposes of consistency we have calculated (see Table \[tab:spinTwo\]) the ${}_2A_{l1}$ eigenvalues for the lowest $l=2,\dots,7$ modes to 10 significant figures and have also compared this with the results of the CFM. In both the AIM and CFM larger $l$ modes require more iterations/recursions to achieve convergence in a given $l$ eigenvalue to the required precision. Care should be taken when comparing the number of iterations in the AIM with that of the number of recursions in the CFM, because one iteration of the AIM is not equivalent to one iteration of the CFM. In fact although we typically need to iterate the improved AIM on average a lesser number of times, the CFM is typically faster for smaller values of c. However, for larger values of $c$ both methods can be faster or slower. The results of the first few $l$ eigenvalues for different values of $c=a\omega$, with $m=1$, are presented in Table \[tab:spinTwo\]. As far as we are aware this is the first time tables of spin-2 spheroids (for general complex $c$) have been presented using the AIM. Further results are presented in Sec. \[sec:5-3\] along with the radial QNMs for the spin two perturbations of the Kerr black hole in Table \[tab:Blake56S\]. ![(Color Online) $D=6$, $g=0$, $(j,m_1,m_2,n_1,n_2)=(0,1,1,0,0)$. A plot of the eigenvalues for various choices of $\epsilon\equiv a_2/a_1$. Note that the dependence on $a_1$ has been scaled into the other quantities. \[fig:evalepsilon\]](eval_wplot_various_epsilonb1.pdf "fig:") ![(Color Online) $D=6$, $g=0$, $(j,m_1,m_2,n_1,n_2)=(0,1,1,0,0)$. A plot of the eigenvalues for various choices of $\epsilon\equiv a_2/a_1$. Note that the dependence on $a_1$ has been scaled into the other quantities. \[fig:evalepsilon\]](eval_wplot_various_epsilonb2.pdf "fig:") ![(Color Online) $D=6$, $\epsilon\equiv a_2/a_1=1/2$, $(j,m_1,m_2,n_1,n_2)=(0,1,1,0,0)$. A plot of the eigenvalues for $g a_1 = 0.5 i, 0, 0.5$, corresponding to deSitter, flat, and anti-deSitter spacetimes respectively. Note that the dependence on $a_1$ has been scaled into the other quantities. \[fig:evalg\]](eval_g_b1.pdf "fig:") ![(Color Online) $D=6$, $\epsilon\equiv a_2/a_1=1/2$, $(j,m_1,m_2,n_1,n_2)=(0,1,1,0,0)$. A plot of the eigenvalues for $g a_1 = 0.5 i, 0, 0.5$, corresponding to deSitter, flat, and anti-deSitter spacetimes respectively. Note that the dependence on $a_1$ has been scaled into the other quantities. \[fig:evalg\]](eval_g_b2.pdf "fig:") Higher Dimensional Scalar Spheroidal Harmonics with two Rotation Parameters {#sec:B} =========================================================================== The two Eqs. (\[eqn:Ryi\]) are in fact the two-rotation generalization of the higher dimensional spheroidal harmonics (HSHs) studied in Ref. [@Berti:2005gp]. In this case, the existence of two rotation parameters leads to a system of two coupled second order ODEs[^5]. In general, one would expect that the generalizations of the HSHs to $\lfloor \frac{D-1}{2}\rfloor$ rotation parameters would lead to even larger systems of equations. While these systems would also be useful generally in studies of MP black holes, here we will only focus on the two rotation case. The angular equations can be written in the Sturm-Liouville form (assuming momentarily that $\omega$ and $b_2$ are real): $$\lambda w(\xi_i) R_{\theta_i}(\xi_i)=-\frac{d}{ d\xi_i}\left(p(\xi_i) \frac{d}{d\xi_i}R_{\theta_i}(\xi_i)\right)+q(\xi_i)R_{\theta_i}(\xi_i)$$ with the weight function $w_1(\xi_i)=\tfrac{1}{4} \xi_i^{(D-5)/2}$, the eigenvalue $\lambda=-b_1$, and $$\begin{aligned} p(\xi_i)&=&\xi_i^{(D-5)/2}\Delta_{\xi_i},\\ q(\xi_i)&=&\frac{1}{4} \xi_i^{(D-7)/2}\left(\frac{(a_1^2-\xi_i)^2(a_2^2-\xi_i)^2}{\Delta_{\xi_i}}\tilde{\omega}_{\xi_i}^2+\frac{a_1^2 a_2^2 j(j+D-7)}{\xi_i} -b_2\right)\; ,\end{aligned}$$ where $\Delta_{\xi_i}$ and $\tilde{\omega}_{\xi_i}$ are defined in the obvious way under the change of coordinates. Since $w(\xi)>0$ we can define the two norm’s: $$\begin{aligned} N_1(R_{\theta_1})&\propto&\int^{a_1^2}_{a_2^2}\xi_1^{(D-5)/2} |R_{\theta_1}|^2 d\xi_1\;,\\ N_2(R_{\theta_2})&\propto& \int^{a_2^2}_{0}\xi_2^{(D-5)/2} |R_{\theta_2}|^2 d\xi_2\;.\end{aligned}$$ For further details see Ref. [@Cho:2011yp]. The regular solutions are found to be: $$\begin{aligned} \label{eqn:modeR1} R_1&\sim& (\xi_1-a_2^2)^{\frac{|m_2|}{2}}(a_1^2-\xi_1)^{\frac{|m_1|}{2}}\Psi_1 \; ;\quad \xi_1\in (a_2^2,a_1^2)\; ,\\ R_2&\sim& \xi_2^{j/2}(a_2^2-\xi_2)^{\frac{|m_2|}{2}}\Psi_2 \; ;\quad \xi_2\in (0,a_2^2) \; .\label{eqn:modeR2}\end{aligned}$$ Now for a given value of $\omega$ we can determine $b_1$ and $b_2$ simply by performing the improved AIM [@Cho:2009cj] on both of the angular equations separately. This will result in two equations in the two unknowns $b_1,b_2$ which we can then be solved using a numerical routine such as the built-in Mathematica functions [NSolve]{} or [FindRoot]{}. More specifically we rewrite Eqs. (\[eqn:Ryi\]) using (\[eqn:modeR1\]) and (\[eqn:modeR2\]) and transform them into the AIM form: $$\begin{aligned} \frac{d^2\Psi_1}{d \xi_1 ^2}&=&\lambda_{01} \frac{d\Psi_1}{d \xi_1} +s_{01} \Psi_{1}\;,\\ \frac{d^2\Psi_2}{d \xi_2 ^2}&=&\lambda_{02} \frac{d\Psi_2}{d \xi_2} +s_{02} \Psi_{2}\;.\end{aligned}$$ The AIM requires that a special point be taken about which the $\lambda_{0i}$ and $s_{0i}$ coefficients are expanded. As was shown in Ref. [@Cho:2009wf] different choices of this point can worsen or improve the speed of the convergence. In the absence of a clear selection criterion we simply choose this point conveniently in the middle of the domains: $$\begin{aligned} \xi_{01}=\frac{a_1^2+a_2^2}{2},\quad \xi_{02}=\frac{a_2^2}{2}\;.\end{aligned}$$ Some results are plotted in Figs. \[fig:evalepsilon\], \[fig:evalg\] above. This method can now be used in the radial QNM equation in Sec. \[sec:6-1\]. 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[^3]: Note that this angular equation is that given by Eq. (\[kerrang\]) for $s=1/2$ and hence using the method in Appendix \[sec:A\], we could solve this numerically. [^4]: Source code and other information for some of the cases presented here can be found on the AIM link at <http://www-het.phys.sci.osaka-u.ac.jp/~naylor/>. [^5]: For the moment we are considering $\omega$ to be an independent parameter.

--- abstract: 'We compute the number of coverings of ${{\mathbb{C}}}P^1\setminus\{0, 1, \infty\}$ with a given monodromy type over $\infty$ and given numbers of preimages of 0 and 1. We show that the generating function for these numbers enjoys several remarkable integrability properties: it obeys the Virasoro constraints, an evolution equation, the KP (Kadomtsev-Petviashvili) hierarchy and satisfies a topological recursion in the sense of Eynard-Orantin.' address: - | Steklov Mathematical Institute\ 8 Gubkin St.\ Moscow 119991 Russia - | St.Petersburg Department of the Steklov Mathematical Institute\ Fontanka 27\ St. Petersburg 191023, and Chebyshev Laboratory of St. Petersburg State University\ 14th Line V.O. 29B\ St.Petersburg 199178 Russia author: - 'M. Kazarian, P. Zograf' title: 'Virasoro constraints and topological recursion for Grothendieck’s dessin counting' --- Introduction and preliminaries ============================== Enumerative problems arising in various fields of mathematics, from combinatorics and representation theory to algebraic geometry and low-dimensional topology, often bear much in common. In many cases the generating functions associated with these problems exhibit similar behavior – in particular, they may satisfy - Virasoro constraints, - Evolution equations of the “cut-and-join” type, - Integrable hierarchy (such as Kadomtsev-Petviashvili (KP), Korteveg-DeVries (KdV) or Toda equations), - Topological recursion (also known as Eynard-Orantin recursion). Simple Hurwitz numbers provide one of the best studied examples of such an enumerative problem – indeed, their generating function satisfies the celebrated cut-and-join equation [@GJ1], the Virasoro constraints (via the ELSV theorem [@ELSV] and the famous Mumford’s Grothendieck-Riemann-Roch formula [@M] it reduces to the Witten-Kontsevich potential), the KP hierarchy [@O], [@KL] or [@K], and the topological recursion [@EMS]. Other examples include the Witten-Kontsevich theory, Mirzakhani’s Weil-Petersson volumes, Gromov-Witten invariants of the complex projective line, invariants of knots, etc. (see [@EO1], [@EO2] for a review). These remarkable integrability properties of generating functions usually result from matrix model reformulations of the corresponding counting problems. However, in this paper we show that for the enumeration of Grothendieck’s [*dessins d’enfants*]{} all these properties follow from pure combinatorics in a rather straightforward way. The origin of Grothendieck’s theory of dessins d’enfants [@G] lies in the famous result by Belyi: [(Belyi, [@B])]{} A smooth complex algebraic curve $C$ is defined over the field of algebraic numbers ${\overline{\mathbb{Q}}}$ if and only if there exist a non-constant meromorphic function $f$ on $C$ (or a holomorphic branched cover $f:C\to{\mathbb{C}P^1}$) that is ramified only over the points $0,1,\infty\in{\mathbb{C}P^1}$. We call $(C,f)$, where $C$ is a smooth complex algebraic curve and $f$ is a meromorphic function on $C$ unramified over ${\mathbb{C}P^1}\setminus\{0,1,\infty\}$, a [*Belyi pair*]{}. For a Belyi pair $(C,f)$ denote by $g$ the genus of $C$ and by $d$ the degree of $f$. Consider the inverse image $f^{-1}([0,1])\subset C$ of the real line segment $[0,1]\subset{\mathbb{C}P^1}$. This is a connected bicolored graph with $d$ edges, whose vertices of two colors are the preimages of 0 and 1 respectively, and the ribbon graph structure is induced by the embedding $f^{-1}([0,1])\hookrightarrow C$. (Recall that a ribbon graph structure is given by prescribing a cyclic order of half-edges at each vertex of the graph.) The following is straightforward (cf. also [@LZ]): \[Gr\][(Grothendieck, [@G])]{} There is a one-to-one correspondence between the isomorphism classes of Belyi pairs and connected bicolored ribbon graphs. A connected bicolored ribbon graph representing a Belyi pair is called Grothendieck’s [*dessin d’enfant*]{}.[^1] Let $(C,f)$ be a Belyi pair of genus $g$ and degree $d$, and let ${\Gamma}=f^{-1}([0,1])\hookrightarrow C$ be the corresponding dessin. Put $k=|f^{-1}(0)|,\;l=|f^{-1}(1)|$ and $m=|f^{-1}(\infty)|$, then we have $2g-2=d-(k+l+m)$. We assume that the poles of $f$ are labeled and denote the set of their orders by $\mu=(\mu_1,\ldots,\mu_m)$, so that $d=\sum_{i\geq 1}\mu_i$. The triple $(k,l,\mu)$ will be called here the [*type*]{} of the dessin ${\Gamma}$, and the set of all dessins of type $(k,l,\mu)$ will be denoted by ${\mathcal{D}}_{k,l;\mu}$. Actually, instead of the dessin ${\Gamma}=f^{-1}([0,1])$ corresponding to a Belyi pair $(C,f)$ it is more convenient to consider the graph ${\Gamma}^*=\overline{f^{-1}(1/2+\sqrt{-1}{\mathbb{R}})}$ dual to $\Gamma$ (where the bar denotes the closure in $C$), see Fig. \[dual\]. The graph ${\Gamma}^*$ is connected, has $m$ ordered vertices of even degrees $2\mu_1,\ldots,2\mu_m$ at the poles of $f$ and inherits a natural ribbon graph structure. Moreover, the boundary components (faces) of ${\Gamma}^*$ are naturally colored: a face is colored in white (resp. in gray) if it contains a preimage of 0 (resp. 1), and every edge of ${\Gamma}^*$ belongs to precisely two boundary components of different color. ![Decomposition of ${\mathbb{C}P^1}$ into two 1-gons.[]{data-label="dual"}](dual){width="5cm"} In this paper we are interested in the weighted count of labeled dessins d’enfants of a given type. Namely, define $$\begin{aligned} N_{k,l}(\mu)=N_{k,l}(\mu_1,\ldots,\mu_m)=\sum_{{\Gamma}\in{\mathcal{D}}_{k,l,\mu}}\frac{1}{|{\rm Aut}_b {\Gamma}|}\;,\end{aligned}$$ where ${\rm Aut}_b {\Gamma}$ denotes the group of automorphisms of ${\Gamma}$ that preserve the boundary componentwise.[^2] Consider the total generating function $$\begin{aligned} \label{gf} F(s,u,v,p_1,p_2,\dots) = \sum_{k,l,m\geq 1}\frac{1}{m!}\sum_{\mu\in{\mathbb{Z}}_+^m} N_{k,l}(\mu) s^{d} u^k v^l\, p_{\mu_1}\ldots p_{\mu_m}\;,\end{aligned}$$ where the second sum is taken over all ordered sets $\mu=(\mu_1,\ldots,\mu_m)$ of positive integers, and $d=\sum_{i=1}^m \mu_i$. The objective of this paper is to show that the generating function $F$ satisfies all four integrability properties listed at the beginning of this section – namely, Virasoro constraints, an evolution equation, the KP (Kadomtsev-Petviashvili) hierarchy and a topological recursion. We prove the Virasoro constraints by a bijective combinatorial argument and derive from them all other properties of $F$.[^3] As a result, we obtain a simpler version of the topological recursion in terms of homogeneous components of $F$. We also revisit the problem of enumeration of the ribbon graphs with a prescribed boundary type. Topological recursion for this problem was first established in [@EO2] (cf. also [@DMSS]). In this paper we give a different, more streamlined proof of it based on the Virasoro constraints and show that the corresponding generating function satisfies an evolution equation and the KP hierarchy as well. These (and other) examples convincingly demonstrate that Virasoro constraints imply topological recursion and are in fact equivalent to it. Additionally, we show how our results can be applied to effectively enumerate orientable maps and hypermaps regardless of the boundary type. In particular, we present a very straightforward derivation of the famous Harer-Zagier recursion [@HZ] for the numbers of genus $g$ polygon gluings from the Walsh-Lehman formula [@WL] (a higher genus generalization of Tutte’s recursion [@T]).[^4] Virasoro constraints ==================== Virasoro constraints for the numbers of dessins ----------------------------------------------- For any integer $n\geq 0$ consider the differential operator $$\begin{gathered} L_n=-\frac{n+1}{s}\frac{\partial}{\partial p_{n+1}}+(u+v)n\frac{\partial}{\partial p_{n}} +\sum_{j=1}^\infty p_j(n+j)\frac{\partial}{\partial p_{n+j}}\\ {}+\sum_{i+j=n}ij\frac{\partial^2}{\partial p_{i} \partial p_{j}}+\delta_{0,n}uv\;.\label{V}\end{gathered}$$ A straightforward check shows that for any integer $m,n\geq 0$ $$\begin{aligned} [L_m, L_n]=(m-n)L_{m+n}\,.\end{aligned}$$ In other words, the operators $L_n$ form (a half of) a representation of the Virasoro (or, rather, Witt) algebra. The main technical statement of this section is the following \[Virasoro\] The partition function $e^F=e^{F(s,u,v,p_1,p_2,\dots)}$ satisfies the infinite system of non-linear differential equations (Virasoro constraints) $$\begin{aligned} \label{cons} L_ne^F=0\;.\end{aligned}$$ The equations  determine the partition function $e^F$ uniquely. The Virasoro constraints  can be re-written as follows: $$\begin{aligned} \frac{n+1}{s}\frac{\partial F}{\partial p_{n+1}}&= \sum_{j=1}^\infty p_j(n+j)\frac{\partial F}{\partial p_{n+j}}+(u+v)n\frac{\partial F}{\partial p_{n}}\nonumber\\ &+\sum_{i+j=n}ij\left(\frac{\partial^2 F}{\partial p_{i} \partial p_{j}}+ \frac{\partial F}{\partial p_{i}} \frac{\partial F}{\partial p_{j}}\right) + \delta_{0,n}uv\;.\label{vc}\end{aligned}$$ Eq. (\[vc\]) for $n+1=\mu_1$ can be further re-written as a recursion relation for the coefficients $N_{k,l}(\mu)$ of $F$: $$\begin{aligned} \mu_1\,N_{k,l}(\mu_1,&\ldots,\mu_m) =\sum_{j=2}^m (\mu_1+\mu_j-1)N_{k,l}(\mu_1+\mu_j-1,\mu_2,\ldots,\widehat{\mu}_j,\ldots,\mu_m)\nonumber\\ &{}+(\mu_1-1)(N_{k-1,l}(\mu_1-1,\mu_2,\ldots,\mu_m)+N_{k,l-1}(\mu_1-1,\mu_2,\ldots,\mu_m))\nonumber\\ &{}+\sum_{i+j=\mu_1-1}ij \bigg(N_{k,l}(i,j,\mu_2,\ldots,\mu_m)\nonumber\\ &\qquad{}+\mathop{\sum_{k_1+k_2=k}}_{l_1+l_2=l}\quad\sum_{I\sqcup J=\{2,\ldots,m\}} N_{k_1,l_1}(i,\mu_I)N_{k_2,l_2}(j,\mu_J)\bigg)\;,\label{vt}\end{aligned}$$ where $\mu_I=\mu_{i_1},\ldots,\mu_{i_k},\;I=\{i_1,\ldots,i_k\}$, and the hat means that the corresponding term is omitted.[^5] This recursion is valid for $\sum_{i=1}^m\mu_i>1$ and expresses the numbers $N_{k,l}(\mu)$ recursively in terms of $N_{1,1}(1)=1$. We prove this recursion similar to [@WL] (cf. also [@DMSS], [@EO2], [@N]) by establishing a direct bijection between dessins counted in the left and right hand sides of . Here it is more convenient to deal with the dual graphs instead. Let ${\Gamma}^*$ be the ribbon graph dual to a dessin ${\Gamma}$ of type $(k,l,\mu)$. There are $2\mu_1$ ways to pick a half-edge incident to the first vertex of ${\Gamma}^*$. Following [@DMSS] we label this half-edge with an arrow (labeling of half-edges allow us to forget about nontrivial automorphisms). When ${\Gamma}$ varies over the set ${\mathcal{D}}_{k,l,\mu}$, this gives twice the number in the l.h.s. of . Let us now express the same number in terms of dessins with one edge less. This can be done by contracting (or expanding) the labeled edges in the dual graphs in a way that preserves the proper coloring of faces. The following possibilities can occur: (i) The labeled edge connects the first vertex with the $j$-th vertex, $j\neq 1$. Contracting this edge we get a ribbon graph with properly bicolored faces of type $(k,l,\mu_1+\mu_j-1,\mu_2,\ldots,\widehat{\mu}_j,\ldots,\mu_m)$, see Fig. \[contract\]. Conversely, given a graph of type $(k,l,\mu_1+\mu_j-1,\mu_2,\ldots,\widehat{\mu}_j,\ldots,\mu_m)$, there are $2(\mu_1+\mu_j-1)$ ways to split its first vertex into two ones of degrees $2\mu_1$ and $2\mu_j$. Since $j$ can vary from 2 to $m$, this gives twice the first sum in the r.h.s. of  . ![Contracting an edge with different endpoints.[]{data-label="contract"}](contract){width="9cm"} (ii) The labeled edge forms a loop that bounds a white 1-gon, see Fig. \[loop\]. Contracting such a loop we reduce both $k$ and $\mu_1$ by 1, leaving $l$ and $\mu_j,\; j=2,\ldots,m,$ unchanged. Conversely, if we have a graph of type $(k-1,l,\mu_1-1,\mu_2,\ldots,\mu_m$, we can insert a loop into any of the $\mu_1-1$ gray sectors at the first vertex in order to get a graph of type $(k,l,\mu_1,\ldots,\mu_m)$, and 2 ways to label one of its half-edges. The case of a loop bounding a gray 1-gon can be treated verbatim, giving twice the second term in the r.h.s. of . ![Contracting a loop that bounds a 1-gon.[]{data-label="loop"}](loop){width="8cm"} (iii) The labeled edge forms a loop whose half-edges are not adjacent relative to the cyclic order of half-edges at the first vertex. Contracting such a loop we split the first vertex into two ones, say, of degrees $2i$ and $2j$, where $i+j=\mu_1-1$, see Fig. \[split\]. Under this operation the graph may remain connected, or may split into two connected components. In the former case we get a graph of type $(k,l,i,j,\mu_2,\ldots,\mu_m)$. Reversing this operation, we join the first two vertices and add a loop. We can place the labeled half-edge of the loop in any of the $2i$ sectors at the first vertex, but its other half-edge can be placed only in one of $j$ sectors of different color at the second vertex (otherwise it will not be compatible with the face coloring). This gives us twice the third term in the r.h.s. of . The latter case when the graph becomes disconnected can be treated similarly. ![Contracting a loop that splits the vertex into two ones of degrees $2i$ and $2j$ with $i+j=\mu_1-1$.[]{data-label="split"}](split){width="8cm"} The operations (i)–(iii) are reversible and compatible with the face coloring, thus establishing a required bijection. This proves the Virasoro constraints . It is also not hard to see that the Virasoro constraints determine the partition function $e^F$ uniquely, since they are equivalent to the recursion . \[evo\] Put $$\begin{aligned} \Lambda_1&=\sum_{i=2}^\infty (i-1)p_i\,\frac{\partial}{\partial p_{i-1}}\;,\nonumber\\ M_1&=\sum_{i=2}^\infty \sum_{j=1}^{i-1} \left((i-1)p_j p_{i-j}\,\frac{\partial}{\partial p_{i-1}} + j(i-j) p_{i+1}\,\frac{\partial^2}{\partial p_j \partial p_{i-j}}\right)\;.\end{aligned}$$ Then the partition function $e^F$ satisfies the evolution equation $$\begin{aligned} \frac{\partial e^F}{\partial s}=((u+v)\Lambda_1+M_1+uvp_1)e^F\;,\label{eveq}\end{aligned}$$ and is uniquely determined by the initial condition $F\left|_{s=0}\right.=0$. In other words, $e^F$ is explicitly given by the formula $$\begin{aligned} e^F=e^{s((u+v)\Lambda_1+M_1+uvp_1)}\,1\end{aligned}$$ were “1” stands for the constant function identically equal to 1. Multiply the both sides of  by $p_{n+1}$ and sum over $n$. We get $$\begin{aligned} \sum_{n=0}^\infty p_{n+1}L_n &=\sum_{n=0}^\infty p_{n+1}\left(-\frac{n+1}{s}\frac{\partial}{\partial p_{n+1}}+(u+v)n\frac{\partial}{\partial p_{n}}\right.\nonumber\\ &\hspace{0.4in}+\sum_{j=1}^\infty p_j(n+j)\frac{\partial}{\partial p_{n+j}} +\sum_{i+j=n}\left. ij\frac{\partial^2}{\partial p_{i} \partial p_{j}}\right)+uvp_1\nonumber\\ &=-\frac{\partial}{\partial s}+(u+v)\Lambda_1+M_1+uvp_1\;,\end{aligned}$$ and the required statement immediately follows from . A different proof of Corollary \[evo\] has recently appeared in [@Z]. \[ps\] Denote by $\psi$ the principal specialization of the partition function $e^F$: $$\begin{aligned} \psi=\psi(s,t,u,v)=e^{F(s,u,v,p_1,p_2,\ldots)}\left|_{p_i=t^i}\right.\;,\end{aligned}$$ where $t$ is a new formal variable. It is not hard to check that $$\begin{aligned} \Lambda_1 e^F\left|_{p_i=t^i}\right.=t\left(t\frac{d}{dt}\right)\psi\,, \qquad M_1 e^F\left|_{p_i=t^i}\right.=t\left(t\frac{d}{dt}\right)^2\psi\,.\end{aligned}$$ Then, with the help of the obvious identity $s\frac{\D\psi}{\D s}=t\frac{\D\psi}{\D t}$, the evolution equation translates into the following equation for the wave function $\psi$: $$\begin{aligned} \frac{1}{st}\left(t\frac{d}{dt}\right)\psi=(u+v)\left(t\frac{d}{dt}\right)\psi+\left(t\frac{d}{dt}\right)^2\psi+uv\psi\;.\end{aligned}$$ It can be further rewritten as the Schrödinger equation $$\begin{aligned} \label{qc} t^2\,\frac{d^2\psi}{dt^2}+\left((u+v+1)\,t-\frac{1}{s}\right)\,\frac{d\psi}{dt}+uv\psi=0\;\end{aligned}$$ Eq.  is often referred to as the [*quantum curve equation*]{} in the literature on topological recursions. Note that the coefficients of $\log\psi$ enumerate dessins with given numbers of white and black vertices, and a given number of edges regardless of genus (or the number of boundary components). Another observation is that the generating function $F=F(s,u,v,p_1,p_2,\dots)$ satisfies an infinite system of non-linear partial differential equations called the KP (Kadomtsev-Petviashvili) hierarchy (this means that the numbers $N_{k,l}(\mu)$ additionally obey an infinite system of recursions). The KP hierarchy is one of the best studied completely integrable systems in mathematical physics. Below are the first few equations of the hierarchy: $$\label{KP} \begin{aligned} &F_{22}=-\frac12\,F_{11}^2+F_{31}-\frac1{12}\,F_{1111}\;,\\ &F_{32}=-F_{11}F_{21}+F_{41}-\frac16F_{2111}\;,\\ &F_{42}=-\frac12\,F_{21}^2-F_{11}F_{31}+F_{51}+\frac18\,F_{111}^2 +\frac1{12}\,F_{11}F_{1111}-\frac14\,F_{3111}+\frac1{120}\,F_{111111}\;,\\ &F_{33}=\frac13\,F_{11}^3-F_{21}^2-F_{11}F_{31}+F_{51} +\frac14\,F_{111}^2+\frac13\,F_{11}F_{1111}-\frac13\,F_{3111}\\ &\hspace{3.5in}+\frac1{45}\,F_{111111}\;, \end{aligned}$$ where the subscript $i$ stands for the partial derivative with respect to $p_i$. The exponential $Z=e^F$ of any solution is called a [*tau function*]{} of the hierarchy. The space of solutions (or the space of tau functions) has a nice geometric interpretation as an infinite-dimensional Grassmannian (called the [*Sato Grassmannian*]{}), see [@MJD] or [@K] for details. In particular, the space of solutions is homogeneous: there is a Lie algebra $\widehat{\mathfrak{gl}(\infty)}$ that acts infinitesimally on the space of solutions, and the action of the corresponding Lie group is transitive. \[tau\] The generating function $F=F(s,u,v,p_1,p_2,\dots)$ satisfies the infinite system of KP equations  with respect to $p_1,p_2,\dots$ for any parameters $s,u,v$. Equivalently, the partition function $Z=e^F$ is a 3-parameter family of KP tau functions. To begin with, we notice that $1$ is obviously a KP tau function. Then, since $p_1, \Lambda_1, M_1\in\widehat{\mathfrak{gl}(\infty)}$ (cf. [@K]), the linear combination $s((u+v)\Lambda_1+M_1 +uvp_1)$ also belongs to $\widehat{\mathfrak{gl}(\infty)}$ for any $s,u,v$. The exponential $e^{s((u+v)\Lambda_1+M_1+uvp_1)}$ therefore preserves the Sato Grassmannian and maps KP tau functions to KP tau functions. Thus, $e^F$ is a KP tau function as well, and $F$ is a solution to KP hierarchy. Corollary \[tau\] was earlier proven in [@GJ2] by a different method. However, [@GJ2] contains no analogs of the Virasoro constraints or the evolution equation. At the end of this subsection we will sketch how to enumerate dessins with $k$ white vertices, $l$ black vertices, $d$ edges and $m$ boundary components regardless of the partition $\mu=(\mu_1,\ldots,\mu_m)$. To these ends, consider the specialization operator $$\begin{aligned} \theta: F\mapsto F|_{p_i=t, i=1,2,\ldots}\label{sp}\end{aligned}$$ and put $f=\theta(F)$. This specialization is more subtle than the one considered in Remark \[ps\], and the coefficients of $f$ do not mix dessins of different genera since by Euler’s formula $(k+l)-d+m=2-2g$. Expanding $f$ into a series in the variables $s$ and $t$, we recompose it as $$\begin{aligned} f(s,t,u,v)=\sum_{g=0}^\infty\sum_{d=2g+1}^\infty \frac{f_{g,d}(t,u,v)}{d}\,s^d\;,\end{aligned}$$ where each coefficient $f_{g,d}(t,u,v)$ is a homogeneous polynomial in $t,u,v$ of degree $d+2-2g$ with integer coefficients. Furthermore, using the Virasoro constraints (\[V\]) with $n=0,1,2$ and the homgeneity equation $$s\frac{\partial F}{\partial s}=\sum_{i=1}^\infty ip_i\frac{\partial F}{\partial p_i}\;,$$ we can express the specializations of partial derivatives of $F$ with respect to the variables $p_1,p_2,p_3$ in terms of $s$-derivatives of $f$. More precisely, a straightforward computation yields We have $$\begin{aligned} &\theta(F_{1})=s^2f'+suv, \\ &\theta(F_{11})=s^2(s^2f'+suv)', \\ &\theta(F_{1111})=s^2(s^2(s^2(s^2f'+suv)')')',\\ &2\theta(F_{2})=(1+s(u+v-t))(s^2f'+suv)-suv, \\ &3\theta(F_{3})=2s(u+v-t)\theta(F_{2})+(1-st)\theta(F_{1})+s\theta(F_{11})+s\theta(F_{1})^2-suv, \\ &\theta(F_{12})=s^2\theta(F_{2})', \\ &\theta(F_{13})=s^2\theta(F_{3})', \\ &2\theta(F_{22})=(1+s(u+v-t))\theta(F_{12})+3s\theta(F_{3})-2s\theta(F_{2}),\end{aligned}$$ where the subscript $i$ stands for the partial derivative with respect to $p_i$, and the prime $'$ denotes the derivative in $s$. Applying the specialization operator $\theta$ to the first KP equation $$\theta(F_{22})=-\frac12\,\theta(F_{11})^2+\theta(F_{31})-\frac1{12}\,\theta(F_{1111})\,,$$ cf. (\[KP\]), and using the above formulas, we get an ordinary differential equation for $f$ as a function of $s$ that translates into the following quadratic recursion: $$\begin{aligned} (d+1)f_{g,d}&=(2d-1)\,a\,f_{g,d-1}+(d-2)\,b\,f_{g,d-2}+(d-1)^2(d-2)f_{g-1,d-2}\nonumber\\ &+\sum_{i=0}^g\sum_{j=1}^{d-3}(4+6j)(d-2-j)f_{i,j}f_{g-i,d-2-j}\;,\label{ad}\end{aligned}$$ where $a=t+u+v,\; b=4(tu+tv+uv)-a^2.$ Starting with $f_{0,1}=tuv$, one can recursively compute the polynomials $f_{g,d}$ for all $g$ and $d$. Finally, let us restrict ourselves to the case of dessins with one boundary component (or bicolored polygon gluings). Denote by $h_{g,d}$ the linear term in $f_{g,d}$ with respect to $t$. Then the recursion (\[ad\]) takes the form $$\begin{aligned} (d+1)h_{g,d}&=(2d-1)(u+v)h_{g,d-1}-(d-2)(u-v)^2 h_{g,d-2}\\ &+(d-1)^2(d-2)h_{g-1,d-2}\;,\end{aligned}$$ and we reproduce the well-known result of [@A] on the enumeration of genus $g$ gluings of a bicolored $2d$-gon with given numbers of white and black vertices (cf. also [@J]). Virasoro constraints for the numbers of ribbon graphs ----------------------------------------------------- A closely related, but somewhat different enumerative problem was considered in [@WL]. Recall that a dessin d’enfant $f^{-1}([0,1])$ is a bicolored connected ribbon graph with vertices “colored" by either 0 or 1 depending on whether $f$ maps the vertex to 0 or 1 in ${{\mathbb{C}}}P^1$. One can similarly try to enumerate all (not necessarily bicolored) connected ribbon graphs, and this is the problem that was addressed in [@WL]. To make it precise, let us label the boundary components of a ribbon graph ${\Gamma}$ (or, equivalently, the vertices of the dual graph ${\Gamma}^*$) by integers from 1 to $m$, and let $\mu_1,\ldots,\mu_m$ be the lengths of the boundary components of ${\Gamma}$ (or the degrees of vertices of ${\Gamma}^*$). Ribbon graphs can naturally be represented by dessins of a special type called [*clean dessins*]{} in [@DMSS]. Namely, color each vertex of a ribbon graph in white and place black vertices at the midpoints of edges. Such a dessin corresponds to a covering of ${\mathbb{C}P^1}$ of even degree $d$ with ramification of type $[2^{d/2}]$ over 1 and arbitrary ramification over 0 and $\infty$. As before, we put $k=|f^{-1}(0)|$ (the number of vertices of the ribbon graph ${\Gamma}$), $l=|f^{-1}(1)|=d/2$ (the number of edges of ${\Gamma}$), and $m=|f^{-1}(\infty)|$ (the number of boundary components of ${\Gamma}$). Clearly, we have $k-d/2+m=2-2g$. Denote by $D_{g,m}(\mu)=D_{g,m}(\mu_1,\ldots,\mu_m)$ the number of genus $g$ ribbon graphs with $m$ labeled vertices of degrees $\mu_1,\ldots,\mu_m$ counted with weights $\frac{1}{|{\rm Aut}_v\,{\Gamma}|}$, where the automorphisms preserve each vertex of ${\Gamma}$ pointwise. Apparently, the same numbers enumerate pure dessins with $m$ labeled boundary components of lengths $(2\mu_1,\ldots,2\mu_m)$. The following recursion for $D_{g,m}(\mu)$ was derived in [@WL], Eq. (6):[^6] $$\begin{aligned} \mu_1 D_{g,m}(\mu_1,\ldots,&\mu_m) =\sum_{j=2}^m (\mu_1+\mu_j-2)D_{g,m-1}(\mu_1+\mu_j-2,\mu_2,\ldots,\widehat{\mu}_j,\ldots,\mu_m)\nonumber\\ &{}+2(\mu_1-2)D_{g,m}(\mu_1-2,\mu_2,\ldots,\mu_m)\nonumber\\ &{}+\sum_{i+j=\mu_1-2}ij \bigg(D_{g-1,m+1}(i,j,\mu_2,\ldots,\mu_m)\nonumber\\ &{}+\sum_{g_1+g_2=g}\quad\sum_{I\sqcup J=\{2,\ldots,m\}} D_{g_1,|I|+1}(i,\mu_I)D_{g_2,|J|+1}(j,\mu_J)\bigg)\;.\label{wl}\end{aligned}$$ Recursion  is valid for all $g\geq 0,\;m\geq 1$, and $\mu$ such that $d=\sum_{i=1}^m\mu_i>2$, whereas for $d=2$ the only nonzero numbers are $D_{0,1}(2)=1/2,\;D_{0,2}(1,1)=1$. Below we give a convenient interpretation of this recursion in terms of PDEs. Similar to (\[gf\]), introduce the generating function $$\begin{aligned} \label{gfM} \tF(s,u,p_1,p_2,\ldots) &=\sum_{g=0}^\infty\sum_{m=1}^\infty\frac{1}{m!} \sum_{\mu\in{{\mathbb{Z}}}_+^m} D_{g,m}(\mu)s^d u^k p_{\mu_1}\ldots p_{\mu_m}\;,\end{aligned}$$ where $d=\sum_{i=1}^m\mu_i$, $k=2-2g-m+d/2$, and $\mu=(\mu_1,\ldots,\mu_m)$ (compared to (\[gf\]), we omit here the trivial factor $v^{d/2}$ that carries no additional information in this case). \[rg\] The generating function $\tF$ enjoys the following integrability properties: (i) Let $$\begin{aligned} \tL_n=-\frac{n+2}{s^2}\frac{\partial}{\partial p_{n+2}} +2\,u\,n\frac{\partial}{\partial p_{n}} +\sum_{j=1}^\infty p_j(n+j)\frac{\partial}{\partial p_{n+j}}\\ {}+\sum_{i+j=n}ij\frac{\partial^2}{\partial p_i \partial p_j}+\delta_{-1,n}u p_1+\delta_{0,n}u^2\;,\end{aligned}$$ where $n\geq -1$. Then the partition function $e^\tF$ satisfies the infinite system of PDE’s (“Virasoro constraints”) $$\begin{aligned} \tL_n e^\tF=0\end{aligned}$$ that determine $\tF$ uniquely. (ii) Put $$\begin{aligned} &\Lambda_2=\sum_{i=3}^\infty (i-2)p_i\,\frac{\partial}{\partial p_{i-2}}+\frac12p_1^2\;,\nonumber\\ &M_2=\sum_{i=2}^\infty \sum_{j=1}^{i-1} \left((i-2)p_j p_{i-j}\,\frac{\partial}{\partial p_{i-2}} + j(i-j) p_{i+2}\,\frac{\partial^2}{\partial p_j \partial p_{i-j}}\right)\;.\end{aligned}$$ Then $e^\tF$ satisfies the evolution equation $$\begin{aligned} \frac{1}{s}\frac{\partial e^\tF}{\partial s}=(2\,u\,\Lambda_2+M_2+u^2p_2)e^\tF\,,\end{aligned}$$ that, together with the initial condition $\tF|_{s=0}=0$, determines $\tF$ uniquely. In other words, $e^\tF$ is explicitly given by the formula $$\begin{aligned} e^\tF=e^{\frac{s^2}{2}(2\,u\,\Lambda_2+M_2+u^2p_2)}\,1\;.\end{aligned}$$ (iii) The partition function $e^\tF$ is a tau function of the KP hierarchy, i.e. its logarithm $\tF(s,u,p_1,p_2,\ldots)$ satisfies  for any $s$ and $u$. Part (i) of the theorem is just a reformulation of the recursions (\[wl\]) for $\mu_1=n+2$. Note that the operators $\tL_n$ obey the commutation relations $[\tL_m,\tL_n]=(m-n)\tL_{m+n}$ for $m>n\geq -1$. To prove (ii) we multiply $\tL_n$ by $p_{n+2}$ and sum over $n$: $$\begin{aligned} \sum_{n=-1}^\infty p_{n+2}\tL_n &=\sum_{n=-1}^\infty p_{n+2}\bigg(-\frac{n+2}{s}\frac{\partial}{\partial p_{n+2}}+2\,u\,n\frac{\partial}{\partial p_{n}}\nonumber\\ &\hspace{0.6in} {}+\sum_{j=1}^\infty p_j(n+j)\frac{\partial}{\partial p_{n+j}} +\sum_{i+j=n}ij\frac{\partial^2}{\partial p_{i} \partial p_{j}}\bigg)+u\,p_1^2+u^2p_2\nonumber\\ &=-\frac{1}{s}\frac{\partial}{\partial s}+2\,u\,\Lambda_2+M_2+u^2 p_2\;.\label{evom}\end{aligned}$$ Part (iii) follows from the fact that $\Lambda_2, M_2$ and $p_2$ belong to $\widehat{\mathfrak{gl}(\infty)}$, cf. Corollary \[tau\]. Similar to Eq.  we can write the quantum curve equation for the wave function $$\widetilde{\psi}=\widetilde{\psi}(s,t,u)=e^{\tF(s,u,p_1,p_2,\ldots)}\left|_{p_i=t^i}\right.\;,$$ that reads in this case as follows: $$t^2\,\frac{d^2\widetilde{\psi}}{dt^2}+\left(2(u+1)\,t-\frac{1}{s^2\,t}\right)\,\frac{d\widetilde{\psi}}{dt}+(u+u^2)\,\widetilde{\psi}=0\;.$$ Note that this equation differs from the one obtained in [@MS]. The reason for that is explained in Footnote \[M\] above. To fix that, put $Z(x,\hbar)=e^{-\frac{\log x}{\hbar}}\,\widetilde{\psi}|_{s=\hbar/x,t=1,u=1/\hbar}$. Then one has $$\left(\hbar^2\frac{d^2}{dx^2}+\hbar\frac{d}{dx}+1\right)Z(x,\hbar)=0$$ precisely like in [@MS]. To complete this section, we will show that the Walsh-Lehman formula (\[wl\]) implies the Harer-Zagier [@HZ] recursion for the numbers of orientable polygon gluings. We will follow the same lines as at the end of the previous subsection. To begin with, put $\tf=\theta(\tF)=\tF|_{p_i=t, i=1,2,\ldots}$, where the specialization operator $\theta$ is defined by Eq. (\[sp\]). The coefficients of $\tf$ enumerate ribbon graphs with given numbers of vertices, edges and boundary components and, therefore, do not mix graphs of different genera. Rearrange the series $f$ as follows: $$\begin{aligned} \tf(s,t,u)=\sum_{g=0}^\infty\sum_{l=2g}^\infty \frac{\tf_{g,l}(t,u)}{2l}\,s^{2l}\;,\end{aligned}$$ where $l=d/2$, and each coefficient $\tf_{g,l}(t,u)$ is a homogeneous polynomial in $t,u$ of degree $l+2-2g$ with integer coefficients. Like in the case of dessins, using the Virasoro constraints of Theorem \[rg\] (i) with $n=-1,0,1$, we can express the specializations of partial derivatives of $\tF$ with respect to the variables $p_1,p_2,p_3$ in terms of $s$-derivatives of $\tf$. A straightforward computation yields We have $$\begin{aligned} &\theta(\tF_{1})=s^3\tf'+s^2tu, \\ &\theta(\tF_{11})=s^3\theta(\tF_{1})'-s^2\theta(\tF_{1})+s^2u, \\ &\theta(\tF_{111})=s^3\theta(\tF_{11})'-2s^2\theta(\tF_{11}),\\ &\theta(\tF_{1111})=s^3\theta(\tF_{111})'-3s^2\theta(\tF_{111}),\\ &2\theta(\tF_{2})=s^3\tf'+s^2u^2, \\ &4\theta(\tF_{22})=s^6\tf''+3s^5\tf'+2s^4u^2, \\ &\theta(\tF_{13})=s^5((2s^2u+1-s^2t)(s\tf')'+(2s^2u+2-s^2t)\tf')+s^6(2tu^2-t^2u)+3s^4u^2, \end{aligned}$$ where, as before, the subscript $i$ stands for the partial derivative with respect to $p_i$, and the prime $'$ denotes the derivative in $s$. Applying the specialization operator $\theta$ to the first KP equation (\[KP\]) and using the above formulas, we get an ordinary differential equation for $\tf$ as a function of $s$ that translates into the following quadratic recursion: $$\begin{aligned} (l+1)\tf_{g,l}&=2(2l-1)(t+u)\tf_{g,l-1}+(2l-1)(2l-3)(l-1)\tf_{g-1,l-2}\nonumber\\ &+3\sum_{i=0}^g\sum_{j=0}^{l-2}(2j+1)(2(l-2-j)+1)\tf_{i,j}\tf_{g-i,l-2-j}\;,\label{hz}\end{aligned}$$ where we put by definition $\tf_{0,0}=u$. This is essentially the formula from [@CC], but derived in a more straightforward way. Note that starting with $\tf_{0,1}=t^2u+tu^2$, one can recursively compute the polynomials $\tf_{g,l}$ for all $g$ and $l$. To enumerate genus $g$ ribbon graphs with one boundary component (or $2l$-gon gluings), it is sufficient to consider the linear terms $\epsilon_{g,l}$ in $\tf_{g,l}$ with respect to $t$. Then Eq. (\[hz\]) turns into the famous Harer-Zagier recursion $$\begin{aligned} (l+1)\epsilon_{g,l}=2(2l-1)u\epsilon_{g,l-1}+(2l-1)(2l-3)(l-1)\epsilon_{g-1,l-2}\;,\end{aligned}$$ cf. [@HZ]. Topological recursion ===================== The generating function $F=\sum_{g,m}F_{g,m}$ of enumerating Belyi pairs $(C,f)$ (or Grothendieck’s dessins) can be naturally decomposed into components with fixed $g$ and $m$, where $g$ is the genus of $C$ and $m$ is the number of poles of $f$: $$\begin{aligned} F_{g,m}(s,u,v,p_1,p_2,\dots) = \frac{1}{m!}\sum_{\mu_1,\dots,\mu_m}\sum_{k+l=d-m+2-2g} N_{k,l}(\mu) s^{d} u^k v^l\, p_{\mu_1}\ldots p_{\mu_m}\label{gm} \end{aligned}$$ (here, as usual, $d=\sum\mu_i$). Another way to collect these numbers into a generating series is to use the *$m$-point correlation functions* $$W_{g,m}(x_1,\dots,x_m) =\sum_{\mu_1,\dots,\mu_m}\sum_{k+l=d-m+2-2g} N_{k,l}(\mu)\,x_1^{\mu_1}\dots x_m^{\mu_m}.$$ *Topological recursion* (cf. [@EO2]) is an “ansatz” that allows to reconstruct the coefficients of certain generating series recursively in $g$ and $m$. Traditionally, it is formulated in terms of correlation functions or, rather, *differentials* $$\begin{aligned} w_{g,m}(x_1,\dots,x_m)&=\frac{\D^mW_{g,m}}{\D x_1\ldots \D x_m} \,dx_1\ldots dx_m\\ &=\sum_{\mu}\sum_{\substack{k,\,l\\k+l=d-m+2-2g}} N_{k,l}(\mu)\prod_{i=1}^m \mu_i x_i^{\mu_i-1} dx_i.\end{aligned}$$ We will present the topological recursion in terms of components $F_{g,m}$ of the generating function $F$. The advantage of this approach is that we need only one set of variables $p_i$ for all $g$ and $n$. The two approaches being equivalent, it proves out, however, that many properties of the recursion become more clear in terms of $p$-variables. One of the nice features of topological recursion is that the generating functions $F_{g,m}$ become polynomilas under a linear change of variables $p_i$. The components $F_{g,m}$ of the total generating function $F$ are infinite formal series in $p_i$, and their polynomiality is far from being an immediate consequence of the Virasoro constraints for $F$. On the other hand, this polynomiality automatically follows from the equations of topological recursion. Another advantage of topological recursion is its universality. For a variety of enumerative problems it takes the same form, differing only in initial conditions. Introduce the formal variables $x$ and $z$ related by $$z(x)=\sqrt{\frac{1-\beta s x}{1-\alpha s x}},\qquad x(z)=\frac{z^2-1}{s \left(\alpha z^2-\beta \right)}, \label{eqzx}$$ where $$\a=(\sqrt{u}-\sqrt{v})^2,\qquad \b=(\sqrt{u}+\sqrt{v})^2. \label{eqabs}$$ We consider  as a formal change of coordinates on the complex projective line ${\mathbb{C}P^1}$ near the point $x=0$ (resp., $z=1$) depending on the parameters $\a,\b,s$ (or $u,v,s$). Put $$T_j(p)=\sum_{i=1}^\infty c_j^{(i)}p_i\,, \label{eqT}$$ where the coefficients $c_j^{(i)}$ are defined by the relation $$\sum_{i=1}^\infty c_j^{(i)}x^i=(z(x))^{j}-1\,.$$ \[th2\] Let $F_{g,m}$ be the infinite series defined by . Then (i) For each $g,m$ with $2g-2+m>0$ there exists a polynomial $G_{g,m}$ of the variables $t_j$, $j\in{\mathbb{Z}}_\odd$, such that $$F_{g,m}(p)=G_{g,m}(t)\bigm|_{t_j=T_j(p)}$$ (i.e. each $F_{g,m}$ is a polynomial in the linear functions $T_{\pm1},T_{\pm3},T_{\pm5},\dots$). (ii) The polynomials $G_{g,m}$ can be recursively computed starting from $G_{0,3}$ and $G_{1,1}$, cf. Eqs. – below. Let us now formulate the recursion for the polynomials $G_{g,m}$ precisely. This can be done in terms of the so-called *spectral curve*. In our case the spectral curve is the projective line ${\mathbb{C}P^1}$ equipped with the globally defined holomorphic involution $z\mapsto -z$ with respect to some affine coordinate $z$. By a [*Laurent form*]{} we understand here a globally defined meromorphic 1-form on ${\mathbb{C}P^1}$ with poles only at $0$ and $\infty$. Denote by $L$ the space of odd Laurent forms relative to the involution $z\mapsto -z$. The forms $d(z^j)=j\,z^{j-1}\,dz$, $j\in{\mathbb{Z}}_\odd$, provide a convenient basis in $L$. Let $P_L$ denote the projector to the space $L$ in the space of all Laurent forms. For a Laurent form $\phi$ its projection $\psi=P_L(\phi)$ to $L$ is uniquely determined by the requirement that the form $\psi-\phi_\odd$ is regular at both $0$ and $\infty$, where $\phi_\odd(z)=\frac12(\phi(z)-\phi(-z))$ is the odd part of $\phi$. More explicitly, the action of $P_L$ is given by the formula $$\begin{aligned} (P_L\phi)(z)&=\sum_{i=0}^\infty\res_{w=0}(\phi(w)w^{2i+1})\;z^{-2i-2}dz- \sum_{i=0}^\infty\res_{w=\infty}(\phi(w)w^{-2i-1})\;z^{2i}dz\nonumber\\ &=\res_{w=0}\left(\phi(w)\frac{w\;dz}{z^2-w^2}\right)+\res_{w=\infty}\left(\phi(w)\frac{w\;dz}{z^2-w^2}\right)\;. \label{res}\end{aligned}$$ Note that for the validity of this definition it will suffice to assume that $\phi$ is defined in a neighborhood of the points $0$ and $\infty$, or even that $\phi$ is a formal Laurent series at these points. On the other hand, the form $P_L\phi\in L$ is always globally defined on ${\mathbb{C}P^1}$. In fact, the recursion applies not to the polynomials $G_{g,m}$ themselves, but to certain $1$-forms $U_{g,m}$. For the set of variables $z$ and $t=(t_{\pm 1},t_{\pm 3},\ldots)$ introduce the differential operator $$\d_{z,t}=\sum_{j\in{\mathbb{Z}}_\odd}jz^{j-1}\pd{}{t_j}\,.$$ For $2g-2+m>0$ put $$\begin{aligned} U_{g,m}(z)&=\d_{z,t} G_{g,m}(z)=\sum_{j\in{\mathbb{Z}}_\odd}jz^{j-1}\pd{G_{g,m}}{t_j}\,.\end{aligned}$$ As we will see later, $U_{g,m}=U_{g,m}(z)dz$ is an odd Laurent form on ${\mathbb{C}P^1}$ that is polynomial in $t_j$. \[remdelta\] The operator $\d_{z,t}$ written in terms of $x$ and $p$-variables becomes $$\d_{x,p}=\frac{dz}{dx}\sum_{i=1}^\infty ix^{i-1}\,\pd{}{p_i}\,,$$ where $x$ is related to $z$ by (\[eqzx\]) and $t_j=T_j(p)$, see . (For brevity we will omit the subscripts ‘$p$’ and ‘$t$’ by $\d$, always associating $p$- and $t$-variables with $x$ and $z$ respectively.) More precisely, assume that a function $f$ of $p$-variables can be expressed as a composition $f=h\circ T$, where $h$ is a function of $t$-variables and $T$ is the linear change . Then we have $\d_xf\,dx=(\d_zh)|_{t_k=T_k(p)}\,dz$. In particular, if $g$ is a polynomial in $t$-variables, then $\d_xfdx$ is a Laurent form in $z$ (with coefficients depending on $p_i$’s). Indeed, the operators on both sides satisfy the Leibnitz rule, and therefore it is sufficient to prove the equality for the case $h=t_k$, that is, $$\d_x T_k(p)dx=k z^{k-1} dz,\qquad z=z(x)\,,$$ which is essentially the definition of the linear functions $T_k(p)$, cf. (\[eqT\]). In the unstable cases (i.e. when $2g-2+m\leq 0$) the definition of $U_{g,m}$ should be modified. Namely, we set $U_{0,1}=0$ and define $U_{0,2}$ by the following formal expansions $$\begin{aligned} U_{0,2}(z)dz&=-\sum_{i=0}^\infty t_{-2i-1}z^{2i}dz=-(t_{-1}+t_{-3}z^2+\dots)\,dz,\quad z\to 0,\\ U_{0,2}(z)dz&=-\sum_{i=0}^\infty t_{2i+1}z^{-2i}d(z^{-1})=(t_1z^{-2}+t_3z^{-4}+\dots)\,dz,\quad z\to\infty,\\ \end{aligned} \label{U02}$$ In general, the homogeneous degree $m$ polynomial $G_{g,m}$ can be recovered form the form $U_{g,m}$ by the Euler formula $$G_{g,m}=\frac{1}{m}\sum t_k\pd{G_{g,m}}{t_k}=\frac1m\,\O(U_{g,m},U_{0,2})\;, \label{eq1}$$ where for odd forms $\phi$ and $\psi$ we set $$\O(\phi,\psi)=-\O(\psi,\phi)=\res_{z=0}\left(\phi\int\psi\right)+\res_{z=\infty}\left(\phi\int\psi\right)\;.$$ The last ingredient needed to write down the topological recursion is the form $$\label{eta} \eta=\eta(z)dz=\frac{\s\,z^2dz}{(z^2-1)^2 (\a\,z^2 -\b)},\qquad \s=(\a-\b)^2=16\,u\,v.$$ This form is odd and has the property that the dual vector field $$\frac{1}{\eta}=\frac{1}{\eta(z)}\,\frac{d}{dz}=\frac{\a\,z^4-(2\,\a+\b)\,z^2+\a +2\,\b -\b\,z^{-2}}{\s}\,\frac{d}{dz}$$ is meromorphic with poles of order $2$ at $z=0$ and $z=\infty$ and regular elsewhere in ${\mathbb{C}P^1}$. The main recursive relation of this paper is $$U_{g,m}=P_L\Biggl(\frac{1}{2\eta}\Biggl(\d U_{g-1,m+1}+\sum_{\substack{g_1+g_2=g,\\m_1+m_2=m+1}}U_{g_1,m_1}U_{g_2,m_2}\Biggr)\Biggr) \label{eq2}$$ (here and below we tacitly assume that $\d f=\d_z f\,dz$). Note that almost all terms in the sum on the right hand side of (\[eq2\]) belong to $L$, so that $P_L$ is identical on these terms. Therefore, (\[eq2\]) can equivalently be rewritten as $$U_{g,m}=\frac{1}{2\eta}\Biggl(\d U_{g-1,m+1}+\mathop{\sum{}^*}_{\substack{g_1+g_2=g,\\m_1+m_2=m+1}} U_{g_1,m_1}U_{g_2,m_2}\Biggr)+P_L\left(\frac1\eta\,U_{g,m-1}U_{0,2}\right)\;,$$ where the star $^*$ by the summation sign means that the terms involving $U_{0,2}$ are excluded (recall that $U_{0,1}=0$ by assumption). This recursion relation is valid for all $g$ and $n$ with $2g-2+m>1$. Moreover, it applies for $(g,m)=(0,3)$ as well: $$U_{0,3}=P_L\left(\frac{U_{0,2}^2}{2\eta}\right). \label{eq3}$$ In the case $(g,m)=(1,1)$ the formula is not applicable since $\d U_{0,2}$ is not defined. This is why we set by definition $$U_{1,1}=U_{1,1}(z)\,dz=P_L\left(\frac{1}{2\eta}\left(\frac{dz}{2z}\right)^2\right)=\frac{1}{2\eta}\left(\frac{dz}{2z}\right)^2 =\frac{1}{8z^2\eta(z)}\,dz\,. \label{eq4}$$ Eqs. – concretize the second statement of Theorem \[th2\]. Below we list the polynomials $U_{g,m}$ and $G_{g,m}$ for small $g$ and $m$: $$\begin{aligned} \s\,U_{0,3}(z)&=\frac{\a}{2}\,t_1^2-\frac{\b}{2}\,t_{-1}^2\,z^{-2},\\[-12pt]\\ \s\,G_{0,3}&=\frac{\a}{6}\,t_1^3+\frac{\b}{6}\,t_{-1}^3,\\[-12pt]\\ \s\,U_{1,1}(z)&=\frac{\a}{8}\,z^2 -\frac{2\a+\b}{8} +\frac{\a+2\b}{8}\,z^{-2}-\frac{\b}{8}\,z^{-4},\\[-12pt]\\ \s\,G_{1,1}&=\frac{\a}{24}\,t_3-\frac{2\a+\b}{8}\,t_1-\frac{\a+2\b}{8}\,t_{-1}+\frac{\b}{24}\,t_{-3},\\[-12pt]\\ \s^2\,U_{0,4}(z)&=\frac{\a^2}{2}t_1^3\,z^2 +\Bigl(\frac{\a^2}{2}\,t_3\, t_1^2-\frac{\a(2\a+\b)}{2}\,t_1^3-\frac{\a\b}{2}\,t_{-1}^2\, t_1\Bigr)\\ &+\Bigl(\frac{\b(\a+2\b)}{2}\,t_{-1}^3+\frac{\a\b}{2}\,t_1^2\, t_{-1}-\frac{\b^2}{2}\,t_{-3}\, t_{-1}^2\Bigr)\,z^{-2} -\frac{\b^2}{2}\,t_{-1}^3\,z^{-4},\\[-12pt]\\ \s^2\,G_{0,4}&=\frac{\a^2}{6}\,t_1^3\, t_3-\frac{\a(2\a+\b)}{8}\,t_1^4-\frac{\a\b}{4}\,t_1^2\, t_{-1}^2 -\frac{\b(\a+2\b)}{8}\,t_{-1}^4+\frac{\b^2}{6}\,t_{-3}\,t_{-1}^3,\\[-12pt]\\ \s^2\,U_{1,2}(z)&=\frac{5\a^2}{8}\,t_1\,z^4+\Bigl(\frac{\a^2}{8}\,t_3-\frac{3\a(2\a+\b)}{4}\,t_1\Bigr)\,z^2\\ &+\Bigl(\frac{10\a^2+16\a\b+\b^2}{8}\,t_1+\frac{\a^2}{8}\,t_5+\frac{\a\b}{2}\,t_{-1}-\frac{\a(2\a+\b)}{4}\,t_3\Bigr)\\ &+\Bigl(-\frac{\a^2+16\a\b+10\b^21}{8}\,t_{-1}+\frac{\b(\a+2\b)}{4}\,t_{-3}-\frac{\a\b}{2}\,t_1-\frac{\b^2}{8}\,t_{-5}\Bigr)\,z^{-2}\\ &+\Bigl(\frac{3\b(\a+2\b)}{4}\,t_{-1}-\frac{\b^2}{8}\,t_{-3}\Bigr)\,z^{-4} -\frac{5\b^2}{8}\,t_{-1}\,z^{-6},\\[-12pt]\\ \s^2\,G_{1,2}&=\frac{\a^2}{8}\,t_1\,t_5+\frac{\a^2}{48}\,t_3^2-\frac{\a(2\a+\b)}{4}\,t_1\,t_3+\frac{\a^2+16\a\b+10\b^2}{16}\,t_{-1}^2\\ &+\frac{\a\b}{2}\,t_{-1}\,t_1+\frac{10\a^2+16\a\b+\b^2}{16}\,t_1^2-\frac{\b(\a+2\b)}{4}\,t_{-3}\,t_{-1}\\ &+\frac{\b^2}{48}\,t_{-3}^2+\frac{\b^2}{8}\,t_{-5}\,t_{-1}\;,\end{aligned}$$ where $\a,\b,\s$ are given by , . This list can be continued further on. Here we compare our form of the topological recursion  with the one that can be found in the literature, see, e.g. [@EO1], [@EO2]. (i) Traditionally, the spectral curve comes with an embedding to (a compactification of) ${\mathbb{C}}^2$ by means of certain meromorphic functions $X,Y$ on ${\mathbb{C}P^1}$. These functions are chosen so that $X$ is even with respect to the involution $z\mapsto -z$, and $\eta=Y\,dX$. In our case we could have set, for example, $$X=\frac1x=s\,\frac{\a z^2-\b}{z^2-1},\qquad Y=-\frac{\a-\b}{2s}\,\frac{z}{\alpha z^2-\beta}.$$ The formulas of the topological recursion, however, involve the coordinates $X$ and $Y$ only in the combination $\eta=Y\,dX$. (ii) The topological recursion is usually formulated in terms of $m$-point correlators. They are related to the homogeneous components $F_{g,m}$ of the generating function $F$ by the formulas $$\begin{aligned} w_{g,m}(x_1,\dots,x_m)&=\d_{x_1}\ldots\d_{x_m} F_{g,m}\,dx_1\ldots dx_m\nonumber\\ &=\d_{x_2}\ldots\d_{x_m} U_{g,m}(x_1)\,dx_1\ldots dx_m\;, \label{eqwgn}\end{aligned}$$ where $\d_{x_j}=\sum_{i=1}^\infty i\,x_j^{i-1}\,\pd{}{p_i}$. Via the change of variables , $w_{g,m}$ can be viewed as a meromorphic $m$-differential on $\left({\mathbb{C}P^1}\right)^m$ that is a Laurent form with respect to each of its arguments provided $2g-2+m>0$. (iii) A version of  holds also for $(g,m)=(0,2)$. Namely, $w_{0,2}(z_1,z_2)=\d_{z_2}U_{0,2}\,dz_2$ is the odd part of the *Bergman kernel* $B(z_1,z_2)=\frac{dz_1\,dz_2}{(z_1-z_2)^2}$: $$w_{0,2}(z_1,z_2)=\d_{z_2}U_{0,2}\,dz_2=\frac12\left(B(z_1,z_2)-B(z_1,-z_2)\right).$$ This can be interpreted as an equality of asymptotic expansions of the left and right hand sides at $z_1\to 0$ and $z_1\to\infty$, cf. : $$w_{0,2}(z_1,z_2)= \begin{cases}-\sum_{i=0}^\infty d(z_2^{-2i-1})\;z_1^{2i}dz_1,&z_1\to 0\;,\medskip\\ -\sum_{i=0}^\infty d(z_2^{2i+1})\;z_1^{-2i}d(z_1^{-1}),& z_1\to\infty\;.\end{cases}$$ (iv) The projector $P_L$, see , is given by the contour integral $$\begin{aligned} (P_L\phi)(z)\,dz=\frac{1}{2\pi\sqrt{-1}}\left(\int_{|w|=\epsilon}\frac{\phi(w)\,w\,dw}{z^2-w^2} +\int_{|w|=1/\epsilon}\frac{\phi(w)\,w\,dw}{z^2-w^2}\right)\,dz\;,\end{aligned}$$ for small $\epsilon > 0$, where $$\frac{w\;dz}{z^2-w^2}=\frac12\int\limits_{-w}^w B(z,\cdot).$$ This explains the appearance of the Bergman kernel in the majority of expositions of the topological recursion. (v) The above items (i)–(iv) demonstrate that the traditional form of the topological recursion in terms of the correlators $w_{g,n}$ is obtainable from  by applying $\d_{z_2}\ldots\d_{z_n}$ to the both sides of it. Proof of the topological recursion ================================== Master Virasoro equation ------------------------ As we will see below, the topological recursion relations are just the equivalently reformulated Virasoro constraints. To begin with, let us collect the Virasoro constraints  into a single equation by multiplying the $n$th equation by $x^n$ (where $x$ is a formal variable) and summing them up: $$\begin{gathered} \sum_{n=0}^\infty x^n\Biggl(-\frac1s(n+1)\pd{F}{p_{n+1}}+(u+v)\,n\pd{F}{p_n}+ \sum_{j=1}^\infty p_j(n+j)\pd{F}{p_{n+j}}\Biggr.\\\Biggl.+ \sum_{i+j=n}ij\left(\pd{^2F}{p_i\D p_j}+\pd{F}{p_i}\pd{F}{p_j}\right)\Biggr)+uv=0.\end{gathered}$$ This equation can be simplified. Notice that $$\begin{aligned} \sum_{n=0}^\infty&\; x^n\left(-\frac1s(n+1)\pd{F}{p_{n+1}}+(u+v)\,n\pd{F}{p_n}\right) =\left(-\frac1s+(u+v)\,x\right)\d_x F\;,\\ \sum_{n=0}^\infty&\; x^n\sum_{i+j=n}ij\left(\pd{^2F}{p_i\D p_j}+\pd{F}{p_i}\pd{F}{p_j}\right)= x^2\left(\d_x^2 F+(\d_x F)^2\right)\;,\end{aligned}$$ where, as in the previous section, $\d_x=\sum_{n=1}^\infty nx^{n-1}\pd{}{p_n}$ (cf. Remark \[remdelta\]). As for the third term in the sum, we use the identity $p_j=\d_y^{-1}(j\,y^{j-1})=\d_y^{-1}d_y(y^j)$, where $y$ is a new independent formal variable and $d_y=\frac{d}{dy}$, to re-write it as follows: $$\begin{aligned} \sum_{n=0}^\infty\; x^n\sum_{j=1}^\infty& p_j(n+j)\pd{F}{p_{n+j}}\\ &=\sum_{n=0}^\infty\;\sum_{i+j=n} x^{i}\,p_j\,n\,\pd{F}{p_n} =\d_y^{-1} d_y\left(\sum_{n=0}^\infty\;\sum_{i+j=n} x^{i}\,y^j\,n\,\pd{F}{p_n}\right)\\ &=\d_y^{-1} d_y\left(\sum_{k=0}^\infty \frac{x^{k+1}-y^{k+1}}{x-y}\, k\,\pd{F}{p_k}\right) =\d_y^{-1} d_y\left(\frac{x^2{\d_x F}-y^2\d_y F}{x-y}\right)\end{aligned}$$ This yields the following *master Virasoro equation* that unifies all Eqs. : $$\begin{aligned} \Bigl(-\frac1s+(u+v)\,x\Bigr)\d_x F&+x^2\left(\d_x^2 F+(\d_x F)^2\right)\nonumber\\ &+\d_y^{-1} d_y\left(\frac{x^2\d_x F-y^2\d_y F}{x-y}\right)+uv=0\;. \label{mve}\end{aligned}$$ Unstable terms and the spectral curve ------------------------------------- Our immediate goal is to extract the homogeneous terms in  contributing to $\d_x F_{g,n}$ for fixed $g$ and $n$. We start with the unstable cases. For $g=0$ and $n=1$ we get $$x^2\,(\d_x F_{0,1})^2+\Bigl(-\frac1{s}+(u+v)\,x\Bigr)\d_x F_{0,1}+u v=0\,.\label{eqspectr}$$ Solving this equation for $x\,\d_x F_{0,1}$, choosing the proper root and expanding it into the Taylor series at $x=0$ we get $$\begin{aligned} x\,\d_x F_{0,1}&=\frac{1}{2} \left(\frac{1}{s x}-u-v-\sqrt{\left(\frac{1}{s x}-u-v\right)^2-4 u v}\right)\\ &=u\,v\,s\,x+u\,v\,(u+v)\,s^2x^2+ u\,v\,\left(u^2+3\,u\,v+v^2\right)\,s^3x^3 +\dots\end{aligned}$$ and $$F_{0,1}=u\,v\,s\,p_1+\frac12u\,v\,(u+v)\,s^2p_2+\frac13u\,v\,\left(u^2+3\,u\,v+v^2\right)\,s^3p_3 +\dots.$$ The *spectral curve* is the Riemann surface of the algebraic function $x\,\d_x F_{0,1}$ in the $x$-variable. In other words, the spectral curve is an algebraic curve such that $x$ and $x\d_x F_{0,1}$ are globally defined univalued meromorphic functions on it. In our case the spectral curve is given by . It is rational (admits a rational parametrization). Let $z$ be an affine coordinate on ${\mathbb{C}P^1}$. Its choice is not important, but, for convenience, we choose it in such a way that the two critical points of the function $x$ on ${\mathbb{C}P^1}$ are $z=0$ (with the critical value $x=\frac{1}{s\b}$) and $z=\infty$ (with the critical value $x=\infty$). The corresponding rational parametrization has the following form: $$\begin{aligned} x&=\frac{z^2-1}{s\,(\a z^2-\b)},& z&=\sqrt{\frac{1-\b s x}{1-\a s x}},\\ x\,\d_x F_{0,1}&=\sqrt{uv}\,\frac{1-z}{1+z},&\d_x F_{0,1}\frac{dx}{dz}&=\frac{8\,u\,v\,z}{(z+1)^2 \left(\alpha z^2-\beta \right)},\end{aligned}$$ where $\a,\b$ are related to $u,v$ by . All functions entering these equalities can be regarded either as rational functions in $z$-variable or as formal power expansions of these functions at $z=1$. We continue with the terms with $g=0$ and $n=2$ in . We have $$\Bigl(-\frac1s+(u+v)x\Bigr)\d_x F_{0,2}+2\,x^2\,\d_x F_{0,1}\,\d_x F_{0,2} +\d_y^{-1} d_y\left(\frac{x^2\,\d_x F_{0,1}-y^2\,\d_y F_{0,1}}{x-y}\right) =0\,,$$ from where we get $$\d_y\,\d_x\,F_{0,2}=\frac{d_y\left(\frac{x^2\,\d_x\, F_{0,1}-y^2\,\d_y\, F_{0,1}}{x-y}\right)} {\frac1s-(u+v)x-2\,x^2\,\d_x F_{0,1}}\;.$$ This equality uniquely determines $\d_y\d_x F_{0,2}dxdy$ as a meromorphic bidifferential on ${\mathbb{C}P^1}\times{\mathbb{C}P^1}$. Substituting the obtained above expressions for $x,\;dx/dz$ and $\d_x F_{0,1}$ into the last formula, after some miraculous cancellations we finally get $$\d_y\d_x F_{0,2}\,dx\,dy=\frac{dz\,dw}{(z+w)^2}=-B(z,-w), \label{ddF02}$$ where $B(z,w)=\frac{dz\,dw}{(z-w)^2}$ is the Bergman kernel, and $w$ is related to $y$ by the same formulas that relate $z$ to $x$. Rational recursion formula -------------------------- Now we look at the homogeneous terms of genus $g$ and degree $m$ with $2g-2+m>1$ in . To begin with, let us extract the unstable terms from the expression $(\d F)^2$ in  in order to re-group them with the other summands. Multiplying  by $x^{-2}$ we get $$\begin{aligned} &2\left(-\frac{1}{2\,s\,x^2}+\frac{u+v}{2\,x}+\d_x F_{0,1}\right)\,\d_x(F-F_{0,1}-F_{0,2})+\d_x^2 F\\ &\quad +\bigl(\d_x(F-F_{0,1}-F_{0,2})\bigr)^2+x^{-2}\d_y^{-1}\,d_y\,\left(\frac{x^2\,\d_x(F-F_{0,1})-y^2\,\d_y(F-F_{0,1})}{x-y}\right)\\ &\hspace{2.4in}+2\d_x(F-F_{0,1})\,\d_x F_{0,2}-(\d_x F_{0,2})^2=0.\end{aligned}$$ Let us rewrite the coefficients of this equation in the $z$-coordinate. It is convenient to put $$\eta=\eta(x)\,dx=-\left(-\frac{1}{2\,s\,x^2}+\frac{u+v}{2\,x}+\d_x F_{0,1}\right)\,dx\;,$$ so that in terms of the coordinate $z$ $$\eta=\eta(z)\,dz=\frac{(\a-\b)^2\,z^2}{(z^2-1)^2 (\a\,z^2 -\b)}\,dz\,.$$ Then, using the already known expressions for $x$, $\frac{dx}{dz}$, $\d_x F_{0,1}$, and $\d_y\d_x F_{0,2}$, we find that $$\begin{aligned} \d_x F_{0,2}=-\d_y^{-1}\,d_y\left(\frac{1}{z+w}\right). \label{delf}\end{aligned}$$ From the above identities we obtain the following equation for $\d_x F$: $$\begin{aligned} \d_x(F-F_{0,1}-F_{0,2})&=\frac{1}{2\,\eta(x)}\Bigl(\d_x^2 F+(\d_x(F-F_{0,1}-F_{0,2}))^2-(\d_xF_{0,2})^2\Bigr)\nonumber\\ +\d_y^{-1}\,d_y&\left(\frac{w}{z^2-w^2}\left(\frac{\d_x(F-F_{0,1})}{\eta(x)}-\frac{\d_y (F-F_{0,1})}{\eta(y)}\right)\right)\, \frac{dz}{dx}\;.\label{eqvirz}\end{aligned}$$ In particular, for the homogeneous components $U_{g,m}=\d_x F_{g,m}$ with $2g-2+m>1$ this equation reads $$\begin{gathered} U_{g,m}(z)=\frac{1}{2\,\eta(z)}\Biggl(\d_z U_{g-1,m+1}(z)+\mathop{\sum{}^{\mathrlap{*}}}_{\substack{g_1+g_2=g,\\m_1+m_2=m+1}} U_{g_1,m_1}(z)\,U_{g_2,m_2}(z)\Biggr)\\ {}+\d_w^{-1}\,d_w\left(\frac{w}{z^2-w^2}\left(\frac{U_{g,m-1}(z)}{\eta(z)}-\frac{U_{g,m-1}(w)}{\eta(w)}\right)\right)\;, \label{eqrat}\end{gathered}$$ where the star $^*$ by the summation sign means that the unstable terms with $2g-2+m\leq 0$ are omitted. We refer to this equation as the *rational recursion formula* for the forms $U_{g,m}=U_{g,m}\,(z)dz$. This equation allows to prove the polynomiality property for the forms $U_{g,m}$ by induction in $g$ and $m$. Indeed, assume that $U_{g',m'}$ is a Laurent form in $z$ with coefficients polynomially depending on $t_k=T_k(p)$, $k\in Z_\odd$, for all $(g',m')$ with $0<2g'-2+m'<2g-2+m$. Then the first summand in  obviously also has this form. Let us examine the second summand. Note that the operator $\d_w^{-1}\,d_w$ is well defined on the space of odd Laurent polynomials in variable $w$, so let us check that this condition always holds. Indeed, the function $\frac{U_{g,m-1}(z)}{\eta(z)}$ is an even Laurent polynomial in $z$, therefore, it can be represented as a Laurent polynomial in $z^2$. Therefore, $\frac1{z^2-w^2}\left(\frac{U_{g,m-1}(z)}{\eta(z)}-\frac{U_{g,m-1}(w)}{\eta(w)}\right)$ is a Laurent polynomial in $z^2$ and $w^2$ regular at $z=\pm w$. Multiplying it by $w$ and applying $d_w$ we obtain an odd Laurent form in $w$. The polynomiality property for the forms $U_{g,m}$ follows now from Remark \[remdelta\]. Utilising Remark \[remdelta\] once again, we obtain the polynomiality property for $F_{g,m}$ with $2g-2+m>0$ as well. This proves the main assertion of Theorem \[th2\] (under the assumption that it is valid for the initial terms $F_{0,3}$ and $F_{1,1}$; this is checked below). The residual formalism ---------------------- Since the both sides of  belong to $L$, it is convenient to equate the projections of the terms entering this equality by applying $P_L$ to each of them. The terms of the first summand on the right hand side already belong to $L$, so $P_L$ is identical on them. Compute the image of the second summand on the right hand side under the projection $P_L$. The key observation is that $P_L$ can be applied to the two terms of this summand separately. In particular, the form $\frac{dz}{z^2-w^2}$ is regular both at $z=0$ and at $z=\infty$ so that it does not contribute to the image. It remains to compute the term $$\begin{aligned} &P_L\!\left(\d_w^{-1}d_w\!\left(\frac{w}{z^2-w^2}\frac{U_{g,m-1}(z)}{\eta(z)}\right)dz\right)\\ &\hspace{2in}=P_L\!\left(\frac{U_{g,m-1}(z)}{\eta(z)}\,\d_w^{-1}\!\left(\frac{(z^2+w^2)}{(z^2-w^2)^2}dz\,dw\right)\right).\end{aligned}$$ The form $\frac{(z^2+w^2)}{(z^2-w^2)^2}\,dz\,dw$ is not Laurent so that $\d_w^{-1}$ is not applicable to it directly. However, what we actually need in order to apply $P_L$ is the expansion of this form at $z=0$ and $z=\infty$. The coefficients of these expansions are Laurent with respect to $w$: $$\begin{aligned} &\frac{(z^2+w^2)}{(z^2-w^2)^2}dz\,dw=-\sum_{i=0}^\infty d(w^{-2i-1})z^{2i}dz= -\d_w\sum_{i=0}^\infty t_{-2i-1}z^{2i}dz,\quad z\to 0,\\ &\frac{(z^2+w^2)}{(z^2-w^2)^2}dz\,dw=-\sum_{i=0}^\infty d(w^{2i+1})z^{-2i}d(z^{-1})\\ &\hspace{2.2in}=-\d_w\sum_{i=0}^\infty t_{2i+1}z^{-2i}d(z^{-1}),\quad z\to \infty.\end{aligned}$$ In other words, the form $\d_w^{-1}\Bigl(\frac{(z^2+w^2)}{(z^2-w^2)^2}\,dz\,dw\Bigr)$ is well defined in some neighborhoods of the points $z=0$ and $z=\infty$ and coincides with the form $U_{0,2}$ defined by . This proves the equality  of the topological recursion. Initial terms of the recursion ------------------------------ In order to finish the proof of Theorem \[th2\], it remains to check it for the initial terms of the recursion, that is, for $U_{1,1}=\d_x F_{1,1}\,dx$ and $U_{0,3}=\d_x F_{0,3}dx$. For the case $g=m=1$, equating the corresponding terms in  we get $$U_{1,1}=\frac{1}{2\eta(x)}\d_x^2F_{0,2}\,dx\,,$$ and using the explicit formula  for $\d^2F_{0,2}$, we obtain the required formula  for $U_{1,1}$. For the case $g=0$, $m=3$ the computations are slightly more involved. Eq.  uniquely determines the form $U_{0,3}$ as $$U_{0,3}= \d_w^{-1}\, d_w\left(\frac{w}{z^2-w^2}\left(\frac{\d_x F_{0,2}(x)}{\eta(x)}-\frac{\d_y F_{0,2}(y)}{\eta(y)}\right)\right)\,dz -\frac{(\d_x F_{0,2})^2}{2\,\eta(x)}\,dx\;,$$ where, as before, $y$ is related to $w$ by the same formulas that relate $x$ to $z$. The form $\d_x F_{0,2}\,dx$ being known, cf. , we directly compute $$U_{0,3}=\frac{1}{32 u v}\left(\a\,t_1^2dz-\b t_{-1}^2\frac{dz}{z^2} \right)=P_L\left(\frac{(U_{0,2}(z))^2}{2\eta(z)}\,dz\right)$$ which agrees with . This completes the proof of Theorem \[th2\]. Topological recursion for ribbon graphs ======================================= Here we discuss the toplogical recursion for the numbers $D_{g,m}$ of genus $g$ ribbon graphs with $m$ labeled boundary components of lengths $\mu_1,\ldots,\mu_m$, see Section 2. A topological recursion for these numbers was first obtained in [@EO1], Theorem 7.3 (it was later rediscovered in [@DMSS]). We present a simple proof of this recursion in terms of the homogeneous components $\tF_{g,m}$ of the generating function  that follows directly from the Virasoro constraints, cf. Theorem \[rg\], (i). In fact, the argument is quite parallel to that of the case of dessins d’enfants. This is why we skip the details paying attention only to the differences between these two enumerative problems. More specifically, we have the same spectral curve ${\mathbb{C}}P^1$ equipped with an affine coordinate $z$ and the involution $z\mapsto-z$. What is different, is the choice of the local coordinate $x$ at the point $z=1$ and the form $\eta$. Consider the linear functions $\tT_k(p)$ given by  with $$\label{zxM} z(x)=\sqrt{\frac{1+2\sqrt{u}\; s\, x}{1-2\sqrt{u}\; s\, x}}$$ \[th3\] Let $\tF_{g,m}$ be the infinite series defined by $$\begin{aligned} \tF_{g,m}(s,u,p_1,p_2,\ldots) &=\frac{1}{m!} \sum_{\mu\in{{\mathbb{Z}}}_+^m} D_{g,m}(\mu)s^d u^k p_{\mu_1}\ldots p_{\mu_m}\;.\end{aligned}$$ Then (i) For each $g,m$ with $2g-2+m>0$ there exist a polynomial $\tG_{g,m}$ of the variables $t_j$, $j\in{\mathbb{Z}}_\odd$, such that $$\tF_{g,m}(p)=\tG_{g,m}(t)\bigm|_{t_j=\tT_j(p)}$$ (i.e. each $\tF_{g,m}$ is a polynomial in the linear functions $\tT_{\pm1},\tT_{\pm3},\tT_{\pm5},\ldots$). (ii) The polynomials $\tG_{g,m}$ can be recursively computed, starting from $\tG_{0,3}$ and $\tG_{1,1}$, by the same Eqs. – with $\eta$ given by the formula $$\eta=\eta(z)\,dz=-\frac{16uz^2dz}{(1-z^2)^3}\;.$$ Like in the case of dessins, we start with the master Virasoro equation that readily follows from Theorem \[th2\], (i): $$\begin{aligned} \Bigl(-\frac1{s^2}+2\,u\,x^2\Bigr)\,\d_x \tF&+x^3\Bigl(\d_x^2 \tF+(\d_x \tF)^2\Bigr)\\ &+\d_y^{-1}\,d_y\left(\frac{x^3\,\d_x \tF-y^3\,d_y \tF}{x-y}\right)+u^2\,x+u\,p_1=0\;.\end{aligned}$$ The spectral curve in this case (an analog of above) is given by the equation $$x^2\,(\d_x\tF_{0,1})^2-x\,\d_x\tF_{0,1}\Bigl(\frac{1}{s^2 x^2}-2 u\Bigr)+u^2=0\,.$$ Solving this equation for $x\,\d_x\tF_{0,1}$ we get the following rational parametrization of the spectral curve $$\begin{aligned} &x(z)=\frac{z^2-1}{2s\sqrt{u}\;(z^2+1)}\;,\\ &x\,\d_x\tF_{0,1}=\frac{1-2\,s^2\,u\,x^2-\sqrt{1-4\,s^2\,u\,x^2}}{2s^2\,x^2}=u\left(\frac{1-z}{1+z}\right)^2\;,\end{aligned}$$ with the inverse change $z(x)$ given by . The genus $0$ two point correlator is the same as in the case of dessins: $$\d_y\d_x\tF_{0,2}\,dx\,dy=\frac{dz\,dw}{(z+w)^2},$$ and instead of the form $\eta$ we have $$\tilde\eta=\tilde\eta(x)dx=\left(-\d_x \tF_{0,1}-\frac{u}{x}+\frac{1}{2\,s^2\,x^3}\right)\,dx=-\frac{16\,u\,z^2}{(1-z^2)^3}\,dz\;.$$ Thus, the master Virasoro equation in $z$-coordinate acquires the same form  and implies the topological recursion  with $\eta$ replaced by $\tilde\eta$. Below are the first few terms of the recursion: $$\begin{aligned} U_{0,3}&=\frac{1}{32\,u}\left(t_1^2-t_{-1}^2z^{-2}\right)\,,\\[-12pt]\\ G_{0,3}&=\frac{1}{96\,u} \left(t_{1}^3+t_{-1}^3\right)\,,\\[-12pt]\\ U_{1,1}&=\frac{1}{128\,u}\left(z^2-3+3\,z^{-2}-z^{-4}\right)\,,\\[-12pt]\\ G_{1,1}&=\frac{1}{384\,u}\left(t_{3}-9 t_{1}-9 t_{-1}+t_{-3}\right)\,,\\[-12pt]\\ U_{0,4}&=\frac{1}{512\,u^2}\left(t_1^3 z^2+(t_3 t_1^2-3 t_1^3-t_{-1}^2 t_1)\right.\\[-12pt]\\ &\hspace{.9in}\left.+(t_1^2 t_{-1}+3 t_{-1}^3-t_{-3} t_{-1}^2)\,z^{-2}-t_{-1}^3\,z^{-4}\right)\,,\\[-12pt]\\ G_{0,4}&=\frac{1}{6144\,u^2}\left(4 t_1^3 t_3-9 t_1^4-6 t_1^2 t_{-1}^2-9 t_{-1}^4+4 t_{-3} t_{-1}^3\right)\,,\\[-12pt]\\ U_{1,2}&=\frac{1}{2048\,u^2}\left(5\,t_1\,z^4+(t_3-18 t_1)\,z^2+(4 t_{-1}+27 t_1-6 t_3+t_5)\right.\\[-12pt]\\ &\hspace{.3in}\left.+(-t_{-5}+6 t_{-3}-27 t_{-1}-4 t_1)\,z^{-2}+(18 t_{-1}-t_{-3})\,z^{-4}-5\,t_{-1}\,z^{-6}\right)\,,\\[-12pt]\\ G_{1,2}&=\frac{1}{12288\,u^2}\left(6 t_1 t_5+t_3^2-36 t_1 t_3+81 t_1^2+24 t_{-1} t_1\right.\\[-12pt]\\ &\hspace{2in}\left.+81 t_{-1}^2-36 t_{-1} t_{-3}+t_{-3}^2+6 t_{-5} t_{-1}\right)\,.\end{aligned}$$ [**Acknowledgments.**]{} The main results of the paper, Theorems 4 and 5, were obtained under support of the Russian Science Foundation grant 14-21-00035. The work of MK was additionally supported by the President of Russian Federation grant NSh-5138.2014.1 and by the RFBR grant 13-01-00383. PZ acknowledges hospitality of the Center for Quantum Geometry of Moduli Spaces at Aarhus University. We thank JSC “Gazprom Neft" for funding short-term visits of MK to the Chebyshev Laboratory at SPbSU. We are grateful to L. Chekhov, B. Eynard, P. Norbury and G. Schaeffer for useful discussions, and to the anonymous referee for correcting a few typos and suggesting several improvements in the text. [00]{} Adrianov, N.: An analog of the Harer–Zagier formula for unicellular bicolored maps, Func. Anal. Appl. [**31**]{}:3, 1–9 (1997). Alexandrov, A., Mironov, A., Morozov, A., Natanzon, S.: On KP-integrable Hurwitz functions. arXiv:1405.1395 (2014). Ambj[ø]{}rn, J., Chekhov, L.: The matrix model for dessins d’enfants. arXiv:1404.4240 (2014). Belyi, G.: On Galois Extensions of a Maximal Cyclotomic Field. 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Canad. J. Math. [**14**]{}, 708–722 (1963). Walsh, T. R. S., Lehman, A. B.: Counting rooted maps by genus. I. J. Combinatorial Theory B [**13**]{}, 192–218 (1972). Zograf, P.: Enumeration of Grothendieck’s dessins and KP hirerarchy. Int. Math. Res. Notices (2015). [^1]: An important observation of Grothendieck that, by Belyi’s theorem, the absolute Galois group ${\rm Gal}({\overline{\mathbb{Q}}}/{\mathbb{Q}})$ naturally acts on dessins, lies beyond the scope of this paper; we refer the reader to [@LZ] for details. [^2]: Equivalently, we can put $N_{k,l}(\mu)=\sum_{{\Gamma}\in{\mathcal{D}}_{k,l,\mu}}\frac{1}{|{\rm Aut}_v {\Gamma}^*|}\;,$ where ${\rm Aut}_v {\Gamma}^*$ is the group of automorphisms of the dual graph ${\Gamma}^*$ preserving each vertex pointwise. A closely related problem of the weighted count of unlabeled dessins ${\Gamma}$ with weights $\frac{1}{|{\rm Aut}\,{\Gamma}|}$ is equivalent to the above one. If one treats $\mu$ as the unordered partition $[1^{m_1}2^{m_2}\ldots]$, where $m_j=\#\{\mu_i=j\}$, then the corresponding number of dessins of type $(k,l,\mu)$ is equal to $\frac{1}{|{\rm Aut}\,\mu|}\,N_{k,l}(\mu)$ with $|{\rm Aut}\,\mu|=m_1!m_2!\ldots\;.$ [^3]: While this paper was in preparation, similar results were independently obtained by matrix integration methods in [@AC] and generalized further in [@AMMN]. [^4]: Recently we came across the paper [@CC] where this result was proven along similar lines. However, the authors of [@CC] do not explicitly use Virasoro constraints that considerably simplify and clarify the proof. [^5]: Formula (\[vt\]) is a “bicolored" analogue of Tutte’s recursion, cf. [@T], Eq. 2.1, for $g=0$ and [@WL], Eq. (6), for any $g\geq 0$ (a more general form of Tutte’s recursion one can find, e. g., in [@EO2]). [^6]: \[M\] This is (a specialization of) Tutte’s recursion for arbitrary $g$, cf. [@EO2]. This formula, undeservedly forgotten, was recently reproduced in [@DMSS], Eq. (3.15). Note that the second term in the r.h.s. of , corresponding to a loop bounding a 1-gon, was inadvertently omitted there. This required some “modification” of the numbers $D_{g,m}(\mu_1,\ldots,\mu_m)$ in [@DMSS].

--- author: - 'S. Barceló Forteza' - 'T. Roca Cortés' - 'A. García Hernández' - 'R.A. García' bibliography: - 'tcb.bib' date: 'Received 8 April 2016; Accepted 24 February 2017' title: 'Evidence of chaotic modes in the analysis of four [[$\delta$ Scuti ]{}]{}stars' --- =1 Introduction {#s:intro} ============ The launch of space telescopes such as MOST, CoRoT, & Kepler satellites [@Walker2003; @Baglin2006; @Borucki2010] announced the beginning of the precise study of the stellar oscillations in stars other than the Sun. Since then, the high quality of the light curves has allowed the precise characterization of the mode parameters of different kinds of stars, and the study of their variation with time and their connection with the stellar structure.\ Although the power-spectral structure of the stars with solar-type oscillations is well known, this is not the case for [[$\delta$ Scuti ]{}]{}stars. The power spectrum of these stars shows a complex structure with dominant peaks of moderate amplitudes and many hundreds of lower amplitude peaks that form a flat plateau [e.g. @Poretti2009], the so-called “grass”. After observation of the “grass”, a long-standing debate about its origin started, including the possibility of it arising from spurious signals produced during the analysis of the data [@Balona2014a].\ A huge theoretical effort has been made to find a possible physical phenomenon behind this power-spectral structure. Some of these arguments are:\ 1. Less effective disc-disc averaging of the flux owing to the geometry of the [[$\delta$ Scuti ]{}]{}star. Therefore, it is possible to find modes with higher degrees than in the spherical symmetric case $l > 4$ [@Balona1999]. Although @Balona2011 find that most of [[$\delta$ Scuti ]{}]{}stars do not seem to have enough density of peaks to support this possibility, several stars seem to show modes with high degrees, up to $l = 20$ [@Kennelly1998; @Poretti2009].\ 2. A granulation background signal due to the effect of a thin outer convective layer [@Kallinger2010]. This effect is found to be more important in cool [[$\delta$ Scuti ]{}]{}stars [@Balona2011a].\ 3. Variations with time that produce sidelobes of the main peak of the spectra. @Balona2011 find that around $\sim$45% of the spectra of Kepler [[$\delta$ Scuti ]{}]{}stars have one-sided sidelobe. They discard effects such as binarity because these yield amplitude-symmetric equal-spaced multiplets [@ShK2012]. However, there are other causes that produce variations with non-symmetric amplitude multiplets, such as resonant mode coupling [RMC; see @BarceloForteza2015 and references therein].\ 4. A magnetic field in a rotating star splits each peak of the rotational multiplet into $(2l+1)$, meaning that one mode is split into $(2l+1)^{2}$ peaks [@Goode1992]. Magnetic fields have been detected in the surface of $\sim$7% of main sequence and pre-main sequence intermediate-mass and massive stars [@Mathis2015]. However, [[$\delta$ Scuti ]{}]{}stars with measurable magnetic fields are not common because only one [[$\delta$ Scuti ]{}]{}star shows a magnetic field [@Neiner2015] and another is suggested to be magnetic from its chemical abundance [@Escorza2016].\ 5. The oblateness of the star produced by high rotation rates is the cause of the appearance of a significant number of chaotic modes [@Lignieres2009].\ The determination of the fundamental structural parameters of these stars such as mass, inclination, rotation rate, and convective efficiency can help us to unveil which of these mechanisms are responsible for this kind of power spectral structure. Four interesting [[$\delta$ Scuti ]{}]{}stars observed by CoRoT and Kepler satellites that have been characterized in this paper are CID 546, CID 3619, CID 8669, and KIC 5892969. Their differences in the power spectra help us in our aim. In Sect. \[s:dScu\] we describe the main characteristics of these kind of stars. The way in which the their oscillations are analysed to obtain the parameters of the modes is presented in Sect. \[s:dSBF\]. Results for each target star are commented in Sect. \[s:4dScu\]. In Sect. \[s:regular\] we estimate their structural parameters. The power-spectral structure is deeply studied and discussed in Sect. \[s:grass\]. In the last section we present our conclusions. [[$\delta$ Scuti ]{}]{}type stars {#s:dScu} ================================= [[$\delta$ Scuti ]{}]{}stars are classical pulsators with oscillation frequencies between $\sim$60 and $\sim$900 $\mu$Hz [e.g., @Zwintz2013]. These stars are located on or slightly off the main sequence, with spectral types between A2 and F5 [@Breger2000]. They are intermediate-mass stars that show fast rotation rates as it is common in stars within their mass domain or of higher mass [@Royer2007]. In fact, one of the reasons that [[$\delta$ Scuti ]{}]{}stars can be separated from RR Lyrae stars is their higher rotational velocity, $v \mathrm{sin}i > 10$ km/s [see @Peterson1996]. Other typical characteristics of [[$\delta$ Scuti ]{}]{}stars are detailed in Table \[t:dScuchar\].\ Characteristic From To ---------------------------------- ------ ------ -- Spectral-type F5 A2 Luminosity class III V $M$ ($M_{\odot}$) 1.5 2.5 $T_{\mathrm{eff}}$ (K) 6300 8600 $\mathrm{log} ~\textit{g}$ (cgs) 3.2 4.3 $v \mathrm{sin}i$ (km/s) 10 250 $\nu$ ($\mu$Hz) 60 930 A (mag) 0.3 : Typical values of the stellar characteristics of [[$\delta$ Scuti ]{}]{}stars by [@Breger2000], [@Aerts2010], and [@Uytterhoeven2011][]{data-label="t:dScuchar"} Hybrid stars {#ss:subgroups} ------------ Several subgroups can be distinguished from the main class of [[$\delta$ Scuti ]{}]{}stars pulsating with nonradial p-modes, such as High Amplitude [[$\delta$ Scuti ]{}]{}stars (HADS), SX Phe variables, or $\delta$ Scu/$\gamma$ Dor hybrid stars [@Breger2000]. This last group comes from the observation of g-modes in [[$\delta$ Scuti ]{}]{}type stars with frequencies typical of [[$\gamma$ Doradus ]{}]{}stars, meaning $\nu \sim \left[6-60 \right] \mu$Hz. @Uytterhoeven2011 point out that a star can be classified as hybrid when all of the three following conditions are accomplished:\ 1) Typical frequencies of both kinds of stars are detected.\ 2) The amplitudes of both domains are comparable, within a factor $\lesssim 5$.\ 3) There are two independent frequencies in both domains with amplitudes higher than 100 parts per million (ppm)\ If the star is hybrid it would be a $\delta$ Scu/$\gamma$ Dor or a $\gamma$ Dor/$\delta$ Scu star depending on which part is the dominant one [@Grigahcene2010]. In this way, it is found that a great amount of [[$\delta$ Scuti ]{}]{}and [[$\gamma$ Doradus ]{}]{}stars are hybrids, $\sim$36%. Other studies suggest that all [[$\delta$ Scuti ]{}]{}stars are hybrids [@Balona2014].\ However, other magnitudes that help us to differentiate [[$\delta$ Scuti ]{}]{}from [[$\gamma$ Doradus ]{}]{}are the convective efficiency ($\Gamma$), $$\Gamma \sim \left(T_{eff}^{3} \mathrm{log} ~\textit{g} \right)^{-\frac{2}{3}}\, , \label{e:conveff}$$ where $g$ is the surface gravity, and the kinetic energy of the waves ($E_{kin}$), $$E_{kin} \sim \left(A_{0} \nu_{0} \right)^{2}\, , \label{e:kine}$$ where $\nu_{0}$ and $A_{0}$ are the frequency and amplitude of the mode with maximum power, respectively. These magnitudes have a dominant value of $\mathrm{log}~\Gamma<-8.1$ and $\mathrm{log}~E_{kin}>10.1$ for [[$\delta$ Scuti ]{}]{}stars when the amplitude is measured in ppm and the frequency in $\mu$Hz [see @Uytterhoeven2011]. Both quantities are related to the convective zone of the star which is more efficient in [[$\gamma$ Doradus ]{}]{}stars. We use all of these tools to find out whether some of our selected stars are hybrid stars and if they present some other differences in their power-spectral structure.\ Rotational effect {#ss:rot} ----------------- Taking into account the effect of rotation on the perturbation analysis of a spherically symmetric star, the modes split into multiplets. For low rotation rates ($\Omega$), these multiplets present $(2l+1)$ symmetric peaks, split approximately a multiple of the rotational splitting ($s$). For higher rotation rates, second-order effects have to be taken into account and the symmetry is broken [@Saio1981; @Dziembowski1992]. Besides the appearance of the asymmetry, the multiplet is also globally shifted. To go even further, a third-order correction has already been studied by @Soufi1998.\ The different contributions can be estimated thanks to two dimensionless magnitudes [@Goupil2000] as $$\epsilon^{2} = \frac{\Omega^{2} R^{3}}{G M} \, , \qquad \mu = \frac{\Omega}{2 \pi \nu} \, , \label{e:eps2}$$ where $\epsilon$ scales the effect of centrifugal force with gravity and $\mu$ scales the effect of the rotational rate to oscillation frequencies. All of these effects are higher for lower frequency g-modes than for higher frequency p-modes. However, they can produce observable shifts up to 1 $\mu$Hz.\ Rotation also produces a deformation of the star [@Cassinelli1987]. Under the assumption that the rotation is uniform and the surface of the star is approximately a Roche surface [@PerezHernandez1999], an averaged effective gravity ($g_{eff}$) can be defined as $$g_{eff} = g - \frac{2}{3} R \Omega^{2}\, , \label{e:geff}$$ where $R$ is the radius of the star with spherical symmetry. With these assumptions it is also possible to obtain the polar radius $R_{p}$, $$R_{p} = \frac{R}{1+\frac{\epsilon^{2}}{3}}\, . \label{e:Rp}$$ Assuming that the volume is constant compared with a spherically symmetric star, the oblateness $O$ of the star is also defined as $$O = 1-\left( 1+ \frac{\epsilon^{2}}{3} \right)^{-\frac{3}{2}} \; . \label{e:oblateness}$$ The oblateness of a star increases with higher $\epsilon$. However, rotation has a maximum limit at which the centrifugal force will destroy the star. This limit is the so-called break-up frequency ($\Omega_{K}$): $$\Omega_{K} = \left( \frac{8 G M}{27 R_{p}^{3}} \right)^{\frac{1}{2}} \label{e:Omegak}$$ Therefore, for a stable [[$\delta$ Scuti ]{}]{}star, $\Omega/\Omega_{K} \leq 1 $ has to be accomplished.\ Another known effect of the rotation is gravity darkening [@vonZeipel1924], meaning an increase of the temperature from the equator to the poles. It follows a potential law as $$\delta T_{eff} = \frac{T_{eff,p}-T_{eff,e}}{T_{eff,p}} \approx 1-\left( 1- \epsilon^{2} \right)^{\frac{\beta}{4}} \; , \label{e:dteff}$$ where the value of $\beta$ depends of the evolutionary stage of the star [@Claret1998]. Taking into account this effect, the measurements of $T_{eff}$ and $\mathrm{log} ~\textit{g}$ will vary depending on the inclination angle of the star ($i$). Therefore, these variations have to be carefully treated to obtain the proper main characteristics of the star. Analysis of CoRoT & Kepler [[$\delta$ Scuti ]{}]{}light curves {#s:dSBF} ============================================================== The data we use were obtained by the CoRoT and *Kepler* satellites. The CoRoT satellite (Convection, Rotation, and planetary Transits) was developed and operated by the French space agency CNES with international contributions of ESA, Austria, Belgium, Brazil, Germany and Spain. The objective of the mission was to search for exoplanets and to perform asteroseismic studies. Two different channels were designed: the exo-channel data are sampled every 512 s and their photometric precision is between 40 and 90 ppm [@Auvergne2009], whereas the seismo-channel has a much shorter cadence of 32 s and a substantially higher photometric precision of between 0.6 and 4 ppm [see @Auvergne2009]. As described in [@Boisnard2006], two kinds of campaigns were planned of different duration: Long Runs (LR) of approximately 150 d, and Short Runs (SR) with durations around 30 d. These runs were carried out in two different fields: one close to the galactic anticentre direction(a) and the other close to the galactic centre direction (c). The light curves of each star can then be classified with the nomenclature &lt;duration&gt;&lt;direction&gt;&lt;number&gt;. For example, LRa01 is the first Long Run to point close to the anticentre of the galaxy.\ The *Kepler* mission was designed by NASA to survey a single region of our own galaxy to detect and characterize Earth-sized planets close to their habitable zones using the transit method [@Thompson2012]. The maximum duration of their light curves are up to four years. Because the original field is no longer observable, the mission was renamed as *K2* [@Howell2014]. There are two different cadences available depending on the star: Long Cadence (LC) of $\sim$29.5 min or the Short Cadence (SC) of $\sim$1 min. Moreover, the data are downloaded in three-months bases called quarters [Q&lt;number&gt;, @Haas2010].\ The asteroseismic analysis was performed by a set of different programs called [[$\delta$ Scuti ]{}]{}Basics Finder ($\delta$SBF) that we built using IDL language programming. We analysed the light curves of [[$\delta$ Scuti ]{}]{}stars with the three stage Method described in detail in @BarceloForteza2015. This iterative method allows us to interpolate the light curve using the information of the subtracted peaks, minimizing the effect of gaps, considerably improving the background noise, and avoiding spurious effects [@Garcia2014]. Thus, we get very accurate results in terms of frequencies, amplitudes, and phases. In addition, we take into account the energy of the signal for each peak $i$: $$E_{i} =\frac{RMS_{i}-RMS_{i+1}}{RMS_{0}} \, , \label{e:esignal}$$ where $RMS_{i}$ is the root mean square of the residual signal after the subtraction of the highest $i$ peaks, and for which i=0 is the original signal.\ The next step in our strategy was to characterize each [[$\delta$ Scuti ]{}]{}star, finding its structural parameters such as mass ($M$) or oblateness ($O$). This is possible thanks to the regularities present in the power-spectral structure of these kind of stars, such as the large separation ($\Delta\nu$) and the rotational splitting ($s$).\ The last step consists in studying each star’s power-spectral structure in order to find any relation between their characteristic parameters, such as density of peaks ($\mathbf{n}_{mean}$), and their structural parameters. These two steps will be further discussed in the sections that follow.\ Characterization of four [[$\delta$ Scuti ]{}]{}stars {#s:4dScu} ===================================================== We tested our method with an already known [[$\delta$ Scuti ]{}]{}star: CID 546. We also analysed the light curves of CID 3619, CID 8669, and KIC 5892969 stars. Figures \[f:S\_4dScu\_546\] to \[f:S\_4dScu\_5893969\] show the four power spectra and their frequency contents. The highest amplitude peaks can be found at Appendix \[ap:PoM\].\ CID 546 {#ss:test} ------- ![From top to bottom: Power-spectral structure of the original light curve for CID 546, and also those after extracting the indicated number of peaks. The contribution of the peaks considered as “grass” is represented in red (see Sect. \[s:grass\]).[]{data-label="f:S_4dScu_546"}](C4_MG546_D.eps "fig:"){width="49.50000%"}\ ![Same as Fig. \[f:S\_4dScu\_546\] for CID 3619.[]{data-label="f:S_4dScu_3619"}](C4_MG3619_D.eps){width="49.50000%"} ![Same as Fig. \[f:S\_4dScu\_546\] for CID 8669.[]{data-label="f:S_4dScu_8669"}](C4_MG8669_D.eps){width="49.50000%"} ![Same as Fig. \[f:S\_4dScu\_546\] for KIC 5892969.[]{data-label="f:S_4dScu_5893969"}](C4_MG5892969_D.eps){width="49.50000%"} The [[$\delta$ Scuti ]{}]{}star CID 546 (HD 50870) is a F0IV star with $M_{v} \sim$1.67 at an approximate distance of 277 pc [$V\sim8.88$; @McCuskey1956], observed by CoRoT close to the anticentre direction of the Galaxy. It has been observed during 114.4 d between 2008 November 13 and 2009 March 8 (LRa02). Its known parameters are listed in the first rows of Table \[t:4dScu\]. Recently, @Mantegazza2012 discovered that CID 546 is a long-period spectroscopic binary star with a cooler companion. Nevertheless, additional observations that cover more of the orbital period are necessary to confirm their results, which they consider to be preliminary. No evidence of binarity is found in the photometric analysis.\ When our analysis is applied to the power spectrum of this star, we obtain 1513 peaks higher than 10 ppm with a signal-to-noise ratio (SNR) greater than or equal to four. These peaks carry 99.77% of the full signal. Seventeen peaks with amplitudes higher than 400 ppm carry 97.37 % of the signal and appear in three different frequency ranges (see Table \[t:peaks546\] and top panel of Fig. \[f:S\_4dScu\_546\]): the main regime from 150 to 200 $\mu$Hz; the second, which is half the value of the previous one, from 80 to 100 $\mu$Hz, approximately; and the third regime close to 400 $\mu$Hz. The analysis also reveals 1446 peaks with amplitudes lower than 130 ppm that only carry 0.95% of the energy of the signal. The flat plateau is clearly visible and could be differentiated from noise after extracting hundreds of peaks (see Fig. \[f:S\_4dScu\_546\]).\ Comparing the results with those obtained by @Mantegazza2012, our method finds all peaks with amplitudes higher than 50 ppm and frequencies higher than 1 $\mu$Hz, with mean relative differences between both results of $5\times 10^{-3}$% in frequency and 10% in amplitude. The differences between frequencies obtained for both methods are $\sim$0.01 $\mu$Hz, one order of magnitude lower than the frequency resolution. Therefore, it can be concluded that the method produces accurate results and allows us to study real light-curves of [[$\delta$ Scuti ]{}]{}stars from both the CoRoT and Kepler satellites.\ CID 3619 {#ss:chocobo} -------- The F0V star CID 3619 (HD 48784) has $M_{v} \sim$1.87 at a distance of approximately 91 pc [$V\sim6.65$; @Charpinet2006] and was observed by CoRoT close to the anticentre direction of the Galaxy. The satellite followed it for 25.3 d during 2008 March 5-31 (SRa01) and also for 40.7 d during 2011 November 29 - 2012 January 9 (SRa05).\ The analysis of the power spectrum of the SRa01 light curve detects 163 peaks down to 5 ppm with a SNR greater than or equal to four. These peaks carry 96.45% of the full signal. The same analysis was performed for the SRa05 light curve obtaining 508 peaks down to 2.5 ppm with the same lower limit for the SNR and carrying 96.8 % of the full signal. We found 37 and 42 peaks with energies higher than 0.1%, respectively. Of all these peaks, only twenty are detected in both runs (see Table \[t:peaks3619\]), which show slight differences in frequency, up to 0.2 $\mu$Hz, but higher changes in amplitude, up to the 73%, and phase, up to 1.5$\pi$.\ The power-spectral structure of this star shows a frequency range that includes the typical regimes of both [[$\gamma$ Doradus ]{}]{}and [[$\delta$ Scuti ]{}]{}stars. Moreover, the ratio between mean amplitudes of both regimes is around $A_{\delta Scu}/ A_{\gamma Dor} \sim$4, which means that CID 3619 is a hybrid $\delta$ Scu/$\gamma$ Dor star candidate. The power spectrum of CID 3619 does not show the flat plateau present for CID 546 (see Figures \[f:S\_4dScu\_546\] and \[f:S\_4dScu\_3619\]).\ CID 8669 {#ss:cactuar} -------- The A5 [[$\delta$ Scuti ]{}]{}star CID 8669 (HD 181555) is of absolute magnitude $M_{v} \sim$2.19 at a distance of approximately 116 pc [$V\sim7.52$; @Charpinet2006] and was observed by CoRoT close to the direction of the centre of the Galaxy. It was observed for 156.6 d between the 2007 May 11 and 2007 October 15 (LRc01).\ The power spectrum of this star shows 3175 peaks higher than 3 ppm with a SNR greater o equal to four. These peaks carry 99.83% of the full signal. Thirty-one peaks with amplitudes higher than 200 ppm carry 95.71 % of the energy of the signal (see Table \[t:peaks8669\]). The analysis also finds 3054 peaks with amplitudes lower than 70 ppm that only carry a 1.75% of the energy of the signal. The flat plateau is also visible and it has a higher density of peaks than CID 546 (see Figures \[f:S\_4dScu\_546\] and \[f:S\_4dScu\_8669\]).\ KIC 5892969 {#ss:tomberi} ----------- The stellar characteristics of the faint [[$\delta$ Scuti ]{}]{}star KIC 5892969, $Kp\sim$12.445, have been studied spectroscopically by @Huber2014. This star has been observed by the Kepler satellite during 1470 d, from Q0 to Q17, in LC and its oscillation modes have been studied in @BarceloForteza2015. Since the Nyquist frequency of KIC 58929’s power spectrum is lower than the typical frequency range of [[$\delta$ Scuti ]{}]{}stars, we cannot ascertain a priori whether there is a flat plateau (see Fig. \[f:S\_4dScu\_5893969\]).\ Searching for possible spectral regularities {#s:regular} ============================================ {width="49.50000%"} {width="49.50000%"} {width="49.50000%"} {width="49.50000%"} As @Suarez2014 stress, the mode organization for [[$\delta$ Scuti ]{}]{}stars includes regularities as the large separation with a negligible variation from the non-rotating case. Using a dense sample of representative models, they obtain the following scaling relation: $$\frac{\Delta\nu}{\Delta\nu_{\odot}} = 0.776\left( \frac{\rho}{\rho_{\odot}}\right)^{0.46}\, , \label{e:lsepsuarez}$$ where $\rho$ is the mean density of the star. This relation is somewhat similar to that found to solar-type oscillators [@Kjeldsen1995]: $$\frac{\Delta\nu}{\Delta\nu_{\odot}} = \left( \frac{\rho}{\rho_{\odot}}\right) ^{\frac{1}{2}}\, . \label{e:lsep}$$ In fact, they point out that the minimum error of these analyses is around 11 to 21% and is due to the stellar deformation. This is in agreement with several previous theoretical studies claiming that regularities in the p-modes of [[$\delta$ Scuti ]{}]{}stars are related to the spherical large separation [e.g. @Pasek2012].\ On the observational side, many [[$\delta$ Scuti ]{}]{}stars show frequency spacings [e.g.: @GarciaHernandez2009; @Zwintz2011a]. Some of these spacings have been interpreted as a combination of frequencies [@Breger2011] or the signal of the rotational splitting [@Zwintz2011]. However, several [[$\delta$ Scuti ]{}]{}stars known to be eclipsing binaries have been analysed [e.g.: @daSilva2014]. As binary stars, it is possible to calculate their stellar characteristics such as mass or radius. With all these data, @GarciaHernandez2015 find a similar relation to the previous one proving that the above scaling relation is independent of the rotation rate.\ Therefore, using previously known parameters, the $\Delta\nu - \rho$ relation, and considering $$\frac{v \sin i}{2 \pi R} \leq s \leq \frac{\Omega_{K}}{2 \pi} \; , \label{e:slim}$$ it is possible to delimit the value of these two regularities. Once we have the mean density of the star and its rotation, the mass and the radius can be estimated using the Stephan-Boltzmann law and the surface gravity acceleration (Eq. \[e:geff\]). Moreover, we can also obtain a mass estimate with the mass-luminosity relation [@Ibanovglu2006].\ We used the following four methods to look for regularities.\ Histogram of differences {#ss:HD} ------------------------ ------------- ---------------- --------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- Star HoD AC TRUFAS ED HoD AC TRUFAS ED KIC 5892969 1.1 $\pm$ 0.2 1.3 $\pm$ 0.1 1.2 $\pm$ 0.3 1.2 $\pm$ 0.1 22.2 $\pm$ 0.5 22.6 $\pm$ 0.9 22.1 $\pm$ 0.2 22.2 $\pm$ 0.4 CID 546 6.4 $\pm$ 0.6 7.5 $\pm$ 0.6 7.0 $\pm$ 1.1 7.1 $\pm$ 0.2 53 $\pm$ 3 54 $\pm$ 1 55 $\pm$ 1 55.4 $\pm$ 0.8 CID 3619 8.6 $\pm$ 0.4 8.1 $\pm$ 0.3 7.6 $\pm$ 0.7 8.1 $\pm$ 0.5 41 $\pm$ 3 40 $\pm$ 2 40 $\pm$ 2 40.3 $\pm$ 0.6 CID 8669 16.3 $\pm$ 0.4 18 $\pm$ 1 16.7 $\pm$ 0.1 16.9 $\pm$ 0.7 55 $\pm$ 1 54 $\pm$ 1 55 $\pm$ 2 55.0 $\pm$ 0.6 ------------- ---------------- --------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- @Breger2009 use a histogram of differences between all the detected modes and find that the radial modes ($l=0$) are not the only kind of mode that allow us to find the large separation. @GarciaHernandez2009 stress that high amplitude modes carry this signature, and including the lower amplitude modes powers other periodicities. These other regularities make it very difficult to determine the large separation. Therefore, we used the histogram of differences between pairs of frequencies within the typical [[$\delta$ Scuti ]{}]{}star range of frequency oscillations and only taking into account the, approximately, 50 highest amplitude peaks (see top left panel in Fig. \[f:S\_C546\]).\ Rotational multiplets can be present in the set of frequencies chosen for the analysis, allowing us to also see the rotational splitting. As we mention in Sect. \[ss:rot\], the higher the rotation rate, the greater the deviation from a symmetric splitting, and the more difficult it is to observe in the histogram [@Goupil2000]. For low rotation rates, $s \sim 1$ $\mu$Hz, the splitting is easily detectable and contributions due to twice and thrice the splitting are also detected. Moderate rotation rates, $s \sim 5$ $\mu$Hz, perturb this structure but the splitting is still dominant. For higher rotation rates, $s \sim 10$ $\mu$Hz, it is not possible to observe a dominant peak, owing to the lack of symmetry between the peaks of the rotational multiplet.\ To obtain an accurate value of these parameters with this method, it is important that the binning of the histogram reaches a compromise between the possible variation with frequency of the periodicities and the accuracy we want to reach. We used a bin of 0.5 $\mu$Hz to look for rotational signatures and bins up to 1.5 $\mu$Hz to find the large separation. The results of this method (see Table \[t:sdnumethod\] and top left panel of Fig. \[f:S\_C546\]) take into account possible multiples of the rotation, multiples or submultiples of the large separation, and their split peaks ($\Delta \nu \pm s$).\ Autocorrelation function {#ss:AC} ------------------------ The autocorrelation function compares a power spectrum with itself as a function of lag. This function has a higher value when the variations of the original spectrum increase and decrease similarly to the shifted spectrum. Then, when the lag coincides with one of the possible regularities of the spectrum, the value of the autocorrelation function increases.\ @Reese2013 test this method with artificial spectra, taking into account a different number of modes and calculating their visibilities with different inclinations and rotation rates. They conclude that it is possible to obtain regularities corresponding to the large separation and half its value, and also the rotation rate and twice its value. These peaks are reinforced when the range of observed frequencies spans a large enough interval and does not include too many modes in the artificial light-curve. This last condition is important to avoid powering other regularities that can be present in the spectrum, as also happens with the histogram of differences. Therefore, we calculated the autocorrelation function of the artificial spectrum that is built, by considering only the approximately 50 highest amplitude modes (see top right panel in Fig. \[f:S\_C546\]).\ Once the autocorrelation function was calculated, we looked for its highest values in a 1 to 100 $\mu$Hz lag interval. The large separation is found by looking for the peak with highest number of consecutive submultiples within its error. Then, we looked for its closest peaks to find possible rotation signatures. The last step was to try to find the rotation signature by looking for one peak within the 1 to 18 $\mu$Hz range that has consecutive multiples within its error. This method usually finds one of the split peaks of the large separation as the dominant peak ($\Delta \nu \pm s$). We correct it by adding or subtracting the rotational splitting (see Table \[t:sdnumethod\] and top right panel of Fig. \[f:S\_C546\]).\ TRUFAS: the spectrum of a subspectrum {#ss:deepground} ------------------------------------- Stars KIC 5892969 CID 546 CID 3619 CID 8669 ------------ --------------------------------- -------------------- ------------------- ------------------- ------------------ Already $T_{eff}$(K) 7560 $\pm$ 250 7600 $\pm$ 200 6990 $\pm$ 140 7000 $\pm$ 200 known $\mathrm{log}~\textit{g}_{eff}$ 3.8 $\pm$ 0.3 3.9 $\pm$ 0.2 4.0 4.3 $\pm$ 0.2 parameters $v \sin i$ (km/s) $\geq$10 17 108 200 Obtained $\Delta \nu$($\mu$Hz) 22.2 $\pm$ 0.4 55.4 $\pm$ 0.8 40.3 $\pm$ 0.6 55.0 $\pm$ 0.6 parameters $s$($\mu$Hz) 1.2 $\pm$ 0.1 7.1 $\pm$ 0.2 8.1 $\pm$ 0.5 16.9 $\pm$ 0.7 $\rho(\rho_{\odot})$ 0.027 $\pm$ 0.001 0.169 $\pm$ 0.005 0.089 $\pm$ 0.005 0.17 $\pm$ 0.01 $M(M_{\odot})$ 2.2 $\pm$ 0.3 1.5 $\pm$ 0.3 2.0 $\pm$ 0.1 2.4 $\pm$ 0.7 $R(R_{\odot})$ 4.2 $\pm$ 0.5 2.09 $\pm$ 0.07 2.8 $\pm$ 0.1 2.4 $\pm$ 0.2 $\epsilon^{2}$(%) 0.5 $\pm$ 0.1 3.0 $\pm$ 0.3 7.4 $\pm$ 0.9 17 $\pm$ 1 $O$(%) 0.27 $\pm$ 0.03 1.5 $\pm$ 0.1 3.6 $\pm$ 0.4 8.0 $\pm$ 0.6 Calculated $R_{p}(R_{\odot})$ 4.2 $\pm$ 0.5 2.07 $\pm$ 0.07 2.7 $\pm$ 0.1 2.3 $\pm$ 0.2 parameters $R_{e}(R_{\odot})$ 4.2 $\pm$ 0.5 2.10 $\pm$ 0.07 2.9 $\pm$ 0.1 2.5 $\pm$ 0.2 $i(^{o})$ - 15 - 35 88 - 90 55 - 90 $\Omega/\Omega_{k}$(%) 14 $\pm$ 7 31 $\pm$ 3 48 $\pm$ 6 70 $\pm$ 7 $\mathrm{log}~\Gamma$ -8.14 $\pm$ 0.08 -8.15 $\pm$ 0.04 -8.09 $\pm$ 0.01 -8.11 $\pm$ 0.04 $\mathrm{log}~E_{kin}$ 11.511 $\pm$ 0.004 12.77 $\pm$ 0.01 9.481 $\pm$ 0.003 11.57 $\pm$ 0.02 $\delta T_{eff}$(%) 0.13 $\pm$ 0.03 0.76 $\pm$ 0.08 0.61 $\pm$ 0.08 4.5 $\pm$ 0.3 This method uses part of the TRUFAS algorithm, originally built to detect p-mode oscillations in solar-like stars as described by @Regulo2002 and later used to find planetary photometric transits [@Regulo2007]. It takes advantage of the properties of the spectrum of the subspectrum $FFT \left\{ S(\nu ) \cdot H(\nu ) \right\}$ for which the spectral signature of the $n>l$ p-modes can be considered as an equally-spaced frequency set of peaks: $$S(\nu) = \sum^{k_{f}}_{k=k_{i}} \delta(\nu-k \Delta \nu) \, , \label{e:pmodesignal}$$ where $k_{i}$ and $k_{f}$ are integers with $k_{i}<k_{f}$; and the window function is $$H (\nu ) = \left\{ \begin{array}{lr} 1 & \nu_{i} \leq \nu \leq \nu_{f} \\ 0 & otherwise\, \end{array} \right. \label{e:hat}$$ where $\nu_{i}$ and $\nu_{f}$ are the frequency limits.\ It is possible to find the large separation by looking for those values with a higher number of peaks in quefrency space with a significant power excess at $q=k/ \Delta \nu$. This process is repeated for values close to the frequency limits of the subspectrum, only varying by a few $\mu$Hz. The number of coincidences for each possible periodicity is then counted (see bottom left panel in Fig. \[f:S\_C546\]).\ Not only can the large separation be found using this method, but also other periodicities such as the rotational splitting [@RocaCortes2001]. The major problem arises when the sought-after periodicity is not exactly uniform (Eq. \[e:pmodesignal\]) because our main assumption is broken. Nevertheless, the better the SNR of the observations, the higher the departure from an uniform frequency spacing that the method will be able to accept [@Regulo2002].\ We considered a frequency range down to approximately three times the highest studied periodicity to clearly detect possible regularities. To achieve a high SNR, we built an artificial light curve. The number of highest amplitude peaks ($I$) taken into account has to reach a compromise to include the spectral regularities and to avoid powering other periodicities or noise. Looking for this compromise, we tested several values of $I$. For most cases, taking into account $I=50$ peaks allowed us to find the rotational splitting, two times its value, the large separation, and half its value (see Table \[t:sdnumethod\] and bottom left panel of Fig. \[f:S\_C546\]).\ Echelle diagram {#ss:ED} --------------- The echelle diagram takes advantage of the regularity of the p-modes (Eq. \[e:pmodesignal\]) and represents the power spectrum in constant slices [@Grec1983]. If the value of the slice is the large separation, the modes with the same degree ($l$) and azimuthal order ($m$) will appear to be aligned (see bottom right panel in Fig. \[f:S\_C546\]). A possible deviation from this regular pattern is produced by the departure from the asymptotic regime and/or a high rotation rate.\ We used this property to delimit the large separation found by previous methods (see Table \[t:sdnumethod\]) within a given frequency range. For close values of the preliminary value, the number of consecutive modes aligned within its error is counted. The limit is found when the number of consecutive modes aligned is lower than a threshold. As happens with other methods, a high number of peaks can power other periodicities. Therefore, only the highest amplitude modes were taken into account.\ Results {#ss:results1} ------- Analysing the power spectra of CID 546 with all the methods already described (see Fig. \[f:S\_C546\]), we find the value for the large separation of $\Delta \nu = 55.4 \pm 0.8$ $\mu$Hz and a splitting of $s = 7.1 \pm 0.2$ $\mu$Hz. All methods detected the signature of the large separation and/or their split peaks as $\sim$47.8 and 61.8 $\mu$Hz. In addition, the differences between split peaks, $\sim$7.1 $\mu$Hz, are compatible with those produced by the the rotational signature or their multiples. Specifically, using the TRUFAS procedure (see bottom left panel in Fig. \[f:S\_C546\]), strong rotational signatures are detected at 6.0 and 7.8 $\mu$Hz, and also at twice (12.2 and 15.6 $\mu$Hz) and at thrice its value (23.3 $\mu$Hz). This departure from symmetric split peaks, around 0.9 $\mu$Hz, is in agreement with a moderate rotation rate.\ @Mantegazza2012 also search for regularities in the power spectrum of CID 546. They found a value for the large separation of $\Delta \nu = 46 \pm 6$ $\mu$Hz and a splitting of $s \sim 6.7$ $\mu$Hz. The value of the large separation is based on half of the highest peak of the FFT of the power spectrum (90.3 $\mu$Hz, see Fig. 17 in their publication). Although this value is compatible with the one we found, it is centred in one of the split peaks. Nevertheless, looking at their figure it is possible to observe three peaks that are consistent with the scenario described above.\ The stellar mass is calculated by @Mantegazza2012 using a grid of models. The value they find, 2.10 to 2.18 $M_{\odot}$, is higher than ours, $1.5\pm 0.3 M_{\odot}$ (see Table \[t:4dScu\] and the beginning of Sect. \[s:regular\] for more details), but their models do not reproduce the expected limits of the modes at the same time as the observed large separation. Nevertheless, both studies find that this star has a moderate rotation rate and low inclination. This is in agreement with its low projected velocity.\ Considering the other three stars, we find that KIC 5892969 has a low rotation rate, $\Omega / \Omega_{k} \approx 0.14$. This is confirmed by the signature of the surface rotation: two high amplitude peaks in the low frequency regime of the power spectra found at 1.235 and 2.465 $\mu$Hz with amplitudes around 100 and 500 ppm, respectively. Therefore, the values of the polar and equatorial radii are similar to the radius of a star with spherical symmetry (see Table \[t:4dScu\]). In addition, the values of the mass and radius are equal to those found by @Huber2014, within errors.\ In contrast, CID 8669 shows a high projected velocity, $v \sin i \sim$200 km/s, suggesting that this star could be a fast rotator with a very high rotation rate, the same as we find with our methodology (see Table \[t:4dScu\]). The high oblateness of this star produces a difference of temperature between the poles and the equator of around 320 K, $\sim4.5$ %.\ The case of CID 3619 has to be differentiated from the others. We confirm that this star might be a hybrid star because its convective efficiency ($\mathrm{log}~\Gamma$) is higher, and the kinetic energy of the waves ($\mathrm{log}~E_{kin}$) is lower, than the typical values for [[$\delta$ Scuti ]{}]{}stars (see Table \[t:4dScu\]). @Claret1998 estimate that stars with this temperature can present a more efficient convective zone, and that the gravity-darkening effect is less effective, $\beta \sim 0.32$ (see Eq. \[e:dteff\]). The variation of its temperature with latitude is then lower than that of CID 546 although its rotation rate and oblateness are higher.\ In depth study of the “*grass*” {#s:grass} =============================== Using an acoustic ray model in a uniformly rotating star, [@Lignieres2009] study the relation between the rotation rate and the power-spectral structure. Depending on the rotation rate regime the spectrum shows several kinds of modes such as the 2- & 6-period island modes that are restricted inside a torus region of the star, whispering gallery modes whose ray trajectories follow the outer boundary thanks to a rotation rate that has not destroyed its torus, and chaotic modes that are produced by rays that are not constrained into a torus.\ Lignières’ & Georgeot’s results (2009) show that chaotic modes are as visible as 2-period island modes and have higher amplitudes than 6-period island modes and whispering gallery modes when the rotation rate is moderate and the star is equator-on. This is caused by a lower disc-averaging cancellation of the chaotic behaviour than of the structured behaviour. The 2-period island modes have higher amplitudes than chaotic modes when the star is pole-on. On the one hand, for lower rotation rates, only 2-period island and whispering gallery modes are present because the chaotic regions are not developed enough. On the other hand, for higher rotations rates, all modes are present except for the 6-period island mode, whose torus has been destroyed.\ Because the four stars we are studying show different rotation rate and oblateness, our sample helps us to analyse how the power-spectral and structural parameters of [[$\delta$ Scuti ]{}]{}stars are modified by rotation. In that way, it allows us to compare our results with those predicted by [@Lignieres2009].\ The power spectrum of a [[$\delta$ Scuti ]{}]{}star is formed by moderate amplitude peaks grouped in bunches forming a power excess, the so-called envelope, and a high number of low amplitude peaks making a flat plateau or grass (e.g. Fig. \[f:S\_4dScu\_546\]). To define this power excess, we used the amount of energy of the observed signal carried by the wave (Eq. \[e:esignal\]) and we assumed that all peaks that fulfil $$E_{i} \gtrsim 0.1 \% \, \label{e:envelope}$$ are part of the envelope. We then estimated different characteristic parameters such as the energy of the power excess or the number of peaks enclosed ($N_{env}$).\ The flat plateau is a nearly-constant amplitude and mode density regime with a significant decrease at a specific frequency [e.g. @Poretti2009; @Mantegazza2012]. We called this the cut-off frequency ($\nu_{c}$) because the higher frequency modes are possibly not reflected as a result of losing their energy through the atmosphere, as predicted for p-modes in the standard theory. The amplitude decrease of the flat plateau ends when it reaches the noise level ($A_{N}$) at the frequency we called “noise frequency” ($\nu_{N}$). We supposed that all peaks that fulfil $$E_{i} \lesssim 0.01 \% \, \label{e:grasscond}$$ are part of the grass. Therefore, we can estimate its energy, and the number of peaks that constitute the grass, $N_{grass}$. To find its characteristic parameters, we can look for the variation of the density of peaks and also the variation in amplitude with frequency.\ We tested these two methods by comparing our results for CID 546 with those obtained by [@Mantegazza2012] (see Sect. \[ss:ndens\] and \[ss:grass\]). Then, we discuss the results for all the stars in Sect. \[ss:results2\] Density of peaks {#ss:ndens} ---------------- ![Density of peaks with frequency of the power spectrum of CID 546. The blue solid line indicates the mean density of peaks, the red dashed lines indicate the cut-off frequency and its density level. The green dotted and purple dashed-dotted lines indicate the frequency at maximum density and the noise frequency, respectively.[]{data-label="f:MoDens546"}](MoDens.eps){width="50.00000%"} ![Mean amplitude per bin of 10 $\mu$Hz of the CID 546 power spectrum after extraction of the peaks considered as envelope. The blue solid line indicates the grass level, and the red dashed lines indicate the cut-off frequency and its amplitude level. The purple dashed-dotted lines indicate the noise frequency and the mean amplitude of noise.[]{data-label="f:PlatLe546"}](PlatLe.eps){width="50.00000%"} The density of peaks can be determined with a histogram of analysed peaks per 10 $\mu$Hz frequency bin (see Fig. \[f:MoDens546\]). The cut-off frequency can then be determined as the frequency whose density value decays more than 1.5 times the standard deviation from the mean density of peaks ($\mathbf{n}_{mean}$). We also calculated the maximum density of peaks and the frequency at maximum density. We note that the separation between higher density peaks is useful in estimating the large separation.\ We find that the density of peaks in the power spectrum of CID 546 decays at a cut-off frequency of 405 $\pm$ 5 $\mu$Hz. This value agrees with that of @Mantegazza2012 found with their analysis.\ Grass level {#ss:grass} ----------- Following the extraction those peaks that are considered as the envelope (see Eq. \[e:envelope\]) from the power spectrum, we calculated the mean amplitude of the grass or grass level ($A_{grass}$; see Fig. \[f:PlatLe546\]). We also find the cut-off frequency as the frequency whose amplitude value decays to more than the standard deviation from the grass level. The noise level is measured as the mean amplitude of the residual power spectrum down to the noise frequency.\ Our analysis reveals that the grass level is an order of magnitude higher than the noise level and that the cut-off frequency is equal to 400 $\pm$ 10 $\mu$Hz. The flat plateau is clearly visible in the last two panels of Fig. \[f:S\_4dScu\_546\] after the extraction of hundreds of peaks. Our results are consistent with those found by @Mantegazza2012, because they also observe the flat plateau after the extraction of hundreds of frequencies with amplitudes down to 12 ppm.\ Results {#ss:results2} ------- ![Mean density of peaks with duration of the studied light curves for KIC 5892969 (red triangles), CID 546 (blue squares), and CID 8669 (purple asterisks). Each line is the linear fit to the data points.[]{data-label="f:REvolution_Dens"}](REvolution_Dens.eps){width="50.00000%"} ![Cut-off frequency (top), maximum amplitude (middle), and mean amplitude of the grass (bottom panel) versus duration of the studied light curve for CID 546 (blue squares), and CID 8669 (purple asterisks). The cut-off frequency of KIC 5892969’s power spectrum is not visible therefore only the maximum amplitude of the grass can be properly calculated (red triangles). Each line is the linear fit to the observed data points. The mean amplitude of the grass for CID 546 has been increased by 20 ppm to properly observe its behaviour.[]{data-label="f:REvolution_Plat"}](REvolution_Plat.eps){width="50.00000%"} Model KIC 5892969 CID 546 CID 8669 --------------------------- ---------------------------------------- ------------- ----------------- ----------------- ----------------- $\Omega / \Omega_{k}$ (%) 59 14 $\pm$ 7 31 $\pm$ 3 70 $\pm$ 7 $ \nu_{c}$ ($\mu$Hz) - 400 $\pm$ 10 310 $\pm$ 7 $\dot{\mathbf{n}}_{mean}$ ($\frac{peaks\, 10^{-2}}{\mu Hz \,d}$) 0.32 $\pm$ 0.03 2.27 $\pm$ 0.25 5 $\pm$ 1 $\mathbf{n}_{mean} \{0\}$ (peaks/$\mu$Hz) 0.16 $\pm$ 0.03 0.89 $\pm$ 0.19 1.38 $\pm$ 1.00 $N_{env}$ (peaks) 34 24 $\pm$ 1 17 $\pm$ 6 29 $\pm$ 3 $N_{grass} \{0\}$ (peaks) 270 $\pm$ 8 13 $\pm$ 6 282 $\pm$ 64 320 $\pm$ 250 The observed power-spectral structure of these [[$\delta$ Scuti ]{}]{}stars consists of a few dominant amplitude modes and a lot of low amplitude peaks with the exception of CID 3619, which is a hybrid star (see Fig \[f:S\_4dScu\_3619\]). Therefore, only considering these three non-hybrid [[$\delta$ Scuti ]{}]{}stars, CID 546, CID 8669, and KIC 5892969 together, we find that the mean density of peaks present in their power spectra increases linearly with the duration of the observing campaign ($\Delta t$; see Fig. \[f:REvolution\_Dens\]). The density of peaks and their increase with time ($\dot{\mathbf{n}}_{mean}$) are higher as the rotation rate is higher too (see Table \[t:revolution\]). Therefore, the mechanism that produces this high number of peaks is related to the rotation rate, and of the increase in frequency content explains the light curve behaviour with time. Taking into account this relation, and also that the subtracted energy of the signal remains constant or slightly decreases with duration, it is not possible that all these peaks are spurious owing to an imperfect subtraction of the signals, as suggested by [@Balona2014a].\ The number of modes not caused by time variations, $N_{grass}\{0\}$, can be estimated with the y-intercept constant of the $n_{mean}$-$\Delta t$ relation while taking into account the observed frequency limits $\nu \in \left[60, \nu_{c} \right]$ $\mu$Hz (see Fig. \[f:REvolution\_Dens\] and Table \[t:revolution\]). Their values are of the same order of magnitude as those chaotic modes estimated by @Lignieres2009, which take into account only axisymmetric modes in a characteristic frequency range for a computed model of a star with rotation rate around $\Omega /\Omega_{K} \sim$0.59. In addition, the observed number of modes in the envelope, $N_{env}$ (those that fulfil Eq. \[e:envelope\]), are also similar to those expected for 2-period island modes. As we can see, a star with higher rotation shows a higher number of chaotic modes because the torus of less-visible modes has been destroyed. The chaotic modes seem to be more visible than the 6-period island modes or the whispering gallery modes due to their irregularity, which makes the cancellation effect less effective.\ Moreover, the maximum amplitude and cut-off frequency in CID 546 and CID 8669 are constant with time (see Fig. \[f:REvolution\_Plat\]). As the number of lower peaks increases, the mean value of the amplitude of the grass decreases. This is also in agreement with a scenario in which initial 2-period island modes and chaotic modes with time variations are present. In agreement with the predicted visibility [@Lignieres2009], CID 546 presents a similar number of 2-period island modes as the other stars in the sample, but they are of higher amplitudes due to its low inclination.\ It is not expected that a low rotation rate [[$\delta$ Scuti ]{}]{}star has chaotic modes. This is in agreement with the initial number of modes that we estimate for KIC 5892969. This star shows a slight decrease of its maximum amplitude of the grass. Therefore, the high number of peaks present in its power spectrum of the whole light curve can be produced by time variations of the 2-period island modes and some whispering gallery modes.\ Finally, although CID 3619 has a higher rotation rate than CID 546, its spectral density is lower and there is not a clear flat plateau. The cause could be CID 3619’s more efficient convective zone. Although @Balona2014 claim that all [[$\delta$ Scuti ]{}]{}stars are hybrids, to identify the star as a hybrid or a non-hybrid star with the criteria specified in Sect. \[s:dScu\] could be of importance to explaining the presence, or absence, of the flat plateau.\ Conclusions {#s:conclusions} =========== Using our own methodology ($\delta$SBF), we analysed the light curves of four [[$\delta$ Scuti ]{}]{}stars, observed by CoRoT and Kepler, from raw data to end products such as the parameters of the modes, the properties of the flat plateau, and possible regularities of the power spectra. We thus determine their observational characteristics producing the best estimates to date of their stellar parameters such as mass, inclination, rotation rate, and convective efficiency. In spite of the high uncertainties in previously known data, the oblateness and the gravity-darkening effect were obtained for all the stars studied. Furthermore, CID 3619 was found to be a hybrid $\delta$ Scu/$\gamma$ Dor star.\ Because these four stars show different rotation rates and oblateness values, our sample allows us to study how the power-spectral and structural parameters of [[$\delta$ Scuti ]{}]{}stars are modified by rotation. We prove that structural parameters such as oblateness, inclination, and convective efficiency can explain the development of the flat plateau. Therefore the power-spectral structure is formed by an envelope constituted of 2-period island modes, and a grass composed of chaotic modes and peaks due to their variation. In this sense, the spurious signal hypothesis is discarded. Our next step is to perform a study of a much larger sample of [[$\delta$ Scuti ]{}]{}stars to provide an in-depth determination of the behaviour of their power-spectral structure.\ #### The authors wish to thank the *CoRoT* and *Kepler* Teams whose efforts made these results possible. The *CoRoT* space mission has been developed and was operated by *CNES*, with contributions from Austria, Belgium, Brazil, ESA (RSSD and Science Program), Germany, and Spain. Funding for *Kepler*’s Discovery mission is provided by NASA’s Science Mission Directorate. S.B.F. wishes to thank E. Michel for encouraging him to study [[$\delta$ Scuti ]{}]{}stars, and the Solar Physics Team of the *Universitat de les Illes Balears* (UIB) for hosting his stay in Majorca. He has received financial support from the Spanish Ministry of Science and Innovation (MICINN) under the grant AYA2010-20982-C02-02. A.G.H. acknowledges support from Fundação para a Ciência e a Tecnologia (FCT, Portugal) through the fellowship SFRH/BPD/80619/2011. R.A.G. acknowledges the financial support from the ANR (Agence Nationale de la Recherche, France) program IDEE (n ANR-12-BS05-0008) “Interaction Des Étoiles et des Exoplanètes” and from the CNES GOLF and PLATO grants at CEA. Obtained parameters of the oscillation modes of [[$\delta$ Scuti ]{}]{}stars {#ap:PoM} ============================================================================ ------- ----------- ----------- -------- -------- -- -- Term Frequency Amplitude Phase Energy i ($\mu$Hz) (ppm) (rad) (%) 0 198.630 12181 -0.213 86.13 1 151.040 2197 1.584 2.80 2 188.078 1879 -1.805 2.05 3 185.213 1487 2.963 1.28 4 94.173 1426 -2.382 1.18 5 158.108 1219 -2.349 0.86 6 82.545 908 -2.782 0.48 7 200.005 836 -2.590 0.41 8 175.679 907 0.706 0.48 9 198.915 642 -0.159 0.24 10 88.663 776 2.009 0.35 11 183.117 523 2.940 0.16 12 183.265 641 -1.309 0.24 13 397.259 476 -1.270 0.13 14 174.049 736 2.657 0.32 15 161.716 509 2.999 0.15 16 162.741 438 -3.077 0.11 Error 0.001 5 0.005 ------- ----------- ----------- -------- -------- -- -- We present the parameters of the highest-amplitude oscillation modes of each star that we obtain with our method. The results for KIC 5892969 were already published in @BarceloForteza2015. Because all of these modes accomplish the condition announced in Eq. \[e:envelope\], they form part of the so-called envelope. Hundreds or thousands of peaks are identified with a SNR higher than four for each light curve (see from Fig. \[f:S\_4dScu\_546\] to Fig. \[f:S\_4dScu\_5893969\]).\ We note that each oscillation mode of CID 3619’s light curve has two different frequencies, one per studied run. Because these runs are separated by approximately four years, the differences in all the parameters might be caused by a modulation mechanism such as RMC. Although the observed frequency variation is of the same order of magnitude as the predicted one [$\delta \nu / \nu \lesssim$0.1 %, see @Moskalik1985], it is not enough to ascertain which mechanism produces the variation of the envelope modes. Nevertheless, the cause of these variations is beyond of the scope of this work.\ ------- --------- --------- ------- ------- ------ ------ ------- ------- Term i 0 185.755 185.751 430.9 434.0 2.23 1.60 23.71 23.15 1 111.340 111.339 477.3 494.3 2.59 0.07 29.10 30.03 2 119.308 119.300 246.0 227.7 5.37 1.42 7.72 6.37 3 15.625 15.615 224.8 111.2 4.34 0.01 6.48 1.52 4 16.101 16.038 231.4 291.2 2.84 4.40 6.82 10.44 5 145.069 145.071 171.7 177.5 2.03 1.88 3.77 3.87 6 118.165 118.168 156.1 156.2 0.29 0.04 3.11 3.00 7 247.298 247.291 133.2 137.9 4.77 5.87 2.27 2.33 9 9.555 9.601 89.0 47.1 0.22 2.96 1.01 0.27 10 21.040 21.114 83.6 55.0 3.19 5.59 0.89 0.37 14 11.266 11.283 74.6 105.1 2.15 4.79 0.71 1.36 15 160.200 160.185 63.2 49.1 2.37 0.45 0.51 0.30 16 14.888 14.915 66.9 51.1 4.39 3.71 0.57 0.32 19 47.455 47.331 48.8 33.1 4.03 2.91 0.31 0.13 24 95.656 95.448 38.3 66.3 5.44 5.07 0.19 0.54 25 28.601 28.620 49.8 38.8 4.46 2.17 0.32 0.19 28 22.349 22.332 37.1 33.1 2.90 5.83 0.18 0.13 29 26.278 26.405 54.9 56.3 2.12 1.51 0.39 0.39 32 21.598 21.658 32.9 29.5 1.29 2.13 0.14 0.11 33 35.508 35.422 36.6 38.7 2.07 5.77 0.17 0.18 Error 0.005 0.003 1.1 0.9 0.01 0.01 ------- --------- --------- ------- ------- ------ ------ ------- ------- ------- ----------- ----------- -------- -------- -- -- Term Frequency Amplitude Phase Energy i ($\mu$Hz) (ppm) (rad) (%) 0 176.4481 3226 1.249 23.12 1 154.5399 2952 2.206 19.35 2 103.6354 2549 0.018 14.43 3 122.0414 1454 1.285 4.69 4 110.3414 1432 -1.988 4.56 5 121.1245 1301 -1.026 3.76 6 89.2400 1087 1.117 2.63 7 100.6208 968 -2.142 2.08 8 88.6171 1100 -1.710 2.69 9 145.1459 1227 -0.043 3.35 10 180.4720 1136 0.984 2.87 11 209.0796 881 1.313 1.73 12 100.0815 1139 -1.952 2.88 13 169.8589 675 1.512 1.01 14 120.0376 702 0.857 1.10 15 98.9628 667 -2.154 0.99 16 87.9891 457 -0.136 0.46 17 211.9782 420 0.685 0.39 18 153.6621 413 1.736 0.38 19 91.2896 502 0.614 0.56 20 117.2402 374 -1.658 0.31 21 100.1849 313 2.748 0.22 23 130.1431 361 2.271 0.29 24 151.2113 402 0.676 0.36 25 125.2902 305 -2.142 0.21 26 170.5608 344 2.131 0.26 27 104.6492 312 1.094 0.22 28 171.9381 252 1.744 0.14 30 142.2407 232 2.521 0.12 31 209.0220 226 -0.989 0.11 32 162.2456 223 -2.442 0.11 Error 0.0007 3 0.005 ------- ----------- ----------- -------- -------- -- --

--- author: - 'Charalampos <span style="font-variant:small-caps;">Skokos</span>$^{1,2,}$[^1], Chris <span style="font-variant:small-caps;">Antonopoulos</span>$^{1,}$[^2], Tassos C.<span style="font-variant:small-caps;">Bountis</span>$^{1,}$[^3] and Michael N. <span style="font-variant:small-caps;">Vrahatis</span>$^{3,}$[^4]' title: 'How does the Smaller Alignment Index (SALI) distinguish order from chaos? ' --- Introduction ============ The evaluation of the [**Smaller Alignment Index (SALI)**]{} is an efficient and simple method to determine the ordered or chaotic nature of orbits in dynamical systems. The SALI was proposed in Ref.  and it has been successfully applied to distinguish between ordered and chaotic motion both in symplectic maps [@Sk01] as well as in Hamiltonian flows.[@GRACM] In order to compute the SALI for a given orbit one has to follow the time evolution of the orbit itself and two deviation vectors which initially point in two different directions. The evolution of these vectors is given by the variational equations for a flow and by the tangent map for a discrete–time system. At every time step the two vectors $\overrightarrow{v_1}(t)$, $\overrightarrow{v_2}(t)$ are normalized and the SALI is computed as: $$SALI(t)= \min \left\{ \left\| \frac{\overrightarrow{v_1}(t)}{\|\overrightarrow{v_1}(t)\|}+ \frac{\overrightarrow{v_2}(t)}{\|\overrightarrow{v_2}(t)\|} \right\| ,\left\| \frac{\overrightarrow{v_1}(t)}{\|\overrightarrow{v_1}(t)\|} -\frac{\overrightarrow{v_2}(t)}{\|\overrightarrow{v_2}(t)\|} \right\| \right\}, \label{eq:SALI}$$ where $t$ is the continuous or the discrete time and $\|\cdot\|$ denotes the Euclidean norm. The properties of time evolution of the SALI clearly distinguish between ordered and chaotic motion as follows: In the case of Hamiltonian flows or $N$–dimensional symplectic maps with $N\geqslant 2$ the SALI fluctuates around a non-zero value for ordered orbits, while it tends to zero for chaotic orbits.[@Sk01; @GRACM] In the case of 2D maps the SALI tends to zero both for ordered and chaotic orbits, following however completely different time rates, which again allows us to separate between these two cases also.[@Sk01] We have recently begun to understand the different behaviors of the SALI in regions of order and chaos. In the latter case, we have been able to connect SALI’s rapid convergence to zero, to the influence of the two largest positive Lyapunov exponents of the motion.[@prep] In the present paper we shall study the behavior of the SALI in the case of ordered orbits. The behavior of the SALI for ordered motion =========================================== Let us try to understand why the SALI does not become zero in the case of ordered motion, by studying in detail the behavior of the deviation vectors. A suitable way to do this for conservative systems is to consider a non-trivial integrable Hamiltonian model whose orbits are bounded and lie on “nested” tori, which foliate all of the available phase space.[@LL] An integrable such Hamiltonian system of 2 degrees of freedom possesses besides the Hamiltonian $H$ a second independent integral $F$, in involution with $H$: $$\{ H,F \}=0, \label{eq:HF}$$ where $\{ \cdot , \cdot \} $ denotes the usual Poisson bracket. In such systems, the motion lies in the intersection of both manifolds $$H=\widetilde{h}, \,\,\, F=\widetilde{f}, \label{eq:man}$$ where $\widetilde{h}$, $\widetilde{f}$ are the constant values of the two integrals. Thus, the orbits in the 4–dimensional phase space move instantaneously on a 2–dimensional “tangent” subspace, which is ‘perpendicular’ to the vectors $$\overrightarrow{\nabla H}= (H_x,H_y,H_{p_x}, H_{p_y}), \,\,\, \overrightarrow{\nabla F}= (F_x,F_y,F_{p_x}, F_{p_y}), \label{eq:grads}$$ $x$, $y$ being the generalized coordinates of the system and $p_x$, $p_y$ their conjugate momenta, while subscripts denote partial derivatives (e. g. $H_x \equiv \frac{\partial H}{\partial x}$). In fact, the motion may be thought of as governed by either one of the Hamiltonian vector fields $$\overrightarrow{f_H}= (H_{p_x}, H_{p_y},-H_x,-H_y), \,\,\, \overrightarrow{f_F}= (F_{p_x}, F_{p_y},-F_x,-F_y). \label{eq:flow}$$ The vectors $\overrightarrow{\nabla H}$, $\overrightarrow{\nabla F}$ (and hence also $\overrightarrow{f_H}$, $\overrightarrow{f_F}$) are linearly independent due to the functional independence of the two integrals at almost all points in phase space. So the corresponding unit vectors $$\widehat{f_H} = \frac{\overrightarrow{f_H}}{\|\overrightarrow{f_H}\|} \bot \widehat{\nabla H}, \,\,\, \widehat{f_F} = \frac{\overrightarrow{f_F}}{\|\overrightarrow{f_F}\|} \bot \widehat{\nabla F}, \,\,\, \mbox{ with } \,\,\, \widehat{\nabla H} = \frac{\overrightarrow{\nabla H}}{\|\overrightarrow{\nabla H}\|}, \,\,\, \widehat{\nabla F} = \frac{\overrightarrow{\nabla F}}{\|\overrightarrow{\nabla F}\|} \label{eq:base}$$ can be used as a basis for the 4–dimensional space where the deviation vectors evolve. This basis is in general not orthogonal as $$\langle \widehat{\nabla H}, \widehat{\nabla F} \rangle = \langle \widehat{f_H}, \widehat{f_F} \rangle = \frac{H_x F_y + H_y F_y +H_{p_x} F_{p_x} + H_{p_y} F_{p_y}} {\|\overrightarrow{\nabla H}\| \,\|\overrightarrow{\nabla F}\|} \label{eq:dots1}$$ is not necessary zero. We note that $\|\overrightarrow{\nabla H}\| =\|\overrightarrow{f_H}\|$, $\|\overrightarrow{\nabla F}\| =\|\overrightarrow{f_F}\|$ and $\langle \cdot , \cdot \rangle$ denotes the usual inner product. Note also that from definitions (\[eq:grads\]) and (\[eq:flow\]) we get $\langle \widehat{\nabla H}, \widehat{f_H} \rangle = \langle \widehat{\nabla F}, \widehat{f_F} \rangle = 0$, while (\[eq:HF\]) yields $\langle \widehat{\nabla H}, \widehat{f_F} \rangle = \langle \widehat{\nabla F}, \widehat{f_H} \rangle = 0.$ So, using vectors (\[eq:base\]) as a basis for studying the evolution of a deviation vector $\overrightarrow{v_1}$, we can write it as $$\overrightarrow{v_1} = a_1 \widehat{f_H} + a_2 \widehat{f_F} + a_3 \widehat{\nabla H} + a_4 \widehat{\nabla F} \label{eq:vector}$$ with $a_1, \, a_2, \, a_3, \, a_4 \in \mathbb{R}$. The values of the coefficients $a_i$, $i=1,2,3,4$, at different times, give us a clear picture for the evolution of $\overrightarrow{v_1}$. In the case of the 2D standard map for example, where ordered orbits lie on an invariant curve (1D torus), it has been shown both numerically and analytically[@Voz] that any deviation vector (considered as a linear combination of the vectors $\widehat{f_H}$, $\widehat{\nabla H}$ using our notation), eventually becomes tangent to the invariant curve, tending to the tangential direction as $n^{-1}$, with $n$ being the number of iterations. Similarly, in the case of an integrable 2D Hamiltonian the deviation vector $\overrightarrow{v_1}$ tends to fall on the “tangent space” of the torus, spanned at each point by $\widehat{f_H}$, $\widehat{f_F} $, meaning that in Eq. (\[eq:vector\]) $a_3 \rightarrow 0$, $a_4 \rightarrow 0$, while the $a_1$, $a_2$ are, in general, different from zero. This is analogous to what has been found for the 2D standard map in Ref. . As a model for studying this behavior let us consider the 2D Van der Waals Hamiltonian [@kn:1] $$H(x,y,p_{x},p_y)= \frac{1}{2}(p_{x}^{2}+p_y^{2})-E(x^2+y^2)+A(x^{6}+y^6)+ B(x^{4} y^2+x^{2} y^{4}), \label{eq:Ham1}$$ where $E$, $A$, $B$ are real parameters. For $B=3A$ and $E\in\mathbb{R}$ the Hamiltonian (\[eq:Ham1\]) is completely integrable and the second integral of motion is given by[@kn:1] $$F(x,y,p_{x},p_y)=(x p_y-y p_{x})^{2}. \label{eq:Fint}$$ In our calculations we consider the integrable case for $A=0.25$, $B=3A=0.75$ and $E=-10^{-8}$. ![ The time evolution of the coefficients $a_1$, $a_2$, $a_3$, $a_4$, of the initial deviation vector with $a_1=1$, $a_2=1$, $a_3=0$, $a_4=1$ (Eq. (\[eq:vector\])). (a) $a_1$ (black line), $a_2$ (gray line). (b) $|a_3|$ (black line), $|a_4|$ (gray line) in log-log scale. The orbit of the Hamiltonian (\[eq:Ham1\]) used, has initial condition $x=-0.6$, $y=0$, $p_x=0$, $p_y=1.99416$. In (b) we also plot the curves $|a_3|=0.107 \cdot t^{-0.962}$, $|a_4|=1.067 \cdot t^{-0.995}$ that fit the data.[]{data-label="fig:1"}](fig1.eps){width="15" height="7.5"} For different initial deviation vectors, we compute the time evolution of the coefficients $a_1$, $a_2$, $a_3$, $a_4$ of Eq. (\[eq:vector\]). We find that in all cases $a_1$, $a_2$ remain different from zero, while $a_3$, $a_4$ tend to zero. A particular example is given in Fig. \[fig:1\]. By fitting the data of Fig. \[fig:1\]b we see that $|a_3|$, $|a_4|$ $\propto t^{-1}$. From Fig. \[fig:1\] we conclude that any vector will eventually fall on the “tangent space” of the torus on which the orbit evolves. This “tangent space” is produced by vectors $\widehat{f_H}$, $\widehat{f_F}$, and so any deviation vector will eventually become a linear combination of these two vectors only. As there is no particular reason for two different initial deviation vectors to end up with the same values of $a_1$, $a_2$, the SALI (\[eq:SALI\]) will in general oscillate around a value different from zero. In other words the two vectors become tangent to the torus and fluctuate quasiperiodically about two different directions. This becomes evident in Fig. \[fig:2\]a where we plot the time evolution of the SALI for an orbit with initial conditions $x=-0.6$, $y=0$, $p_x=0$, $p_y=1.99416$ marked by a black point in the Poincaré Surface of Section (PSS) of the system seen in Fig. \[fig:2\]b. The initial deviation vectors used are $\overrightarrow{v_1}= \widehat{f_H} + \widehat{f_F} + \widehat{\nabla F}$ (the time evolution of which is given in Fig. \[fig:1\]) and $\overrightarrow{v_2}=\widehat{f_H} + \widehat{f_F} + \widehat{\nabla H} $. ![The time evolution of the SALI (a) for the ordered orbit marked by a black point in the PSS $y=0$ of the system (b).[]{data-label="fig:2"}](fig2.ps){width="15" height="7.5"} Conclusions =========== In this paper we have analyzed the behavior of the SALI in regions of ordered motion, by studying the evolution of deviation vectors in the case of an integrable 2D Hamiltonian system. Using a suitable basis of 4 vectors (\[eq:base\]) we have shown that any pair of arbitrary deviation vectors tends to the tangential space of the torus, following a $t^{-1}$ time evolution and having in general 2 different directions. This explains why for ordered orbits the SALI oscillates quasiperiodically about values that are different from zero. The same result is observed to hold in the case of ordered motion in a stability region of a non–integrable system,[@GRACM] where the presence of “islands” implies the existence of an additional approximate integral $F$, independent of the Hamiltonian. For Hamiltonian systems of more than 2 degrees of freedom we expect similar results. The only difference is that the “tangent space” is of higher dimension generated by the vectors $\widehat{f_H}, \,\,\widehat{f_{F_1}},\,\,\widehat{f_{F_2}}, \,\ldots$, with $F_1$, $F_2, \, \ldots$, being the additional (approximate or not) integrals of the motion. Acknowledgements {#acknowledgements .unnumbered} ================ Ch. Skokos was supported by the ‘Karatheodory’ post–doctoral fellowship No 2794 of the University of Patras and by the Research Committee of the Academy of Athens (program No 200/532). Ch.Antonopoulos was supported by the ‘Karatheodory’ graduate student fellowship No 2464 of the University of Patras. [99]{} Ch. Skokos, . Ch. Skokos, Ch. Antonopoulos, T. C. Bountis and M. N. Vrahatis, in *Proceedings of the 4th GRACM Congress on Computational Mechanics*, ed. D. T. Tsahalis, (Univ. Patras, Patras, 2002), Vol. IV, p. 1496; in *Libration Point Orbits and Applications*, eds. G. Gómez, M. W. Lo and J. J. Masdemont, (World Scientific, 2003), in press, nlin.CD/0210053. Ch. Skokos, Ch. Antonopoulos, T. C. Bountis and M. N. Vrahatis, 2003, in preparation. M. A. Lieberman and A. J. Lichtenberg, *Regular and Chaotic Dynamics* (Springer Verlag, 1992). Ch. L. Vozikis, . K. Ganesan and M. Lakshmanan, . [^1]: E-mail: hskokos@cc.uoa.gr [^2]: E-mail: antonop@math.upatras.gr [^3]: E-mail: bountis@math.upatras.gr [^4]: E-mail: vrahatis@math.upatras.gr

--- abstract: 'We study explosive percolation (EP) on Erdös-Rényi network for product rule (PR) and sum rule (SR). Initially, it was claimed that EP describes discontinuous phase transition, now it is well-accepted as a probabilistic model for thermal continuous phase transition (CPT). However, no model for CPT is complete unless we know how to relate its observable quantities with those of thermal CPT. To this end, we define entropy, specific heat, re-define susceptibility and show that they behave exactly like their thermal counterparts. We obtain the critical exponents $\nu, \alpha, \beta$ and $\gamma$ numerically and find that both PR and SR belong to the same universality class and they obey Rushbrooke inequality.' author: - 'M. K. Hassan and M. M. H. Sabbir' title: 'Product-Sum universality and Rushbrooke inequality in explosive percolation ' --- The notion of percolation is omnipresent in many seemingly disparate natural and man-made systems [@ref.Stauffer]. Examples include spread of forest fire, flow of fluid through porous media, spread of biological and computer viruses etc. [@ref.saberi; @ref.Newman_virus; @ref.Moore_virus]. Besides such direct applications, percolation is best known as a paradigmatic model for phase transition. One of the simplest models for percolation is the classical random percolation (RP) on Erdös-Rényi (ER) network in which one starts with $N$ labeled nodes that are initially all isolated [@ref.erdos]. Then at each step a link, say $e_{ij}$, is picked at random from all the possible pair of links and occupy it to connect nodes $i$ and $j$. As the number of occupied links $n=tN$ increases from zero we find that clusters, i.e. contiguous nodes connected by occupied links, are formed and on the average grown. In the process, the largest cluster $s_{{\rm max}}$ undergoes a transition across $t_c=0.5$ from minuscule size ($s_{{\rm max}}\sim \log N$) to giant size ($s_{{\rm max}} \sim N$). The emergence of such threshold value $t_c$ is found to be accompanied by a sudden change in the order parameter $P$, the ratio of the largest cluster to the network size, such that $P=0$ at $t\leq t_c$ and $P>0$ at $t>t_c$ in the limit $N\rightarrow \infty$. This is reminiscent of the second order or continuous phase transition (CPT). In 2009, Achlioptas [*et al.*]{} proposed a class of percolation model in which two links are picked randomly instead of one at each step [@ref.Achlioptas] . However, ultimately only one of the links, that results in the smaller clustering, is occupied and the other is discarded for future picking. One of the key features of this rule, which is now known as the Achlioptas process (AP), is that it discourages the growth of the larger clusters and encourages the smaller ones which inevitably delays the transition. Eventually, when it reaches near the critical point it is so unstable that occupation of one or two links triggers an explosion of growth. It leads to the emergence of a giant cluster with a bang and hence it is called “explosive percolation" (EP). Indeed, the corresponding $P$, in contrast to its classical counterpart, undergoes such an abrupt transition that it was at first mistaken as a discontinuity and suggested to exhibit the first order or discontinuous transition. Their results jolted the scientific community through a series of claims, unclaims and counter-claims [@ref.Friedman; @ref.ziff_1; @ref.radicchi_1; @ref.Costa_2; @ref.souza; @ref.cho_1; @ref.ara; @ref.da_Costa; @ref.Grassberger; @ref.Bastas]. It is now well settled that the explosive percolation transition is actually continuous but with first order like finite-size effects [@ref.Grassberger; @ref.Bastas; @ref.Riordan; @ref.bastas_review; @ref.Choi]. In general, scientists use theoretical model, just like architects use geometric model before building large expensive structure, because it provides useful insights into the real-world systems. The real systems that percolation represent is complex as it often involves quantum and many particle interaction effects. However, modeling is only useful if we know how to relate its various observable quantities to those of the real-world systems. To this end, Kasteleyn and Fortuin used the mapping of the percolation problem onto the $q$-state Potts model in order to relate its observable to the thermal quantities of the Potts model [@ref.Kasteleyn]. Owing to that mapping we know that $P$ is the order parameter, mean cluster size $\langle s\rangle$ is the susceptibility etc. but not equivalent counterpart of entropy. In thermal CPT, the entropy $S$ and the order parameter (OP) complement each other as $S$, that measures the degree of disorder, is maximum where OP is zero and OP, that measures the extent of order, is maximum where $S$ is zero. A similar behaviour in percolation is also expected in order to elucidate whether it is also an order-disorder transition or not. Universality is another aspect that we find common in the thermal CPT and in the random percolation. In the case of EP, we are yet to find universality of any type or any kind. Another interesting aspect of thermal CPT is that its critical exponents $\alpha, \beta$ and $\gamma$ obey the Rushbrooke inequality $\alpha+2\beta+\gamma\geq 2$ which reduces to equality under static scaling hypothesis [@ref.Stanley]. Whether it holds in explosive percolation or not, is also an interesting issue. In this article, we investigate EP on the ER networks for product rule (PR) and sum rule (SR) and find their critical exponents numerically. First, we define susceptibility $\chi$ as the ratio of the successive jump $\Delta P$ of $P$ and the magnitude of successive intervals $\Delta t$ instead of using the mean cluster size $\langle s\rangle$ as susceptibility. Then we obtain the critical exponents $\nu$ of the correlation length, $\gamma$ of $\chi$, and $\beta$ of $P$. Note that $\langle s\rangle$ exhibits the expected divergence only if the largest cluster size is excluded from it and even then it gives too large a value of $\gamma$. Realizing these drawbacks, many researchers are already considering alternative definitions [@ref.radicchi_1; @ref.ziff_3; @ref.qian]. Second, we define entropy $H$ for EP and find that it is continuous across the whole spectrum of the control parameter $t$ which clearly reveals that EP transition is indeed continuous in nature. We then define the specific heat as $C=q{{dH}\over{dq}}$ where $q=(1-t)$ and find that it diverges with positive critical exponent $\alpha$. The most intriguing and unexpected findings of this work is that PR and SR belong to the same universality class. Besides, we find that the elusive Rushbrooke inequality holds in EP. Recently, using the the same definitions for entropy, specific heat and susceptibility we have shown that the Rushbrooke inequality holds in the random percolation too [@ref.hassan_didar]. Finding that RI also holds in EP on random network provides a clear testament of how robust our results are. Percolation is all about clusters as every observable quantity of it is related, this way or another, to the clusters by virtue of definition. Initially, all the labeled nodes are considered isolated so that every node is a cluster of its own size. The process starts by picking two distinct links, say $e_{ij}$ and $e_{kl}$, randomly at each step. To apply the PR, we then calculate the products, $\Pi_{ij}=s_i\times s_j$ and $\Pi_{kl}= s_k \times s_l$, of the size of the clusters that the two nodes on either side of each link contain. The link with the smaller value of the products $\Pi_{ij}$ and $\Pi_{kl}$ is occupied. On the other hand, if we find $\Pi_{ij}=\Pi_{kl}$ then we occupy one of the two links at random with equal probability. In the case of SR, we take the sum $\Sigma_{ij}=s_i+s_j$ and $\Sigma_{kl}=s_k+s_l$ instead of the product and do the rest exactly in the same way as we did for PR. Each time we occupy a link, either the size of an existing cluster grows due to occupation of an inter-cluster link or the cluster size remains the same due to addition of an intra-cluster link. In either case, the growth of large clusters are always disfavoured which is in sharp contrast to its RP counterpart. Thus, the emergence of a giant cluster is considerably slowed down but eventually when it happens, it happens abruptly but without discontinuity. We first investigate not the $P$ itself but its successive jump $\Delta P$ within successive interval $\Delta t=1/N$. The idea of successive jump size $\Delta P$ was first introduced by Manna [@ref.manna]. We use it to define susceptibility as $$\chi(t)={{\Delta P}\over{\Delta t}},$$ which essentially becomes the derivative of $P$ in the limit $N\rightarrow \infty$. In Figs. (\[fig:1a\]) and (\[fig:1b\]) we show plots of $\chi$ versus $t$ for both PR and SR model. According to the finite-size scaling (FSS) hypothesis, the susceptibility $\chi_{{\rm max}}$ at $t=t_c$ increases following a power-law $\chi_{{\rm max}}\sim N^{\gamma/\nu}$. In an attempt to verify this we plot $\log(\chi_{{\rm max}})$ vs $\log(N)$, see insets of Figs. (\[fig:1a\]) and (\[fig:1b\]), and find straight lines with slopes $\gamma/\nu=0.480(3)$ for PR and $\gamma/\nu=0.475(4)$ for SR. Following the procedures in Ref. [@ref.Hassan_Rahman_1] we also get a rough estimate of the exponent $1/\nu = 0.535(5)$ for PR and $1/\nu = 0.537(1)$ for SR. The FSS theory further suggests that if we now plot $\chi N^{-\gamma/\nu}$ vs $(t_c-t)N^{1/\nu}$, all the distinct plots of Figs. (\[fig:1a\]) and (\[fig:1b\]) should collapse into their respective universal curves. Indeed, by tuning ${\gamma/\nu}$ and $1/\nu$ we find excellent data collapse, see Figs. (\[fig:1c\]) and (\[fig:1d\]), if we use $\gamma/\nu=0.478$ and $1/\nu=0.535$ for PR and SR respectively. Note that $t_c$ value also affects the data collapse and hence tuning the initial estimates for $t_c$ we get the best data-collapse if we use $t_c=0.88850$ for PR and $t_c= 0.86018$ for SR. The quality of data collapse itself provides a clear testament to the extent of accuracy of these values. What is most noteworthy, however, is that both PR and SR share the same value for the exponents $\gamma$ and $\nu$. Using now the relation $N\sim (t-t_c)^{-\nu}$ in $\chi\sim N^{\gamma/\nu}$ we find that $$\chi\sim (t-t_c)^{-\gamma},$$ where $\gamma=0.893$ for both PR and SR rules within the acceptable limit of error. We find that the susceptibility now diverges even without the exclusion of the largest cluster and that too with the same $\gamma$. Now, we consider the order parameter $P$ itself and plot it as a function of $t$ in Figs. (\[fig:2a\]) and (\[fig:2b\]) for PR and SR respectively. We follow the same standard procedure as in Ref. [@ref.Hassan_Rahman_1; @ref.Hassan_Rahman_explosive] and find $\beta/\nu=0.045$ for both the variants. It is well-known that $P(t,N)$ exhibits finite-size scaling. One way of testing it is to plot $PN^{\beta/\nu}$ vs $(t-t_c)N^{1/\nu}$ and check if all the distinct curves of $P$ vs $t$ curves collapse or not. Indeed, Figs. (\[fig:2c\]) and (\[fig:2d\]) suggest that they all collapse superbly with $\beta/\nu=0.045$ and $1/\nu=0.535$ values regardless of whether it is PR or SR. Substituting the relation $N\sim (t-t_c)^{-\nu}$ in $P\sim N^{-\beta/\nu}$ we get $$P(t)\sim (t-t_c)^\beta.$$ This is exactly how the order parameter behaves near critical point in the thermal CPT as well. We once again find that both PR and SR rules share the same exponent $\beta =0.084$ within the acceptable limits of error. Such unusually low value of $\beta$ compared to that of the RP on ER where $\beta=1$ is the hallmark of EP transition [@ref.mori]. Note also that Grassberger [*et al.*]{} obtained $1/\nu=0.5$ and $\beta=0.0861(5)$ for PR on ER [@ref.Grassberger]. Our values are quite close to their values; however little differences are there which marks a significant improvement in the quality of data-collapse. Phase transitions always entail a change in entropy and hence no model for phase transition is complete without a proper definition for it. To this end, we find that the most suitable choice for entropy in percolation is the Shannon entropy which is defined as $$\label{eq:shannon_entropy} H(t)=-K\sum_i^m \mu_i\log \mu_i,$$ where we choose $K=1$ since it merely amounts to a choice of a unit of measure of entropy [@ref.shannon]. Although there is no explicit restriction per se on the choice of $\mu_i$ there exist some implicit restrictions. The text-book definitions of thermal entropy $S$ and the specific heat $c$ suggest that the $S$ vs $T$ plot must always have a sigmoidal shape with positive slope [@ref.Stanley]. Recently, Vieira [*et al.*]{} used the probability $w_s$, that a node picked at random belongs to a cluster exactly of size $s$, in Eq. (\[eq:shannon\_entropy\]) to measure Shannon entropy for explosive percolation and found that the entropy increases from zero at $t=0$ to reach its maximum value at $t_c$ followed by sharp decrease above $t_c$ [@ref.Vieira]. We also know that the order parameter is also zero at $t=0$. It means that the system is ordered and disordered at the same time which is not possible. Besides, the bell-shaped like entropy curve also violates the second law of thermodynamics. The problem lies in the fact that the sum in Eq. (\[eq:shannon\_entropy\]) is over each individual cluster not over a class of cluster of size $s$ and hence one cannot use $w_s$ to measure entropy. Note that the Shannon entropy measures how much information is contained in each cluster like in each message in the information theory. To find the appropriate probability $\mu_i$ for Eq. (\[eq:shannon\_entropy\]), we assume that for a given $t$ there are $m$ distinct and disjoint labeled clusters $i=1,2,...,m$ of size $s_1,s_2,....,s_m$ respectively. We then propose a labeled cluster picking probability (CPP) $\mu_i$, that a node picked at random belongs to cluster $i$, and assume that it depends on the size $s_i$ of the cluster $i$ itself, so that $\mu_i=s_i/\sum_i s_i$ where $\sum_i s_i=N$. Incorporating $\mu_i=s_i/N$ in Eq. (\[eq:shannon\_entropy\]) we obtain entropy for explosive percolation. To visualize we plot it in Figs. (\[fig:3a\]) and (\[fig:3b\]) as a function of $q=1-t$ for PR and SR respectively. We observe that the maximum entropy occurs at $q=1$ where $\mu_i=1/N$ $\ \forall \ i$ which means that every node has the same probability to be picked if we hit one at random. This is exactly like the state of the isolated ideal gas since here too all accessible microstates are equally probable. The $q=1$ state is thus the most confused or disordered state. Now as we lower the $q$ value, we see that entropy decreases slowly but as we approach towards $q_c=1-t_c$ we observe a dramatic decrease in entropy. This is because as we approach $q_c$ from higher $q$ value we find that many moderately large sized clusters get accumulated as the AP discourages growth of large clusters and encourages the smaller ones. Eventually the crowding of the moderately large clusters reaches a critical state at $q=q_c$ where addition of a few links triggers the growth of the largest cluster in an explosive fashion. We find that at $q=0$ the entropy $H$ is minimally low but the order parameter $P$ is maximally high and hence it is clearly the ordered phase. We thus see that at $q=1$ the entropy is maximally high but the order parameter $P=0$ and hence it correspond to the disordered phase. On the other hand at $q=0$, the order parameter is maximally high and entropy is minimally low. The term percolation therefore refers to the transition from ordered phase characterized by vanishingly small entropy at $q<q_c$ to disordered phase characterized by $P=0$ at $q>q_c$ as one tunes the control parameter $q$. We thus find that in percolation too, like in the thermal CPT, entropy $H$ and order parameter $P$ compliments each other. Once we know the entropy, we can find specific heat as we can define it as $$C(t)=(1-t){{dH}\over{d(1-t)}}$$ in analogy with the definition of its thermal counterpart. Taking differentiation of $H$ from first principles and multiplying that value with the corresponding value of $(1-t)$, we can immediately obtain $C(t)$. We then plot $C(t)$ in Figs. (\[fig:4a\]) and (\[fig:4b\]) as a function of $t$ for PR and SR respectively. To compute the corresponding critical exponent $\alpha$ once again we use the FSS hypothesis and find $\alpha/\nu=0.535$ for both PR and SR. Finally, we plot $CL^{-\alpha/\nu}$ vs $(t-t_c)L^{1/\nu}$ and obtain a perfect data-collapse with $\alpha/\nu=0.535$ and $1/\nu=0.535$ for both PR and SR as shown in Figs. (\[fig:4c\]) and (\[fig:4d\]). We then use the relation $L\sim (t-t_c)^{-\nu}$ in $C(t)\sim L^{\alpha/\nu}$ and immediately find that the specific heat diverges like $$C(t)\sim (t-t_c)^{-\alpha},$$ where $\alpha=1$ for both PR and SR. The quality of data-collapse is a clear testament of the accuracy of $\alpha$ value. Classifications of any system into universality classes is always an interesting proposition. To this end, finding that PR and SR of explosive percolation belongs to the same universality class is a significant development especially when we know that the most expected site-bond universality breaks down even in the lattice. To check whether the Rushbrooke inequality holds in EP or not, we substitute our values of $\alpha=1$, $\gamma=0.893$ and $\beta=0.084$ in the Rushbrooke relation and find $\alpha+2\beta+\gamma=2.061$. Thus, we find that the Rushbrooke inequality not only holds but also its value is close to equality, within the acceptable range of errors. Recently, we applied the same approach to the square and weighted planar stochastic (WPS) lattices where we found that RI holds in RP for both the lattices albeit they belong to different universality classes [@ref.hassan_didar]. Moreover, in both the cases, we find that RI holds almost as an equality like in the thermal CPT. Thus, finding that RI holds for three different universality classes that include a class as exotic as EP provides sufficient confidence in our results. It implies that explosive percolation is indeed a paradigmatic model for continuous phase transition with some unusual finite-size behaviours since we find hysteresis loops in its forward and reverse processes, doublehump in the distributions of the order parameter $P$, which, however, disappears in the thermodynamic limit [@ref.Grassberger; @ref.Bastas; @ref.Riordan; @ref.bastas_review; @ref.Choi]. Besides, we also know that the time difference $\Delta=t_2-t_2$ between the last step $t_2$ for which the largest cluster $C <N^{1/2}$ and the first step $t_1$ for which $C>0.5N$ is not extensive while it is extensive for RP on ER . For all these reasons explosive percolation is indeed a non-trivial paradigmatic model for CPT. To summarize, we have used our recently defined entropy, specific heat and re-defined susceptibility in explosive percolation on random network. Until now we could only quantify the extent of order in percolation by measuring the order parameter $P$. Thanks to the definition of entropy, we can now quantify the other phase too. It is so high in the phase where $P=0$ that we can regard it as disordered phase. It implies that the high-$q$ phase is more disordered, i.e., has a higher symmetry than the low-$q$ phase thus revealing that percolation is an order-disorder transition like ferromagnetic transition. We have also shown that the specific heat and susceptibility diverge at the critical point without having to exclude the largest cluster which is in sharp contrast to the mean cluster size which also diverges at the critical point but only if we exclude the largest cluster from it. We obtained the critical exponents $\alpha$, $\beta$, $\gamma$ and $\nu$ numerically and found that their values for PR and SR are the same, revealing that they belong to the same universality class. Such PR-SR universality is highly intriguing and unexpected especially against the background of the breakdown of the usual site-bond universality even in the lattice. We have also shown that the value of the critical exponents $\alpha$, $\beta$, $\gamma$ obey the Rushbrooke inequality. 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--- abstract: 'We have modelled X-ray burst oscillations observed with the Rossi X-ray Timing Explorer (RXTE) from two low mass X-ray binaries (LMXB): 4U 1636-53 with a frequency of 580 Hz, and 4U 1728-34 at a frequency of 363 Hz. We have computed least squares fits to the oscillations observed during the rising phase of bursts using a model which includes emission from either a single circular hot spot or a pair of circular antipodal hot spots on the surface of a neutron star. We model the spreading of the thermonuclear hot spots by assuming that the hot spot angular size grows linearly with time. We calculate the flux as a function of rotational phase from the hot spots and take into account photon deflection in the relativistic gravitational field of the neutron star assuming the exterior space-time is the Schwarzschild metric. We find acceptable fits with our model in a $\chi^2$ sense, and we use these to place constraints on the compactness of the neutron stars in these sources. For 4U 1636-53, in which detection of a 290 Hz sub-harmonic supports the two spot model, we find that the compactness (i.e., mass/radius ratio) is constrained to be $M/R < 0.163$ at 90 % confidence ($G = c = 1$). This requires a relatively stiff equation of state (EOS) for the stellar interior. For example, if the neutron star has a mass of $1.4 M_{\odot}$ then its radius must be $> 12.8$ km. Fits using a single hot spot model are not as highly constraining. We discuss the implications of our findings for recent efforts to calculate the EOS of dense nucleon matter and the structure of neutron stars.' author: - 'Nitya R. Nath, Tod E. Strohmayer & Jean H. Swank' title: 'Bounds on Compactness for LMXB Neutron Stars from X-ray Burst Oscillations' --- Introduction ============ X-ray brightness oscillations with frequencies in the 300 - 600 Hz range have now been observed during thermonuclear X-ray bursts from 10 LMXB systems (see Strohmayer 2001 for a recent review). Substantial evidence suggests that rotational modulation of a localized hot spot or a pair of antipodal spots is responsible for the observed oscillations, especially during the rising phase (see for example Strohmayer, Zhang & Swank 1997; Heise 2000). As the mass to radius ratio, $M/R$ or “compactness”, of a neutron star increases, the deflection of photons by its relativistic gravitational field becomes stronger and consequently a greater fraction of the stellar surface is visible to an observer at any given time. This effect weakens the spin modulation pulsations produced by a rotating hot spot on the neutron star surface. Because of this effect, Strohmayer et al. (1997) suggested that modelling of the burst oscillation amplitude could in principle provide a constraint on the neutron star compactness. Strohmayer, Zhang & Swank (1997) investigated the temporal evolution of the amplitude of burst oscillations from 4U 1728-34 and showed that a simple model of an expanding hot spot on a neutron star was in qualitative agreement with the data. Miller & Lamb (1998) performed a study of the dependence of the oscillation amplitude from a point-like hot spot on the stellar compactness, the surface rotational velocity, and the spectrum of the surface emission, and showed that if two antipodal spots are present, the resulting limits on the compactness can be highly constraining. Weinberg, Miller, & Lamb (2000) have recently performed similar calculations but allow for hot spots of finite size. Psaltis, Ozel, & DeDeo (2001) have also recently investigated the effects of relativistic photon deflection on the inferred properties of thermally emitting neutron stars. Miller (1999) reported the detection of a 290 Hz sub-harmonic of the stronger 580 Hz oscillation frequency in a study of 5 bursts from 4U 1636-53. This led him to suggest that the neutron star spin frequency is actually 290 Hz in this source and that two antipodal hot spots produce the 580 Hz modulation. The observation of a pair of high frequency quasi-periodic oscillations (QPO) with a frequency separation of $\sim 251$ Hz in this source (Mendez, van der Klis, & van Paradijs 1998), has also been interpreted, in the context of a beat frequency model for the high frequency QPO, as evidence for a neutron star spin frequency of $\sim 290$ Hz rather than 580 Hz (see Miller, Lamb & Psaltis 1998). We note, however, that recent efforts to confirm the sub-harmonic detection in subsequent bursts from 4U 1636-53 have not been successful (Strohmayer 2001). Strohmayer et al. (1998a) reported very large amplitude oscillations at 580 Hz during the rising phase of some bursts from 4U 1636-53. This combination of large measured amplitudes near burst onset and the evidence that two hot spots may produce the modulation, make 4U 1636-53 perhaps the best source currently known in which to constrain the neutron star mass and radius based on the properties of burst oscillations. Here we report on our efforts to do this by detailed modelling of the burst oscillations observed during the rising phase of bursts. We focus on 4U 1636-53 because if the two hot spot conjecture is correct for this object then our results place strong constraints on the neutron star compactness. However, we also summarize our results for 4U 1728-34, a source which has also shown strong oscillations during the rising phase of bursts. The plan of this paper is as follows. In §2 we discuss the basic features and assumptions of our model. In §3 we outline the method of calculation. In §4 we describe our model fitting procedures and our results for both single and antipodal hot spot models. We also summarize the results of fits to data from 4U 1636-53 and 4U 1728-34. In §5 we summarize our results and discuss them in the context of recent efforts to constrain the EOS of neutron star matter. We also discuss future steps we will take to improve the hot spot model. Model Assumptions ================= Both spectral and temporal evidence indicate that the X-ray emission near the onset of at least some thermonuclear bursts is localized to a “hot spot” which spreads in some fashion until eventually encompassing all of the neutron star surface (see for example Strohmayer, Zhang & Swank 1997). This likelihood was also recognized early on in theoretical studies of thermonuclear bursts (Joss 1978). Motivated by this we model the burst rise by assuming that all the burst emission comes from either one or a pair of circular hot spots which expand linearly in angular size with time. The rest of the neutron star surface is assumed dark. Photon trajectories are computed assuming the Schwarzschild metric describes the space-time exterior to the star. This is a reasonable approximation since the influence of the neutron star’s rotation on the space-time only affects the oscillation amplitude to second order (Miller & Lamb 1996). For the present work we shall only investigate bolometric modulations across the full $\sim 2 - 90$ keV bandpass of the RXTE Proportional Counter Array (PCA). We shall also ignore Doppler shifts and relativistic aberration produced by the rotational motion of the hot spot (see for example Miller 1999; Chen & Shaham 1989). We discuss later the likely influence on our results of this approximation. Our model is uniquely characterized by seven parameters: (1) an overall source intensity or normalization, $S$, which can be thought of as the flux leaving unit surface area of the neutron star. (2) neutron star compactness, $\beta = M/R$, where $M$ and $R$ are the stellar mass and radius, respectively, (3) initial angular size of the spot (half of the subtended angle), $\alpha_0$, (4) angular growth rate of the hot spot, $\dot\alpha$, (5) initial rotational phase, $\delta_0$, (6) latitude of the spot center, $\theta_s$, measured from the rotational equator, and (7) latitude of the observers line of sight, $\theta_{obs}$, also measured from the rotational equator. One of our primary goals is to determine an upper bound on the compactness. To do this within the context of our model we set the hot spot latitude and observation latitude to zero. That is, both the hot spots and the line of sight to the observer are centered on the rotational equator. This geometry produces the largest possible modulation amplitude. Since any observed modulation must be equal to or less than this limit, and since the modulation amplitude decreases with increasing compactness, the upper limit follows. For completeness, we also investigate the influence of moving the hot spot and the line of sight off the rotational equator. The geometry of our model is illustrated in Figure 1. Related hot spot models have been worked out by Pechenick, Ftaclas, & Cohen (1983) and Strohmayer (1992). Method of Calculation ===================== The geometry of a photon trajectory in relation to the observers line of sight $\vec r_{obs}$ is shown in Figure 1. The figure is drawn with $\theta_s = \theta_{obs} = 0$. For any single point on the hot spot with radius vector $\vec r$, the path of a photon reaching the observer lies in the plane of $\vec r$ and $\vec r_{obs}$, and is asymptotically parallel to $\vec r_{obs}$ with impact parameter $b$. The two angles, $\phi$ (between $\vec r$ and $\vec r_{obs}$) and $\psi$ (the emission angle with respect to the surface normal), complete the description. For non-zero $\theta_s$ and $\theta_{obs}$, the deflection geometry remains the same, only the plane in which the desired trajectory lies (the plane of $\vec r$ and $\vec r_{obs}$) changes. The angle $\phi$ can be expressed as, $$\phi = \int_0^{\sin^{-1}(\hat b)} \left [ 1 - 2(M/R)(1-\sin^3 y / \hat b ) / (1 - \sin^2 y) \right ] ^{-1/2} dy,$$ where $\hat b = b/b_{max}$ is the reduced impact parameter, $b_{max} = R(1-2(M/R))^{-1/2}$, and $M$ and $R$ are the stellar mass and radius respectively. This form for the angle $\phi$ is somewhat non-standard compared to previous work. More commonly $\phi$ is expressed as $$\phi = \int_0^{M/R} \left [u_b^2 - (1-2u)u^2 \right ] ^{-1/2} du,$$ where $u_b \equiv M/b$ (see for example, Pechenick, Ftaclas & Cohen 1983; Miller & Lamb 1998). Our rationale for rewriting the integrand is twofold; first, to explicitly show what parameters $\phi$ depends on, and second to remove singular behavior of the integrand to facilitate numerical evaluation of the integral. Changing variables in (2) to $u = (M/R) x$ results in the following expression; $$\phi = \int_0^1 \left [\hat b ^{-2} (1-\frac{2M}{R}) - (1-\frac{2M}{R}x)x^2 \right ] ^{-1/2} dx.$$ As $M/R$ becomes small this integral has the form, $$\phi = \int_0^1 \left [\hat b ^{-2} - x^2 \right ] ^{-1/2} dx,$$ which has singular behavior as $\hat b$ and $x$ approach unity. The second change of variables to $y=sin^{-1}\hat b x$ is motivated by the form of equation (4) above, whose solution corresponds to the inverse sine function. With this final substitution we arrive at the expression in equation (1), which explicitly shows the dependence of $\phi$ on $M/R$ and $\hat b$, and is well defined and non singular. As $\hat b$ varies from 0 to 1, $\phi$ varies from 0 to $\phi_{max}$, the maximum value of $\phi$, which is attained when a photon is emitted tangentially to the stellar surface. We note several interesting limiting cases; for $M/R = 0$, 0.284, 0.331, 0.33333, we have $\phi_{max} = \pi/2$, $\pi$, $2\pi$, and $\infty$, respectively. The last case, $\phi_{max} = \infty$, corresponds to the bound photon orbit at $M/R = 1/3$. To compute the flux as a function of rotational phase we first invert $\phi\; (\hat b, M/R)$ numerically to obtain tables of $\hat b$ as a function of $\phi$ and $M/R$. We use Gaussian quadratures to solve the integral numerically. The method is fast and converges quickly. For a given $M/R$ and each $\phi = \cos^{-1} (\vec r \cdot \vec r_{obs})$ we then find $\hat b$ and compute $\cos\psi = (1 - \hat b^2 )^{1/2}$. The observed flux is then given by $\int I_{\nu} \cos\psi\; d\Omega$, where $I_{\nu}$ is the local specific intensity at the surface of the neutron star, and the integral is carried out over the hot spot or spots. For the specific intensity we use both an isotropic emission function, $I_{\nu} = 1$ and an angular dependent beaming function consistent with emission from a grey scattering atmosphere, $I_{\nu} = 3/5 \cos\psi + 2/5$ (see Chandrasekhar 1960). Such a function should be appropriate for bursting neutron star atmospheres which are dominated by Thomson scattering (London, Taam, & Howard 1986). Figure 2 shows several examples of light-curves computed with our model using one hot spot and different values of $M/R$. The decrease in modulation amplitude with increasing compactness is clearly evident. Data Analysis Procedures and Results ==================================== We searched the available RXTE data from 4U 1636-53 and 4U 1728-34 for bursts and selected for analysis four from 4U 1636-53 and two from 4U 1728-34 which showed particularly strong oscillations during the rising phase. The data are in the form of X-ray event times recorded with 125 $\mu$s resolution across the full 2 - 90 keV PCA bandpass. In order to fit our model we first break up the rising interval from each burst into a number, $n_{interval}$, of contiguous subintervals. Within each subinterval we epoch fold the data into $n_{bin}$ phase bins using the oscillation frequency determined from a power spectral analysis of the entire rising interval. We then perform a $\chi^2$ minimization by computing $\chi^2 = \sum_{i=1}^N (O_i - M_i)^2 / \sigma^2 $. Here $O_i$ and $M_i$ are the numbers of observed and predicted counts, respectively, in the $i$$^{\rm th}$ data bin. For $\sigma^2$ we use the Poisson variance, which is simply equal to the number of counts in the bin. In general we also add a constant background level to the model as a way of modeling the pre-burst, accretion driven flux, which we assume is not associated with the burst. This also implies a tacit assumption that the accretion driven flux is not significantly altered by the burst. This quantity is well determined by the pre-burst data, so typically we do not treat it as a model parameter. In general the total length of data that we fit does not extend all the way to the peak of the burst for a number of reasons. The oscillation has usually dropped below our detection threshold before the peak is reached and often episodes of radius expansion also begin before the count rate reaches a maximum. In general, our assumptions regarding the growth of the hot spot should be most valid the closer we remain to the onset of the burst. This also tends to maximize our signal to noise ratio in data from a given burst since the modulation amplitude is largest near burst onset. We minimize $\chi^2$ using the Marquardt-Levenberg method and we can simultaneously vary all seven model parameters. Our choice regarding the number of data bins is a tradeoff between having sufficient counts in each bin and the need to have enough time resolution to adequately model the rise of the burst and hence constrain the hot spot spreading speed, $\dot\alpha$. In general we found that $n_{region} = 8$ and $n_{bin} = 8$ gave the best results. With this choice we have a total of 64 data bins. We also restrict $M/R \le 0.284$, the limit beyond which photons from a given point on the stellar surface can reach the observer along more than one unique path. In general we find acceptable fits using both one and two hot spots for both sources. In the remainder we will summarize our results and discuss the implications for neutron star compactness, concentrating on the two spot fits for 4U 1636-53 for the reasons outlined above. Antipodal Hot Spot Models ------------------------- Our best fitting models for bursts from 4U 1636-53 using two antipodal hot spots and the grey atmosphere intensity function are summarized in Table 1, where for each burst we give the observation date, the length of the time interval in which we fit the data, the best fitting model parameters and the minimum $\chi^2$. For these fits we have fixed to zero both the spot latitude, $\theta_s$ and the observers latitude, $\theta_{obs}$, and we used 64 data bins. With 5 free parameters we therefore have 59 degrees of freedom. Our minimum $\chi^2$ values are all statistically acceptable, indicating that the simple rotating hot spot model is consistent with the data. In Figure 3 we show the two spot fits for each of the four bursts from 4U 1636-53. Each panel shows the count rate in the PCA for the rising interval of a burst. The bursts are labelled by date. The vertical dashed lines denote the region in which we fit our model. The solid curve shows the best fitting model [*extrapolated*]{} to the time at which the entire surface of the neutron star is covered by the hot spots. The time resolution in these plots is not sufficient to resolve the oscillations, rather, this figure is meant to give the reader an assessment of how well the model does in describing the gross time evolution of each burst. There are several things to note from Figure 3. First, the fits [*within*]{} each interval are quite good, and they also extrapolate beyond the fitting interval rather well over a limited portion of the burst rise. The deviations at later times are not unexpected since in several of these bursts episodes of photospheric radius expansion begin at about the same time as the model begins to deviate from the burst rise. Indeed the burst on 08/20/98 did not show radius expansion and in this case the model extrapolates rather well for most of the rise. All the other bursts show radius expansion near the time that the model deviates from the data. Second, the maximum count rates inferred from our model for bursts 12/28/96 and 08/19/98 are quite similar. Since these bursts were quite similar in their peak fluxes, the model normalizations, which can be thought of as an averaged description of the thermonuclear burning, should also be similar and indeed they are. Note that though these two bursts have similar peak fluxes they do not have similar rise times, and our model succesfully accounts for this difference. Although the models are clearly inadequate to describe the details of the [*entire*]{} burst rise, they do better the closer one stays to the burst onset, and this behavior is the most relevant with regard to fitting the oscillations and constraining $M/R$. Figure 4 graphically illustrates how well the model can fit the observed oscillations by comparing the best fit model and data for several different fits to the 12/28/96 burst from 4U 1636-53. Shown are the best fitting two spot model with $\theta_s = \theta_{obs} = 0$ (solid); the best fitting one spot model with $\theta_s = \theta_{obs} = 0$ (dashed); and the best fitting one spot model with all parameters free to vary (dotted). Since fits to the other bursts all look very similar we did not feel it was essential to show similar plots for each individual burst. The derived best-fit compactness for the four bursts from 4U 1636-53 span a rather tight range from $\beta = 0.075$ to $0.134$. In Figure 5 we show the best-fit values of $M/R$ and their uncertainties. We fit a constant, $\beta_{avg}$, to the four values and find they are consistent with a single value for the compactness of $\beta_{avg} = 0.126$ (solid horizontal line in Figure 5). The $\chi^2$ per degree of freedom for the fit was 0.2. In order to derive a firm upper limit on the compactness we investigated the confidence region for $\beta_{avg}$ and found the values of $\beta$ which increased $\chi^2$ by 2.71 (for $90\%$ confidence) and 6.63 (for $99\%$ confidence). These values are also shown in Figure 5 as the dashed ($90\%$) and dot-dashed ($99\%$) horizontal lines in Figure 5. The derived upper limits are $\beta_{90\%} = 0.163$ and $\beta_{99\%} = 0.183$. Since we do not in general know the orientation of the binary systems in which the neutron stars reside we performed the fitting under the assumption that the hot spot and observer are both in the plane of the rotational equator. This geometry gives the maximum rotational modulation. Thus each fitted value for $\beta$ from a different burst gives a measure of the maximum allowable compactness of the neutron star. However, each individual measurement has associated with it a rather large uncertainty. Thus, our methodology in deriving an upper limit on $\beta$ is to combine a number of these independent measurements in order to reduce the overall uncertainty. In this way $\beta_{avg}$ is our best estimate of how large the compactness of the neutron star is, but this estimate too is not exact and has a confidence region associated with it. It is the confidence region on $\beta_{avg}$ that we use to determine a final upper limit. This may not be a unique statistical methodology, but we feel it is reasonable given the nature of the other model assumptions we have made. We will discuss the implications of our compactness limits for the neutron star EOS in the next section. We also computed fits allowing the two angles $\theta_s$ and $\theta_{obs}$ to vary. As might be expected we find the inclusion of the additional parameters improves the fits, but only marginally. With these parameters free to vary we find that $M/R$ tends to decrease, and both $\theta_s$ and $\theta_{obs}$ move off the rotational equator. We find, however, no stationary solutions in $M/R$ with all seven parameters varying. These results serve to illustrate the basic correlation between compactness and the hot spot and viewing geometries. If the spot moves or is viewed away from the rotational equator then the inferred value of $M/R$ must decrease in order to make up for the loss of modulation amplitude produced by a less than favorable geometry. Since realistic neutron star EOSs cannot support stars with arbitrarily small $M/R$, if the two spot model is correct, then our results suggest that the hot spots must be relatively near the rotational equator in order to achieve the high observed amplitudes. If the hot spots are linked to the poles of a magnetic field in 4U 1636-53 (see Miller 1999), then this would suggest that the magnetic axis would have to be nearly perpendicular to the rotation axis. If we assume the surface emission is isotropic, the fits for all the bursts are very similar, but the $M/R$ values are systematically lower, with the weighted mean dropping to $M/R = 0.05$. This is as expected, since isotropic emission produces a lower amplitude than the grey atmosphere beaming function. Our results for 4U 1728-34 are quite similar to those derived for 4U 1636-53. The results of the two spot fits for bursts from 4U 1728-34, with $\theta_s$ and $\theta_{obs}$ fixed at zero and with beamed emission are also shown in Table 1. The weighted average of the two fits yields the value $M/R = 0.121$, with $90\%$ and $99\%$ confidence upper limits of $0.171$ and $0.199$, respectively. These are similar to the limits derived for 4U 1636-53. Although no sub-harmonic has been detected for this source, the closeness of the derived $M/R$ limits for the two sources is striking, and may be an indication that, irrespective of the model, the actual compactness of the two sources is similar. One Spot Models --------------- For one spot models we generally find there are no strong constraints on the compactness for either source. This results from the fact that stars even as compact as our computational limit, $M/R = 0.284$, can still produce a sufficiently large modulation amplitude to match the data. For example, the best fits for the 4U 1636-53 bursts with four parameters varying (ie., $M/R$ fixed at 0.284, $\theta_s$ and $\theta_{obs}$ fixed at zero), and with beamed emission, give $\chi^2$ = 69.4, 66.2, 70.6, and 75.0, for each burst respectively. These values are marginally higher than for the corresponding two spot fits, however, from a statistical point of view they are still formally acceptable. For the one spot fits we find that $\chi^2$ monotonically decreases as $M/R$ increases from 0 to 0.284, but never reaches a minimum. In other words we find no meaningful upper limit to the compactness, at least within the confines of our model assumptions. A comparison of the $\chi^2$ values between the two spot and one spot fits at first glance seems to suggest that the two spot fits are better, however, this is misleading because the one spot fits are not stationary in $M/R$, that is they have not converged to a minimum. Discussion and Summary ====================== We have shown that if two hot spots produce the observed modulation at 580 Hz in 4U 1636-53 then the large amplitude of oscillations near burst onset provide a strong constraint on the compactness. In Figure 6 we show in the mass - radius plane our 90 and 99 % confidence upper limits on the compactness $\beta=M/R$ for 4U 1636-53 from our two hot spot fits. The shaded region denotes the ranges of $M$ and $R$ which satisfy our compactness constraint and have $M > 1.4 M_{\odot}$, which we take as a reasonable estimate of the minimum mass of the neutron star in these old accreting systems. We also show several theoretical neutron star EOSs which span a range of stiffnesses based on current uncertainties in the exact composition of neutron star matter and our incomplete knowledge of the nucleon - nucleon interaction. Also shown in Figure 6 is our computational limit at $M/R \le 0.284$ (solid diagonal line). As can be seen our results tend to favor moderately stiff to very stiff EOSs. For example, our limits are comfortably consistent with EOS L (Pandharipande & Smith 1975). However, the most recent theoretical calculations of neutron star EOSs which are consistent with the currently available nucleon scattering data are generally not as stiff as this EOS (see for example Akmal, Pandharipande & Ravenhall 1998). For example, the best EOS of Akmal, Pandharipande & Ravenhall (1998), which is denoted APR in Figure 6, is barely consistent with our $99\%$ limit. However, these modern EOSs are still not rigorously self consistent, and become “superluminal” (the sound speed exceeds the speed of light) above some density. Modifications to the EOS can be made in an ad hoc manner by setting the sound speed equal to the speed of light above some critical or “matching” density (see for example Heiselberg & Hjorth-Jensen 1999) . This has the effect of stiffening the EOS. Recently, Olson (2001) has investigated changes to the high density EOS of neutron star matter required by constraints derived from relativistic kinetic theory. In Figure 6 we show two of these modifications to the APR EOS. The thick dashed lines show the APR EOS modified by the kinetic theory constraints for two different matching densities, 0.316 fm$^{-3}$ (APR-Kin1) and 0.270 fm$^{-3}$ (APR-Kin2) (see Olson 2001 for a detailed discussion). With the kinetic theory assumptions the APR EOS is now reasonably consistent with our limits. Recently, Lattimer & Prakash (2000) have argued that measurements of the neutron star radius to about $10\%$ precision should be sufficient to usefully constrain the neutron star EOS. They showed that as long as extreme softening of the EOS does not occur in the vicinity of nuclear matter equilibrium density then the stellar radius is almost independent of the mass. Since observed neutron star masses cluster rather closely around 1.4 $M_{\odot}$ they argued that the more important quantity in terms of constraining the EOS is the stellar radius. Since the neutron stars in LMXBs are upwards of $10^8$ yr old and they have been accreting most of their lifetime, it is very likely that they are at least more massive than the $1.4 M_{\odot}$ typically found for younger neutron stars (Thorsett & Chakrabarty 1999). If this is the case, then our results place a rather firm lower limit on $R$ of about 11.5 km. Such a limit is consistent with the notion that extreme softening of the EOS, as can be produced by pion, kaon or other hyperon condensates, does not occur in neutron star cores (see Lattimer & Prakash 2000). Since these inferences depend crucially on the two hot spot hypothesis for the burst oscillations from 4U 1636-53, it is vital to try and settle this issue in the near future. We have generally tried to employ the simplest assumptions consistent with maintaining the essential physics of the model and the observed properties of the bursts. For this work we have neglected the Doppler shifts and relativistic aberration produced by the rapid motion of the hot spots. Although we do not know the rotational velocity precisely because of our uncertainty in the stellar radius and the number of hot spots, it is likely that the velocity on the rotational equator is $\le 0.1\; c$. Miller & Lamb (1998) investigated the effects of the rotational velocity of a point spot on the bolometric and energy dependent amplitude and showed that although such a velocity can have important effects on the amplitude measured at particular photon energies, they also showed that the effect on the bolometric amplitude of the rotational velocity is very modest (see their Figure 1d; see also Weinberg, Miller & Lamb 2000). The calculations of Miller & Lamb (1998) were for point-like spots and hence represent upper limits to the size of any rotational effect. Since our model uses spots of a finite and growing angular size, the rotational effects, which represent an integral of the line of sight rotational velocity over the hot spot, must be less than the estimates computed by Miller & Lamb (1998). The amplitude of higher harmonics is more sensitive to the rotational velocity; however, the present RXTE data are not very sensitive to the shape of the pulses, i.e., we do not detect any higher harmonics, nor do we know of any published reports of significant harmonics of burst oscillations. Based on this and because we only investigate the bolometric amplitude we believe we are justified in neglecting the Doppler effects for the present work. However, by not investigating the energy dependent effects we are indeed ignoring some useful information which can eventually help provide more powerful constraints on $M$ and $R$. We plan to improve our model by including these energy dependent effects and will report the results from such a study in a sequel. Using our model we have also begun to investigate the constraints that can be obtained with data of a higher statistical precision than presently available. We have found that the present RXTE data is essentially insufficient for constraining the hot spot and viewing geometry. However, if the count rate were increased by a factor of 10 - 20 times the RXTE rate then our simulations suggest that it will be possible to simultaneously constrain both the stellar compactness and the hot spot and viewing geometries. Thus future large area timing experiments, such as the proposed Timing of Extreme X-ray Astrophysical Sources (TEXAS) experiment, will be extremely powerful tools for probing the structure of neutron stars. We thank Cole Miller, Craig Markwardt and Tim Olson for many helpful discussions and comments on the manuscript. We thank Cole Miller for providing some of the mass - radius relations for the equations of state shown in Figure 6. We also thank Tim Olson for providing the mass radius relations based on kinetic theory constraints to the APR equation of state. Akmal, A., Pandharipande, V. R. & Ravenhall, D. G. 1998, Phys. Rev. C, 58, 1804 Chandrasekhar, S. 1960, Radiative Transfer (New York: Dover) Chen, K. & Shaham, J. 1989, ApJ, 339, 279 Heise, J. et al. 2000, Talk presented at AAS HEAD meeting, Honolulu, HI Heiselberg, H. & Hjorth-Jensen, M. 1999, ApJ, 525, L45 Joss, P. C. 1978, ApJ, 225, L123 Lattimer, J. M. & Prakash, M. 2000, ApJ, in press, (astro-ph/0002232) London, R. A., Taam, R. E. & Howard, W. M. 1986, ApJ, 306, 170 Miller, M. C. & Lamb, F. K. 1996, ApJ, 470, 1033 Miller, M.C. & Lamb, F.K. 1998, ApJ, 499, L37 Miller, M.C. 1999, ApJ, 515, L77 Miller, M. C., Lamb, F. K. & Psaltis, D. 1998, ApJ, 508, 791 Mendez, M., van der Klis, M. & van Paradijs, J. 1998, ApJ, 506, L117 Olson, T. S. 2001, Phys. Rev. C, 63, 015802 Pandharipande, V. R., & Smith, R. A. 1975, Phys. Lett., 59B, 15 Pechenick, K. R., Ftaclas, C. & Cohen, J. M. 1983, ApJ, 274, 846 Psaltis, D. Ozel, F. & DeDeo, S. 2001, ApJ, in press, (astro-ph/0004387) Strohmayer, T. E. 2001, Advances Sp. Res. submitted, (astro-ph/0012516) Strohmayer, T. E., Zhang, W., Swank, J. H., White, N. E. & Lapidus, I. 1998a, ApJ, 498, L135 Strohmayer, T. E., Jahoda, K., Giles, A. B. & Lee, U. 1997, ApJ, 486, 355 Strohmayer, T.E., Zhang, W., & Swank, J.H. 1997, ApJ, 487, L77 Strohmayer, T. E., Zhang, W., Swank, J. H., Smale, A. P., Titarchuk, L., Day, C. & Lee, U. 1996, ApJ, 469, L9 Strohmayer, T. E. 1992, ApJ, 388, 138 Thorsett, S. E. & Chakrabarty, D. 1999, ApJ, 512, 288 Weinberg, N., Miller, M. C. & Lamb, D. Q. 2000, ApJ, submitted, (astro-ph/0001544) Figure Captions =============== {width="6in" height="7.2in"} Figure 1: Geometry for calculation of the flux from a hot spot on a rotating neutron star. Here the hot spot is situated on the rotational equator. See the text for a description of the relationship between the angles $\phi$, $\psi$ and the impact parameter, $b$. {width="6in" height="7in"} Figure 2: Light-curves generated with the rotating hot spot model for different values of the neutron star compactness. Notice the decrease in amplitude with increasing compactness. Note also the decrease in amplitude and increase in flux as the hot spot spreads to encompass the entire surface. These models were computed with one hot spot assuming isotropic emission from the surface. The top three curves have been displaced vertically for clarity. The curves were computed using isotropic emission from the surface, the qualitative behavior of the amplitude with compactness, $M/R$, using the grey atmosphere beaming function is the same, only the modulation amplitudes differ slightly. {width="5in" height="6in"} 10 pt Figure 3: Model fits to the bursts from 4U 1636-53. Each panel shows the data (histogram) and model (thick solid curve) fit to the rising portion of a burst. The dashed vertical lines denote the time interval in which we fit the hot spot model. The extent of each model curve covers the total time it takes for the hot spots to envelope the entire neutron star surface. The bursts are labelled by date. {width="6in" height="6in"} Figure 4: Data and best fit models for several fits to the December 28th, 1996 burst from 4U 1636-53. Shown are fits using two hot spots with $\theta_s = \theta_{obs} = 0$ (solid); one hot spot with $M/R$ fixed at 0.284 and $\theta_s = \theta_{obs} = 0$ (dashed); and one hot spot with $\theta_s$ and $\theta_{obs}$ free to vary (dotted). All the fits shown were computed with the grey atmosphere beaming function. {width="6in" height="5in"} 10 pt Figure 5: Compactness constraints for the four bursts from 4U 1636-53. The solid horizontal line is the best fitting constant value of compactness, $\beta_{avg}$. The dashed and dot-dashed lines are the 90 and $99\%$ confidence upper limits on $\beta_{avg}$. The burst number corresponds to their position in Table 1. {width="6in" height="5in"} Figure 6: Summary of mass radius constraints from fits to bursts from 4U 1636-53 using the two spot model and the grey atmosphere beaming function. The diagonal dashed lines show the 90 and 99 % confidence upper limits for $M/R$ from the four fits in Table 1 (see also Figure 5). The shaded region is the allowed range of $M$ and $R$ which satisfies the compactness constraints and has $M > 1.4 M_{\odot}$. The solid diagonal line corresponds to our computational limit, $M/R = 0.284$. The other curves show mass - radius relations for equations of state FPS (Lorenz et al. 1993), L (Pandharipande & Smith 1975b), and APR (Akmal, Pandharipande & Ravenhall 1998), which range from very soft (FPS) to very stiff (L). We also show two different modifications to the APR EOS based on the relativistic kinetic theory constraints of Olson (2001) (thick dashed curves). The two curves correspond to the use of different matching densities for the high density kinetic theory constraints (see §5 and Olson 2001). The results favor stiffer equations of state with $R > 11.5$ km for a 1.4 $M_{\odot}$ neutron star. -55pt [cccccccc]{} 12/28/96 at 22:39:34 & 0.276 & $0.134\pm 0.037$ & $1641.6$ & $6.41\pm 1.6$ & $222.9\pm 39.6$ & $116.7$ & 53.6\ 08/19/98 at 11:47:07 & 0.303 & $0.133\pm 0.056$ & $1716.1$ & $14.38\pm 3.8$ & $130.7\pm 54.3$ & 81.0 & 63.7\ 08/20/98 at 05:16:35 & 0.496 & $0.075\pm 0.072$ & 882.1 & $8.66\pm 3.0$ & $69.5\pm 36.0$ & -3.0 & 65.3\ 06/18/99 at 23:50:10 & 0.460 & $0.127\pm 0.037$ & 1272.3 & $6.70\pm 1.4$ & $102.4\pm 22.2$ & 14.8 & 61.5\ 02/16/96 at 10:00:49 & 0.221 & $0.113\pm 0.042$ & 1238.7 & $9.1\pm 2.2$ & $370.8\pm 39.6$ & 102.7 & 69.7\ 09/21/97 at 18:10:56 & 0.354 & $0.130\pm 0.043$ & 1001.9 & $7.4\pm 2.0$ & $181.8\pm 32.4$ & 142.1 & 54.6\

--- abstract: 'In this paper we study the topic of signal restoration using complexity regularization, quantifying the compression bit-cost of the signal estimate. While complexity-regularized restoration is an established concept, solid practical methods were suggested only for the Gaussian denoising task, leaving more complicated restoration problems without a generally constructive approach. Here we present practical methods for complexity-regularized restoration of signals, accommodating deteriorations caused by a known linear degradation operator of an arbitrary form. Our iterative procedure, obtained using the alternating direction method of multipliers (ADMM) approach, addresses the restoration task as a sequence of simpler problems involving $ \ell _2$-regularized estimations and rate-distortion optimizations (considering the squared-error criterion). Further, we replace the rate-distortion optimizations with an arbitrary standardized compression technique and thereby restore the signal by leveraging underlying models designed for compression. Additionally, we propose a shift-invariant complexity regularizer, measuring the bit-cost of all the shifted forms of the estimate, extending our method to use averaging of decompressed outputs gathered from compression of shifted signals. On the theoretical side, we present an analysis of complexity-regularized restoration of a cyclo-stationary Gaussian signal from deterioration by a linear shift-invariant operator and an additive white Gaussian noise. The theory shows that optimal complexity-regularized restoration relies on an elementary restoration filter and compression spreading reconstruction quality unevenly based on the energy distribution of the degradation filter. Nicely, these ideas are realized also in the proposed practical methods. Finally, we present experiments showing good results for image deblurring and inpainting using the JPEG2000 and HEVC compression standards.' author: - | Yehuda Dar, Michael Elad, and Alfred M. Bruckstein\ [^1] bibliography: - 'IEEEabrv.bib' - 'complexity\_regularized\_inverse\_problems\_\_refs.bib' title: Restoration by Compression --- [ ]{}  \   Complexity regularization, rate-distortion optimization, signal restoration, image deblurring, alternating direction method of multipliers (ADMM). Introduction ============ restoration methods are often posed as inverse problems using regularization terms. While many solutions can explain a given degraded signal, using regularization will provide signal estimates based on prior assumptions on signals. One interesting regularization type measures the complexity of the candidate solution in terms of its compression bit-cost. Indeed, encoders (that yield the bit cost) rely on signal models and allocate shorter representations to more likely signal instances. This approach of complexity-regularized restoration is an attractive meeting point of signal restoration and compression, two fundamental signal-processing problems. Numerous works [@saito1994simultaneous; @natarajan1995filtering; @chang1997image; @mihcak1999low; @rissanen2000mdl; @chang2000adaptive; @liu2001complexity] considered the task of denoising a signal corrupted by an additive white Gaussian noise using complexity regularization. In [@natarajan1995filtering; @liu2001complexity], this idea is translated to practically estimating the clean signal by employing a standard lossy compression of its noisy version. However, more complex restoration problems (e.g., deblurring, super resolution, inpainting), involving non-trivial degradation operators, do not lend themselves to a straightforward treatment by compression techniques designed for the squared-error distortion measure. Moulin and Liu [@moulin2000statistical] studied the complexity regularization idea for general restoration problems, presenting a thorough theoretical treatment together with a limited practical demonstration of Poisson denoising based on a suitably designed compression method. Indeed, a general method for complexity-regularized restoration remained as an open question for a long while until our recent preliminary publication [@dar2016image], where we presented a generic and practical approach flexible in both the degradation model addressed and the compression technique utilized. Our strategy for complexity-regularized signal restoration relies on the alternating direction method of multipliers (ADMM) approach [@boyd2011distributed], decomposing the difficult optimization problem into a sequence of easier tasks including $ \ell_2 $-regularized inverse problems and standard rate-distortion optimizations (with respect to a squared-error distortion metric). A main part of our methodology is to replace the rate-distortion optimization with standardized compression techniques enabling an indirect utilization of signal models used for efficient compression designs. Moreover, our method relates to various contemporary concepts in signal and image processing. The recent frameworks of Plug-and-Play Priors [@venkatakrishnan2013plug; @sreehari2016plug] and Regularization-by-Denoising [@romano2017little] suggest leveraging a Gaussian denoiser for more complicated restoration tasks, achieving impressive results (see, e.g., [@venkatakrishnan2013plug; @sreehari2016plug; @romano2017little; @dar2016postprocessing; @dar2016reducing; @rond2016poisson]). Essentially, our approach is the compression-based counterpart for denoising-based restoration concepts from [@venkatakrishnan2013plug; @sreehari2016plug; @romano2017little]. Commonly, compression methods process the given signal based on its decomposition into non-overlapping blocks, yielding block-level rate-distortion optimizations based on block bit-costs. The corresponding complexity measure sums the bit-costs of all the non-overlapping blocks, however, note that this evaluation is shift sensitive. This fact motivates us to propose a shift-invariant complexity regularizer by quantifying the bit-costs of all the overlapping blocks of the signal estimate. This improved regularizer calls for our restoration procedure to use averaging of decompressed signals obtained from compressions of shifted signals. Our shift-invariant approach conforms with the Expected Patch Log-Likelihood (EPLL) idea [@zoran2011learning], where a full-signal regularizer is formed based on a block-level prior in a way leading to averaging MAP estimates of shifted signal versions. Our extended method also recalls the cycle spinning concept, presented in [@coifman1995translation] for wavelet-based denoising. Additional resemblance is to the compression postprocessing techniques in [@nosratinia2001enhancement; @nosratinia2003postprocessing] enhancing a given decompressed image by averaging supplementary compression-decompression results of shifted versions of the given image, thus, our method generalizes this approach to any restoration problem with an appropriate consideration of the degradation operator. Very recent works [@beygi2017compressed; @beygi2017efficient] suggested the use of compression techniques for compressive sensing of signals and images, but our approach examines other perspectives and settings referring to restoration problems as will be explained below. In this paper we extend our previous conference publication [@dar2016image] with improved algorithms and new theoretical and experimental results. In [@dar2016image] we implemented our concepts in procedures relying on the half quadratic splitting optimization technique, in contrast, here we present improved algorithms designed based on the ADMM approach. The new ADMM-based methods introduce the following benefits (with respect to using half quadratic splitting as in [@dar2016image]): significant gains in the restoration quality, reduction in the required amount of iterations, and an easier parameter setting. In addition, in this paper we provide an extensive experimental section. While in [@dar2016image] we experimentally examined only the inpainting problem, in this paper we present new results demonstrating the practical complexity-regularized restoration approach for image deblurring. While deblurring is a challenging restoration task, we present compelling results obtained using the JPEG2000 method and the image compression profile of the HEVC standard [@RefWorks:112]. An objective comparison to other deblurring techniques showed that the proposed HEVC-based implementation provides good deblurring results. Moreover, we also extend our evaluation given in [@dar2016image] for image inpainting, where here we use the JPEG2000 and HEVC compression standards in our ADMM-based approach to restore images from a severe degradation of 80% missing pixels. Interestingly, our compression-based image inpainting approach can be perceived as the dual concept of inpainting-based compression of images and videos suggested in, e.g., [@galic2008image; @schmaltz2009beating; @andris2016proof] and discussed also in [@adam2017denoising]. Another prominent contribution of this paper is the new theoretical study of the problem of complexity-regularized restoration, considering the estimation of a cyclo-stationary Gaussian signal from a degradation procedure consisting of a linear shift-invariant operator and additive white Gaussian noise. We gradually establish few equivalent optimization forms, emphasizing two main concepts for complexity-regularized restoration: the degraded signal should go through a simple inverse filtering procedure, and then should be compressed so that the decompression components will have a varying quality distribution determined by the degradation-filter energy-distribution. We explain how these ideas materialize in the practical approach we propose, thus, establishing a theoretical reasoning for the feasible complexity-regularized restoration. This paper is organized as follows. In section \[sec:Complexity-Regularized Restoration\] we overview the settings of the complexity-regularized restoration problem. In section \[sec:Proposed Methods\] we present the proposed practical methods for complexity-regularized restoration. In section \[sec:Rate-Distortion Theoretic Analysis for the Gaussian Case\] we theoretically analyze particular problem settings where the signal is a cyclo-stationary Gaussian process. In section \[sec:Experimental Results\] we provide experimental results for image deblurring and inpainting. Section \[sec:Conclusion\] concludes this paper. Complexity-Regularized Restoration: Problem Settings {#sec:Complexity-Regularized Restoration} ==================================================== Regularized Restoration of Signals {#subsec:Regularized-Restoration Optimization} ---------------------------------- In this paper we address the task of restoring a signal $ {\mathbf{x}}_0 \in\nolinebreak \mathbb{R}^N $ from a degraded version, $ {\mathbf{y}} \in \mathbb{R}^M $, obeying the prevalent deterioration model: [rCl]{} \[eq:corruption model\] = \_0 + where $ {\mathbf{H}} $ is a $M\times N$ matrix being a linear degradation operator (e.g., blur, pixel omission, decimation) and ${\mathbf{n}}\in \mathbb{R}^M$ is a white Gaussian noise vector having zero mean and variance $\sigma _n ^2$. Maximum A-Posteriori (MAP) estimation is a widely-known statistical approach forming the restored signal, $\hat{ {\mathbf{x}}}$, via [rCl]{} \[eq:MAP estimation\] = \_ p( [|]{} ) where $ p\left( {{{\mathbf{x}}}|{{\mathbf{y}}}} \right) $ is the posterior probability. For the above defined degradation model (\[eq:corruption model\]), incorporating additive white Gaussian noise, the MAP estimate reduces to the form of [rCl]{} \[eq:log MAP estimate - AWGN model\] = \_ \_2\^2 - p ( ) where $p({\mathbf{x}})$ is the prior probability that, here, evaluates the probability of the candidate solution. Another prevalent restoration approach, embodied in many contemporary techniques, forms the estimate via the optimization [rCl]{} \[eq:restoration using a general prior\] = \_ \_2\^2 + s() where $s({\mathbf{x}})$ is a general regularization function returning a lower value for a more likely candidate solution, and $\mu\ge 0$ is a parameter weighting the regularization effect. This strategy for restoration based on arbitrary regularizers can be interpreted as a generalization of the MAP approach in (\[eq:log MAP estimate - AWGN model\]). Specifically, comparing the formulations (\[eq:restoration using a general prior\]) and (\[eq:log MAP estimate - AWGN model\]) exhibits the regularization function $s({\mathbf{x}})$ and the parameter $ \mu $ as extensions of $( - \log p \left( {{{\mathbf{x}}}} \right)) $ and the factor $2\sigma ^2_n$, respectively. Among the various regularization functions that can be associated with the general restoration approach in (\[eq:restoration using a general prior\]), we explore here the class of complexity regularizers measuring the required number of bits for the compressed representation of the candidate solution. The practical methods presented in this section focus on utilizing existing (independent) compression techniques, implicitly employing their underlying signal models for the restoration task. Operational Rate-Distortion Optimization {#subsec:Operational Rate-Distortion Optimization} ---------------------------------------- The practical complexity-regularized restoration methods in this section are developed with respect to a compression technique obeying the following conceptual design. The signal is segmented to equally-sized non-overlapping blocks (each is consisted of $ N_b $ samples) that are independently compressed. The block compression procedure is modeled as a general variable-rate vector quantizer relying on the following mappings. The compression is done by the mapping $ Q : \mathbb{R}^{N_b} \rightarrow \mathcal{W} $ from the $ N_b $-dimensional signal-block domain to a discrete set $ \mathcal{W} $ of binary compressed representations (that may have different lengths). The decompression procedure is associated with the mapping $ F : \mathcal{W} \rightarrow \mathcal{C} $, where $ \mathcal{C} \subset \mathbb{R}^{N_b} $ is a finite discrete set (a codebook) of block reconstruction candidates. For example, consider the block ${\mathbf{x}}_{block} \in \mathbb{R}^{N_b}$ that its binary compressed representation in $ \mathcal{W} $ is given via $ b = Q\left( {\mathbf{x}}_{block} \right) $ and the corresponding reconstructed block in $ \mathcal{C} $ is $ \hat{{\mathbf{x}}}_{block} = F\left( b \right) $. Importantly, it is assumed that shorter codewords are coupled with block reconstructions that are, in general, more likely. The signal $ {\mathbf{x}} $ is compressed based on its segmentation into a set of blocks $\{{\mathbf{x}}_i\}_{i\in\mathcal{B}}$ (where $ \mathcal{B}$ denotes the index set of blocks in the non-overlapping partitioning of the signal). In addition we introduce the function $ r({\mathbf{z}}) $ that evaluates the bit-cost (i.e., the length of the binary codeword) for the block reconstruction $ {\mathbf{z}}\in\mathcal{C} $. Then, the operational rate-distortion optimization corresponding to the described architecture and a squared-error distortion metric is [rCl]{} \[eq:rate-distortion optimization - blocks\] {\_i}\_[i]{} = \_[{\_i}\_[i]{} ]{} \_[i]{}[ \_2\^2]{} + \_[i]{}[r(\_i)]{} , where $ \lambda \ge 0 $ is a Lagrange multiplier corresponding to some total compression bit-cost. Importantly, the independent representation of non-overlapping blocks allows solving (\[eq:rate-distortion optimization - blocks\]) separately for each block [@shoham1988efficient; @ortega1998rate]. Our mathematical developments require the following algebraic tools for block handling. The matrix $ {\mathbf{P}}_i $ is defined to provide the $ i^{th} $ block from the complete signal via the standard multiplication ${\mathbf{P}}_i{\mathbf{x}} = {\mathbf{x}}_i$. Note that $ {\mathbf{P}}_i $ can extract any block of the signal, even one that is not in the non-overlapping grid $ \mathcal{B} $. Accordingly, the matrix $ {\mathbf{P}}_i^T $ locates a block in the $ i^{th} $ block-position in a construction of a full-sized signal and, therefore, lets to express the a complete signal as ${\mathbf{x}} = \mathop \sum\limits_{i\in\mathcal{B}} {\mathbf{P}}_i^T{\mathbf{x}}_i$. Now we can use the block handling operator $ {\mathbf{P}}_i $ for expressing the block-based rate-distortion optimization in its corresponding full-signal formulation: [rCl]{} \[eq:rate-distortion optimization - full signal\] = \_[ \_ ]{} [ \_2\^2]{} + r\_[tot]{}() . where $ \mathcal{C}_{\mathcal{B}} $ is the full-signal codebook, being the discrete set of candidate reconstructions for the full signal, defined using the block-level codebook $ \mathcal{C} $ as [rCl]{} \[eq:group of solutions based on nonoverlapping blocks\] \_ = . Moreover, the regularization function in (\[eq:rate-distortion optimization - full signal\]) is the total bit cost of the reconstructed signal defined for $ {\mathbf{v}} \in \mathcal{C}_{\mathcal{B}} $ as $ r_{tot}({\mathbf{v}}) \triangleq \nolinebreak \sum\limits_{i\in \mathcal{B}} r({\mathbf{P}}_i {\mathbf{v}})$. Complexity-Regularized Restoration: Basic Optimization Formulation {#subsec:Complexity-Regularized Restoration} ------------------------------------------------------------------ While the regularized-restoration optimization in (\[eq:restoration using a general prior\]) is over a continuous domain, the operational rate-distortion optimization in (\[eq:rate-distortion optimization - full signal\]) is a discrete problem with solutions limited to the set $ \mathcal{C}_{\mathcal{B}} $. Therefore, we extend the definition of the block bit-cost evaluation function such that it is defined for any $ {\mathbf{z}} \in \mathbb{R}^{N_b} $ via [rCl]{} \[eq:extended block bit-cost\] |r () = { [\*[20]{}[c]{}]{} [r( )]{}&[, ]{}\ &[, ]{} . , and the corresponding extension of the total bit-cost $\bar{r}_{tot}({\mathbf{x}})\triangleq\nolinebreak \sum\limits_{i\in \mathcal{B}} \bar r({\mathbf{P}}_i {\mathbf{x}})$ is defined for any $ {\mathbf{x}} \in \mathbb{R}^{N} $. Now we define the complexity regularization function as [rCl]{} \[eq:complexity regularization function\] s()=|[r]{}\_[tot]{}() and the corresponding restoration optimization is [rCl]{} \[eq:complexity-regularized restoration - full signal\] = \_ \_2\^2 + |[r]{}\_[tot]{}(). Due to the definition of the extended bit-cost evaluation function, $\bar{r}_{tot}({\mathbf{x}})$, the solution candidates of (\[eq:complexity-regularized restoration - full signal\]) are limited to the discrete set $ \mathcal{C}_{\mathcal{B}} $ as defined in (\[eq:group of solutions based on nonoverlapping blocks\]). Examining the complexity-regularized restoration in (\[eq:complexity-regularized restoration - full signal\]) for the Gaussian denoising task, where $ {\mathbf{H}} = {\mathbf{I}} $, shows that the optimization reduces to the regular rate-distortion optimization in (\[eq:rate-distortion optimization - full signal\]), namely, the compression of the noisy signal $ {\mathbf{y}} $. However, for more complicated restoration problems, where $ {\mathbf{H}} $ has an arbitrary structure, the optimization in (\[eq:complexity-regularized restoration - full signal\]) is not easy to solve and, in particular, it does not correspond to standard compression designs that are optimized for the regular squared-error distortion metric. Proposed Methods {#sec:Proposed Methods} ================ In this section we present three restoration methods leveraging a given compression technique. The proposed algorithms result from two different definitions for the complexity regularization function. While the first approach regularizes the total bit-cost of the non-overlapping blocks of the restored signal, the other two refer to the total bit-cost of all the overlapping blocks of the estimate. Regularize Total Complexity of Non-Overlapping Blocks {#subsec:Total Complexity of Non-Overlapping Blocks} ----------------------------------------------------- Here we establish a practical method addressing the optimization problem in (\[eq:complexity-regularized restoration - full signal\]) based on the alternating direction method of multipliers (ADMM) approach [@boyd2011distributed] (for additional uses see, e.g., [@venkatakrishnan2013plug; @sreehari2016plug; @dar2016postprocessing; @rond2016poisson; @afonso2010fast]). The optimization (\[eq:complexity-regularized restoration - full signal\]) can be expressed also as [rCl]{} \[eq:complexity-regularized restoration - expressing blocks\] = \_ \_2\^2 + \_[i]{} |r(\_i), where the degradation matrix $ {\mathbf{H}} $, having a general structure, renders a block-based treatment infeasible. Addressing this structural difficulty using the ADMM strategy [@boyd2011distributed] begins with introducing the auxiliary variables $ \left\lbrace {\mathbf{z}}_i \right\rbrace_{i\in\mathcal{B}} $, where $ {\mathbf{z}}_i $ is coupled with the $ i^{th} $ non-overlapping block. Specifically, we reformulate the problem (\[eq:complexity-regularized restoration - expressing blocks\]) into [rCl]{} \[eq:complexity-regularized restoration - with splitting - constrained form\] &&( , \_i \_[i]{} ) = \_[,\_i \_[i]{}]{} \_2\^2 + \_[i]{} [|[r]{}(\_i)]{}\ &&       \_i = \_i   i . Then, reformulating the constrained optimization (\[eq:complexity-regularized restoration - with splitting - constrained form\]) using the augmented Lagrangian (in its scaled form) and the method of multipliers (see [@boyd2011distributed Ch. 2]) leads to the following iterative procedure [rCl]{} \[eq:complexity-regularized restoration - with splitting\] ( \^[(t)]{}, \_i\^[(t)]{} \_[i]{} ) &=& \_[,\_i \_[i]{}]{} \_2\^2 + \_[i]{} |r(\_i)       \ && + \_[i]{} [ \_2\^2]{}\ \_i\^[(t+1) ]{} & = & \_i\^[(t) ]{} + ( \_i \^[(t)]{} - \_i\^[(t)]{} )    i , where $ t $ is the iteration number, $ \beta $ is a parameter originating in the augmented Lagrangian, and $ {\mathbf{u}}_i^{(t) } \in \mathbb{R}^{N_b} $ is the scaled dual variable corresponding to the $ i^{th} $ block (where $ i\in \mathcal{B} $). Each of the optimization variables in (\[eq:complexity-regularized restoration - with splitting\]) participates only in part of the terms of the cost function and, therefore, employing one iteration of alternating minimization (see [@boyd2011distributed Ch. 2]) leads to the ADMM form of the problem, where the included optimizations are relatively simple. Accordingly, the $ t^{th} $ iteration of the proposed iterative solution is [rCl]{} \[eq:complexity-regularized restoration - iterative solution - inversion\] && \^[(t)]{} = \_ \_2\^2 + \_[i]{} [ \_2\^2]{}    \ \[eq:complexity-regularized restoration - iterative solution - compression\] && \_i\^[(t)]{} = \_[\_i]{} + |r(\_i),    i\ \[eq:complexity-regularized restoration - iterative solution - beta update\] && \_i\^[(t+1)]{} = \_i\^[(t)]{} + ( \_i \^[(t)]{} - \_i\^[(t)]{} ) ,    i , where $ {\tilde{{\mathbf{z}}}}_{i}^{(t)} \triangleq {\hat{{\mathbf{z}}}}_{i}^{(t-1)} - {\mathbf{u}}_i^{(t) } $ and $ {\tilde{{\mathbf{x}}}}_{i}^{(t)} \triangleq {\mathbf{P}}_i \hat{{\mathbf{x}}}^{(t)} + {\mathbf{u}}_i^{(t)} $ for $ i\in\mathcal{B} $. The analytic solution of the first stage optimization in (\[eq:complexity-regularized restoration - iterative solution - inversion\]) is [rCl]{} \[eq:complexity-regularized restoration - iterative solution - inversion - analytic form\] \^[(t)]{} = ( \^[T]{} + )\^[-1]{} ( \^T + \_[i]{}[ \_i\^T \_[i]{}\^[(t)]{} ]{} )\ rendering this stage as a weighted averaging of the deteriorated signal with the block estimates obtained in the second stage of the previous iteration. While the analytic solution (\[eq:complexity-regularized restoration - iterative solution - inversion - analytic form\]) explains the underlying meaning of the $ \ell_2 $-constrained deconvolution stage (\[eq:complexity-regularized restoration - iterative solution - inversion\]), it includes matrix inversion that, in general, may lead to numerical instabilities. Accordingly, in the implementation of the proposed method we suggest to address (\[eq:complexity-regularized restoration - iterative solution - inversion\]) via numerical optimization techniques (for example, we used the biconjugate gradients method). The optimizations in the second stage of each iteration (\[eq:complexity-regularized restoration - iterative solution - compression\]) are rate-distortion optimizations corresponding to each of the non-overlapping blocks of the signal estimate $ \hat{ {\mathbf{x}}}^{(t)} $ obtained in the first stage. Accordingly, the set of block-level optimizations in (\[eq:complexity-regularized restoration - iterative solution - compression\]) can be interpreted as a single full-signal rate-distortion optimization with respect to a Lagrange multiplier value of $ {\lambda} = \frac{2\mu}{\beta} $. We denote the compression-decompression procedure that replaces (\[eq:complexity-regularized restoration - iterative solution - compression\]) as [rCl]{} \[eq:compression-decompression function\] \^[(t)]{}=CompressDecompress\_ ( \^[(t)]{} ), where $ \tilde{{\mathbf{x}}}^{(t)} \triangleq \mathop \sum\limits_{i\in\mathcal{B}} {\mathbf{P}}_i^T \tilde{{\mathbf{x}}}_i^{(t)} $ is the signal to compress, assembled from all the non-overlapping blocks, and $ \hat{{\mathbf{z}}}^{(t)} $ is the corresponding decompressed full signal. Moreover, by defining a full-sized scaled dual variable ${{\mathbf{u}}}^{(t)} \triangleq \mathop \sum\limits_{i\in\mathcal{B}} {\mathbf{P}}_i^T{{\mathbf{u}}}_i^{(t)} $ we get that $ \tilde{{\mathbf{x}}}^{(t)} = \hat{{\mathbf{x}}}^{(t)} + {{\mathbf{u}}}^{(t)}$. Then, using the definitions established here we can translate the block-level computations (\[eq:complexity-regularized restoration - iterative solution - inversion\])-(\[eq:complexity-regularized restoration - iterative solution - beta update\]) into the full-signal formulations described in Algorithm \[Algorithm:Proposed Method Non-overlapping\]. We further suggest using a standardized compression method as the compression-decompression operator (\[eq:compression-decompression function\]). While many compression methods do not follow the exact rate-distortion optimizations we have in our mathematical development, we still encourage utilizing such techniques as an approximation for (\[eq:complexity-regularized restoration - iterative solution - compression\]). Additionally, since many compression methods do not rely on Lagrangian optimization, their operating parameters may have different definitions such as quality parameters, compression ratios, or output bit-rates. Accordingly, we present the suggested algorithm with respect to a general compression-decompression procedure with output bit-cost directly or indirectly affected by a parameter denoted as $ \theta $. These generalizations are also implemented in the proposed Algorithm \[Algorithm:Proposed Method Non-overlapping\]. In Section \[sec:Experimental Results\] we elaborate on particular settings of $ \theta $ that were empirically found appropriate for utilization of the HEVC and the JPEG2000 standard. In cases where the compression method significantly deviates from a Lagrangian optimization form, it can be useful to appropriately update the compression parameter in each iteration (this is the case for JPEG2000 as explained in Section \[sec:Experimental Results\]). Importantly, Algorithm \[Algorithm:Proposed Method Non-overlapping\] does not only restore the deteriorated input image, but also provides the signal estimate in a compressed form by employing the output of the compression stage of the last iteration. Inputs: ${\mathbf{y}}$, $ \beta $, $ \theta $. Initialize $ {\hat{{\mathbf{z}}}}^{(0)} $ (depending on the deterioration type). $t = 1$ and $ {{\mathbf{u}}}^{(1)} = {\mathbf{0}} $ $\tilde{{\mathbf{z}}}^{(t)} = {\hat{{\mathbf{z}}}}^{(t-1)} - {\mathbf{u}}^{(t)}$ Solve the $ \ell_2 $-constrained deconvolution:$~~~~~~~~\hat{ {\mathbf{x}}}^{(t)} = \mathop {\text{argmin}}\limits_{{\mathbf{x}}} \left\| { {\mathbf{H}} {\mathbf{x}} - {\mathbf{y}} } \right\|_2^2 + \frac{\beta}{2} {\left\| { {\mathbf{x}} - \tilde{{\mathbf{z}}}^{(t)} } \right\|_2^2} $ $\tilde{{\mathbf{x}}}^{(t)} = {\hat{{\mathbf{x}}}}^{(t)} + {\mathbf{u}}^{(t)}$ $ \hat{{\mathbf{z}}}^{(t)} = {CompressDecompress}_{\theta}\left( \tilde{{\mathbf{x}}}^{(t)} \right) $ ${\mathbf{u}}^{(t+1)} = {\mathbf{u}}^{(t)} + \left( \hat{{\mathbf{x}}}^{(t)} - \hat{{\mathbf{z}}}^{(t)} \right)$ $ t \gets t + 1$ Regularize Total Complexity of All Overlapping Blocks {#subsec:Complexity of Overlapping Blocks} ----------------------------------------------------- Algorithm \[Algorithm:Proposed Method Non-overlapping\] emerged from complexity regularization measuring the total bit-cost of the estimate based on its decomposition into non-overlapping blocks (see Eq. (\[eq:complexity-regularized restoration - expressing blocks\])), resulting in a restored signal available in a compressed form compatible with the compression technique in use. Obviously, the above approach provides estimates limited to the discrete set of signals supported by the compression architecture, thus, having a somewhat reduced restoration ability with respect to methods providing estimates from an unrestricted domain of solutions. This observation motivates us to develop a complexity-regularized restoration procedure that provides good estimates from the continuous unrestricted domain of signals while still utilizing a standardized compression technique as its main component. As before, our developments refer to a general block-based compression method relying on a codebook $ \mathcal{C} $ as a discrete set of block reconstruction candidates. We consider here the segmentation of the signal-block space, $ \mathbb{R}^{N_b} $, given by the voronoi cells corresponding to the compression reconstruction candidates, namely, for each $ {\mathbf{c}} \in \mathcal{C} $ there is a region [rCl]{} \[eq:voronoi cell of codebook element\] V\_ \^[N\_b]{}    |[   = \_ - \_2\^2 ]{} defining all the vectors in $ \mathbb{R}^{N_b} $ that $ {\mathbf{c}} $ is their nearest member of $ \mathcal{C} $. We use the voronoi cells in (\[eq:voronoi cell of codebook element\]) for defining an alternative extension to the bit-cost evaluation of a signal block (i.e., the new definition, $\bar{r}_{v}({\mathbf{z}})$, will replace $\bar r({\mathbf{z}})$ given in (\[eq:extended block bit-cost\]) that was used for the development of Algorithm \[Algorithm:Proposed Method Non-overlapping\]). Specifically, we associate a finite bit-cost to any $ {\mathbf{z}} \in \mathbb{R}^{N_b} $ based on the voronoi cell it resides in, i.e., [rCl]{} \[eq:voronoi extended block bit-cost\] |[r]{}\_[v]{}() = r() V\_ where $ r({\mathbf{c}}) $ is the regular bit-cost evaluation defined in Section \[subsec:Operational Rate-Distortion Optimization\] only for blocks in $\mathcal{C}$. The method proposed here emerges from a new complexity regularization function that quantifies the total complexity of all the overlapping blocks of the estimate. Using the extended bit-cost measure $ \bar{r}_{v}(\cdot) $, defined in (\[eq:voronoi extended block bit-cost\]), we introduce the full-signal regularizer as [rCl]{} \[eq:complexity regularizer of all overlapping blocks\] s\^[\*]{}()=\_[i\^\*]{} |[r]{}\_v (\_i) where $ {\mathbf{x}} \in \mathbb{R}^N $, and $ \mathcal{B}^* $ is a set containing the indices of all the overlapping blocks of the signal. The associated restoration optimization is [rCl]{} \[eq:complexity-regularized restoration - overlapping blocks\] = \_ \_2\^2 + \_[i\^\*]{} |[r]{}\_v (\_i). Importantly, in contrast to the previous subsection, the function $ s^{*}({\mathbf{x}}) $ evaluates the complexity of any $ {\mathbf{x}} \in \mathbb{R}^N $ with a finite value and, thus, does not restrict the restoration to the discrete set of codebook-based constructions, $ \mathcal{C}_{\mathcal{B}} $, defined in (\[eq:group of solutions based on nonoverlapping blocks\]). While the new regularizer in (\[eq:complexity-regularized restoration - overlapping blocks\]) is not separable into complexity evaluation of non-overlapping blocks, the ADMM approach can accommodate it as well. This is explained next. We define the auxiliary variables $ \left\lbrace {\mathbf{z}}_i \right\rbrace_{i\in\mathcal{B}^*} $, where each $ {\mathbf{z}}_i $ is coupled with the $ i^{th} $ overlapping block. Then, the optimization (\[eq:complexity-regularized restoration - overlapping blocks\]) is expressed as [rCl]{} \[eq:complexity-regularized restoration - overlapping blocks - with splitting - constrained form\] &&( , \_i \_[i\^\*]{} ) = \_[,\_i \_[i\^\*]{}]{} \_2\^2 + \_[i\^\*]{} [|[r]{}\_v (\_i)]{}\ &&       \_i = \_i   i\^\* . As in Section \[subsec:Total Complexity of Non-Overlapping Blocks\], employing the augmented Lagrangian (in its scaled form) and the method of multipliers results in an iterative solution provided by the following three steps in each iteration (as before $ t $ denotes the iteration number): [rCl]{} \[eq:complexity-regularized restoration - overlapping blocks - iterative solution - inversion\] && \^[(t)]{} = \_ \_2\^2 + \_[i\^[\*]{}]{} [ \_2\^2]{}    \ \[eq:complexity-regularized restoration - overlapping blocks - iterative solution - compression\] && \_i\^[(t)]{} = \_[\_i]{} + |r(\_i),    i\^[\*]{}\ \[complexity-regularized restoration - overlapping blocks - iterative solution - beta update\] && \_i\^[(t+1)]{} = \_i\^[(t)]{} + ( \_i \^[(t)]{} - \_i\^[(t)]{} ) ,    i\^[\*]{} , where $ {\mathbf{u}}_i^{(t)} $ is the scaled dual variable for the $ i^{th} $ block, $ {\tilde{{\mathbf{z}}}}_{i}^{(t)} \nolinebreak \triangleq \nolinebreak {\hat{{\mathbf{z}}}}_{i}^{(t-1)} - {\mathbf{u}}_i^{(t) } $ and $ {\tilde{{\mathbf{x}}}}_{i}^{(t)} \triangleq {\mathbf{P}}_i \hat{{\mathbf{x}}}^{(t)} + {\mathbf{u}}_i^{(t)} $ for $ i\in\mathcal{B}^{*} $. While the procedure above resembles the one from the former subsection, the treatment of overlapping blocks has different interpretations to the optimizations in (\[eq:complexity-regularized restoration - overlapping blocks - iterative solution - inversion\]) and (\[eq:complexity-regularized restoration - overlapping blocks - iterative solution - compression\]). Indeed, note that the block-level rate-distortion optimizations in (\[eq:complexity-regularized restoration - overlapping blocks - iterative solution - compression\]) are not discrete due to the extended bit-cost evaluation $ \bar{r}_v(\cdot) $ defined in (\[eq:voronoi extended block bit-cost\]). Due to the definition of $ \bar{r}_v(\cdot) $, the rate-distortion optimizations (\[eq:complexity-regularized restoration - overlapping blocks - iterative solution - compression\]) can be considered as continuous relaxations of the discrete optimizations done by the practical compression technique. Since we intend using a given compression method without explicit knowledge of its underlying codebook, we cannot construct the voronoi cells defining $ \bar{r}_v(\cdot) $ and, thus, it is impractical to accurately solve (\[eq:complexity-regularized restoration - overlapping blocks - iterative solution - compression\]). Consequently, we suggest to approximate the optimizations (\[eq:complexity-regularized restoration - overlapping blocks - iterative solution - compression\]) by the discrete forms of [rCl]{} \[eq:complexity-regularized restoration - overlapping blocks - iterative solution - compression - discrete\] && \_i\^[(t)]{} = \_[\_i]{} + |[r]{}(\_i),   i\^\* where $ \bar{r}(\cdot) $ is the discrete evaluation of the block bit-cost, defined in (\[eq:extended block bit-cost\]), letting to identify the problems as operational rate-distortion optimizations of the regular discrete form. Each block-level rate-distortion optimization in (\[eq:complexity-regularized restoration - overlapping blocks - iterative solution - compression - discrete\]) is associated with one of the overlapping blocks of the signal. Accordingly, we interpret this group of optimizations as multiple applications of a full-signal compression-decompression procedure, each associates to a shifted version of the signal (corresponding to different sets of non-overlapping blocks). Specifically, for a signal $ {\mathbf{x}} $ and a compression block-size of $ N_b $ samples, there are $ N_b $ shifted grids of non-overlapping blocks. For mathematical convenience, we consider here cyclic shifts such that the $ j^{th} $ shift ($ j=1,...,N_b $) corresponds to a signal of $ N $ samples taken cyclically starting at the $ j^{th} $ sample of $ {\mathbf{x}} $ (in practice other definitions of shifts may be used, e.g., see Section \[sec:Experimental Results\] for a suggested treatment of two-dimensional signals). We denote the $ j^{th} $ shifted signal as $ shift_{j}\left\lbrace {\mathbf{x}} \right\rbrace $. Moreover, we denote the index set of blocks included in the $ j^{th} $ shifted signal as $ \mathcal{B}^{j} $ (noting that $ \mathcal{B}^{1} = \mathcal{B} $), hence, $ \mathcal{B}^{*} = \cup_{j=1}^{N_{b}}\mathcal{B}^{j} $. Therefore, the decompressed blocks $\left\lbrace{\hat{{\mathbf{z}}}_i^{(t)}}\right\rbrace_{i\in\mathcal{B}^*}$ can be decomposed into $ N_{b} $ subsets, $\left\lbrace{\hat{{\mathbf{z}}}_i^{(t)}}\right\rbrace_{i\in\mathcal{B}^j}$ for $ j=1,...,N_b $, each contains non-overlapping blocks corresponding to a different shifted grid. Moreover, the $ j^{th} $ set of blocks, $\left\lbrace{\hat{{\mathbf{z}}}_i^{(t)}}\right\rbrace_{i\in\mathcal{B}^j}$, is associated with the full signal $ \hat{{\mathbf{z}}}^{j,(t)} \triangleq \sum\limits_{i\in \mathcal{B}^j}{ {\mathbf{P}}_i^T \hat{{\mathbf{z}}}_{i}^{(t)}} $. Then, the set of full signals $\left\lbrace{\hat{{\mathbf{z}}}^{j,(t)}}\right\rbrace_{j=1}^{N_b}$ can be obtained by multiple full-signal compression-decompression applications, namely, for $ j\nolinebreak=\nolinebreak 1,...,N_b $: $ \hat{{\mathbf{z}}}_{shifted}^{j,(t)}\nolinebreak=\nolinebreak CompressDecompress_{ \lambda} \left( \tilde{{\mathbf{x}}}^{j,(t)}_{shifted} \right) $, where the Lagrangian multiplier value is $ {\lambda} = \frac{2\mu}{\beta} $ and [rCl]{} \[eq:complexity-regularized restoration - overlapping blocks - shifted compression input - definition\] && \^[j,(t)]{}\_[shifted]{} shift\_[j]{}\^[(t)]{} + \^[j,(t)]{} is the compression input formed as the $ j^{th} $ shift of $ \hat{{\mathbf{x}}}^{(t)} $ combined with the full-sized dual variable defined via [rCl]{} \[eq:complexity-regularized restoration - overlapping blocks - shifted dual variable - definition\] && \^[j,(t)]{} \_[i\^j]{}[ \_i\^T \_[i]{}\^[(t)]{}]{} assembled from the block-level dual variables corresponding to the $ j^{th} $ grid of non-overlapping blocks. Notice that inverse shifts are required for obtaining the desired signals, i.e., [rCl]{} \[eq:complexity-regularized restoration - overlapping blocks - shifted decompression output - definition\] && \^[j,(t)]{} = shift\_[j]{}\^[-1]{}\_[shifted]{}\^[j,(t)]{} where $ shift^{-1}_{j} \left\lbrace \cdot \right\rbrace$ is the inverse shift operator that (cyclically) shifts back the given full-size signal by $ j $ samples. The deconvolution stage (\[eq:complexity-regularized restoration - overlapping blocks - iterative solution - inversion\]) of the iterative process can be rewritten as [rCl]{} \[eq:complexity-regularized restoration - overlapping blocks - iterative solution - inversion - full signals\] && \^[(t)]{} = \_ \_2\^2 + \_[j=1]{}\^[N\_b]{} [ \_2\^2]{}     where the regularization part (the second term) considers the distance of the estimate from the $ N_b $ full signals defined via [rCl]{} \[eq:complexity-regularized restoration - overlapping blocks - iterative solution - inversion - full signals - auxiliary definition\] && \^[j,(t)]{} \^[j,(t)]{} - \^[j,(t)]{} for $ j=1,...,N_b $, where $ {\hat{{\mathbf{z}}}}^{j,(t)} $ and $ {{\mathbf{u}}}^{j,(t)} $ were defined above. The analytic solution of the optimization (\[eq:complexity-regularized restoration - overlapping blocks - iterative solution - inversion - full signals\]) is [rCl]{} \[eq:complexity-regularized restoration - overlapping blocks - inversion - analytic form - full signals\] &&\^[(t)]{} = ( \^[T]{} + N\_b )\^[-1]{} ( \^T + \_[j=1]{}\^[N\_b]{}[ \^[j,(t)]{} ]{} ) ,    showing that the first stage of each iteration is a weighted averaging of the given deteriorated signal with all the decompressed signals (and the dual variables) obtained in the former iteration. It should be noted that the analytic solution (\[eq:complexity-regularized restoration - overlapping blocks - inversion - analytic form - full signals\]) is developed here for showing the essence of the $ \ell_2 $-constrained deconvolution part of the method. Nevertheless, the possible numerical instabilities due to the matrix inversion appearing in (\[eq:complexity-regularized restoration - overlapping blocks - inversion - analytic form - full signals\]) motivate the practical direct treatment of (\[eq:complexity-regularized restoration - overlapping blocks - iterative solution - inversion\]) via numerical optimization techniques. Algorithm \[Algorithm:Proposed Method Overlapping Blocks\] summarizes the practical restoration method for a compression technique operated by the general parameter $ {\theta} $ for determining the bit-cost (see details in Section \[subsec:Total Complexity of Non-Overlapping Blocks\]). The computational cost of Algorithm \[Algorithm:Proposed Method Overlapping Blocks\] stems from its reliance on repeated applications of compressions, decompressions, and $ \ell_2 $ - constrained deconvolution procedures. While the actual run-time depends on the computational complexity of the utilized compression technique, we can generally state that the total run-time will be of at least the run-time of compression and decompression processes for a total number of applications equal to the product of the number of iterations and the number of shifts considered. Inputs: $ {\mathbf{y}} $, $\beta$, $ \theta $. Initialize $ \left\lbrace {\hat{{\mathbf{z}}}}^{j,(0)} \right\rbrace_{j=1}^{N_b} $ (depending on the deterioration type). $t = 1$ and $ {{{\mathbf{u}}}}^{j,(1)} = {\mathbf{0}} $ for $ j = 1,...,{N_b} $. ${\tilde{{\mathbf{z}}}}^{j,(t)} = {\hat{{\mathbf{z}}}}^{j,(t-1)} - {\mathbf{u}}^{j,(t) } \text{~~for~}j=1,...,N_b $ Solve the $ \ell_2 $-constrained deconvolution:$~~~~\hat{ {\mathbf{x}}}^{(t)} = \mathop {\text{argmin}}\limits_{{\mathbf{x}}} \left\| { {\mathbf{H}} {\mathbf{x}} - {\mathbf{y}} } \right\|_2^2 + \frac{\beta}{2} \sum\limits_{j=1}^{N_b} {\left\| { {\mathbf{x}} - {\tilde{{\mathbf{z}}}}^{j,(t)}} \right\|_2^2} $ $\tilde{{\mathbf{x}}}_{shifted}^{j,(t)} = shift_{j}\left\lbrace \hat{{\mathbf{x}}}^{(t)} + {{\mathbf{u}}}^{j,(t)} \right\rbrace$ $ \hat{{\mathbf{z}}}_{shifted}^{j,(t)} = {CompressDecompress}_{\theta}\left( \tilde{{\mathbf{x}}}_{shifted}^{j,(t)} \right) $ $\hat{{\mathbf{z}}}^{j,(t)} = shift^{-1}_{j}\left\lbrace { \hat{{\mathbf{z}}}_{shifted}^{j,(t)} } \right\rbrace$ ${\mathbf{u}}^{j,(t+1)} = {\mathbf{u}}^{j,(t)} + \left( \hat{{\mathbf{x}}}^{(t)} - \hat{{\mathbf{z}}}^{j,(t)} \right) $ $ t \gets t + 1$ The ADMM is known for promoting distributed optimization structures [@boyd2011distributed]. In Algorithm \[Algorithm:Proposed Method Overlapping Blocks\] the distributed nature of the ADMM is expressed in the separate optimization of each of the shifted block-grids (see stages 8-11). In particular, the dual variables $ \left\lbrace {\mathbf{u}}^{j,(t)} \right\rbrace_{j=1}^{N_b} $, associated with the various grids (see stages 5,8, and 11 in Algorithm \[Algorithm:Proposed Method Overlapping Blocks\]), are updated independently in stage 11 such that each considers only its respective $ \hat{{\mathbf{z}}}^{j,(t)} $. However, the dual variables $ \left\lbrace {\mathbf{u}}^{j,(t)} \right\rbrace_{j=1}^{N_b} $ essentially refer to the same data based on different block-grids. Accordingly, we suggest to merge the independent dual variables to form a single, more robust, dual variable defined as [rCl]{} \[eq:complexity-regularized restoration - overlapping blocks - robust dual variable\] && \_[total]{}\^[(t)]{} = \_[j=1]{}\^[N\_b]{}[ \^[j,(t)]{} ]{} where the averaging tends to reduce particular artifacts that may appear due to specific block-grids. We utilize the averaged dual variable (\[eq:complexity-regularized restoration - overlapping blocks - robust dual variable\]) to extend Algorithm \[Algorithm:Proposed Method Overlapping Blocks\] into Algorithm \[Algorithm:Proposed Method Overlapping Blocks with Robust Dual Variables\]. Notice stages 5,8, and 13 of Algorithm \[Algorithm:Proposed Method Overlapping Blocks with Robust Dual Variables\], where the averaged dual variable is used instead of the independent ones. Inputs: $ {\mathbf{y}} $, $\beta$, $ \theta $. Initialize $ \left\lbrace {\hat{{\mathbf{z}}}}^{j,(0)} \right\rbrace_{j=1}^{N_b} $ (depending on the deterioration type). $t = 1$ and $ {\mathbf{u}}_{total}^{(1)} = {\mathbf{0}} $. ${\tilde{{\mathbf{z}}}}^{j,(t)} = {\hat{{\mathbf{z}}}}^{j,(t-1)} - {\mathbf{u}}_{total}^{(t)} \text{~~for~}j=1,...,N_b $ Solve the $ \ell_2 $-constrained deconvolution:$~~~~\hat{ {\mathbf{x}}}^{(t)} = \mathop {\text{argmin}}\limits_{{\mathbf{x}}} \left\| { {\mathbf{H}} {\mathbf{x}} - {\mathbf{y}} } \right\|_2^2 + \frac{\beta}{2} \sum\limits_{j=1}^{N_b} {\left\| { {\mathbf{x}} - {\tilde{{\mathbf{z}}}}^{j,(t)}} \right\|_2^2} $ $\tilde{{\mathbf{x}}}_{shifted}^{j,(t)} = shift_{j}\left\lbrace \hat{{\mathbf{x}}}^{(t)} + {\mathbf{u}}_{total}^{(t)} \right\rbrace$ $ \hat{{\mathbf{z}}}_{shifted}^{j,(t)} = {CompressDecompress}_{\theta}\left( \tilde{{\mathbf{x}}}_{shifted}^{j,(t)} \right) $ $\hat{{\mathbf{z}}}^{j,(t)} = shift^{-1}_{j}\left\lbrace { \hat{{\mathbf{z}}}_{shifted}^{j,(t)} } \right\rbrace$ ${\mathbf{u}}^{j,(t+1)} = {\mathbf{u}}^{j,(t)} + \left( \hat{{\mathbf{x}}}^{(t)} - \hat{{\mathbf{z}}}^{j,(t)} \right) $ ${\mathbf{u}}_{total}^{(t+1)} = \frac{1}{N_b} \sum\limits_{j=1}^{N_b}{ {\mathbf{u}}^{j,(t+1)} }$ $ t \gets t + 1$ In Section \[sec:Experimental Results\] we further discuss practical aspects of the proposed Algorithms \[Algorithm:Proposed Method Non-overlapping\]-\[Algorithm:Proposed Method Overlapping Blocks with Robust Dual Variables\] and evaluate their performance for deblurring and inpainting of images. Rate-Distortion Theoretic Analysis for the Gaussian Case {#sec:Rate-Distortion Theoretic Analysis for the Gaussian Case} ======================================================== In this section we theoretically study the complexity-regularized restoration problem from the perspective of rate-distortion theory. While our analysis is focused on the particular settings of a cyclo-stationary Gaussian signal and deterioration caused by a linear shift-invariant operator and additive white Gaussian noise, the results clearly explain the main principles of complexity-regularized restoration. In general, theoretical studies of rate-distortion problems for the Gaussian case provide to the signal processing practice optimistic beliefs about which design concepts perform well for the real-world non-Gaussian instances of the problems (see the excellent discussion in [@donoho1998data Sec. 3]). Moreover, theoretical and practical solutions may embody in a different way the same general concepts. Therefore, one should look for connections between theory and practice in the form of high-level analogies. The optimal solution presented in this section considers the classical framework of rate-distortion theory and a particular, however, important case of a Gaussian signal and a linear shift-invariant degradation operator. Our rate-distortion analysis below will show that the optimal complexity-regularized restoration consists of the following two main ideas: pseudoinverse filtering of the degraded input, and compression with respect to a squared-error metric that is weighted based on the degradation-filter squared-magnitude (considering a processing in the Discrete Fourier Transform (DFT) domain). In Subsection \[subsec:Rate-Distortion Theoretic - Conceptual Relation to The Proposed Approach\] we explain how these two concepts connect to more general themes having different realizations in the practical approach proposed in Section \[sec:Proposed Methods\]. In this section, consider the signal $ {\mathbf{x}} \in \mathbb{R}^N $ modeled as a zero-mean cyclo-stationary Gaussian random vector with a circulant autocorrelation matrix $ {\mathbf{R}}_{{\mathbf{x}}} $, i.e., ${\mathbf{x}} \sim \mathcal{N}\left(0,{\mathbf{R}}_{{\mathbf{x}}}\right) $. The degradation model studied remains [rCl]{} \[eq:theoretic Gaussian analysis - degradation model\] = + , where here $ {\mathbf{H}} $ is a real-valued $N\times N$ circulant matrix representing a linear shift-invariant deteriorating operation and ${\mathbf{n}} \sim \mathcal{N}\left(0,\sigma_n^2{\mathbf{I}}\right) $ is a length $N$ vector of white Gaussian noise. Clearly, the degraded observation $ {\mathbf{y}} $ is also a zero-mean cyclo-stationary Gaussian random vector with a circulant autocorrelation matrix $ {\mathbf{R}}_{{\mathbf{y}}}\nolinebreak=\nolinebreak{\mathbf{H}} {\mathbf{R}}_{{\mathbf{x}}} {\mathbf{H}}^* + \sigma_n^2{\mathbf{I}} $. Prevalent Restoration Strategies {#subsec:Alternative Restoration Strategies} -------------------------------- We precede the analysis of the complexity-regularized restoration with mentioning three well-known estimation methods. The restoration procedure is a function [rCl]{} \[eq:theoretic Gaussian analysis - restoration function\] = f ( ), where $ f $ maps the degraded signal $ {\mathbf{y}} $ to an estimate of $ {\mathbf{x}} $ denoted as $ \hat{{\mathbf{x}}} $. In practice, one gets a realization of $ {\mathbf{y}} $ denoted here as $ {\mathbf{y}}_r $ and forms the corresponding estimate as $ {\hat{{\mathbf{x}}}}_r \nolinebreak = \nolinebreak f \left( {\mathbf{y}}_r \right) $. ### Minimum Mean Squared Error (MMSE) Estimate This restoration minimizes the expected MSE of the estimate, i.e., [rCl]{} \[eq:theoretic Gaussian analysis - MMSE estimate optimization\] f\_[MMSE]{} =   E \_2\^2 , yielding that the corresponding estimate is the conditional expectation of $ {\mathbf{x}} $ given $ {\mathbf{y}} $ [rCl]{} \[eq:theoretic Gaussian analysis - MMSE estimate\] \_[MMSE]{} = f\_[MMSE]{} ( ) = E . Nicely, for the Gaussian case considered in this section, the MMSE estimate (\[eq:theoretic Gaussian analysis - MMSE estimate\]) reduces to a linear operator, presented below as the Wiener filter. ### Wiener Filtering The Wiener filter is also known as the Linear Minimum Mean Squared Error (LMMSE) estimate, corresponding to a restoration function of the form [rCl]{} \[eq:theoretic Gaussian analysis - Wiener restoration function\] = f\_[Wiener]{} ( ) = + , optimized via [rCl]{} \[eq:theoretic Gaussian analysis - Wiener estimate optimization\] =   E \_2\^2 . In our case, where $ {\mathbf{x}} $ and $ {\mathbf{y}} $ are zero mean, $ \hat{{\mathbf{b}}} = {\mathbf{0}} $ and [rCl]{} \[eq:theoretic Gaussian analysis - Wiener filter matrix\] = \_ \^\* ( \_ \^\* + \_n\^2 )\^[-1]{} . If the distributions are Gaussian, this linear operator coincides with the optimal MMSE estimator. ### Constrained Deconvolution Filtering This approach considers a given degraded signal $ {\mathbf{y}}_r = {\mathbf{H}} {\mathbf{x}}_0 + {\mathbf{n}}_r $, with the noise vector a realization of a random process while the signal ${\mathbf{x}}_0$ is considered as a deterministic vector, with perhaps some known properties. Then, the restoration is carried out by minimizing a carefully-designed penalty function, $ g $, that assumes lower values for ${\mathbf{x}}$ vectors that fit the prior knowledge on ${\mathbf{x}}_0$. Note that for a sufficiently large signal dimension we get that $ \left\| { {\mathbf{y}}_r - {\mathbf{H}} {\mathbf{x}}_0 } \right\|_2^2 = \left\| {\mathbf{n}}_r \right\|_2^2 \approx N \sigma_n^2 $. The last result motivates to constrain the estimate, $ \hat{{\mathbf{x}}} $, to conform with the known degradation model (\[eq:theoretic Gaussian analysis - degradation model\]), by demanding the similarity of $ { {\mathbf{y}}_r - {\mathbf{H}} \hat{{\mathbf{x}}} } $ to the additive noise term via the equality relation $ \left\| { {\mathbf{y}}_r - {\mathbf{H}} \hat{{\mathbf{x}}} } \right\|_2^2 = N \sigma_n^2 $. The above idea is implemented in an optimization of the form [rCl]{} \[eq:theoretic Gaussian analysis - constrained deconvolution filtering\] & & & [ g( ) ]{}\ & & & \_2\^2 = N \_n\^2 . Our practical methods presented in Section \[sec:Proposed Methods\] emerge from an instance of the constrained deconvolution optimization (\[eq:theoretic Gaussian analysis - constrained deconvolution filtering\]), in its Lagrangian version, where the penalty function $ g $ is the cost in bits measuring the complexity in describing the estimate $ \hat{{\mathbf{x}}} $. In the remainder of this section, we study the complexity-regularized restoration problem from a statistical perspective. The Complexity-Regularized Restoration Problem and its Equivalent Forms {#subsec:Reformulations of the Problem} ----------------------------------------------------------------------- Based on rate-distortion theory (e.g., see [@cover2012elements]), we consider the estimate of $ {{\mathbf{x}}} $ as a random vector $ \hat{{\mathbf{x}}} \in \mathbb{R}^N $ with the probability density function (PDF) $ p_{\hat{{\mathbf{x}}}}\left( {{\hat{x}}} \right) $. The estimate characterization, $ p_{\hat{{\mathbf{x}}}}\left( {{\hat{x}}} \right) $, is determined by optimizing the conditional PDF $ p_{\hat{{\mathbf{x}}} | {\mathbf{y}} }\left( {{\hat{x}}} | {{y}} \right) $, statistically representing the mapping between the given data $ {\mathbf{y}} $ and the decompression result $ \hat{{\mathbf{x}}} $. Moreover, the rate is measured as the mutual information between $\hat{{\mathbf{x}}}$ and $ {\mathbf{y}} $, defined via [rCl]{} \[eq:theoretic Gaussian analysis - mutual information definition\] I( ; ) = . Then, the basic form of the complexity-regularized restoration optimization is expressed as \[problem: basic form\] [rCl]{} & & & [ I( ; ) ]{}\ & & & E \_2\^2 = N \_n\^2 . Here the estimate rate is minimized while maintaining suitability to the degradation model (\[eq:theoretic Gaussian analysis - degradation model\]) using a distortion constraint set to achieve an a-priori known total noise energy. In general, Problem \[problem: basic form\] is complicated to solve since the distortion constraint considers $ \hat{{\mathbf{x}}} $ through the degradation operator $ {\mathbf{H}} $, while the rate is directly evaluated for $ \hat{{\mathbf{x}}} $. The shift invariant operator $ {\mathbf{H}} $ is a circulant $ N \times N $ matrix, thus, diagonalized by the $ N\times N $ Discrete Fourier Transform (DFT) matrix $ {\mathbf{F}} $. The $ (k,l) $ component of the DFT matrix ($ k,l=0,...,N-1$) is ${\mathbf{F}}_{k,l}= W_N^{kl} $ where $ W_N \triangleq \frac{1}{\sqrt{N}} e^{-i2\pi/N} $. Then, the diagonalization of $ {\mathbf{H}} $ is expressed as [rCl]{} \[eq:theoretic Gaussian analysis - diagonlization of H\] \^\* = \_H , where $ {\mathbf{\Lambda}}_H $ is a diagonal matrix formed by the components $ h^F_k $ for $ k=0,...,N-1 $. Using $ {\mathbf{\Lambda}}_H $ we define the pseudoinverse of $ {\mathbf{H}} $ as [rCl]{} \[eq:theoretic Gaussian analysis - pseudoinverse of H\] \^[+]{} = \^\* \_H\^[+]{} , where $ {\mathbf{\Lambda}}_H^{+} $ is the pseudoinverse of $ {\mathbf{\Lambda}}_H $, an $ N \times N $ diagonal matrix with the $ k^{th} $ diagonal element: [rCl]{} \[eq:theoretic Gaussian analysis - pseudoinverse of Lambda\_H - diagonal elements\] h\^[F,+]{}\_k = { [\*[20]{}[c]{}]{} &[, h\^F\_k0 ]{}\ 0&[, h\^F\_k = 0 . ]{} . We denote by $N_{H}$ the number of nonzero diagonal elements in $ {\mathbf{\Lambda}}_H $, the rank of $ {\mathbf{H}} $. The first main result of our analysis states that Problem \[problem: basic form\], being the straightforward formulation for complexity-regularized restoration, is equivalent to the next problem. \[problem: pseudoinverse filtered input\] [rCl]{} & & & [ I( ; ) ]{}\ & & & E ( [ - ]{} ) \_2\^2 = N\_[H]{} \_n\^2 , where [rCl]{} \[eq:theoretic Gaussian analysis - pseudoinverse filtered input definition\] = \^[+]{} is the pseudoinverse filtered version of the given degraded signal $ {\mathbf{y}} $. One should note that Problem \[problem: pseudoinverse filtered input\] has a more convenient form than Problem \[problem: basic form\] since the distortion is an expected weighted squared error between the two random variables determining the rate. The equivalence of Problems \[problem: basic form\] and \[problem: pseudoinverse filtered input\] is proved in Appendix \[appendix:Equivalence of Problems 1 and 2\]. In this section, $ {\mathbf{x}} $ is a cyclo-stationary Gaussian signal, hence, having a circulant autocorrelation matrix $ {\mathbf{R}}_{{\mathbf{x}}} $. Consequently, and also because $ {\mathbf{H}} $ is circulant, the deteriorated signal $ {\mathbf{y}} $ is also a cyclo-stationary Gaussian signal. Moreover, $ {\mathbf{H}}^{+} $ is also a circulant matrix, thus, by (\[eq:theoretic Gaussian analysis - pseudoinverse filtered input definition\]) the pseudoinverse filtering result, $ {\tilde{{\mathbf{y}}}} $, is also cyclo-stationary and zero-mean Gaussian. Specifically, the autocorrelation matrix of $ {\tilde{{\mathbf{y}}}} $ is [rCl]{} \[eq:theoretic Gaussian analysis - pseudoinverse filtered input - autocorrelation matrix\] \_ & = & \^[+]{} \_ [\^[+]{}]{}\^\*\ & = & \^[+]{} \_ \^\* [\^[+]{}]{}\^\* + \_n\^2 \^[+]{} [\^[+]{}]{}\^\* , and, as a circulant matrix, it is diagonalized by the DFT matrix yielding the eigenvalues [rCl]{} \[eq:theoretic Gaussian analysis - pseudoinverse filtered input - autocorrelation eigenvalues\] \^[()]{}\_k = { [\*[20]{}[c]{}]{} [\^[()]{}\_k + ]{}&[, h\^F\_k0 ]{}\ 0&[, h\^F\_k = 0 . ]{} . The DFT-domain representation of $ {\tilde{{\mathbf{y}}}} $ is [rCl]{} \[eq:theoretic Gaussian analysis - pseudoinverse filtered input - in DFT domain\] \^F = , consisted of the coefficients $ \left\lbrace {\tilde{y}}^F_k \right\rbrace_{k=0}^{N-1} $, being independent zero-mean Gaussian variables with variances corresponding to the eigenvalues in (\[eq:theoretic Gaussian analysis - pseudoinverse filtered input - autocorrelation eigenvalues\]). Transforming Problem \[problem: pseudoinverse filtered input\] to the DFT domain, where $ {\tilde{{\mathbf{y}}}} $ becomes a set of independent Gaussian variables to be coded under a joint distortion constraint, simplifies the optimization structure to the following separable form (see proof sketch in Appendix \[appendix:Equivalence of Problems 2 and 3\]). \[problem: Separable Form in DFT domain\] [rCl]{} & & & [ \_[k=0 ]{}\^[N-1]{} I( [\^F\_k]{}; \^F\_k ) ]{}\ & & & \_[k=0]{}\^[N-1]{} [| h\^F\_k | \^2]{} E = N\_[H]{} \_n\^2 , where $ \left\lbrace {\hat{x}}^F_k \right\rbrace_{k=0}^{N-1} $ are the elements of $ {\hat{{\mathbf{x}}}}^F = {\mathbf{F}}{\hat{{\mathbf{x}}}} $. Nicely, the separable distortion in Problem \[problem: Separable Form in DFT domain\] considers each variable using a squared error that is weighted by the squared magnitude of the corresponding degradation-filter coefficient. The rate-distortion function of a single Gaussian variable with variance $ \sigma^2 $ has the known formulation [@cover2012elements]: [rCl]{} \[eq:theoretic Gaussian analysis - rate-distortion function of a Gaussian variable\] R(D) = \_+ evaluating the minimal rate for a squared-error allowed reaching up to $ D \ge 0 $. In addition, the operator $ \left[\cdot\right]_+ $ is defined for real scalars as $ \left[\alpha\right]_+ \triangleq \max\left\{\alpha, 0\right\} $, hence, $ R \left( D \right) = 0 $ for $ D \ge \sigma^2 $. Accordingly, the rate-distortion function of the Gaussian variable ${\tilde{y}^F_k}$ is [rCl]{} \[eq:theoretic Gaussian analysis - rate-distortion function of DFT component\] R\_k(D\_k) = \_+ where $ D_k $ denotes the maximal squared-error allowed for this component. Now, similar to the famous case of jointly coding independent Gaussian variables with respect to a regular (non-weighted) squared-error distortion [@cover2012elements], we explicitly express Problem \[problem: Separable Form in DFT domain\] as the following distortion-allocation optimization. \[problem: Gaussian distortion allocation\] [rCl]{} \[eq:theoretic Gaussian analysis - Gaussian distortion allocation\] & & & \_[k=0]{}\^[N-1]{} [ \_+ ]{}\ & & & \_[k=0]{}\^[N-1]{} [ | h\^F\_k |\^2 D\_k ]{} = N\_[H]{} \_n\^2\ & & & D\_k 0    , k=0,...,N-1. The optimal distortion-allocation satisfying the last optimization is [rCl]{} \[eq:theoretic Gaussian analysis - optimal distortion allocation\] D\_k\^[opt]{} = { [\*[20]{}[c]{}]{} &[, h\^F\_k0 ]{}\ 0&[, h\^F\_k = 0 ]{} . and the associated optimal rates are [rCl]{} \[eq:theoretic Gaussian analysis - optimal rate allocation\] R\_k\^[opt]{} = { [\*[20]{}[c]{}]{} [ ( | h\^F\_k |\^2 + 1 )]{}&[, h\^F\_k0 ]{}\ 0&[, h\^F\_k = 0 . ]{} . Results (\[eq:theoretic Gaussian analysis - optimal distortion allocation\]) and (\[eq:theoretic Gaussian analysis - optimal rate allocation\]) are proved in Appendix \[appendix:Solution of Problem 4\]. Demonstration of The Explicit Results {#subsec:Rate-Distortion Theoretic - Demonstration of The Explicit Results} ------------------------------------- Let us exemplify the optimal rate-distortion results (\[eq:theoretic Gaussian analysis - optimal distortion allocation\])-(\[eq:theoretic Gaussian analysis - optimal rate allocation\]) for a cyclo-stationary Gaussian signal, $ {\mathbf{x}} $, having the circulant autocorrelation matrix presented in Fig. \[fig:signal\_autocorrelation\_matrix\], corresponding to the eigenvalues $ \{ {\lambda}^{({\mathbf{x}})}_k \} _{k=0}^{N-1} $ (Fig. \[fig:signal\_autocorrelation\_matrix\_DFT\_decomposition\]) obtained by a DFT-based decomposition. We first examine the denoising problem, where the signal-domain degradation matrix is $ {\mathbf{H}} = {\mathbf{I}} $ (Fig. \[fig:gaussian\_denoising\_\_degradation\_filter\_signal\_domain\]) and its respective DFT-domain spectral representation consists of $ h^F_k = 1 $ for any $ k $ (see Fig. \[fig:gaussian\_denoising\_\_degradation\_filter\_DFT\_coefficient\_magnitudes\]). The additive white Gaussian noise has a sample variance of $ \sigma_n^2 = 5 $. Fig. \[fig:gaussian\_denoising\_\_effective\_waterfilling\_demonstration\] exhibits the optimal distortion allocation using a reverse-waterfilling diagram, where the signal-energy distribution $ \{ {\lambda}^{({\mathbf{x}})}_k \} _{k=0}^{N-1} $ (black solid line) and the additive noise energy (the light-red region) defining together the noisy-signal energy level (purple solid line) corresponding to $\lambda^{\left(\tilde{{\mathbf{y}}}\right)}_k = \lambda^{({\mathbf{x}})}_k + \sigma_n^2$. The blue dashed line in Fig. \[fig:gaussian\_denoising\_\_effective\_waterfilling\_demonstration\] shows the water level associated with the uniform distortion allocation. The optimal rate-allocation, corresponding to Fig. \[fig:gaussian\_denoising\_\_effective\_waterfilling\_demonstration\] and Eq. (\[eq:theoretic Gaussian analysis - optimal rate allocation\]), is presented in Fig. \[fig:gaussian\_denoising\_\_optimal\_rate\_allocation\] showing that more bits are spent on components with higher signal-to-noise ratios. Another example considers the same Gaussian signal described in Fig. \[Fig:Gaussian signal for demonstration\] and the noise level of $ \sigma_n^2 = 5 $, but here the degradation operator is the circulant matrix shown in Fig. \[fig:gaussian\_restoration\_\_degradation\_filter\_signal\_domain\] having a DFT-domain representation given in magnitude-levels in Fig. \[fig:gaussian\_restoration\_\_degradation\_filter\_DFT\_coefficient\_magnitudes\] exhibiting its frequency attenuation and amplification effects. The waterfilling diagram in Fig. \[fig:gaussian\_restoration\_\_effective\_waterfilling\_demonstration\] includes the same level of signal energy (black solid line) as in the denoising experiment, but the effective additive noise levels and the allocated distortions are clearly modulated in an inversely proportional manner by the squared magnitude of the degradation operator. For instance, frequencies corresponding to degradation-filter magnitudes lower than 1 lead to increase in the effective noise-energy addition and in the allocated distortion. The optimal rate allocation (Fig. \[fig:gaussian\_restoration\_\_optimal\_rate\_allocation\]) is affected by the signal-to-noise ratio and by the squared-magnitude of the degradation filter (see also Eq. (\[eq:theoretic Gaussian analysis - optimal rate allocation\])), e.g., components that are attenuated by the degradation operator get less bits in the rate allocation. Conceptual Relation to The Proposed Approach {#subsec:Rate-Distortion Theoretic - Conceptual Relation to The Proposed Approach} -------------------------------------------- As explained at the beginning of this section, theoretical and practical solutions may include different implementations of the same general ideas. Accordingly, connections between theory and practice should be established by pointing on high-level analogies. Our rate-distortion analysis (for a Gaussian signal and a LSI degradation operator) showed that the optimal complexity-regularized restoration relies on two prominent ideas: pseudoinverse filtering of the degraded input, and compression with respect to a squared-error metric that is weighted based on the degradation-filter squared-magnitude (considering the DFT-domain procedure). We will now turn to explain how these two concepts connect to more general themes having different realizations in the practical approach proposed in Section \[sec:Proposed Methods\] [^2]. $\bullet$ ***Design Concept \#1:** Apply simple restoration filtering*. The general idea of using an elementary restoration filter is implemented in the Gaussian case as pseudoinverse filtering. Correspondingly, our practical approach relies on a simple filtering mechanism, extending the pseudoinverse filter as explained next. Stage 6 of Algorithm \[Algorithm:Proposed Method Non-overlapping\] is an $ \ell_2 $-constrained deconvolution filtering that its analytic solution can be rewritten, using the relation $ {\mathbf{H}}^*\left( {\mathbf{I}} - {\mathbf{H}}{\mathbf{H^*}} \right) \nolinebreak=\nolinebreak {\mathbf{0}} $, as (see proof in Appendix \[appendix:Equivalent Forms of Stage 4 of Algorithm 1\]) [rCl]{} \[eq:theory-practice relation - interpreting first stage\] \^[(t)]{} = ( \^\* + )\^[-1]{} ( \^\* + \^[(t)]{} ) .  As before, $ \tilde{{\mathbf{y}}} = {\mathbf{H}}^+ {\mathbf{y}} $, i.e., the pseudoinverse-filtered version of $ {\mathbf{y}} $. The expression (\[eq:theory-practice relation - interpreting first stage\]) can be interpreted as an initial pseudoinverse filtering of the degraded input, followed by a simple weighted averaging with $\tilde{{\mathbf{z}}}^{(t)}$ (that includes the decompressed signal obtained in the last iteration). Evidently, the filtering in (\[eq:theory-practice relation - interpreting first stage\]) is determined by the $\beta$ value, specifically, for $ \beta=0 $ the estimate coincides with the pseudoinverse filtering solution and for a larger $\beta$ it is closer to $\tilde{{\mathbf{z}}}^{(t)}$. $\bullet$ ***Design Concept \#2:** Compress by promoting higher quality for signal-components matching to higher $ h $-operator magnitudes*. This principle is realized in the theoretic Gaussian case as weights attached to the squared-errors of DFT-domain components (see Problems \[problem: Separable Form in DFT domain\] and \[problem: Gaussian distortion allocation\]). Since the weights, ($ \left\lvert h_0^F \right\rvert ^2 , ..., \left\lvert h_{N-1}^F \right\rvert ^2 $), are the squared magnitudes of the corresponding degradation-filter coefficients, in the compression of the pseudoinverse-filtered input the distortion is spread unevenly being larger where the degradation filter-magnitude is lower. Remarkably, this concept is implemented differently in the proposed procedure (Algorithm \[Algorithm:Proposed Method Non-overlapping\]) where regular compression techniques, optimized for the squared-error distortion measure, are applied on the filtering result of the preceding stage. We will consider the essence of the effective compression corresponding to these two stages together. Let us revisit (\[eq:theory-practice relation - interpreting first stage\]), expressing stage 6 of Algorithm \[Algorithm:Proposed Method Non-overlapping\]. Assuming $ {\mathbf{H}} $ is a circulant matrix, we can transform (\[eq:theory-practice relation - interpreting first stage\]) into its Fourier domain representation [rCl]{} \[eq:theory-practice relation - interpreting first stage - circulant H\] \^[F,(t)]{} = ( \_H\^\* \_H + )\^[-1]{} ( \_H\^\* \_H \^[F]{} + \^[F,(t)]{} )\ where $ \hat{ {\mathbf{x}}}^{F,(t)} $ and $ \tilde{{\mathbf{z}}}^{F,(t)} $ are the Fourier representations of $ \hat{ {\mathbf{x}}}^{(t)} $ and $ \tilde{{\mathbf{z}}}^{(t)} $, respectively. Furthermore, (\[eq:theory-practice relation - interpreting first stage - circulant H\]) reduces to the componentwise formulation [rCl]{} \[eq:theory-practice relation - interpreting first stage - circulant H - component\] \^[F,(t)]{}\_k = where $ \hat{ {x}}^{F,(t)}_k $ and $ \tilde{z}^{F,(t)}_k $ are the $ k^{th} $ Fourier coefficients of $ \hat{ {\mathbf{x}}}^{F,(t)} $ and $ \tilde{{\mathbf{z}}}^{F,(t)} $, respectively. Equation (\[eq:theory-practice relation - interpreting first stage - circulant H - component\]) shows that signal elements (of the pseudoinverse-filtered input) corresponding to degradation-filter components of weaker energies will be retracted more closely to the respective components of $ \tilde{z}^{F,(t)}_k $ – thus, will be farther from $\tilde{y}^{F}_k$, yielding that the corresponding components in the standard compression applied in the next stage of this iteration will be of a relatively lower quality with respect their matching components of $\tilde{y}^{F}_k$ (as they were already retracted relatively far from them in the preceding deconvolution stage). To conclude this section, we showed that the main architectural ideas expressed in theory (for the Gaussian case) appear also in our practical procedure. The iterative nature of our methods (Algorithms \[Algorithm:Proposed Method Non-overlapping\]-\[Algorithm:Proposed Method Overlapping Blocks with Robust Dual Variables\]) as well as the desired shift-invariance property provided by Algorithms \[Algorithm:Proposed Method Overlapping Blocks\]-\[Algorithm:Proposed Method Overlapping Blocks with Robust Dual Variables\] are outcomes of treating real-world scenarios such as non-Gaussian signals, general linear degradation operators, and computational limitations leading to block-based treatments – these all relate to practical aspects, hence, do not affect the fundamental treatment given in this section. Experimental Results {#sec:Experimental Results} ==================== In this section we present experimental results for image restoration. Our main study cases include deblurring and inpainting using the image-compression profile of the HEVC standard (in its BPG implementation [@hevc_software_bpg]). We also provide evaluation of our method in conjunction with the JPEG2000 technique for the task of image deblurring. We empirically found it sufficient to consider only a part of all the shifts, i.e., a portion of $ \mathcal{B}^* $. When using HEVC, the limited amount of shifts is compensated by the compression architecture that employs inter-block spatial predictions, thus, improves upon methods relying on independent block treatment. The shifts are defined by the rectangular images having their upper-left corner pixel relatively close to the upper-left corner of the full image, and their bottom-right corner pixel coincides with that of the full image. This extends the mathematical developments in Section \[sec:Proposed Methods\] as practical compression handles arbitrarily sized rectangular images. Many image regularizers have visual interpretation, for example, the classical image-smoothness evaluation. In our framework, the regularization part in (\[eq:complexity-regularized restoration - full signal\]) measures the complexity in terms of the compression bit-cost with respect to a specific compression architecture, designed based on some image model. Our complexity regularization also has a general visual meaning since, commonly, low bit-cost compressed images tend to be overly smooth or piecewise-smooth. Image Deblurring {#subsec:Deblurring} ---------------- Here we consider two deterioration settings taken from [@danielyan2012bm3d]. The first setting, denoted here also as ‘Set. 1’, considers a noise variance $ \sigma_n^2 = 2 $ coupled with a blur operator defined by the two-dimensional point-spread-function (PSF) $ h(x_1,x_2)=1/{(1+x_1^2+x_2^2)} $ for $ x_1,x_2=-7,...,7 $, and zero-valued otherwise. The second setting, denoted here also as ‘Set. 2’ (named in [@danielyan2012bm3d] as ‘Scenario 3’), considers a noise variance $ \sigma_n^2 \approx 0.3 $ joint with a blur operator defined by the two-dimensional uniform blur PSF of size $ 9\times 9 $. We precede the deblurring experiments with empirical evaluations of four important aspects of the proposed method. ### Iterative Reduction of the Fundamental Restoration Cost In Section \[sec:Proposed Methods\] we established the basic optimization problems for restoration by regularizing the bit-costs of the non-overlapping and the overlapping blocks of the estimate (see (\[eq:complexity-regularized restoration - expressing blocks\]) and (\[eq:complexity-regularized restoration - overlapping blocks\]), respectively). As explained above, these two fundamental optimization tasks cannot be directly addressed and, therefore, we developed the ADMM-based Algorithms \[Algorithm:Proposed Method Non-overlapping\]-\[Algorithm:Proposed Method Overlapping Blocks with Robust Dual Variables\], that iteratively employ simpler optimization problems. Figures \[fig:empirical\_analysis\_cost\_evolution\_deblurring\_shift1\_cameraman\] and \[fig:empirical\_analysis\_cost\_evolution\_deblurring\_shift3\_cameraman\] demonstrate that, for appropriate parameter settings, the fundamental optimization cost reduces in each iteration. The provided figures also show the fidelity term, $ \left\| { {\mathbf{H}} {\mathbf{x}} - {\mathbf{y}} } \right\|_2^2 $, and the regularizing bit-cost (multiplied by $ \mu $) of each iteration. ### Iterative Improvement of the Restored Image The fundamental optimization costs in (\[eq:complexity-regularized restoration - expressing blocks\]) and (\[eq:complexity-regularized restoration - overlapping blocks\]) include the fidelity term $ \left\| { {\mathbf{H}} {\mathbf{x}} - {\mathbf{y}} } \right\|_2^2 $ that considers the candidate estimate ${\mathbf{x}}$ and the given degraded signal $ {\mathbf{y}} $. However, the ultimate goal of the restoration process is to produce an estimate ${\mathbf{x}}$ that will be close to the original (unknown!) signal ${\mathbf{x}}_0$. It is common to evaluate proposed methods in experiment settings where ${\mathbf{x}}_0$ is known and used only for the evaluation of the squared error $ \left\| { {\mathbf{x}} - {\mathbf{x}}_0 } \right\|_2^2 $ or its corresponding PSNR. Accordingly, it is a desired property that our iterative methods will provide increment in the PSNR along the iterations and, indeed, Figures \[fig:empirical\_analysis\_psnr\_evolution\_deblurring\_shift1\_cameraman\] and \[fig:empirical\_analysis\_psnr\_evolution\_deblurring\_shift3\_cameraman\] show that this is achievable for appropriate parameter settings (the use of improper parameters may lead to unwanted decrease of the PSNR starting at some unknown iteration that, however, can be detected in many cases by heuristic divergence rules based on the dual variables used in the ADMM process). Interestingly, for some parameter settings, the PSNR may increase with the iterations, whereas the fundamental restoration cost will not necessarily consistently decrease. The last behavior may result from the fact the our optimization problem (with respect to a standard compression technique) is discrete, non-linear, and usually not convex and, therefore, the convergence guarantees of the ADMM [@boyd2011distributed] do not hold here for the fundamental restoration cost. ### The Optimal Compression Parameter Another question of practical importance is the value of the parameter $ \theta $, determining the compression level of the standard technique utilized in the proposed Algorithms \[Algorithm:Proposed Method Non-overlapping\]-\[Algorithm:Proposed Method Overlapping Blocks with Robust Dual Variables\]. Recall that the ADMM-based developments in Section \[sec:Proposed Methods\] led to an iterative procedure including a stage of Lagrangian rate-distortion optimization operated for a Lagrange multiplier $ \lambda \triangleq \frac{2\mu}{\beta} $ and, then, we replaced this optimization with application of a standard compression-decompression technique operated based on a parameter $ \theta $. It is clear that $ \theta $ is a function of $ \lambda $. In the particular case where the standard compression has the Lagrangian form from our developments, then $ \theta = \lambda $, however, this is not the general case. For an arbitrary compression technique, we assume that its parameter $ \theta $ has $ K $ possible values $ \theta_1,...,\theta_K $ (for example, the HEVC standard supports 52 values for its quantization parameter), then, for a given $ \lambda \triangleq \frac{2\mu}{\beta} $ the required $ \theta $ value in stage 8 of Algorithm \[Algorithm:Proposed Method Non-overlapping\] can be determined via [rCl]{} \[eq:experiments - general setting of theta - rate-distortion optimization\] \^[(t)]{}\_[,opt]{} = \_[\_1,...,\_K ]{} [ \_2\^2]{} + r\_[tot,]{} . where $ \hat{{\mathbf{z}}}^{(t)}_{\theta } = {CompressDecompress}_{\theta}\left( \tilde{{\mathbf{x}}}^{(t)} \right) $ is the decompressed signal and $ r_{tot,\theta} $ is the associated compression bit-cost. We present here experiments (see Figs. \[Fig:empirical\_analysis\_for\_deblurring\_shift1\_cameraman\] and \[Fig:empirical\_analysis\_for\_deblurring\_shift3\_cameraman\]) for Algorithm \[Algorithm:Proposed Method Non-overlapping\] and \[Algorithm:Proposed Method Overlapping Blocks\] that in each iteration optimize the $ \theta $ value corresponding to $ \lambda \triangleq \frac{2\mu}{\beta} $ based on procedures similar to (\[eq:experiments - general setting of theta - rate-distortion optimization\]). Nicely, it is shown in Figures \[fig:empirical\_analysis\_compression\_factor\_evolution\_deblurring\_shift1\_cameraman\] and \[fig:empirical\_analysis\_compression\_factor\_evolution\_deblurring\_shift3\_cameraman\] that, for the HEVC compression used here, the best $ \theta $ values along the iterations are nearly the same (for a specific restoration task). This important property may be a result of the fact that HEVC extensively relies on Lagrangian rate-distortion optimizations (although in much more complex forms than those presented in Section \[sec:Proposed Methods\]). Accordingly, in order to reduce the computational load, in the experiments shown below we will use a constant compression parameter given as an input to our methods. Interestingly, when examining the optimal compression parameters (compression ratios in this case) for the JPEG2000 method that applies wavelet-based transform coding, there is a decrease in the optimal compression ratio along the iterations (see Fig. \[fig:empirical\_analysis\_deblurring\_ADMM\_JPEG2000\_compression\_ratios\_cameraman\]). Accordingly, in order to reduce the computational load in the experiments below, we employed a predefined rule for reducing the JPEG2000 compression ratio along the iterations. Importantly, when we use the sub-optimal predefined rules for setting the compression parameter $ \theta $ values (see Table \[table:Experimental Results - Parameter Setting\] for the settings used in our evaluation comparisons in Table \[table:Deblurring - PSNR Comparison\]) we do not longer need to set a value for $ \mu $ (since, in these cases, $ \mu $ is practically unused). \[\] \[table:Experimental Results - Parameter Setting\] \[\] --------------------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- Set. 1 Set. 2 Set. 1 Set. 2 Set. 1 Set. 2 Set. 1 Set. 2 Input PSNR 22.23 20.76 25.61 24.11 27.25 25.84 23.34 22.49 ForWaRD [@neelamani2004forward] 28.99 28.10 32.96 33.67 33.30 32.81 27.03 26.51 SV-GSM [@guerrero2008image] 29.68 28.09 34.25 33.15 - - 30.19 27.56 BM3DDEB [@dabov2008image] **30.42** 29.10 **34.93** 34.96 **35.20** 33.81 **31.14** **28.35** TVMM [@oliveira2009adaptive] 29.64 29.30 33.59 34.50 33.61 33.31 26.44 25.98 CGMK [@chantas2010variational] 30.03 29.91 33.92 34.86 34.01 33.70 25.79 26.04 IDD-BM3D [@danielyan2012bm3d] **31.08** **31.21** **35.56** **37.00** **35.22** **34.75** **30.98** **28.54** EPLL [@zoran2011learning] 29.40 29.54 33.88 35.87 **34.64** 34.39 27.62 26.78 28.99 27.48 33.55 33.89 34.11 32.56 26.42 25.14 29.28 28.10 33.07 34.01 34.38 32.60 26.31 25.18 30.19 **30.20** 33.17 **36.47** 33.32 **34.40** 29.83 **27.74** **30.35** **30.14** **34.37** **36.57** 34.55 **34.41** **30.20** 27.72 --------------------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ### Restoration Improvement for Increased Number of Image Shifts In our experiments we noticed that the restoration quality improves as more shifts are used, however, at some point the added gain due to the added shifts becomes marginal. As an example to the benefits due to shifts see Figs. \[fig:empirical\_analysis\_psnr\_evolution\_deblurring\_shift1\_cameraman\] and \[fig:empirical\_analysis\_psnr\_evolution\_deblurring\_shift3\_cameraman\], where the PSNR obtained for deblurring the Cameraman image using Algorithm 2 and 9 shifts is about 2dB higher than the PSNR obtained using Algorithm 1 (i.e., without additional shifts). In our main evaluation, we examined the proposed Algorithms 2 and 3 for image deblurring in conjunction with the JPEG2000 and HEVC compression techniques (see the parameter settings in Table \[table:Experimental Results - Parameter Setting\]). Table \[table:Deblurring - PSNR Comparison\] shows a comparison between various deblurring methods tested in the above two settings for four grayscale images[^3]. In 7 out of the 8 cases, the proposed Algorithms 2 or 3 utilizing the HEVC standard provided one of the best three results. Visual results are presented in Figures \[Fig:Deblurring results - cameraman\] and \[Fig:Deblurring results - barbara\]. Image Inpainting {#subsec:Inpainting} ---------------- We presented in [@dar2016image] experimental results for the inpainting problem, in its noisy and noiseless settings. Here we focus on the noiseless inpainting problem, where only pixel erasure occurs without an additive noise. The degradation is represented by a diagonal matrix $ {\mathbf{H}} $ of $ N\times N $ size with main diagonal values of zeros and ones, indicating positions of missing and available pixels, respectively. Then, the product $ {\mathbf{H}}{\mathbf{x}} $ equals to an $ N $-length vector where its $ k^{th} $ sample is determined by $ {\mathbf{H}} $: if $ {\mathbf{H}}[k,k] = 0$ then it is zero, and for $ {\mathbf{H}}[k,k] = 1$ it equals to the corresponding sample of $ {\mathbf{x}} $. The structure of the pixel erasure operator let us to simplify the optimization in step 6 of Algorithm \[Algorithm:Proposed Method Overlapping Blocks\]. We note that $ {\mathbf{H}} $ is a square diagonal matrix and, therefore, $ {\mathbf{H}}^T = {\mathbf{H}} $ and $ {\mathbf{H}}^T {\mathbf{y}} $ is equivalent to a vector $ {\mathbf{y}} $ with zeroed components according to $ {\mathbf{H}} $’s structure. Additional useful relation is $ {\mathbf{H}}^T {\mathbf{H}} = {\mathbf{H}} $. Consequently, step 6 of Algorithm \[Algorithm:Proposed Method Overlapping Blocks\] facilitates a componentwise computation that is interpreted to form the $ k^{th} $ sample of $ \hat{{\mathbf{x}}}^{(t)} $ as [rCl]{} \[eq:noiseless inpainting - k-th component of x\] \^[(t)]{} \[k\] = { [ll]{}        \[k,k\]=1\ \_[j=1]{}\^[N\_b]{} \^[j,(t)]{}\[k\]            \[k,k\]=0    . We initialize the shifted images $ \left\lbrace {\hat{{\mathbf{z}}}}^{j,(0)} \right\rbrace _{j=1}^{N_b} $ as the given image with the missing pixels set as the corresponding local averages of the available pixels in the respective $ 7\times 7 $ neighborhoods. When the iterative processing ends, we use the fact that the available pixels are noiseless and set them in the reconstructed image. The rest of the procedure remains as before. We present here implementations of Algorithms \[Algorithm:Proposed Method Overlapping Blocks\] and \[Algorithm:Proposed Method Overlapping Blocks with Robust Dual Variables\] utilizing the JPEG2000 and HEVC image compression (the parameter settings are described in Table \[table:Experimental Results - Parameter Setting\]). We consider the experimental settings from [@ram2013image], where 80% of the pixels are missing (see Fig. \[fig:inpainting - barbara - degraded image\] and \[fig:inpainting - house - degraded image\]). Five competing inpainting methods are considered: cubic interpolation of missing pixels via Delaunay triangulation (using Matlab’s ’griddata’ function); inpainting using sparse representations of patches of $ 16\times 16 $ pixels based on an overcomplete DCT (ODCT) dictionary (see method description in [@elad2010sparse Ch. 15]); using patch-group transformation [@li2008patch]; based on patch clustering [@yu2012solving]; and via patch reordering [@ram2013image]. The PSNR values of images restored using the above methods (taken from [@ram2013image]) are provided in Table \[table:Experimental Results - Inpainting\] together with our results. For two images our HEVC-based implementation of Algorithm 3 provides the highest PSNR values. Visually, Figures \[fig:inpainting - barbara - proposed restoration\] and \[fig:inpainting - house - proposed restoration\] exhibit the effectiveness of our method in repairing the vast amount of absent pixels. \[\] \[table:Experimental Results - Inpainting\] Image ODCT ------- ------- ------- ------- ----------- ------- ------- ------- ------- ----------- 30.25 29.97 31.62 32.22 31.96 30.31 30.96 32.14 **32.55** 22.88 27.15 25.40 **30.94** 29.71 24.25 24.83 26.06 28.80 29.21 29.69 32.87 33.05 32.71 29.49 30.50 32.42 **33.10** Conclusion {#sec:Conclusion} ========== In this paper we explored the topic of complexity-regularized restoration, where the likelihood of candidate estimates are determined by their compression bit-costs. Using the alternating direction method of multipliers (ADMM) approach we developed three practical methods for restoration using standard compression techniques. Two of the proposed methods rely on a new shift-invariant complexity regularizer, evaluating the total bit-cost of the signal shifted versions. We explained few of the main ideas of our approach using an insightful theoretical-analysis of complexity-regularized restoration of a cyclo-stationary Gaussian signal from deterioration of a linear shift-invariant operator and additive white Gaussian noise. Experiments for deblurring and inpainting of images using the JPEG2000 and HEVC technique showed good results. Proofs for the theory section ============================= Equivalence of Problems \[problem: basic form\] and \[problem: pseudoinverse filtered input\] {#appendix:Equivalence of Problems 1 and 2} --------------------------------------------------------------------------------------------- We start by showing the equality between the distortion constraints of Problems \[problem: basic form\] and \[problem: pseudoinverse filtered input\]. We develop the distortion of Problem \[problem: basic form\] as follows: [rCl]{} \[eq:appendix - proofs - equivalence of problem 1 and 2 - distortions\] \_2\^2 & = & \_2\^2\ & = & \_2\^2\ & = & \_2\^2 + \_2\^2\ && + ( \^[+]{} - )\^[\*]{} \^[\*]{} ( - \^[+]{} )\ && + \^[\*]{} ( - \^[+]{} )\^[\*]{} ( \^[+]{} - )\ & = & \_2\^2 + \_2\^2      where the last equality follows from [rCl]{} \[eq:appendix - proofs - equivalence of problem 1 and 2 - distortions - auxiliary result\] \^[\*]{} ( - \^[+]{} ) = that can be easily proved, e.g., by using the DFT-based diagonalization of $ {\mathbf{H}} $ and $ {\mathbf{H}}^{+} $. The first term in (\[eq:appendix - proofs - equivalence of problem 1 and 2 - distortions\]) can be further developed: [rCl]{} \[eq:appendix - proofs - equivalence of problem 1 and 2 - distortions - first part - deterministic\] \_2\^2 & = & \_2\^2\ & = & \_2\^2\ & = & \_2\^2\ & = & \_[k: h\_k\^F = 0]{} y\_k\^F \^2 =\ & = & \_[k: h\_k\^F = 0]{} n\_k\^F \^2 where $ {\mathbf{y}}^{F} \triangleq {\mathbf{F}} {\mathbf{y}} $ is the DFT-domain representation of $ {\mathbf{y}} $ (correspondingly, we use these notations to any vector), and the last equality is implied from the DFT-component relation $ y_k^F = h_k^F x_k^F + n_k^F $ that reduces to $ y_k^F = n_k^F $ for components with $ h_k^F = 0 $. Consequently, [rCl]{} \[eq:appendix - proofs - equivalence of problem 1 and 2 - distortions - first part\] E \_2\^2 & = & E\_[k: h\_k\^F = 0]{} n\_k\^F \^2\ & = & (N - N\_[H]{}) \_n\^2 where $ N_{H} $ was defined in Section \[subsec:Reformulations of the Problem\] as the rank of $ {\mathbf{H}} $. Accordingly, and also using (\[eq:appendix - proofs - equivalence of problem 1 and 2 - distortions\]), the distortion constraint of Problem \[problem: basic form\], i.e., [rCl]{} \[eq:appendix - proofs - equivalence of problem 1 and 2 - distortion of problem 1\] E \_2\^2 = N \_n\^2 equals to (recall that $ \tilde{{\mathbf{y}}} = {\mathbf{H}}^{+} {\mathbf{y}} $) [rCl]{} \[eq:appendix - proofs - equivalence of problem 1 and 2 - distortion of problem 2\] E ( [ - ]{} ) \_2\^2 = N\_[H]{} \_n\^2 , that is, the distortion constraint of Problem \[problem: pseudoinverse filtered input\]. We now turn to prove the equivalence of Problems \[problem: basic form\] and \[problem: pseudoinverse filtered input\]. Our proof sketch conforms with common arguments in rate-distortion function proofs (see [@cover2012elements]): first, we lower bound the mutual information $ I\left( {\mathbf{y}}; \hat{{\mathbf{x}}} \right) $, which is the cost function of Problem \[problem: basic form\]; then, we provide a statistical construction achieving the lower bound while obeying the distortion constraint. The proposed lower bound for $ I\left( {\mathbf{y}}; \hat{{\mathbf{x}}} \right) $ is established by noting that $ \tilde{{\mathbf{y}}} = {\mathbf{H}}^{+} {{\mathbf{y}}} $ and, therefore, the data processing inequality [@cover2012elements] implies here that [rCl]{} \[eq:appendix - proofs - equivalence of problem 1 and 2 - data processing inequality\] I( ; ) I( ; ) , where $ I\left( \tilde{{\mathbf{y}}} ; \hat{{\mathbf{x}}} \right) $ is the cost function of Problem \[problem: pseudoinverse filtered input\]. The relation in (\[eq:appendix - proofs - equivalence of problem 1 and 2 - data processing inequality\]) is known to be attained with equality when $ {\mathbf{y}} $ and $\hat{{\mathbf{x}}} $ are independent given $ \tilde{{\mathbf{y}}} $. The next construction shows that this is indeed the case. We will now show the achievability of the lower bound in (\[eq:appendix - proofs - equivalence of problem 1 and 2 - data processing inequality\]) by describing a two-stage setting that statistically represents $ \tilde{{\mathbf{y}}} $ as an outcome of $ \hat{{\mathbf{x}}} $, and $ {{\mathbf{y}}} $ as a consequence of $ \tilde{{\mathbf{y}}} $. This layout is an instance of the construction concept known as the (backward) test channel [@cover2012elements]. The first stage of our construction is based on [rCl]{} \[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - x \] & \~& ( , \^[+]{} \_ \^[+\*]{} - \_n\^2 \^[+]{}\^[+\*]{} )\ \[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - z \] & \~& ( , \_n\^2 \^[+]{}\^[+\*]{} ) where $ \hat{{\mathbf{x}}} $ and $ {\mathbf{z}} $ are independent. Consequently, we define [rCl]{} \[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - y\_tilde as a sum \] = + , implying $ \tilde{{\mathbf{y}}} \sim \mathcal{N} \left( {\mathbf{0}}, {\mathbf{H}}^{+} {\mathbf{R}}_{{\mathbf{y}}} {\mathbf{H}}^{+*} \right) $ that, indeed, conforms with $ \tilde{{\mathbf{y}}} = {\mathbf{H}}^{+} {\mathbf{y}} $ where ${\mathbf{y}} \sim \mathcal{N} \left( {\mathbf{0}}, {\mathbf{R}}_{{\mathbf{y}}} \right)$. Moreover, the construction (\[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - x \])-(\[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - y\_tilde as a sum \]) yields [rCl]{} \[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - distortion constraint satisfies\] E ( [ - ]{} ) \_2\^2 & = & E \_2\^2\ & = & E\^[\*]{} \^[\*]{}\ & = & ETrace\^[\*]{} \^[\*]{}\ & = & ETrace \^[\*]{} \^[\*]{}\ & = & Trace \_ \^[\*]{}\ & = & \_n\^2 Trace \^[+]{}\^[+\*]{} \^[\*]{}\ & = & \_n\^2 Trace\_[[H]{}]{} \_[[H]{}]{}\^[\*]{}\ & = & \_n\^2 N\_H where $ {\mathbf{P}}_{{H}} \triangleq {\mathbf{H}} {\mathbf{H}}^{+} $ is the matrix projecting onto the range of $ {\mathbf{H}} $, note it is also a circulant matrix diagonalized by the DFT matrix to the diagonal matrix $ {\mathbf{\Lambda}}_{{\mathbf{P}}_{{H}}} \triangleq {\mathbf{\Lambda}}_{H} {\mathbf{\Lambda}}_{H}^{+} $. The last computation of the trace is due to the structure of ${\mathbf{\Lambda}}_{{\mathbf{P}}_{H}}$, having ones in main-diagonal entries corresponding to the DFT-domain indices of the range of $ {\mathbf{H}} $, and zeros elsewhere. The result in (\[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - distortion constraint satisfies\]) shows that the distortion constraint (\[eq:appendix - proofs - equivalence of problem 1 and 2 - distortion of problem 2\]) is satisfied. Let us consider the second stage of the construction, awaiting to prove that $ {\mathbf{y}} $ and $\hat{{\mathbf{x}}} $ are independent given $ \tilde{{\mathbf{y}}} $. We precede the construction with examining the following decomposition of $ {\mathbf{y}} $ [rCl]{} \[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - decomposition of y\] & = & \_[H]{} + ( - \_[H]{} )\ & = & \^[+]{} + ( - \_[H]{} ) ( + )\ & = & + ( - \_[H]{} ) where the second equality uses the degradation model, and the third equality is due to $ \left( {\mathbf{I}} - {\mathbf{P}}_{H} \right) {\mathbf{H}} = {\mathbf{0}} $. Importantly, Eq. (\[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - decomposition of y\]) describes $ {\mathbf{y}} $ as a linear combination of two independent random vectors: $ \tilde{{\mathbf{y}}} $ and $ \left( {\mathbf{I}} - {\mathbf{P}}_{H} \right) {\mathbf{n}} $. Since $ \tilde{{\mathbf{y}}} $ and $ \left( {\mathbf{I}} - {\mathbf{P}}_{H} \right) {\mathbf{n}} $ are Gaussian random vectors, their independence is proved by showing they are uncorrelated via [rCl]{} \[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - uncorrelated backward variables - new \] && E( - \_[H]{} ) \^[\*]{}\ && = E( - \_[H]{} ) \^[\*]{} \^[+\*]{}\ && = E( - \^[\*]{} \_[\_[H]{}]{} ) \^[\*]{} \^[\*]{} \_[H]{}\^[+\*]{}\ && = E\^[\*]{} ( - \_[\_[H]{}]{} ) ( )\^[\*]{} \_[H]{}\^[+\*]{}\ \[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - uncorrelated backward variables - new - explain\] && = \^[\*]{} E( - \_[\_[H]{}]{} ) \^[F]{} ( \_[H]{}\^[+]{} \^[F]{} )\^[\*]{}\ && = \^[\*]{}\ && = where in (\[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - uncorrelated backward variables - new - explain\]) we used the facts that $\left( {\mathbf{I}} - {\mathbf{\Lambda}}_{{\mathbf{P}}_{H}} \right) {\mathbf{n}}^{F}$ is a DFT-domain vector with zeros in components corresponding to the range of $ {\mathbf{H}} $, and $ {\mathbf{\Lambda}}_{H}^{+} {\mathbf{y}}^{F} $ is a DFT-domain vector with zeros in entries corresponding to the nullspace of $ {\mathbf{H}} $, hence, these zero patterns yield the outer-product matrix which is all zeros. The decomposition in (\[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - decomposition of y\]) motivates us to consider $ {\mathbf{y}} $ to emerge from $ \tilde{{\mathbf{y}}} $ via the statistical relation [rCl]{} \[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - y from y\_tilde \] = + where $ {\mathbf{w}} \sim \mathcal{N} \left( {\mathbf{0}}, \sigma_n^2 \left({\mathbf{I}} - {\mathbf{P}}_{H} \right) \left({\mathbf{I}} - {\mathbf{P}}_{H} \right)^{*} \right) $ and is independent of $ \tilde{{\mathbf{y}}} $ (and also of $ {\mathbf{z}} $). Note that $ {\mathbf{w}} $ takes the role of $ \left( {\mathbf{I}} - {\mathbf{P}}_{H} \right) {\mathbf{n}} $ appearing in (\[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - decomposition of y\]), e.g., they have the same distribution. One can further examine the suitability of the construction (\[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - y from y\_tilde \]) to the considered problem by noting it satisfies the distortion constraint of Problem \[problem: basic form\], namely, [rCl]{} \[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - distortion constraint satisfies - Problem 2\] E \_2\^2 & = & E \_2\^2\ & = & E \_2\^2\ & = & E \_2\^2\ & = & E \_2\^2 + E \_2\^2\ & = & \_n\^2 Trace \^[+]{}\^[+\*]{} \^[\*]{}\ && + \_n\^2 Trace( - \_[H]{} ) ( - \_[H]{} )\^[\*]{}\ & = & \_n\^2 Trace\_[H]{} \_[H]{}\^[\*]{}\ && + \_n\^2 Trace( - \_[H]{} ) ( - \_[H]{} )\^[\*]{}\ & = & \_n\^2 N\_H + \_n\^2 ( N - N\_H )\ & = & N \_n\^2 as required. Furthermore, the constructed $ {\mathbf{y}} $ indeed obeys ${\mathbf{y}} \nolinebreak \sim \nolinebreak \mathcal{N} \left( {\mathbf{0}}, {\mathbf{R}}_{{\mathbf{y}}} \right)$. Specifically, its autocorrelation matrix stems from the calculation [rCl]{} \[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - Problem 2 - y autocorrelation\] E \^[\*]{} & = & E( [ + ]{} ) ( [ + ]{} )\^[\*]{}\ & = & E( [ ]{} ) ( [ ]{} )\^[\*]{} + E\^[\*]{}\ & = & \^[+]{} \_ \^[+\*]{} \^[\*]{} + \_\ & = & \_[H]{} ( \_ \^[\*]{} + \_n\^2 ) \_[H]{}\^[\*]{} + \_\ & = & \_[H]{} \_ \^[\*]{} \_[H]{}\^[\*]{} + \_n\^2 \_[H]{} \_[H]{}\^[\*]{} + \_\ & = & \_ \^[\*]{} + \_n\^2 \_[H]{} \_[H]{}\^[\*]{} + \_n\^2 ( - \_[H]{} ) ( - \_[H]{} )\^[\*]{}\ & = & \_ \^[\*]{} + \_n\^2\ & = & \_ where we used the auxiliary result [rCl]{} \[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - Problem 2 - y autocorrelation - auxiliary result\] && \_[H]{} \_[H]{}\^[\*]{} + ( - \_[H]{} ) ( - \_[H]{} )\^[\*]{}\ && = \^[\*]{} \_[\_[H]{}]{} \_[\_[H]{}]{}\^[\*]{} + \^[\*]{} ( - \_[\_[H]{}]{} ) ( - \_[\_[H]{}]{} )\^[\*]{}\ && = \^[\*]{} \_[\_[H]{}]{} + \^[\*]{} ( - \_[\_[H]{}]{} )\ && = . Joining the two stages of the construction, presented in (\[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - y\_tilde as a sum \]) and (\[eq:appendix - proofs - equivalence of problem 1 and 2 - backward channel construction - y from y\_tilde \]), exhibits $ \hat{{\mathbf{x}}} \rightarrow \tilde{{\mathbf{y}}} \rightarrow {\mathbf{y}} $ as a Markov chain and, therefore, $ {\mathbf{y}} $ and $\hat{{\mathbf{x}}} $ are independent given $ \tilde{{\mathbf{y}}} $. This evident construction turns (\[eq:appendix - proofs - equivalence of problem 1 and 2 - data processing inequality\]) into [rCl]{} \[eq:appendix - proofs - equivalence of problem 1 and 2 - mutual information equality\] I( ; ) = I( ; ) that completes proving the equivalence of Problems \[problem: basic form\] and \[problem: pseudoinverse filtered input\]. Equivalence of Problems \[problem: pseudoinverse filtered input\] and \[problem: Separable Form in DFT domain\] {#appendix:Equivalence of Problems 2 and 3} --------------------------------------------------------------------------------------------------------------- The rate-distortion function for a Gaussian source with memory (i.e., correlated components) is usually derived in the Principle Component Analysis (PCA) domain where the components are independent Gaussian variables (see, e.g., [@berger1971rate]). In our case, where the signal is cyclo-stationary, the PCA is obtained using the DFT matrix. As in the usual case, [rCl]{} \[eq:appendix - proofs - equivalence of problem 2 and 3 - mutual information equality under DFT\] I( ; ) & = & I( \^[F]{} ; \^[F]{} )\ \[eq:appendix - proofs - equivalence of problem 2 and 3 - mutual information reduction to sum\] & = & \_[k=0 ]{}\^[N-1]{} I( [\^F\_k]{} ; \^F\_k ) where (\[eq:appendix - proofs - equivalence of problem 2 and 3 - mutual information equality under DFT\]) emerges from the reversibility of the transformation, and (\[eq:appendix - proofs - equivalence of problem 2 and 3 - mutual information reduction to sum\]) is due to the independence of the $ \left\lbrace {\tilde{y}^F_k} \right\rbrace_{k=0}^{N-1} $ components [@cover2012elements] and the backward-channel construction (see Appendix \[appendix:Equivalence of Problems 1 and 2\]) that can be translated into a form of independent DFT-domain component-level channels. The main difference from the well-known rate-distortion analysis is that here, in Problem \[problem: pseudoinverse filtered input\], the distortion constraint is not a regular squared error – but a weighted one, that will be developed next. Since DFT is a unitary transformation, its energy preservation property yields [rCl]{} \[eq:appendix - proofs - equivalence of problem 2 and 3 - distortion energy preservation\] E ( [ - ]{} ) \_2\^2 & = & E \_[H]{} ( [ \^F - \^F ]{} ) \_2\^2\ & = & \_[k=0]{}\^[N-1]{} [ | h\^F\_k |\^2 E| [ \^F\_k - \^F\_k ]{} |\^2 ]{} where the last equality is due to the diagonal structure of $ {\mathbf{\Lambda}}_{H} $. Hence, we got that the two expected-distortion expressions in Problems \[problem: pseudoinverse filtered input\] and \[problem: Separable Form in DFT domain\] are equal. Solution of Problem \[problem: Gaussian distortion allocation\] {#appendix:Solution of Problem 4} --------------------------------------------------------------- For a start, the transition between Problem \[problem: Separable Form in DFT domain\] and Problem \[problem: Gaussian distortion allocation\] is analogous to the familiar case of jointly coding a set of independent Gaussian variables [@cover2012elements]. Accordingly, and also due to lack of space, we do not elaborate here on this problem-equivalence proof. Problem \[problem: Gaussian distortion allocation\] is compelling as it is a distortion-allocation optimization, where the distortion levels $ \left\lbrace D_k \right\rbrace_{k=0}^{N-1} $ are allocated under the joint distortion constraint. We address Problem \[problem: Gaussian distortion allocation\] via its Lagrangian form (temporarily ignoring the constraints of non-negative distortions) [rCl]{} \[eq:appendix - proofs - solution of problem 4 - Lagrangian form\] \_[k=0]{}\^[N-1]{} [ \_[+]{} ]{} + \_[k=0]{}\^[N-1]{} [ | h\^F\_k |\^2 D\_k ]{}\ where $ \mu \ge 0 $ is the Lagrange multiplier. Recalling that some components may correspond to $ h^F_k = 0 $ and, by (\[eq:theoretic Gaussian analysis - pseudoinverse filtered input - autocorrelation eigenvalues\]), also $\lambda^{\left(\tilde{{\mathbf{y}}}\right)}_k = 0$ – meaning they are deterministic variables. These deterministic components do not need to be coded (i.e., $ R_k \nolinebreak = \nolinebreak 0 $) while still attaining $ D_k = 0 $. Accordingly, the Lagrangian optimization (\[eq:appendix - proofs - solution of problem 4 - Lagrangian form\]) is updated into [rCl]{} \[eq:appendix - proofs - solution of problem 4 - Lagrangian form - only components with nonzero h\_k\^F\] \_[k: h\^F\_k 0]{} [ ( ) ]{} + \_[k: h\^F\_k 0]{} [ | h\^F\_k |\^2 D\_k ]{}\ where we used the expression from (\[eq:theoretic Gaussian analysis - pseudoinverse filtered input - autocorrelation eigenvalues\]), and assumed that the distortions are small enough such that the operator $ \left[\cdot\right]_+ $ can be omitted (a correct assumption as will be later shown). Now, the optimal $ D_k $ value can be determined by equating the respective derivative of the Lagrangian cost to zero, leading to allocated distortion (still as a function of $ \mu $) [rCl]{} \[eq:appendix - proofs - solution of problem 4 - optimal distortion allocation - function of mu\] D\_k\^[opt]{} = k: h\^F\_k 0 and by setting the $ \mu $ satisfying the total distortion constraint from Problem \[problem: Gaussian distortion allocation\] we get [rCl]{} \[eq:appendix - proofs - solution of problem 4 - optimal distortion allocation\] D\_k\^[opt]{} = k: h\^F\_k 0. Expressing a nonuniform distortion-allocation (for components with nonzero $ h^F_k $), being inversely proportional to the weights $ \{ \left| h^F_k \right|^2 \} _{k=0}^{N-1} $. One should note that the assumption on small-enough distortions is satisfied as $ D_k^{opt} \le \lambda^{\left(\tilde{{\mathbf{y}}}\right) }_k $ for any $ k $ obeying $ h^F_k \ne 0 $, and that all the distortions are non-negative as required. The optimal distortions established here (for $ k $ where $ h^F_k \ne 0 $) are set in the rate formula (\[eq:theoretic Gaussian analysis - rate-distortion function of a Gaussian variable\]), providing the optimal rate allocation [rCl]{} \[eq:appendix - proofs - solution of problem 4 - optimal rate allocation\] R\_k\^[opt]{} = { [\*[20]{}[c]{}]{} [ ( | h\^F\_k |\^2 + 1 )]{}&[, h\^F\_k0 ]{}\ 0&[, h\^F\_k = 0 . ]{} . Equivalent Form of Stage 6 of Algorithm \[Algorithm:Proposed Method Non-overlapping\] {#appendix:Equivalent Forms of Stage 4 of Algorithm 1} ------------------------------------------------------------------------------------- The analytic solution of Stage 6 of Algorithm \[Algorithm:Proposed Method Non-overlapping\] is considered here with the conjugate-transpose operator, $ ^* $, extending the regular transpose: [rCl]{} \[eq:appendix - proofs - Equivalent Forms of Stage 4 of Algorithm 1 - signal domain\] \^[(t)]{} & = & ( \^[\*]{} + )\^[-1]{} ( \^\* + \^[(t)]{} ) . We note that [rCl]{} \[eq:appendix - proofs - Equivalent Forms of Stage 4 of Algorithm 1 - H\^\*y\] \^\* & = & \^\* ( \^[+]{} + ( - \^[+]{} ) )\ & = & \^[\*]{} + \^[\*]{} ( - \^[+]{} )\ & = & \^[\*]{} where the last equality results from the relation $ {\mathbf{H}}^{*} \left({\mathbf{I}} - {\mathbf{H}} {\mathbf{H}}^{+} \right) = {\mathbf{0}} $. Consequently, (\[eq:appendix - proofs - Equivalent Forms of Stage 4 of Algorithm 1 - signal domain\]) becomes [rCl]{} \[eq:appendix - proofs - Equivalent Forms of Stage 4 of Algorithm 1 - signal domain - 2\] \^[(t)]{} & = & ( \^[\*]{} + )\^[-1]{} ( \^[\*]{} + \^[(t)]{} ) , which is the form presented in (\[eq:theory-practice relation - interpreting first stage\]). [^1]: The authors are with the Department of Computer Science, Technion, Israel. E-mail addresses: {ydar, elad, freddy}@cs.technion.ac.il. [^2]: Since the differences between Algorithm \[Algorithm:Proposed Method Non-overlapping\] and Algorithms \[Algorithm:Proposed Method Overlapping Blocks\] and \[Algorithm:Proposed Method Overlapping Blocks with Robust Dual Variables\] are for a shift-invariance purpose, an issue that we do not concern in this section, we compare our theoretic results only to Algorithm \[Algorithm:Proposed Method Non-overlapping\]. [^3]: The results in Table \[table:Deblurring - PSNR Comparison\] for the methods from [@neelamani2004forward; @guerrero2008image; @dabov2008image; @oliveira2009adaptive; @chantas2010variational; @danielyan2012bm3d] were taken as is from [@danielyan2012bm3d].

--- abstract: 'The Hamiltonian Mean Field (HMF) model has a low-energy phase where $N$ particles are trapped inside a cluster. Here, we investigate some properties of the trapping/untrapping mechanism of a single particle into/outside the cluster. Since the single particle dynamics of the HMF model resembles the one of a simple pendulum, each particle can be identified as a high-energy particle (HEP) or a low-energy particle (LEP), depending on whether its energy is above or below the separatrix energy. We then define the trapping ratio as the ratio of the number of LEP to the total number of particles and the “fully-clustered” and “excited” dynamical states as having either no HEP or at least one HEP. We analytically compute the phase-space average of the trapping ratio by using the Boltzmann-Gibbs stable stationary solution of the Vlasov equation associated with the $N \to \infty$ limit of the HMF model. The same quantity, obtained numerically as a time average, is shown to be in very good agreement with the analytical calculation. Another important feature of the dynamical behavior of the system is that the dynamical state changes transitionally: the “fully-clustered” and “excited” states appear in turn. We find that the distribution of the lifetime of the “fully-clustered” state obeys a power law. This means that clusters die hard, and that the excitation of a particle from the cluster is not a Poisson process and might be controlled by some type of collective motion with long memory. Such behavior should not be specific of the HMF model and appear also in systems where [*itinerancy*]{} among different “quasi-stationary” states has been observed. It is also possible that it could mimick the behavior of transient motion in molecular clusters or some observed deterministic features of chemical reactions.' author: - 'Hiroko Koyama[^1]' - 'Tetsuro Konishi[^2]' - 'Stefano Ruffo[^3]' title: 'Clusters die hard: Time-correlated excitation in the Hamiltonian Mean Field model' --- introduction ============ In systems with long-range interactions [@yellowbook] it is quite common that particle dynamics leads to the formation of clusters. This happens for instance in self-gravitating systems [@binney], where massive particles interacting with Newtonian potential, initially put in a homogeneous state, can create patterns made of many clusters. This phenomenon can be observed in simplified models, like the one-dimensional self-gravitating systems (sheet models) [@sheetmodel]. For this model an itinerant behavior [@itinerancy] between “quasi-equilibria” and “transient” states has been observed in the long-time evolution [@TGK]. In the “quasi-equilibrium” states particles are clustered, as at equilibrium [@rybicki], but with different energy distributions. In the “transient” states one particle emitted from the cluster bears the highest energy throughout the lifetime of the state. The authors of Ref. [@TGK] also claimed that averaging over a sufficiently long time, which includes many quasi-equilibrium and transient states, should give approximately thermal equilibrium. Motion over several quasi-stationary states is observed also in other Hamiltonian systems, like globally coupled symplectic map systems [@KK], or even in realistic systems of anisotropically interacting molecules [@oomine]. This shows that thermal equilibrium is not the only possible asymptotic behavior of Hamiltonian dynamics. For such cases approaches other than standard statistical mechanics would be needed. Coming back to one-dimensional self-gravitating systems, the generation of high-energy particles plays an important role in dynamical evolution. However, a difficulty of the model is that the definition of high-energy particle is ambiguous, which is an obstacle to precisely define “quasi-stationary” and “transient” states. A time continuous Hamiltonian model for which particle clustering has been studied both from the statistical and the dynamical point of view is the Hamiltonian Mean Field Model (HMF) [@IK; @AR], which describes the motion of fully coupled particles on a circle with attractive/repulsive cosine potential. Recent reviews discussing this model can be found in Refs. [@hmf-review-2002; @chavanis]. This model has a second order phase transition and, in the ordered low energy phase, particles are clustered. However, when the number of particles is finite, some particles can leave the cluster and acquire a high energy. Hence, the “fully-clustered” state has a finite lifetime and an “excited” state appears where at least one particle does not belong to the cluster [@AR; @nakagawa-kaneko-2000-jpsj]. Therefore, below the critical energy, we can observe a similar itinerant behavior as for one-dimensional self-gravitating systems, between a “fully-clustered” state and an “excited” state. In this paper we investigate and characterize the intermittent transitions between these states during a long-time evolution for the HMF model. The main advantage of studying this phenomenon for the HMF model is that the ambiguity to define the dynamical states can be resolved. In fact, the equations of motion of each HMF particle can be represented as those of a perturbed pendulum. An ordinary simple pendulum shows two types of motion: libration and rotation. It shows libration when the phase-point is inside the separatrix, and rotation when it is outside the separatrix. We then define High-Energy Particles (HEP) of the HMF model as those particles which are outside the separatrix, and Low-Energy Particles (LEP) as those which are inside the separatrix. This allows us to define a “trapping ratio” which takes the value $1$ for the “fully-clustered” state and is strictly smaller than $1$ in the “excited” state. Contrary to an ordinary simple pendulum, the value of the separatrix energy is not constant in time and hence the trapping ratio can fluctuate in time. Here, we show that the numerically computed time-averaged trapping ratio agrees with that obtained by a statistical average performed for the Boltzmann-Gibbs stable stationary solution of the Vlasov equation associated to the HMF model [@IK; @AR]. However, we find numerically that the probability distribution of the lifetime of the “fully-clustered” state is not exponential but follows instead a power law. Therefore, although an average trapping ratio exist, there appear to be no typical trapping ratio in the probabilistic sense. This paper is organized as follows. In Sec. \[sec:model\], we review the HMF model and define the dynamical states of the system. In Sec. \[sec:scs\] we estimate analytically the trapping ratio, using a Vlasov equation approach and compare it with the value obtained from numerical simulations. In Sec. \[sec:lifetime\] we numerically compute the probability distribution of the lifetime of the “fully-clustered” state in order to show that it obeys a power law. The final section is devoted to summary and discussion. Model and definition of dynamical states {#sec:model} ======================================== In this section we introduce the HMF model and define the dynamical states of the system. The Hamiltonian of the HMF model [@AR] is $$\label{eq:hamiltonian} H=K+V=\sum_{i=1}^{N}\frac{p_i^2}{2}+\frac{\varepsilon}{2N}\sum_{i,j=1}^N[1-\cos(\theta_i-\theta_j)].$$ The model describes a system of $N$ particles moving on a circle, each characterized by an angle $\theta_i$ and possessing momentum $p_i$. The interaction force between each pair of particles is attractive or repulsive, for $\varepsilon>0$ or $\varepsilon<0$, respectively. In the following we will consider only the attractive case, with $\varepsilon=1$. In this case, the model displays a second order phase transition at the energy density $U=H/N=3/4$ from a “clustered” phase at low energy (where particles are clumped) to a “gas” phase at high energy (where particles are homogeneously distributed on the circle). The HMF model is a globally coupled pendulum system, and the equations of motion can be expressed as those of a perturbed pendulum, $$\label{pendulum} \ddot{\theta}_i=-M\sin(\theta_i-\phi),$$ where $M$ (the order parameter of the phase transition) and the phase $\phi$ are defined as $$\begin{aligned} \label{eq:m} M&\equiv&\sqrt{M_x^2+M_y^2},\nonumber\\ \tan\phi&\equiv&\frac{M_y}{M_x},\nonumber\\ (M_x,M_y)&\equiv& \frac{1}{N}\left(\sum_{j=1}^{N}\cos\theta_j,\sum_{j=1}^{N}\sin\theta_j\right).\end{aligned}$$ The single particle energy is $$\label{eq:ei} e_i=\frac{p_i^2}{2}+[1-M\cos(\theta_i-\phi)].$$ Then, the separatrix energy $E_{sep}$ is $$\label{eq:sep} E_{sep}=1+M,$$ and the resonance width is $2\sqrt{M}$. An ordinary simple pendulum shows two types of motion: libration and rotation. It shows libration when the phase point is inside the separatrix, and rotation when it is outside the separatrix. We define High-Energy Particles (HEP) of the HMF model as those that lie outside the separatrix, i.e. their energy is larger than the separatrix energy, $e_i>E_{sep}$ . Low-Energy Particles (LEP) lie instead inside the separatrix, and have energy $e_i<E_{sep}$. Contrary to the simple pendulum, each particle of HMF model can go from inside to outside the separatrix and vice versa, because $M$ and $\phi$ are time dependent. Next, we define the trapping ratio $R$ as $$R \equiv \frac{N_{LEP}}{N}, \label{def:tp}$$ where $N_{LEP}$ is the number of LEP. Finally, we define the dynamical states of the system. We say that the system is “fully-clustered” if all the particles are LEP. Otherwise, if at least one particle, among the $N$, is HEP, we say the system is “excited”. The value of $R$ is $1$ if the system is in the “fully-clustered” state, it is less than one if the system is in the “excited” state. Let us discuss an example where these states appear. The system has $N=8$ particles and total energy density $U=0.4$ (below the phase transition energy). In the initial condition particles are uniformly distributed in a square rectangle $[-\theta_0,\theta_0]\times[-p_0,p_0]$ of the single-particle phase-space, with $\theta_0$ and $p_0$ conveniently chosen in order to get the energy $U$. This is the so-called “water-bag” initial distribution. As shown in Fig. \[fig:zahyou\] the system shows both the “fully-clustered” (panel (a)) and the “excited” state (panel (b)) at different time instances, and can switch from one to the other. In the “fully-clustered” state positions of all the particles fluctuate around a given angle. In the “excited” state one HEP has escaped from the cluster and rotates on the circle (all the others remain clustered). The momentum extracted by the HEP is compensated by an opposite momentum acquired by the cluster. This is a typical situation that appears in the low-energy phase of the model. It may be that more than one particle escapes from the cluster, expecially if the energy is increased. From this example it is clear that the trapping ratio $R$ is a time fluctuating quantity. In the next section, we will show that the Boltzmann-Gibbs equilibrium solution of the Vlasov equation associated to the HMF model allows us to compute analytically the time average of $R$ as a function of the energy density $U$. The result will be successfully compared with numerical simulations performed at finite $N$. Another quantity of interest is the time duration of the “fully-clustered” state. We define the lifetime $\tau$ of the “fully-clustered” state as the time interval from the absorption of a HEP to form the full cluster to the excitation of a particle from the cluster again. We will study numerically the properties of the probability distribution of $\tau$. Calculation of the average trapping ratio {#sec:scs} ========================================= In this section we introduce the Vlasov equation corresponding to the $N\to\infty$ limit of Hamiltonian (\[eq:hamiltonian\]) [@IK], and we derive the stationary stable solution corresponding to Boltzmann-Gibbs equilibrium. We then show how the knowledge of this solution allows us to derive the average trapping ratio. The Vlasov equation for the HMF model is $$\label{eq:Vlasov} \frac{\partial f}{\partial t}+p\frac{\partial f}{\partial \theta} -\frac{\partial V}{\partial \theta}\frac{\partial f}{\partial p}=0,$$ $$\label{eq:VlasovV} V(\theta) \equiv \frac{1}{2} \int_0^{2\pi}\int_{-\infty}^{\infty} \left[1-\cos(\theta-\theta')\right]f(\theta',p')d\theta'dp',$$ where $f(\theta,p,t)$ is a single particle distribution function and $V(\theta)$ is the self-consistent potential. According to Ref. [@I93], this Vlasov equation has the following Boltzmann-Gibbs stable stationary solution $$\label{eq:s-odf} f_{BG}(\theta,p) =\Theta(\theta)\rho(p)= \sqrt{\frac{\beta}{2\pi}}\exp\left(-\frac{\beta p^2}{2}\right) \frac{1}{2\pi I_0(\beta \langle M\rangle)} \exp(\beta \langle M\rangle \cos\theta),$$ where $I_0$ is the zero-order modified Bessel function and $\beta$ is the inverse temperature $\beta =1/T$ ($k_B=1$). Temperature is twice the kinetic energy $T=2K$. $\langle M\rangle$ is the solution of the self-consistency equation $$\begin{aligned} \label{eq:mbar} \langle M\rangle =\left|\frac{I_1( \beta \langle M\rangle)}{I_0( \beta \langle M\rangle)}\right|.\end{aligned}$$ We easily find that Eq. (\[eq:mbar\]) has a non-zero solution only if the temperature is sufficiently low, i.e., $T < T_c=1/2$. $T_c=1/2$ (corresponding to $U=3/4$) is the temperature of the second order phase transition. The averaged potential energy is $$\begin{aligned} \langle V\rangle&=&\int_0^{2\pi}V(\theta)\Theta(\theta)d\theta =\frac{1}{2}(1-\langle M\rangle^2),\end{aligned}$$ and the total energy density is $$\begin{aligned} \label{eq:u} U=\frac{T}{2}+\frac{1}{2}(1-\langle M\rangle^2).\end{aligned}$$ In Ref. [@AR], the energy dependence of the time average trapping ratio $\bar{R}(U)$ has been numerically calculated. The authors claimed that this quantity remains close to unity up to $U=U_b\sim 0.3$ and that it quickly decreases to zero as soon as $U \sim U_c$. Here, we analytically compute the statistically averaged trapping ratio $\langle R\rangle(U)$, using the Boltzmann-Gibbs distribution function (\[eq:s-odf\]). The main idea of how to perform this calculation is that of associating the average trapping ratio to the integral of the single particle distribution function performed inside the phase-space region $\Omega$ bounded by the upper and lower separatrices of the pendulum motion (\[pendulum\]). $$\langle R\rangle(U)= \int_\Omega f_{BG}(\theta,p)dpd\theta. \label{idea}$$ Using formula (\[eq:s-odf\]), we obtain $$\label{eq:tp} \langle R\rangle(U)= \frac{1}{2\pi I_0(\beta \langle M\rangle)} \int_0^{2\pi}{\rm Erf}\left(\sqrt{\langle M\rangle\beta(1+\cos\theta)}\right) \exp(\beta\langle M\rangle\cos\theta)d\theta,$$ where ${\rm Erf}$ is the error function. The integral in this equation has been performed numerically to obtain the $\langle R\rangle(U)$ function plotted in Fig. \[fig:tp\]. Numerical values of $\langle R\rangle(U)$ are also reported in Table 1. The values we obtain for $\langle R\rangle(U)$ are consistent with the time averaged quantity $\bar{R}(U)$ , first computed numerically in Ref. [@AR]. However, we have decided to repeat these numerical calculations for $N=100$ at various energy densities: the corresponding results are plotted in Fig. \[fig:tp\]. The agreement between $\langle R\rangle(U)$ and $\bar{R}(U)$ is extremely good. In order to characterize the finite $N$ fluctuations of the “fully-clustered” state, which may create HEP even below the critical temperature, we will analyze in the next section the probability distribution of the lifetime of the clustered state. Power-law distribution of the lifetime of the fully-clustered state {#sec:lifetime} =================================================================== In this section, we will further characterize the properties of the fully-clustered state, describing in more detail the trapping-untrapping process for a small number of particles (a study that has already been partially done in Ref. [@AR]). Below the critical energy density, $U<U_c$, HEP are repeatedly excited from and absorbed into the cluster in an intermittent fashion. The fully-clustered and excited states appear in turn, and the static picture predicted by the Vlasov equation, where particles that are within the separatrices remain there forever, is never observed. Nevertheless, quite surprisingly, this intermittent state produces, as we have discussed in the previous section, a time average trapping ratio which is in good agreement with Vlasov equation predictions. We want therefore to study the statistical properties of the lifetime of the cluster, as defined above for a finite number of particles, beginning with small systems of $N=8$ particles. If the trapping-untrapping transition process were a Poisson process, the probability distribution of the lifetime of the fully-clustered state would be exponential. The results of our numerical simulations are shown in Fig. \[fig:ltq\]. The distribution is definitely not exponential but, rather, it obeys a power law, with an exponent that appears to be slightly dependent on $U$ and to be close to $-1$. At fixed $U$, as the number of particle increases, the distribution is cut-off at large times. This effect is less evident in Fig. \[fig:ltq\]a than in Fig. \[fig:ltq\]b. We think that this is due to the difference in energy between the two cases: at the smaller energy of Fig. \[fig:ltq\]a one would probably need a larger value of $N$ to make the cut-off visible. Indeed, the presence of such a cut-off time is compatible with the fact that the average trapping ratio is finite, as shown in the previous section. In order to get an estimate of the number of particles needed to produce the cut-off, we propose the following heuristic argument. The single-particle untrapping probability per unit time can be assumed to be proportional to $1-\langle R\rangle(U)$. In order to destroy the fully-clustered state, it’s enough that one particle untrap, hence the probability per unit time that the fully clustered state is destroyed is proportional to $N(1-\langle R\rangle(U))$. When this probability is of order $1$, the fully-clustered state is destroyed. A preliminary numerical study of the behavior of $\langle R\rangle$ for small $U$ gives the non perturbative behavior $\langle R\rangle =1-\exp (-2/U)$. Therefore, the number of particle needed to create the cut-off diverges as $\exp (2/U)$ in the limit $U \to 0$, a growth that is faster than any power. The power-law behavior of the lifetime distribution could in principle extend to infinite time in this limit. This argument could be tested numerically and will be the subject of future investigations. summary ======= In this paper, we have investigated the dynamical behavior of the Hamiltonian Mean Field (HMF) model, focusing in particular on the mechanisms of particle trapping and untrapping from the cluster that is formed in the low-energy phase. Using the notion of separatrix for the related pendulum dynamics, we have been able to define precisely the dynamical state of “full-clustering”, and the “excitation” of a single particle. We have defined the [*trapping ratio*]{} as the ratio of the number of particles that are trapped in the cluster (with energy smaller than the separatrix energy) to the total number of particles. Using the Boltzmann-Gibbs stable stationary solution of the Vlasov equation, and performing a phase space integral within the separatrix region, we have been able to compute analytically the phase-space average of the trapping ratio. This quantity has been also obtained from numerical simulations of the $N$-body dynamics of the HMF model and it has been shown to be in perfect agreement with the Vlasov equation analytical calculation. Below the critical energy, when the number of particles is finite, the dynamical state of the system changes transitionally: the “fully-clustered” and “excited” states appear in turn. That is, high-energy particles (HEP) are excited from and absorbed into the cluster intermittently. We have numerically computed the probability distribution of the lifetime of the “fully-clustered” state, finding that it obeys a power law. This shows that the excitation of a particle below the critical energy of the HMF model is not a Poisson process. The discovery of the power law in this system is quite interesting and important. Its existence implies that, at the moment of ejection of an HEP, the system still “remembers” when the previous HEP had been swallowed into the cluster. One might think that the system would tend to behave “thermally” as the number of particles is increased, producing an exponential cut-off of the lifetime probability distribution. Although a cut-off at large times is present in the numerical simulations, the power law is observed over many decades and its extension increases quite rapidly with the number of particles. A heuristic calculation based on the Vlasov equation gives a non-perturbative fast increase of the number of particles necessary to observe the cut-off. It will be interesting to investigate the physical origin of the strong temporal correlations which give rise to the power-law behavior of the lifetime of the fully-clustered state: a possible collective particle motion is a candidate. Moreover, the power law behavior of the lifetime of clustered states should not be specific of the HMF model. It could appear also in systems showing [*chaotic itinerancy*]{} among “quasi-stationary” states, like those mentioned in the Introduction [@itinerancy] and in models of interacting molecules [@oomine]. Finally, if we look at the HMF model as representing an “abstract molecule”, then the excitation and decay of HEP could be considered as a chemical reaction. Then, the power-law type behavior of the lifetime discovered in this paper implies the possibility of non-standard behavior of chemical reactions: the reaction rate might be dynamically governed by the coherent motion of the reactants [@acp]. H.K. would like to thank Naoteru Gouda for fruitful discussions. H.K. is supported by a JSPS Fellowship for Young Scientists. T.K. is supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology. S.R. thanks JSPS for financial support and the Italian MIUR for funding this research under the grant PRIN05 [*Dynamics and thermodynamics of systems with long range interactions*]{}. [10]{} T. Dauxois et al., [*Lecture Notes in Physics*]{} [**602**]{} (2002) 458–487 (Springer-Verlag). J. Binney and S. Tremaine, [*Galactic Dynamics*]{}, Princeton University Press (1987). M. Lecar, in [*The Theory of Orbits in the Solar System and in Stellar Systems*]{}, IAU Symp. [**25**]{} 46 (1966), G. I. Kontopoulos Ed.. K. Ikeda, K. Matsumoto and K. Ohtsuka, [*Prog. Theor. Phys. Suppl.*]{}, [**99**]{}, 295 (1989),\ I. Tsuda, [*Neurocomputers and attention*]{}, A. V. Holden and V. I. Kryukov Eds., Manchester Univ. Press, 430, (1991),\ K. Kaneko, [*Phys. Rev. Lett.*]{}, [**63**]{}, 219, (1989). K. Kaneko and I. Tsuda Eds., [*Focus Issue: Chaotic Itinerancy*]{}, [*CHAOS*]{} (2003). T.Tsuchiya, N.Gouda and T.Konishi, [*Astrophys. Space Sci.*]{}, [**257**]{}, 319 (1998) G. B. Rybicki, [*Astrophys. Space Sci.*]{}, [**14**]{}, 56 (1971). T. Konishi and K. Kaneko, [*J. Phys. A*]{}, [**25**]{}, 6283 (1992),\ K. Kaneko and T. Konishi, [*Physica D*]{}, [**71**]{}, 146 (1994). H. Tanaka and I. Ohmine, [*J. Chem. Phys.*]{} [**87**]{}, 6128 (1987),\ M. Sasai, I. Ohmine and R. Ramaswamy, [*J. Chem. Phys.*]{}, [**96**]{}, 3045 (1992). S.Inagaki and T.Konishi, [*Publ. Astron. Soc. Japan*]{}, [**45**]{}, 733 (1993). M.Antoni and S.Ruffo, , [**52**]{}, 2361 (1995). T. Dauxois et al., “The Hamiltonian Mean Field Model: from Dynamics to Statistical Mechanics and back", in [@yellowbook]. P. H. Chavanis, J. Vatteville and F. Bouchet, [*Euro. Phys. J. B*]{}, [**46**]{}, 61 (2005). N. Nakagawa and K. Kaneko, [*J. Phys. Soc. Jpn.*]{}, [**69**]{} , 1255 (2000). S.Inagaki, [*Progress of Theoretical Physics*]{}, [**90**]{}, 577 (1993). M. Toda et al. eds., “Geometric Structures of Phase Space in Multi-Dimensional Chaos", [*Adv. Chem. Phys.* ]{} [**130**]{}(2005). ![a) Time evolution of the positions of the particles on the circle for the HMF model in a “fully-clustered” state; b) same time evolution for an “excited” state. Number of particles is $N=8$, system energy density is $U=0.4$ and the initial condition is a “water-bag”. The two states coexist in a given orbit of the system and appear at different times.[]{data-label="fig:zahyou"}](zahyou1-8-0p4-8-nonhep-mono2.eps "fig:"){width="12cm"} ![a) Time evolution of the positions of the particles on the circle for the HMF model in a “fully-clustered” state; b) same time evolution for an “excited” state. Number of particles is $N=8$, system energy density is $U=0.4$ and the initial condition is a “water-bag”. The two states coexist in a given orbit of the system and appear at different times.[]{data-label="fig:zahyou"}](zahyou1-8-0p4-8-hep-mono2.eps "fig:"){width="12cm"} ![Average trapping ratio vs. energy density. The full line is the result of the analytical calculation of the statistical average $\langle R\rangle(U)$. The points correspond to the numerical calculation of the time average $\bar{R}(U)$.[]{data-label="fig:tp"}](ur6.eps){width="12cm"} ![The distribution of the lifetime of the fully-clustered state with total energy $U=0.3$ (panel (a)) and $U=0.4$ (panel (b)). Numbers of particles are $8$, $16$, $32$, $64$ and $128$. The power index is well-fitted by $-0.8$ (panel (a)) and $-0.9$ (panel (b)).[]{data-label="fig:ltq"}](ltq-0p3-8-16-32-64-128-10_7-mono2.eps "fig:"){width="12cm"} ![The distribution of the lifetime of the fully-clustered state with total energy $U=0.3$ (panel (a)) and $U=0.4$ (panel (b)). Numbers of particles are $8$, $16$, $32$, $64$ and $128$. The power index is well-fitted by $-0.8$ (panel (a)) and $-0.9$ (panel (b)).[]{data-label="fig:ltq"}](ltq-0p4-8-16-32-64-128-10_7-mono2.eps "fig:"){width="12cm"} $U$ $M$ $T=1/\beta$ $\langle R\rangle(U)$ ------ ------------ ------------- ----------------------- 0.1 0.947209 0.0972051 1.0 0.2 0.887109 0.186963 0.999871 0.3 0.815506 0.26505 0.996731 0.4 0.728459 0.330652 0.982999 0.5 0.621782 0.386613 0.949673 0.6 0.485422 0.435635 0.880594 0.7 0.282056 0.479556 0.712613 0.71 0.252424 0.483718 0.679443 0.72 0.21873 0.487843 0.63799 0.73 0.178692 0.491931 0.582471 0.74 0.126424 0.495983 0.496192 0.75 0.00680319 0.500046 0.0136051 : Statistical equilibrium values of order parameter, temperature and average trapping ratio at different energy densities[]{data-label="tab:num"} [^1]: koyama@gravity.phys.waseda.ac.jp [^2]: tkonishi@r.phys.nagoya-u.ac.jp [^3]: stefano.ruffo@unifi.it

--- abstract: 'With growing consumer adoption of online grocery shopping through platforms such as Amazon Fresh, Instacart, and Walmart Grocery, there is a pressing business need to provide relevant recommendations throughout the customer journey. In this paper, we introduce a production within-basket grocery recommendation system, RTT2Vec, which generates real-time personalized product recommendations to supplement the user’s current grocery basket. We conduct extensive offline evaluation of our system and demonstrate a 9.4% uplift in prediction metrics over baseline state-of-the-art within-basket recommendation models. We also propose an approximate inference technique 11.6x times faster than exact inference approaches. In production, our system has resulted in an increase in average basket size, improved product discovery, and enabled faster user check-out.' address: 'Walmart Labs, Sunnyvale, California, USA' bibliography: - 'strings.bib' - 'refs.bib' title: 'A Large-Scale Deep Architecture for Personalized Grocery Basket Recommendations' --- Recommender System, Personalization, Representation Learning Introduction {#sec:intro} ============ A critical component of a modern day e-commerce platform is a user-personalized system for serving recommendations. While there has been extensive academic research for recommendations in the general e-commerce setting, user personalization in the online groceries domain is still nascent. An important characteristic of online grocery shopping is that it is highly personal. Customers show both regularity in purchase types and purchase frequency, as well as exhibit specific preferences for product characteristics, such as brand affinity for milk or price sensitivity for wine. One important type of grocery recommender system is a within-basket recommender, which suggests grocery items that go well with the items in a customer’s shopping basket, such as milk with cereals or pasta with pasta sauce. In practice, customers often purchase groceries with a particular intent, such as for preparing a recipe or stocking up for daily necessities. Therefore, a within-basket recommendation engine needs to consider both item-to-item compatibility within a shopping basket as well as user-to-item affinity, to generate efficient product recommendations that are truly user-personalized. In this paper, we introduce Real-Time Triple2Vec, **RTT2Vec**, a real-time inference architecture for serving within-basket recommendations. Specifically, we develop a representation learning model for personalized within-basket recommendation task, and then convert this model into an approximate nearest neighbour (ANN) retrieval task for real-time inference. Further, we also discuss some of the scalability trade-offs and engineering challenges when designing a large-scale, deep personalization system for a low-latency production application. For evaluation, we conducted exhaustive offline experiments on two grocery shopping datasets and observe that our system has superior performance when compared to the current state-of-the-art models. Our main contributions can be summarized as follows: - We introduce an approximate inference method which transforms the inference phase of a within-basket recommendation system into an Approximate Nearest Neighbour (ANN) embedding retrieval. - We describe a production real-time recommendation system which serves millions of online customers, while maintaining high throughput, low latency, and low memory requirements. Related Work {#sec:relatedwork} ============ Collaborative Filtering (CF) based techniques have been widely adopted in academia and industry for both user-item [@hu2008collaborative] and item-item recommendations [@linden2003amazon]. Recently,this approach has been extended to the within-basket recommendation task. The factorization-based models, **BFM** and **CBFM** [@le2017basket], consider multiple associations between the user, the target item, and the current user-basket to generate within-basket recommendations. Even though these approaches directly optimize for task specific metrics, they fail to capture non-linear user-item and item-item interactions. Due to the success of using latent representation of words (such as the **skip-gram** technique [@mikolov2013distributed; @mikolov2013efficient]) in various NLP applications, representation learning models have been developed across other domains. The **word2vec** inspired **CoFactor** [@liang2016factorization] model utilizes both Matrix Factorization (MF) and item embeddings jointly to generate recommendations. **Item2vec** [@barkan2016item2vec] was developed to generate item embeddings on itemsets. Using these, item-item associations can be modeled within the same itemset (basket). **Prod2vec** and **bagged-prod2vec** [@grbovic2015commerce] utilize the user purchase history to generate product ads recommendations by learning distributed product representations. Another representation learning framework, **metapath2vec** [@dong2017metapath2vec], uses meta-path-based random walks to generate node embeddings for heterogenous networks, and can be adapted to learn latent representations on a user-item interaction graph. By leveraging both basket and browsing data jointly, **BB2vec** [@trofimov2018inferring] learns dual vector representations for complementary recommendations. Even though the above skip-gram based approaches are used in wide areas of applications such as digital advertising and recommendation systems, they fail to jointly optimize for user-item and item-item compatibility. {width="\textwidth" height="6cm"} There has also been significant research to infer functionally complementary relations for item-item recommendation tasks. These models focus on learning compatibility [@veit2015learning], complementarity [@zhang2018quality; @kang2019complete; @DBLP:journals/corr/abs-1904-12574], and complementary-similarity [@mcauley2015inferring; @mane2019complementary] relations across items and categories from co-occurrence of items in user interactions. Method {#sec:method} ====== In this section, we explain the modeling and engineering aspects of a production within-basket recommendations system. First, we briefly introduce the state-of-the-art representation learning method for within-basket recommendation tasks, triple2vec. Then, we introduce our Real-Time Triple2Vec (RTT2Vec) system inference formulation, production algorithm, and system architecture. **Problem Definition**: Consider $m$ users $\mathfrak{U}$ = $\{u_1, u_2, .....,u_m\}$ and $n$ items $\mathfrak{I}$ = $\{i_1,i_2,...i_n\}$ in the dataset. Let ${\mathfrak{B}}_u$ denote a basket corresponding to user $u \in \mathfrak{U} $, where basket refers to a set of items $\{i^{'} | i^{'} \in \mathfrak{I}\}$. The goal of the within-basket recommendation task is given ($u$, ${\mathfrak{B}}_u$) generate top-k recommendations $\{i^{*} | i^{*} \in \mathfrak{I}\setminus {\mathfrak{B}}_u\}$ where $i^*$ is complementary to items in ${\mathfrak{B}}_u$ and compatible to user $u$. Triple2vec model {#ssec:T2V} ---------------- We utilize the triple2vec [@wan2018representing] model for generating personalized recommendations. The model employs (user $u$, item $i$, item $j$) triples, denoting two items ($i$, $j$) bought by the user $u$ in the same basket, and learns representation $h_u$ for the user $u$ and a dual set of embeddings ($p_i, q_j$) for the item pair ($i$, $j$). $$\label{cohesion_score} \begin{split} s_{i,j,u} = p_i^T q_j + p_i^T h_u + q_j^T h_u \end{split}$$ The cohesion score for a triple ($u,i,j$) is defined by Eq. \[cohesion\_score\]. It captures both user-item compatibility ($p_i^T h_u $, $q_j^T h_u$) as well as item-item complementarity ($p_i^T q_j$). The embeddings are learned by maximizing the co-occurrence log-likelihood of each triple as: $$\label{likelihood} \begin{split} L = \sum_{\forall (i,j,u)}\log{P(i|j,u)}+\log{P(j|i,u)}+\log{P(u|i,j)} \end{split}$$ where $P(i|j,u)=\frac{\exp(s_{i,j,u})}{\sum_{i^{'}} \exp(s_{i^{'},j,u})}$. Similarly, $P(j|i,u)$ and $P(u|i,j)$ can be obtained by interchanging ($i$,$j$) and ($i$,$u$) respectively. In accordance with most skip-gram models with negative sampling, the softmax function in Eq. \[likelihood\] is approximated by the Noise Contrastive Estimation (NCE) loss function, using TensorFlow [@abadi2016tensorflow]. A log-uniform (Zipf) distribution is used to sample negative examples. RTT2Vec: Real-Time Model Inference {#ssec:RTI} ---------------------------------- Serving a personalized basket-to-item recommendation system is challenging in practice. In conventional production item-item or user-item recommendation systems, model recommendations are precomputed offline via batch computation, and cached in a database for static lookup in real-time. This approach cannot be can’t be applied to basket-to-item recommendations, due to the exponential number of possible shopping baskets. Additionally, model inference time increases with basket size (number of items), making it challenging to perform real-time inference within production latency requirements. $$\label{argmax1} \begin{split} \operatorname*{arg\,max}_{j}(p_i^T q_j + p_i^T h_u + q_j^T h_u) =\operatorname*{arg\,max}_{j}{(\underbrace{[p_i\quad h_u]}_\text{Query Vector}}^T \underbrace{[q_j\quad q_j]}_\text{ANN Index}) \end{split}$$ We transform the inference phase of triple2vec (Section \[ssec:T2V\]) into a similarity search of dense embedding vectors. For a given user $u$ and anchor item $i$, this can be achieved by taking $argmax$ of the cohesion score (Eq. \[cohesion\_score\]) and adjusting it as shown in Eq. \[argmax1\]. The first term, the **query vector**, depends on the inputs $u$ and $i$, and the second term, the **ANN index**, only depends on $j$, thus transforming our problem into a similarity search task. $$\label{argmax2} \begin{split} \operatorname*{arg\,max}_{j}(\frac{(p_i^T q_j + p_i^T h_u + q_j^T h_u) + (p_j^T q_i + p_j^T h_u + q_i^T h_u)}{2}) \\ = \operatorname*{arg\,max}_{j}(p_i^T q_j + q_j^T h_u + p_j^T q_i + p_j^T h_u) \\ = \operatorname*{arg\,max}_{j}{(\underbrace{[p_i\quad h_u \quad q_i \quad h_u]}_\text{Query Vector}}^T \underbrace{[q_j \quad q_j \quad p_j \quad p_j]}_\text{ANN Index}) \end{split}$$ Further, we speed up the similarity search of the inference problem by using an off-the-shelf Approximate Nearest Neighbour (ANN) indexing library, such as FAISS [@JDH17], ANNOY [@annoy_lib], or NMSLIB [@naidan2016non; @boytsov2013engineering], to perform approximate dot product inference efficiently at large-scale. We also observe that model performance improves by interchanging the dual item embeddings and taking the average of the cohesion scores, as shown in Eq. \[argmax2\]. RTT2Vec: Production Algorithm {#ssec:RRT2vec} ----------------------------- The RTT2Vec algorithm used for generating top-k within-basket recommendations in production consists of three principal tasks: basket-anchor set selection, model inference, and post-processing. These steps are described below in detail: **Basket-anchor set selection:** To generate personalized within-basket recommendations, we replace the item embeddings $p_i$ and $q_i$ with the average embedding of all the items in the shopping basket. This approach works very well for baskets with smaller sizes, but in practice, a typical family’s shopping basket of groceries contains dozens of items. Taking the average of such large baskets results in losing information about the individual items in the basket. For larger baskets, we deploy a sampling algorithm which randomly selects 50% of items in the basket as a basket-anchor set. **Model Inference:** For each item in the basket-anchor set, we create the query vector $ [p_i\quad h_u \quad q_i \quad h_u ] $ using the pre-trained user embedding $h_u$ and item embeddings $p_i$ and $q_i$ (refer Eq. \[argmax2\]). Then, we search the query vector in the Approximate Nearest Neighbour (ANN) index to retrieve the top-k recommendations. The ANN index is created from the concatenation of the dual item embeddings $[q_j \quad q_j \quad p_j \quad p_j] \forall $ j $\in \mathfrak{I}$. The ANN index and embeddings are stored in memory for fast lookup. In practice, the inference can be further speed up by performing a batch lookup in the ANN index instead of performing a sequential lookup for each item in the basket-anchor set. After the top-k recommendations are retrieved for each anchor item in the basket-anchor set, a recommendation aggregator module is used to blend all the recommendations together. The aggregator uses several factors such as number of distinct categories in the recommendation set, the individual item scores in the recommendations, taxonomy-based weighting, and business rules to merge the multiple recommendation sets, and filter to a top-k recommendation set. **Post-processing:** Once the top-k recommendation set is generated, an additional post-processing layer is applied. This layer incorporates diversification of items, removes blacklisted items and categories, utilizes market-basket analysis association rules for taxonomy-based filtering, and applies some business requirements to generate the final top-k recommendations for production serving. RTT2Vec: Production System Architecture --------------------------------------- In this section, we provide a high level overview of our production recommendation system as illustrated in Figure \[fig:arch\]. This system is comprised of both offline and online components. The online system consists of a memcached distributed cache, streaming system, a real time inference engine, and a front-end client. The offline system encompasses a data store, a feature store serving all the recommendation engines at Walmart, and an offline model training framework deployed on a cluster of GPUs. At Walmart Grocery, we deal with a large volume of customer interactions, streaming in at various velocities. We use the Kafka streaming engine to capture real-time customer data without delay and store the data in a Hadoop-based distributed file system. For offline model training, we construct training examples by extracting features from our feature store through Hive and Spark jobs. Then, the training examples are input into an offline deep learning model, which is trained on a GPU cluster, generating user and dual-item embeddings. These embeddings are then stored in an embedding store (distributed cache) to facilitate online retrieval by the real-time inference engine. The primary goal of deploying a real-time inference engine is to provide personalized recommendations, while ensuring very high throughput and providing a low-latency experience to the customer. The real-time inference engine utilizes a Approximate Nearest Neighbor (ANN) index, constructed from the trained embeddings, and deployed as a micro-service. This engine interacts with the front-end client to obtain user and basket context and generates personalized within-basket recommendations in real-time. Experiments {#sec:exp} =========== Datasets -------- Our experimental evaluation is performed on one public dataset and one proprietary dataset. Both datasets are split into train, validation, and test sets. The public Instacart dataset is already split into prior, train and test sets. For the Walmart Grocery dataset, the train, validation, and test sets comprise of one year, the next 15 days, and the next one month of transactions respectively. - **Instacart:** We use the open-source grocery dataset published by Instacart [@instacart_data], containing approximately 206k users and 50k items with 3.4m total interactions. The average basket size is 10. - **Walmart:** We use a subset of a proprietary online Walmart Grocery [@walmart_grocery] dataset for these experiments. The dataset contains approximately 3.5m users and 90k items with 800m interactions. \[tab:within-basket\] **Dataset** **Method** **Recall@20** **NDCG@20** ------------- ----------------- -------------------- -------------------- ItemPop 0.1137 0.1906 BB2vec 0.0845 0.1258 item2vec 0.0810 0.1356 triple2vec (NP) 0.0794 0.1709 triple2vec 0.1354\* 0.1876\* RTT2Vec **0.1481** **0.2391** Improv.% $\textbf{9.37\% }$ $\textbf{21.53\%}$ ItemPop 0.0674 0.1318 BB2vec 0.0443 0.0740 item2vec 0.0474 0.0785 triple2vec (NP) 0.0544 0.0988 triple2vec 0.0685\* 0.1142\* RTT2Vec **0.0724** **0.1245** Improv.% $\textbf{5.75\% }$ $\textbf{9.01\%}$ : Within-Basket Recommendations \[table:1\] Evaluation ---------- **Metrics**: We evaluate the performance of models with the metrics: **Recall@K** and **NDCG@K**. **Recall@K** measures the fraction of relevant items successfully retrieved when the top-K items are recommended. **NDCG@K** (Normalized Discounted Cumulative Gain) is a ranking metric which uses position in the recommendation list to measure gain. Metrics are reported at K=20. For the within-basket recommendation task, given a subset of the basket, the goal is to predict the remaining items in the basket. Let the basket be split into two sets $B_T$ and $B_{T'}$, where $B_T=\{i_1, i_2, .....,i_m\}$ denotes the subset of items in basket used for inference, and $B_{T'}=B \setminus{B_T} = \{j_1, j_2, .....,j_n\}$ denotes the remaining set of items in the basket. Let $S_K=\{r_1, r_2, .....,r_K\}$ denote the top-K recommendation list generated using $B_T$. Then: $$\label{recall_k} {\rm Recall@K} = \frac{|S_K \cap B_{T'}|}{|B_{T'}|}$$ $$\label{ndcg_k} {\rm NDCG@K} = \sum_{p=1}^{k} \frac{\mathbbm{1}{[l_p \in B_{T'}]}}{\log_{2}{(p+1)}}$$ where $p$ denotes the rank of the item in the recommended list, and $ \mathbbm{1}{}$ is the indicator function indicating if $l_p \in B_{T'}$. ![System Latency Comparison[]{data-label="fig:latencygraph"}](SystemArch_latest.png){width="\textwidth"} Baseline Models {#baseline_models} --------------- Our system is evaluated against the following models: - **ItemPop**: The Item Popularity (ItemPop) model selects the top-K items based on their frequency of occurrence in the training set. The same set of items are recommended for each test basket for each user. - **item2vec**: The item2vec [@barkan2016item2vec] model uses Skip-Gram with Negative Sampling (SGNS) to generate item embeddings on itemsets. We apply this model on within-basket item-sets to learn co-occurrence of items in the same basket. - **BB2vec**: The BB2vec [@trofimov2018inferring] model learns vector representations from basket and browsing sessions. For a fair comparison with other models, we have adapted this method to only use basket data and ignore view data. - **triple2vec (NP)**: This is a non-personalized variation of triple2vec (as explained in Section \[ssec:T2V\]), where we only use the dual item embeddings and ignore user embeddings during inference. The cohesion score here (Eq. \[cohesion\_score\]) can be re-written as: $s_{i,j} = p_i^T q_j$ - **triple2vec**: The state-of-the-art triple2vec model (as explained in Section \[ssec:T2V\]) employs the Skip-Gram model with Negative Sampling (SGNS), applied over (user, item, item) triples in the test basket to generate within-basket recommendations. **Parameter Settings**: We use an embedding size of 64 for all skip-gram based techniques, along with the Adam Optimizer with a initial learning rate of 1.0, and the noise-contrastive estimation (NCE) of softmax as the loss function. A batch size of 1000 and a maximum of 100 epochs are used to train all skip-gram based models. We use 5 million triples to train the Instacart dataset and 200 million triples for the Walmart dataset. Results ------- We next evaluate our model predictive performance and system latency. The models are trained on an NVIDIA K80 GPU cluster, each consisting of 48 CPU cores. For evaluation and benchmarking, we use an 8-core x86\_64 CPU with 2-GHz processors. **Predictive Performance**: We compare the performance of our system, RTT2Vec, against the models described in Section \[baseline\_models\] on the within-basket recommendation task. For each basket in the test set, we use 80% of the items as input and the remaining 20% of items as the relevant items to be predicted. As displayed in Table \[table:1\], we observe that our system outperforms all other models on both Instacart and Walmart datasets, improving Recall@20 and NDCG@20 by 9.37% (5.75%) and 21.5% (9.01%) for Instacart (Walmart) datasets when compared to the current state-of-the-art model triple2vec. **Real-Time Latency**: Further, we test real-time latency for our system using exact and approximate inference methods as discussed in Section \[sec:method\]. Figure \[fig:latencygraph\] displays system latency (ms) versus basket size. To perform exact inference based on Eq. \[argmax2\], we use ND4J [@nd4j] and for approximate inference (as discussed in Section \[ssec:RTI\]), we test Faiss, Annoy, and NMSLIB libraries. ND4J is a highly-optimized scientific computing library for the JVM. Faiss is used for efficient similarity search of dense vectors that can scale to billions of embeddings, Annoy is an approximate nearest neighbour library optimized for memory usage and loading/saving to disk ,and NMSLIB is a similarity search library for generic non-metric spaces. On average, ND4J adds 186.5ms of latency when performing exact real-time inference. For approximate inference, Faiss, Annoy, and NMSLIB libraries add an additional 29.3ms, 538.7ms, and 16.07ms of system latency respectively. Faiss and NMSLIB provide an option to perform batch queries on the index, therefore latency is much lower than Annoy. Faiss and NMSLIB are 6-10 times faster than the exact inference method using ND4J. In practice, we use NMSLIB in our production system as it provides better overall performance. Conclusion and Future Work {#sec:conclusion} ========================== In this paper, we propose a state-of-the-art real-time user-personalized within-basket recommendation system, RTT2vec, to serve personalized item recommendations at large-scale within production latency requirements. As per our knowledge, this study is the first description of a large-scale production grocery recommendation system in the industry. Our approach outperforms all baseline models on evaluation metrics, while respecting low-latency requirements when serving recommendations at scale. Due to the increasing adoption of online grocery shopping and the associated surge in data size, there is an increase in the training time required for deep embedding models for personalized recommendations. Future work includes investigating the performance tradeoff of different sampling methodologies during model training. We are also exploring the introduction of additional content and contextual embeddings for improving model predictions further.

--- abstract: 'This paper provides a sample of a LaTeX document which conforms, somewhat loosely, to the formatting guidelines for ACM SIG Proceedings.[^1]' author: - Ben Trovato - 'G.K.M. Tobin' - 'Lars Th[ø]{}rv[ä]{}ld' - 'Lawrence P. Leipuner' - Sean Fogarty - Charles Palmer - John Smith - 'Julius P. Kumquat' bibliography: - 'sample-bibliography.bib' subtitle: Extended Abstract title: SIG Proceedings Paper in LaTeX Format --- &lt;ccs2012&gt; &lt;concept&gt; &lt;concept\_id&gt;10010520.10010553.10010562&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computer systems organization Embedded systems&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10010520.10010575.10010755&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computer systems organization Redundancy&lt;/concept\_desc&gt; &lt;concept\_significance&gt;300&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10010520.10010553.10010554&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computer systems organization Robotics&lt;/concept\_desc&gt; &lt;concept\_significance&gt;100&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10003033.10003083.10003095&lt;/concept\_id&gt; &lt;concept\_desc&gt;Networks Network reliability&lt;/concept\_desc&gt; &lt;concept\_significance&gt;100&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;/ccs2012&gt; [^1]: This is an abstract footnote

--- abstract: 'A Hermitian symplectic manifold is a complex manifold endowed with a symplectic form $\omega$, for which the bilinear form $\omega(I\cdot,\cdot)$ is positive definite. In this work we prove ${dd^c}$-lemma for 1- and (1,1)-forms for compact Hermitian symplectic manifolds of dimension 3. This shows that Albanese map for such manifolds is well-defined and allows one to prove Kählerness if the dimension of the Albanese image of a manifold is maximal.' author: - Grigory Papayanov date: 2015 title: Cohomological properties of Hermitian sympletic threefolds --- [**Cohomological properties of Hermitian\ symplectic threefolds** ]{}\ Grigory Papayanov\ Introduction {#introduction .unnumbered} ============ A Hermitian symplectic manifold is a complex manifold $(M,I)$ together with a symplectic form $\omega$, for which the bilinear form $\omega(I\cdot,\cdot)$ is positive definite (that is, $\omega(IX,X)>0$ for any vector field $X$ on $M$). Any Kähler manifold is obviously Hermitian symplectic, and it is an open problem whether there exist other examples of Hermitian symplectic manifolds. Hermitian symplectic manifolds were studied by Streets and Tian in [@Streets_Tian:pluriclosed] and [@Streets_Tian:flow]; they constructed an appropriate Ricci flow on Hermitian symplectic manifolds, and studied its convergency properties. Since then, many people searched for non-trivial examples of Hermitian symplectic manifolds. The search for non-Kähler examples of Hermitian symplectic manifolds was vigorous, but ultimately unsuccessful. All common sources of examples of non-Kähler manifolds were tapped at some point. For complex dimension 2, Hermitian symplectic structures are all Kähler. This was shown by Streets and Tian in [@Streets_Tian:pluriclosed]. Another proof could be obtained from the Lamari ([@Lamari]) result about existence of positive, exact $(1,1)$-current on any non-Kähler complex surface. In [@Peternell], it was shown that any non-Kähler Moishezon manifold admits an exact, positive $(n-1,n-1)$-current; therefore, Moishezon manifolds which are Hermitian symplectic are also Kähler. In [@Enrietti_Fino_Vezzoni] it was shown that no complex nilmanifold can admit a Hermitian symplectic structure, and in [@Fino_Kasuya_Vezzoni] this result was extended to all complex solvmanifolds and Oeljeklaus-Toma manifolds. Existence of Kähler metric implies some restrictions on the cohomology of a manifold: for example the Frölicher spectral sequence of Kähler manifold always degenerates at the first page. Results of Cavalcanti ([@Cavalcanti:SKT]) show that the Frölicher spectral sequence for Hermitian symplectic manifolds degenerates at the first page. In this work we define some Laplacian-like operators, kernels of which conjecturally isomorphic to the spaces of cohomology, and, with the help of these operators, prove ${dd^c}$-lemma for (1,1)-forms on Hermitian symplectic threefolds. Argument of Gauduchon ([@Gauduchon]) shows that ${dd^c}$-lemma for (1,1)-forms is equivalent to the equality $b^1=2h^{0,1}$. It follows that the Albanese map is well-defined and, if its image is not a point, the generic fiber of ${\operatorname{Alb}}$ is Kähler. The question of existence of special (e.g. Kähler or balanced) metrics on total spaces of maps with Kähler base and fibers is studied, for example, in [@HL] and [@Michelsohn]. Using the Albanese map, we are able to prove that if a Hermitian symplectic threefold $M$ has ${\operatorname{dim}}{\operatorname{Alb}}(M)=3$, then it admits a Kähler metric, and if ${\operatorname{dim}}{\operatorname{Alb}}(M)=1$, $M$ is balanced. If $dd^c$-lemma holds for $(2,2)$-forms, then by [@HL] ${\operatorname{dim}}{\operatorname{Alb}}(M)=2$ would imply that $M$ is Kähler, but, unfortunately, we have not proven ${dd^c}$-lemma in full generality yet. [**Acknowledgements.**]{} The author would like to thank M.Verbitsky for many extremely helpful discussions. Work on sections 1–3 was supported by RSCF, grant number 14-21-00053, within the Laboratory of Algebraic Geometry. Work on section 4 was supported by RFBR 15-01-09242. Preliminaries ============= [ ]{}Let $M$ be a smooth manifold of dimension 2n, $I:TM {{\:\longrightarrow\:}}TM$ an integrable complex structure, ${\mathcal{A}}^{p,q}$ the corresponding Hodge decomposition on the bundle of differential forms: ${\mathcal{A}}^n\otimes {{\Bbb C}}=\bigoplus_{n=p+q}{\mathcal{A}}^{p,q}$, ${\omega^{1,1}}$ a form in ${\mathcal{A}}^{1,1}$. We will say that ${\omega^{1,1}}$ is [*Hermitian*]{} if the tensor $h(\cdot,\cdot):={\omega^{1,1}}(\cdot,I\cdot)$ is a Riemannian metric on $M$, and we will say that ${\omega^{1,1}}$ is [*Hermitian symplectic*]{} if there exists a symplectic form $\omega$ such that ${\omega^{1,1}}$ is the (1,1)-component in the Hodge decomposition of $\omega$. If $M$ is endowed with such ${\mathcal{I}}$ and ${\omega^{1,1}}$, we will call it a Hermitian symplectic manifold. For a Hermitian symplectic manifold $(M,I,\omega)$, let $d: {\mathcal{A}}^\bullet{{\:\longrightarrow\:}}{\mathcal{A}}^{\bullet+1}$ be the usual de Rham differential acting on forms, $d^c:=IdI^{-1}: {\mathcal{A}}^{\bullet}{{\:\longrightarrow\:}}{\mathcal{A}}^{\bullet+1}$ the twisted differential, $L: A^\bullet{{\:\longrightarrow\:}}A^{\bullet+2}$ the operator of (left) multiplication by $\omega$, $L(\eta):= \omega\wedge \eta$, $\Lambda: {\mathcal{A}}^{\bullet}{{\:\longrightarrow\:}}{\mathcal{A}}^{\bullet-2}$ the adjoint operator ([@Yau_Tseng]). In the local Darboux coordinates $p_i, q_i$ where $\omega=\sum dp_i\wedge dq_i$, operator $\Lambda$ looks like $\sum i_{\!\frac{{\partial}}{{\partial}p_i}}i_{\!\frac{{\partial}}{{\partial}q_i}}$. We will denote by $L^{1,1}$ the operator of multiplication by the hermitian form ${\omega^{1,1}}$, and by $\Lambda^{1,1}$ the adjoint operator to $L^{1,1}$. [ ]{}\[SKT\] The form ${\omega^{1,1}}$ is the SKT form, that is, ${\partial}{\overline}{\partial}{\omega^{1,1}}=0$. [**Proof:**]{} Let $\omega={\omega^{1,1}}+\alpha$, where $\alpha$ lies in ${\mathcal{A}}^{2,0}\oplus{\mathcal{A}}^{0,2}$. Since $d\omega=0$, ${\partial}{\omega^{1,1}}=-{\overline}{\partial}\alpha$ and ${\partial}{\overline}{\partial}{\omega^{1,1}}={\overline}{\partial}^2\alpha=0$. [ ]{}Let $\alpha$ be a differential form on $M$. We will say that $\alpha$ is [*primitive with respect to $\omega$*]{} if $\Lambda\alpha=0$, and that $\alpha$ is primitive with respect to ${\omega^{1,1}}$ if $\Lambda^{1,1}\alpha=0$. [ ]{}(The Weil identities). Let $B^{p,q}$ be a primitive with respect to ${\omega^{1,1}}$ $(p,q)$-form, $p+q=r$. Then the following formula holds ([@Voisin Proposition 6.29]): $$*B^{p,q}=(-1)^{\frac{r(r+1)}{2}}(\sqrt{-1})^{p-q}\frac{1}{(n-r)!}({\omega^{1,1}})^{n-k}\wedge B^{p,q}.$$ [ ]{}An operator $\Delta$ defined as double graded commutator, $\Delta:=\{d,\{d^c,\Lambda^{1,1}\}\}$ is called [*the Hermitian symplectic*]{} Laplacian. [ ]{}$\Delta$ is not a Laplacian associated to the Riemannian metric $h$. Nevertheless they differ by a differential operator of first order (see e.g. [@Liu_Yang] for the exact formula), therefore they have equal symbols, so $\Delta$ is elliptic. Recall the graded Jacobi identity for the graded commutator: $$\{a,\{b,c\}\}=\{\{a,b\},c\}+(-1)^{deg(a)deg(b)}\{b,\{a,c\}\}.$$ [ ]{}\[commutators\] $\Delta=\{d^c,\{d,\Lambda^{1,1}\}\}$. Therefore $\Delta$ commutes with $d$ and with $d^c$. [**Proof:**]{} Follows simply from the Jacobi identity. [ ]{}\[spectral\] (Spectral theorem). Let $(M, I,\omega)$ be a compact Hermitian symplectic manifold. Then the space of differential forms decomposes as a topological direct sum of generalized eigenspaces of $\Delta$: ${\mathcal{A}}^\bullet(M)=\bigoplus_{\lambda_i}{\mathcal{A}}^\bullet_{\lambda_i}(M)$, each component of this decomposition is finite-dimensional and preserved by $d$, $d^c$ and $\delta$. [**Proof:**]{} Decomposition is in fact proven in [@BGV Proposition 2.36] ($\Delta$ is a generalized laplacian in their terminology); one has to apply spectral theorem for compact operators: compact operator on Hilbert space has a canonical Jordan form with finite-dimensional generalized eigenvalues ([@Conway]). By \[commutators\], $\Delta$ commutes with $d$ and $d^c$, so all generalized eigenspaces are in fact subcomplexes. [ ]{}Let $\alpha$ be a closed form in $\bigoplus_{\lambda_i\ne 0}{\mathcal{A}}^\bullet_{\lambda_i}(M)$. Then $\alpha$ is exact. [**Proof:**]{} When restricted to $\bigoplus_{\lambda_i\ne 0}{\mathcal{A}}^\bullet_{\lambda_i}(M)$, Laplacian $\Delta$ has an inverse, $\Delta^{-1}$. So $$\alpha=\Delta\Delta^{-1}\alpha=(\pm dd^c\Lambda \pm d\Lambda d^c)\Delta^{-1}\alpha.$$ Forms on a Hermitian symplectic manifold ======================================== In this section $M$ is assumed to be compact. [ ]{}\[ddc1\] ($dd^c$-lemma for 1-forms). Let $\alpha$ be a $d$-exact, $d^c$-closed (or $d^c$-exact and $d$-closed) 1-form. Then $\alpha=0$. [**Proof:**]{} Suppose $\alpha$ is $d$-exact, $\alpha=df$. Then $dd^cf=0$. By Hopf maximum principle ([@_Gilbarg_Trudinger_]), $f$ is constant, hence $\alpha=df=0$. We will now investigate whether holomorphic forms on $M$ are closed. [ ]{}Let the $n$ be the complex dimension of $M$. Then every holomorphic $n-2$-form is closed. [**Proof:**]{} Let $\alpha$ be a holomorphic $n-2$-form, $\alpha \in {\mathcal{A}}^{n-2,0}$, ${\overline}{\partial}\alpha=0$. Then $d\alpha={\partial}\alpha$ is primitive with respec to ${\omega^{1,1}}$, by dimension reasons. So, by Weil identities, $$||d\alpha||^2=\int d\alpha\wedge d{\overline}\alpha \wedge {\omega^{1,1}}=\int {\partial}\alpha\wedge {\overline}{\partial}{\overline}\alpha \wedge {\omega^{1,1}}= \alpha \wedge {\overline}\alpha \wedge {\partial}{\overline}{\partial}{\omega^{1,1}}=0.$$ Hence $\alpha$ is closed. [ ]{}\[holoforms\] Obviously, on any compact complex manifold of complex dimension $n$, every holomorphic function and every holomorphic $n$-form is closed. Every holomorphic $n-1$-form is also closed, as the simple argument with the integration shows. So, any holomorphic form on a Hermitian symplectic threefold is closed. $dd^c$-lemma for (1,1)-forms ============================ Recall that by \[spectral\] every differential form $\alpha$ decomposes by generalized eigenspaces of $\Delta$: $\alpha=\alpha_0 + \alpha_{\ne 0}$, where $\Delta^N(\alpha_0)=0$ for some $N$, and $\alpha_{\ne 0}=\Delta\Delta^{-1}\alpha_{\ne 0}$. Suppose that $\alpha$ is $d$-exact and $d^c$-closed. Then $\alpha_0$ and $\alpha_{\ne 0}$ are also $d$-exact and $d^c$-closed. [ ]{}In notations as above, $\alpha_{\ne 0}$ is $dd^c$-exact. [**Proof:**]{} by \[commutators\], $\Delta^{-1}$ commutes with $d$ and $d^c$, so $\Delta\Delta^{-1}\alpha_{\ne 0}=dd^c\Lambda^{1,1}\Delta^{-1}\alpha_{\ne 0}=\alpha$. [ ]{}\[primitivness\] Suppose exact (1,1)-form $\eta=d\gamma$ lies in the kernel of $\Delta^{1,1}$. Then $\eta$ is primitive (with respect both to $\omega$ and to $\omega^{1,1}$). [**Proof:**]{} $\Delta\eta=dd^c\Lambda^{1,1}\eta=0$, so, by Hopf maximum principle [@_Gilbarg_Trudinger_] $\Lambda^{1,1}\eta=c$, where $c$ is some constant. It means that $\Lambda\eta$ also equals $c$. If $\Lambda\eta=c,$ then $\eta=c\omega+B$, where $B$ is a primitive form. Since $\eta=d\gamma$, the cohomology classes of $c\omega$ and $B$ are equal, but the cohomology class of a symplectic form cannot be represented by a primitive form. Indeed, $\omega\wedge\omega^{n-1}$ is a volume form, hence nonzero in cohomology, but $B\wedge\omega^{n-1}=0$. So $c=0$ and $\eta$ is primitive. [ ]{}\[vanish\] Suppose ${\operatorname{dim}}(M)=3$, $\eta=dd^cf$ is $dd^c$-exact primitive $(1,1)$-form. Then $\eta=0$. [**Proof:**]{} Note first that, since $\eta$ is primitive with respect to $\omega$, it is primitive with respect to $\omega^{1,1}$, so, by Weil identities, $*\eta=\eta\wedge(\omega^{1,1})^{\wedge n-2}$, where $*$ is the Hodge star operator associated with the Hermitian metric $h$ with corresponding 2-form equal to $\omega^{1,1}$ ([@Griffiths_Harris]). Then $h(\eta,\eta)=$ $$=\int \eta\wedge*\eta=\int \eta\wedge\eta\wedge(\omega^{1,1})^{\wedge n-2}=\int f\eta\wedge dd^c(\omega^{1,1})^{\wedge n-2}.$$ But $dd^c\omega^{1,1}=0$ on a Hermitian symplectic manifold, so the integral vanishes. Since $h$ is a hermitian metric, $\eta$ also equals to zero. [ ]{}Let ${\operatorname{dim}}(M)=3$. Suppose that an exact (1,1)-form $\eta=d\gamma$ lies in the kernel of $(\Delta)^n,$ $n>1$. Then $\eta$ lies in the kernel of $(\Delta)^{n-1}.$ [**Proof:**]{} $(\Delta)^{n-1}\eta$ is an exact (1,1)-form lying in the kernel of $\Delta$, so, by \[primitivness\] it is primitive. Since $d\eta=d^c\eta=0, (\Delta)^{n-1}\eta=(dd^c\Lambda)^{n-1}\eta$, it is $dd^c$-exact, therefore, by \[vanish\], it vanishes. In order to complete the proof of ${dd^c}$-lemma for (1,1)-forms on Hermitian symplectic manifolds, we have to prove that an exact, primitive (1,1)-form vanishes. [ ]{} Let $M$ be a Hermitian symplectic manifold of dimension 3, $\eta$ be an exact, primitive (1,1)-form on $M$. Then $\eta=0$. [**Proof:**]{} Square of Hermitian norm of $\eta$ is equal to $\int \eta\wedge\eta\wedge\omega^{1,1}$, but in dimension 3 we have the equality $\eta\wedge\eta\wedge\omega^{1,1}=\eta\wedge\eta\wedge\omega$; the latter form is exact, therefore $\eta=0$. [ ]{}\[ddc11\] Let $M$ be a compact Hermitian symplectic threefold, $\alpha$ is a $d$-closed, $d^c$-exact $(1,1)$-form. Then $\alpha=dd^cf$ for some function $f$. Applications ============ [ ]{}(Gauduchon, [@Gauduchon]). For a complex manifold $M$, $dd^c$-lemma for $(1,1)$-forms is equivalent to the equality $b^1=2h^{1,0}$. [**Proof:**]{} Consider the cohomology sequence associated to the short exact sequence of sheaves of the form $0 {{\:\longrightarrow\:}}\sqrt{-1}{{\Bbb R}}{{\:\longrightarrow\:}}\mathcal{O} \stackrel{Re}{{{\:\longrightarrow\:}}} {{\mathcal{H}}}{{\:\longrightarrow\:}}0$, where ${{\mathcal{H}}}$ is the sheaf of pluriharmonic functions. The relevant piece looks like $$... \stackrel{0}{{{\:\longrightarrow\:}}} H^1(M,\sqrt{-1}{{\Bbb R}}) {{\:\longrightarrow\:}}H^1(M,\mathcal{O}) {{\:\longrightarrow\:}}H^1(M,{{\mathcal{H}}}) {{\:\longrightarrow\:}}H^2(M,\sqrt{-1}{{\Bbb R}}) {{\:\longrightarrow\:}}...$$ It is well-known that $$H^1(M,{{\mathcal{H}}})=\frac{{\operatorname{Ker}}d:{\mathcal{A}}^{1,1} {{\:\longrightarrow\:}}{\mathcal{A}}^3}{{\operatorname{Im}}dd^c: {\mathcal{A}}^0 {{\:\longrightarrow\:}}{\mathcal{A}}^{1,1}}.$$ So $dd^c$-lemma for $(1,1)$-forms holds if and only if the third arrow is an isomorphism, and, by exactness, if and only if the first arrow is an isomorphism. So, the equality $b^1=2h^{1,0}$ holds on compact Hermitian symplectic threefolds. It follows that we have the Hodge decomposition on the first cohomology of $M$: $H^1(M,{{\Bbb C}})=H^{0,1}(M) \oplus H^{1,0}(M)$, and ${\operatorname{dim}}H^{0,1}={\operatorname{dim}}H^{1,0}$. So the rank of the abelian group $H^1(M,{{\Bbb Z}})$ is equal to the dimension of the real vector space $H^{0,1}(M)$. It follows that the Albanese torus is defined correctly and we have the Albanese map ${\operatorname{Alb}}: M {{\:\longrightarrow\:}}H^{0,1}(M)^*/H_1(M,{{\Bbb Z}})$. Its image ${\operatorname{Alb}}(M)$ is a subvariety (possibly singular) of a torus. [ ]{}Suppose ${\operatorname{dim}}{\operatorname{Alb}}(M)=3$. Then $M$ is Kähler. [**Proof:**]{} If ${\operatorname{Alb}}(M)$ is smooth, then ${\operatorname{Alb}}$ is an immersion, and pullback of the Kähler form ${\operatorname{Alb}}^*\omega$ is the Kähler form on $M$. Otherwise, we can desingularize the morphism ${\operatorname{Alb}}$ to obtain the Kähler metric on some manifold $\tilde{M}$ bimeromorphic to $M$ ($M$ is then a manifold in the Fujiki class C). On the other hand, $M$ admits an SKT structure (\[SKT\]). From the theorem of Chiose ([@Chiose]) it follows that $M$ is Kähler. [ ]{}It would be interesting to know what one can extract from the Albanese map if ${\operatorname{dim}}{\operatorname{Alb}}(M)=1$ or $2$. For example, if ${\operatorname{Alb}}(M)$ is a smooth curve $C$, fibers of ${\operatorname{Alb}}(M)$ are Hermitian symplectic (and therefore Kähler) surfaces, and the pullback of the volume form ${\operatorname{Alb}}^* {\operatorname{Vol}}_C$ is a closed, non-exact $(1,1)$-form on $M$. By $dd^c$-lemma for $(1,1)$-forms and \[holoforms\], it could not be cohomologous to a form of type $(2,0)+(0,2)$. By a theorem of Michelsohn ([@Michelsohn]), in that situation there exists a [*balanced*]{} metric on $M$, that is, a Hermitian form $\omega$ such that $d\omega^{{\operatorname{dim}}M - 1}=0$. Actually, the smoothness of $C$ is not necessary, because a manifold bimeromorphic to a balanced manifold is balanced itself ([@AB]). [100]{} L. Alessandrini, G. Bassanelli, [*Modifcations of compact balanced manifolds*]{}, In: C.R. Acad. Sci. Paris Math. [**320**]{} (1995), 1517–1522 Nicole Berline, Ezra Getzler, and Michèle Vergne, [*Heat Kernels and Dirac Operators*]{}, Grundlehren Math. Wiss., vol. 298, Springer-Verlag, New York, 1992. Brylinski, Jean-Luc; [*A differential complex for Poisson manifolds*]{}, In: J. Differential Geom. Volume 28, Number 1 (1988), 93–114. Gil R. Cavalcanti, [*Hodge theory and deformations of SKT manifolds*]{}, arXiv:1203.0493 Chiose, I.; [*Obstructions to the existence of Kähler structures on compact complex manifolds*]{}, In: Proc. of the Amer. Math. Soc., [**142**]{} (2014), no. 10, 3561–3568. John B. Conway, [*A course in Functional Analisys*]{}, Graduate Texts in Mathematics 96, Springer 1990. Deligne, Pierre; Griffiths, Phillip; Morgan, John; Sullivan, Dennis; [*Real homotopy theory of Kähler manifolds*]{}, In: Invent. Math. [**29**]{} (1975), no. 3, 245–274. Enrietti, Nicola; Fino, Anna; Vezzoni, Luigi; [*Tamed symplectic forms and SKT metrics,*]{} arxiv:1002.3099. Fino, Anna; Kasuya, Hisashi; Vezzoni, Luigi; [*Tamed complex structures on solvmanifolds*]{}, arxiv:1302.5569. Gauduchon, Paul; [*La 1-forme de torsion d’une variété hermitienne compacte*]{}, Math. Ann. 267, 495-518 (1984). D. Gilbarg,N. S. Trudinger, [*Elliptic Partial Differential Equations of Second Order,*]{} Springer-Verlag, 1983 Griffiths, Phillip; Harris, Joe; [*Principles of Algebraic Geometry*]{}, Wiley-Interscience, New York, 1978. Harvey, Reese; Lawson, H. Blaine; [*An intrinsic characterization of Kähler manifolds*]{}, Invent. math. [**74**]{}, 169–198 (1983). A. Lamari. *Courants kählériens et surfaces compactes* (French) \[Kähler currents and compact surfaces\]. Ann. Inst. Fourier (Grenoble) **49** (1999), no. 1, 263–285. Kefeng Liu, Xiaokui Yang; *Geometry of Hermitian manifolds,* arXiv:1011.0207 Michelsohn, M. L., [*On the existence of special metrics in complex geometry.*]{} Acta Math. 149 (1982), no. 3-4, 261–295. Peternell, Th., [*Algebraicity criteria for compact complex manifolds.*]{} Math. Ann. 275 (1986), no. 4, 653-672. Streets, Jeffrey; Tian, Gang; [*Symplectic curvature flow*]{}, arXiv:1012.2104. Streets, Jeffrey; Tian, Gang [*A parabolic flow of pluriclosed metrics*]{}, Int. Math. Res. Not. IMRN 2010, no. 16, 3101-3133. Voisim, Claire; [*Hodge Theory and Complex Algebraic Geometry I: Volume 1*]{}, Cambridge Studies in Advanced Mathematics. Tseng, Li-Sheng; Yau, Shing-Tung; [*Cohomology and Hodge Theory on Symplectic Manifolds: I*]{}, arXiv:0909.5418. [: [Laboratory of Algebraic Geometry,\ National Research University HSE,\ Department of Mathematics, 7 Vavilova Str. Moscow, Russia,]{}\ datel@mail.ru]{}.

--- abstract: 'We prove that almost every finite collection of matrices in $GL_d( \mathbb{R} )$ and $SL_d({{\mathbb R}})$ with positive entries is Diophantine. Next we restrict ourselves to the case $d=2$. A finite set of $SL_2({{\mathbb R}})$ matrices induces a (generalized) iterated function system on the projective line $\RP^1$. Assuming uniform hyperbolicity and the Diophantine property, we show that the dimension of the attractor equals the minimum of 1 and the critical exponent.' address: - 'Yuki Takahashi, Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel' - 'Boris Solomyak, Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel' author: - BORIS SOLOMYAK - YUKI TAKAHASHI bibliography: - 'bib.bib' title: 'Diophantine property of matrices and attractors of projective iterated function systems in $\RP^1$' --- Introduction and main results ============================= Diophantine property of matrices -------------------------------- Recently there has been interest in Diophantine properties in non-Abelian groups. The following is a variant of [@GJS1999 Definition 4.2]. Let $\Ak = \{A_i\}_{i\in \Lam}$ be a finite subset of a topological group $G$ equipped with a metric $\varrho$. Write $A_\bi = A_{i_1}\cdots A_{i_n}$ for $\bi = i_1\ldots i_n$. We say that the set $\Ak$ is [*Diophantine*]{} if there exists a constant $c>0$ such that for every $n\in \N$, we have $$\label{Dioph1} \bi,\bj\in \Lam^n,\ A_\bi\ne A_\bj \implies \varrho(A_\bi,A_\bj) > c^n.$$ The set $\Ak$ is [*strongly Diophantine*]{} if there exists $c>0$ such that for all $n\in \N$, $$\label{Dioph2} \bi,\bj\in \Lam^n,\ \bi\ne \bj \implies \varrho(A_\bi,A_\bj) > c^n.$$ Clearly, $\Ak$ is strongly Diophantine if and only if it is Diophantine and generates a free semigroup. Gamburd, Jacobson, and Sarnak [@GJS1999 Definition 4.2] gave a definition of a Diophantine set, which is equivalent to ours, except that they always consider symmetric sets (that is, $g\in \Ak\ \Rightarrow\ g^{-1}\in \Ak$). Diophantine-type questions in groups arise in connection with spectral gap estimates, see [@GJS1999; @Bourgain2014]. See [@ABRS2015; @ABRS2018] for a recent discussion of Diophantine properties in groups and related problems. In [@ABRS2018] a Lie group $G$ is called Diophantine, if almost every $k$ elements of $G$, chosen independently at random according to the Haar measure, together with their inverses, form a Diophantine set in $G$. Gamburd et al. [@GJS1999] conjectured that $SU_2({{\mathbb R}})$ is Diophantine. More generally, it is conjectured that semi-simple Lie groups are Diophantine. Kaloshin and Rodnianski [@KR2001] proved a weaker Diophantine-type property: for a.e. $(A,B) \in SO_3({{\mathbb R}})\times SO_3({{\mathbb R}})$, there exists $c>0$ such that for any $n{\geqslant}1$ and any two distinct words $W_1, W_2$ over the set $\Ak=\{A,B,A^{-1},B^{-1}\}$ of length $n$, $$\|W_1-W_2\|{\geqslant}c^{n^2}.$$ It is mentioned in [@KR2001] that their method is general, and applies to $SU_2({{\mathbb R}})$ as well, and also to $m$-tuples of matrices for any $m{\geqslant}2$. Next we state our first result. For any collection of linearly independent vectors $v_1,\ldots,v_{d}$ in ${{\mathbb R}}^{d}$ consider the simplicial cone $$\label{cone} \Sig=\Sig_{v_1,\ldots,v_{d}} = \{x_1 v_1 + \cdots + x_{d} v_{d}:\ x_1,\ldots,x_{d}{\geqslant}0\}.$$ If a matrix $A\in GL_{d}({{\mathbb R}})$ satisfies $$A({\Sig}{\smallsetminus}\{0\}) \subset \Sig^\circ,$$ we say that $\Sig$ is [*strictly invariant*]{} for $A$. Given a cone $\Sig=\Sig_{v_1,\ldots,v_{d}}$, denote by $\Xk_{\Sig,m}$ (respectively, $\Yk_{\Sig,m}$) the set of all $GL_{d}({{\mathbb R}})$ (respectively, $SL_{d}({{\mathbb R}})$) $m$-tuples of matrices for which $\Sig$ is strictly invariant. We consider $\Xk_{\Sig,m}$ as an open subset of ${{\mathbb R}}^{d^2m}$ and $\Yk_{\Sig,m}$ as a $(d^2-1)m$-dimensional manifold. \[main\_thm\] Let $\Sig=\Sig_{v_1,\ldots,v_{d}}$ be a simplicial cone in ${{\mathbb R}}^{d}$ and $m{\geqslant}2$. [(i)]{} For a.e. $\mathcal{A} \in \mathcal{X}_{\Sig, m}$, the $m$-tuple $\mathcal{A}$ is strongly Diophantine. In particular, a.e. $m$-tuple of positive $GL_{d}({{\mathbb R}})$ matrices is strongly Diophantine. [(ii)]{} For a.e. $\mathcal{A} \in \mathcal{Y}_{\Sig, m}$, the $m$-tuple $\mathcal{A}$ is strongly Diophantine. In particular, a.e. $m$-tuple of positive $SL_{d}({{\mathbb R}})$ matrices is strongly Diophantine. *1. Unfortunately, our results do no cover any example of a symmetric set, since the strict invariance property cannot hold for a matrix $A$ and $A^{-1}$ simultaneously.* 2\. Every $m$-tuple of matrices with algebraic entries is Diophantine (but not necessarily strongly Diophantine), see, e.g., [@GJS1999 Prop.4.3]. 3\. It is well-known that Diophantine numbers in ${{\mathbb R}}$ form a set of full measure, which is, however, meagre in Baire category sense (its complement contains a dense $G_\delta$ set). Baire category genericity of non-Diophantine $m$-tuples in $SU_2({{\mathbb R}})$ has been pointed out in [@GJS1999]. In $G=SL_d({{\mathbb R}})$ the situation is different, since there are, for example, open sets of $m$-tuples in $G\times G$ which satisfy (\[Dioph2\]). For instance, if ${{\mathbb R}}^d_+$ is mapped by $A,B$ into closed cones that are disjoint, except at the origin, then (\[Dioph2\]) holds for $\{A,B\}$. On the other hand, there are open sets in $(SL_d({{\mathbb R}}))^m$ in which non-Diophantine pairs are dense. For instance, the set of elliptic matrices in $SL_2({{\mathbb R}})$ is open, and a standard argument shows that a generic $m$-tuple that contains an elliptic matrix is not Diophantine. The scheme of the proof of Theorem \[main\_thm\] is as follows. We consider the induced action of the matrices on the projective space, and show that, given a non-degenerate family of $m$-tuples strictly preserving an open set, depending on a parameter real-analytically, for all parameters outside an exceptional set of zero Hausdorff dimension, the induced iterated function system (IFS) satisfies a version of the “exponential separation condition”. This property implies the strong Diophantine condition for the matrices. We then locally foliate the space of $m$-tuples of matrices and apply Fubini’s Theorem. The result on the zero-Hausdorff dimensional set of exceptions uses the notion of [*order-$k$ transversality*]{}, which is a modified version of that which appeared in the work of Hochman [@Hochman2014; @Hochman2015]. The strict open set preservation property is needed to ensure that the induced IFS is contracting (uniformly hyperbolic). Projective IFS and linear cocycles ---------------------------------- Let $\Ak = \{A_i\}_{i\in \Lam}$ be a finite collection of $SL_d({{\mathbb R}})$ matrices. The linear action of $SL_d({{\mathbb R}})$ on ${{\mathbb R}}^{d}$ induces an action on the projective space $\RP^{d-1}$, and thus $\Ak$ defines an IFS $\Phi_\Ak=\{\varphi_A\}_{A\in \Ak}$ on $\RP^{d-1}$, called a [*(real) projective IFS*]{}. Such IFS were studied by Barnsley and Vince [@BV2012], and by De Leo [@Leo2015a; @Leo2015b]. Following [@BV2012], we say that the IFS $\Phi_\Ak$ has an attractor $K$ if for every nonempty compact set $B$ in a neighborhood of $K$, we have $\lim_{k\to \infty}\Phi_\Ak^k(B) = K$ in the Hausdorff metric, where $\Phi_\Ak(B) = \bigcup_{A\in \Ak} \varphi_A(B)$. Let $\Ak = \{A_i\}_{i\in \Lam}$ be a finite collection of $GL_d({{\mathbb R}})$ matrices and $\Phi_\Ak$ the associated IFS on $\RP^{d-1}$. An alternative, but closely related viewpoint, is to consider the linear cocycle $A: \Lam^{{\mathbb Z}}\to SL_d({{\mathbb R}})$ over the shift on $\Lam^{{\mathbb Z}}$, defined by $A(\bi) = A_{i_1}$. Strict contractivity of the projective IFS turns out to be equivalent to [*uniform hyperbolicity*]{} of the cocycle [@BG2009]. Here we restrict ourselves to the case of $d=2$, which was investigated in great detail by Avila, Bochi, and Yoccoz [@ABY2010]. There is a natural identification between $[0, \pi)$ and the projective space $\RP^1$. Below we use this identification freely, and whenever necessary we view $[0, \pi)$ as $\mathbb{R} / \pi \mathbb{Z}$. For $A\in GL_2({{\mathbb R}})$ denote the action of $A$ on $[0, \pi)\cong \RP^1$ by the symbol $\varphi_A$. Denote by $d_{\Prob}$ the metric on $\RP^1$ induced from the identification with ${{\mathbb R}}/\pi{{\mathbb Z}}$. Below we work with $m$-tuples of $SL_2({{\mathbb R}})$-matrices, since the action of $GL_2({{\mathbb R}})$ factors through the $SL_2({{\mathbb R}})$ action in the obvious way, via $A\mapsto (\det A)^{-1}A$. A [*multicone*]{} is a proper nonempty open subset $U$ of $\RP^1$, having finitely many connected components with disjoint closures. In the following theorem we extracted the results relevant for us from [@ABY2010; @BV2012] (note that [@BV2012] considers real projective IFS of any dimension). \[th-unihyp\] Let $\Ak = \{A_i\}_{i\in \Lam}$ be a family of $SL_2({{\mathbb R}})$ matrices and let $\Phi_\Ak$ be the associated IFS on $\RP^1$. The following are equivalent: [(i)]{} the IFS $\Phi_\Ak$ has an attractor $K \ne \RP^1$; [(ii)]{} the associated linear cocycle over $\Lam^{{\mathbb Z}}$ is uniformly hyperbolic; [(iii)]{} there is a multicone $U$, such that $\Phi_\Ak(\ov{U}) \subset U$; [(iv)]{} there is nonempty open set $V\subset \RP^1$ such that $\Phi_\Ak$ is contractive on $\ov{V}$, with respect to a metric equivalent to $d_{\Prob}$. Following [@ABY2010], we will call a multicone satisfying $\Phi_\Ak(\ov{U})\subset U$, a [*strictly invariant multicone*]{} for the family of matrices and for the IFS. There are examples, see [@ABY2010], which show that one may need a multicone having $k$ components, for any given $k{\geqslant}2$, even for a pair of $SL_2({{\mathbb R}})$ matrices $\{A_1,A_2\}$. Our next result concerns the dimension of the attractor. Following De Leo [@Leo2015a], consider the $\zeta$-function $$\zeta_\Ak(t) = \sum_{n{\geqslant}1} \sum_{\bi \in \Lam^n} \|A_\bi\|^{-t},$$ and define the [*critical exponent*]{} of $\Ak$ by \[critic\] s\_= \_[t0]{} {t: \_(t) = }. \[thm-attr\] Let $\Ak= \{A_i\}_{i\in \Lam}$ be a finite set of $SL_2({{\mathbb R}})$ matrices which has a strictly invariant multicone (or satisfies any of the equivalent conditions from Theorem \[th-unihyp\]), and let $K$ be the attractor of the associated IFS $\Phi_\Ak$ on $\RP^1$. Assume that at least two of the maps $\varphi_{A_i}$ have distinct attracting fixed points. If $\Ak$ is strongly Diophantine, then $\dim_H(K) = \min\{1,\half s_\Ak\}$, where $s_\Ak$ is the critical exponent (\[critic\]). In the special case when the IFS $\Phi_\Ak$ satisfies the Open Set Condition, this result is due to De Leo [@Leo2015a Th.4]. Recall that the strong Diophantine condition holds, in particular, when $\Ak$ generates a free semigroup and all the entries of $A_i$ are algebraic. \[rem-DeLeo\] [*It is further shown in [@Leo2015a] that for $\Ak$ hyperbolic (and in some parabolic cases), $$s_\Ak = \lim_{r\to \infty} \frac{N_{\Ak}(r)}{\log r},$$ where $N_{\Ak}(r)$ is the number of elements of norm ${\leqslant}r$ of the semigroup generated by $\Ak$. An analogy is pointed out with the classical results on Kleinian and Fuchsian groups, see, e.g., [@Sullivan1984].* ]{} Let $\Phi=\Phi_\Ak$. An alternative way to express the dimension, and one we actually use in the proof, is in terms of [*Bowen’s pressure formula*]{} \[Bowen1\] P\_(s) = 0,where $P_\Phi(\cdot)$ is the pressure function associated with the IFS $\Phi$. Throughout the paper we use the notation $$\varphi_\bi = \varphi_{i_1}\ldots \varphi_{i_n}.$$ The pressure is defined by \[Bowen0\] P\_(t) = \_[n]{} \_[\^n]{} \_’\^t, where $\|\cdot\|$ is the supremum norm on $\ov{U}$. As will be clear from the Bounded Distortion Property, the definition of $P_\Phi(t)$ does not depend on the choice of strictly invariant multicone $U$, and moreover, \[critic2\] 2s = s\_. It is a classical result, going back to Bowen [@Bowen1979] and Ruelle [@Ruelle1982], see also [@Falconer_Tech], that if $\{\varphi_i\}_{i\in \Lam}$ is a hyperbolic IFS on ${{\mathbb R}}$ of smoothness $C^{1+{{\varepsilon}}}$, satisfying the Open Set Condition, then the dimension of the attractor $K$ is given by the Bowen’s equation. In the case that the maps $\varphi_i$ are affine, $s>0$ is the unique solution of $$\sum_{i\in \Lam} r_i^s = 1,$$ where $r_i \in (0,1)$ is the contraction ratio of $\varphi_i$. For an IFS with overlaps this is not necessarily true. In [@SSU1], Simon, Solomyak, and Urbański showed that for a one-parameter family of nonlinear IFS with overlaps (hyperbolic and some parabolic) satisfying the [*order-1 transversality condition*]{}, for Lebesgue-a.e. parameter the dimension of the attractor is given by $$\label{dimul} \dim_H (K) = \min\{1,s\},$$ where $s$ is from (\[Bowen1\]) and the pressure is given by (\[Bowen0\]). \[def-sep\] Let ${{\mathcal F}}= \{f_i\}_{i\in \Lam}$ be an IFS on a metric space $(X,\varrho)$, that is, $f_i:X\to X$. We say that ${{\mathcal F}}$ satisfies the *exponential separation condition* on a set $J\subset \Xk$ if there exists $c > 0$ such that for all $n\in \N$ we have $$\label{exp_sep} \sup_{x \in J} \varrho(f_\bi(x), f_\bj(x) ) > c^{n},\ \ \mbox{for all}\ \bi,\bj\in \Lam^{n}\ \ \mbox{with}\ \ i_1\ne j_1\ \ \mbox{and}\ \ f_\bi\not \equiv f_\bj.$$ If, in addition, the semigroup generated by ${{\mathcal F}}$ is free, that is, $f_\bi \equiv f_\bj \ \Longleftrightarrow\ \bi=\bj$, we say that ${{\mathcal F}}$ satisfies the [*strong exponential separation condition*]{}. If these properties hold for infinitely many $n$, then we say that ${{\mathcal F}}$ satisfies the [*(strong) exponential separation condition on $J$ along a subsequence*]{}. It is rather straightforward to show that the (strong) Diophantine condition for an $m$-tuple in $SL_2({{\mathbb R}})$ matrices is equivalent to the (strong) exponential separation condition for the associated projective IFS (see Lemma \[lem-Dioph\] below). In [@Hochman2014 Cor. 1.2], Hochman proved (\[dimul\]) for an affine IFS $\mathcal{F} = \{ f_i \}_{i \in \Lambda}$ satisfying the exponential separation condition on $J=\{0\}$ along a subsequence. Thus our Theorem \[thm-attr\] is, in a sense, a generalization of Hochman’s result to the case of contractive projective IFS. [*In fact, Hochman [@Hochman2014] used the condition (\[exp\_sep\]) without the requirement $i_1\ne j_1$. However, for an IFS $\{f_i\}_{i\in \Lam}$ on an interval $J\subset {{\mathbb R}}$, such that $$\inf_{x\in J, i\in \Lam}|f_i'(x)|{\geqslant}r_{\min}>0,$$ requiring $i_1\ne j_1$ in (\[exp\_sep\]) does not weaken the exponential separation condition — it only affects the constant $c$. This follows from the estimate $$|f_\bi(x) -f_\bj(x)| =|f_{(\bi\wedge \bj)\bu}(x) - f_{(\bi\wedge \bj)\bv}(x) |{\geqslant}r_{\min}^n |f_\bu(x) - f_\bv(x)|, \ \ \bi,\bj\in \Lam^n,$$ where $\bi\wedge \bj$ is the common initial segment of $\bi$ and $\bj$, so that $u_1 \ne v_1$.* ]{} IFS of linear fractional transformations ---------------------------------------- It is well-known that the action of $GL_2({{\mathbb R}})$ on $\RP^1$ can be expressed in terms of linear fractional transformations. For $$A = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \in GL_2(\mathbb{R}),$$ let $f_A(x) = (ax + b)/(cx + d)$, and define $\psi : [0, \pi) \to \mathbb{R^*}$ by $\psi (\theta) = \cos\theta/\sin\theta$, where ${{\mathbb R}}^* = {{\mathbb R}}\cup \{\infty\}$. It is easy to see that the following diagram commutes: \(m) \[matrix of math nodes, row sep=3em, column sep=4em, minimum width=2em\] [ \[0, ) & [\[0, )]{}\ &\ ]{}; (m-1-1) edge node \[left\] [$\psi$]{} (m-2-1) edge node \[above\] [$\varphi_A$]{} (m-1-2) (m-2-1.east|-m-2-2) edge node \[above\] [$f_A$]{} (m-2-2) (m-1-2) edge node \[left\] [$\psi$]{} (m-2-2); Observe that $\psi$ is smooth, and on any compact subset of $(0,\pi)$ the derivatives of $\psi$ and $\psi^{-1}$ are bounded. The following is then an immediate corollary of Theorem \[thm-attr\]. \[cor-ifs\] Let $\mathcal{F} = \{ f_i \}_{ i \in \Lambda }$ be a finite collection of linear fractional transformations with real coefficients. Assume that there exists $U\subset {{\mathbb R}}$, a finite union of bounded open intervals with disjoint closures, such that $f_i( \overline{U} ) \subset U$ for all $i \in \Lambda$. If $\mathcal{F}$ satisfies the strong exponential separation condition on $\ov{U}$, then we have $ \dim_H (K) = \min\{ 1, s \}, $ where $s > 0$ is the unique zero of the pressure function $P_{{\mathcal F}}$. Furstenberg measure ------------------- Let $\mathcal{A} = \{ A_i \}_{i \in \Lambda}$ be a finite collection of $SL_2(\mathbb{R})$ matrices, and let $p = ( p_i )_{i \in \Lambda}$ be a probability vector. Assume that $p_i > 0$ for all $i \in \Lambda$ (we always assume this for any probability vector). We consider the finitely supported probability measure $\mu$ on $SL_2({{\mathbb R}})$: \[def-mu\] = \_[i]{} p\_i \_[A\_i]{}. Our standing assumption is that $\mathcal{A}$ generates an unbounded and totally irreducible subgroup (i.e., does not preserve any finite set in $\RP^1$). Then there exists a unique probability measure $\nu$ on $\RP^1$ satisfying $\mu\cdot\nu = \nu$, that is, $$\label{Furst1} \nu = \sum_{i \in \Lambda} p_i A_i \nu,$$ where $A_i \nu$ is the push-forward of $\nu$ under the action of $A_i$, see [@Furstenberg1963]. The measure $\nu$ is the [*stationary measure*]{}, or the [*Furstenberg measure*]{}, for the random matrix product $A_{i_n}\cdots A_{i_1}$ where the matrices are chosen i.i.d. from $\Ak$ according to the probability vector $p$. The properties of the Furstenberg measure for $SL_2({{\mathbb R}})$ random matrix products, such as absolute continuity, singularity, Hausdorff dimension, etc., were studied by many authors, including [@Ledrappier1983; @BL1985]. In [@Pincus1994; @Lyons2000; @SSU2] this investigation was linked with the study of IFS consisting of linear fractional transformations. The reader is referred to [@HS2017] for a discussion of more recent applications. We will recall the main result of [@HS2017], since it will be the main tool in proving Theorem \[thm-attr\]. Let $\chi_{\mathcal{A}, p}$ be the *Lyapunov exponent*, which is the almost sure value of the limit $$\label{Lyap1} \lim_{n \to \infty} \frac{1}{n} \log \| A_{i_1 \cdots i_n } \|,$$ where $i_1, i_2, \cdots \in \Lambda$ is a sequence chosen randomly according to the probability vector $p = ( p_i )_{ i \in \Lambda}$. The Lyapunov exponent is usually defined as the almost sure value of the limit $$\label{Lyap2} \lim_{n \to \infty} \frac{1}{n} \log \| A_{i_n \cdots i_1 } \|,$$ but it is easy to see that (\[Lyap1\]) and (\[Lyap2\]) define the same value (e.g., by Egorov’s Theorem). Under the standing assumptions, the limit exists almost surely and is positive [@Furstenberg1963]. The Hausdorff dimension of a measure $\nu$ is defined by $$\dim_H(\nu) = \inf\{\dim_H (E):\ \nu(E^c)=0\}.$$ For a probability vector $p = ( p_i )_{i \in \Lambda}$, we denote the *entropy* $H(p)$ by $$H(p) = - \sum_{i \in \Lambda} p_i \log p_i.$$ \[thmHS\] Let $\mathcal{A} = \{ A_i \}_{i \in \Lambda}$ be a finite collection of $SL_2(\mathbb{R})$ matrices. Assume that $\mathcal{A}$ is strongly Diophantine and generates an unbounded and totally irreducible subgroup. Let $p = ( p_i )_{i \in \Lambda}$ be a probability vector, and let $\nu$ be the associated Furstenberg measure. Then we have $$\label{eq-Fursten} \dim_H (\nu) = \min \Big\{ 1, \frac{H( p )}{ 2 \chi_{ \mathcal{A}, p } } \Big\}.$$ Theorem \[main\_thm\] implies, in particular, that the dimension formula (\[eq-Fursten\]) holds for the Furstenberg measure associated with a.e. finite family of positive matrices (independent of the probability vector). Next we address the question: what is the Hausdorff dimension of the support of the Furstenberg measure? Sometimes, the support is all of $\RP^1$, in which case the answer is trivially one. The definition (\[Furst1\]) implies that the support is invariant under the IFS $\Phi$ induced by $\Ak$. Thus, Theorem \[thm-attr\] has the following immediate corollary: Let $\Ak= \{A_i\}_{i\in \Lam}$ be a Diophantine set of $SL_2({{\mathbb R}})$ matrices which has a strictly invariant multicone, $\mu$ a finitely supported measure defined by (\[def-mu\]), and $\nu$ the associated Furstenberg measure. Then $\dim_H({\mathrm{supp} \,}\nu) = \min\{1,\half s_\Ak\}$, where $s_\Ak$ is the critical exponent of $\Ak$. Denote by $\Hk_m$ the set of $m$-tuples in $SL_2({{\mathbb R}})$ which have a strictly invariant multicone. Avila (see [@Yoccoz2004 Prop.6]) proved that the interior of the complement of $\Hk_m$ in $(SL_2({{\mathbb R}}))^m$ is ${{\mathcal E}}_m$, where ${{\mathcal E}}_m$ is the set of $m$-tuples which generate a semigroup containing an elliptic matrix. Observe that if an elliptic matrix is conjugate to an irrational rotation, then certainly the invariant set (support of the Furstenberg measure) is all of $\RP^1$. On the other hand, if it is conjugate to a rational rotation, then the semigroup generated by $\Ak$ contains the identity and the strong Diophantine property fails. We expect that our methods can be extended to cover strongly Diophantine families on the boundary of $\Hk_m$, which include parabolic systems. Structure of the paper ---------------------- The rest of the paper is organized as follows. In the next section we prove Theorem \[main\_thm\]. In Section 3 we consider projective IFS and prove Theorem \[thm-attr\]. Finally, in Section 4 we include proofs of some standard technical results for the reader’s convenience. Diophantine property of $GL_{d+1}(\mathbb{R})$ and $SL_{d+1}(\mathbb{R})$ matrices ================================================================================== For notational reasons it is convenient to consider $GL_{d+1}({{\mathbb R}})$ instead of $GL_d({{\mathbb R}})$. $GL_{d+1}({{\mathbb R}})$ actions --------------------------------- Let $A\in GL_{d+1}({{\mathbb R}})$ be a matrix that strictly preserves a cone $\Sig=\Sig_{v_1,\ldots,v_{d+1}}\subset {{\mathbb R}}^{d+1}$. Without loss of generality, we can assume that $\Sig{\smallsetminus}\{0\}$ is contained in the halfspace $\{x\in {{\mathbb R}}^{d+1}: x_{d+1}>0\}$. It is convenient to represent the induced action of $A$ on $\RP^{d}$ on the affine hyperplane $\{x\in {{\mathbb R}}^{d+1}: x_{d+1}=1\}$, and consider the corresponding action on ${{\mathbb R}}^{d}$. To be precise, for $x= (x_1,\ldots,x_{d})\in {{\mathbb R}}^{d}$, we consider $(x,1) = (x_1,\ldots,x_{d},1)\in {{\mathbb R}}^{d+1}$ and let $$f_A(x) = {\rm P}_{d}\Bigl(\frac{A(x,1)}{A(x,1)_{d+1}}\Bigr),\ \ \mbox{when}\ \ A(x,1)_{d+1}\ne 0,$$ where ${\rm P}_{d}$ is the projection onto the first $d$ coordinates. The components of $f_A$ are rational functions, which are, of course, real-analytic on their domain. Consider $$\ov{V} := {\rm P}_{d}(\Sig \cap \{x\in {{\mathbb R}}^{d+1}: x_{d+1}=1\}).$$ By assumption, $f_A$ is well-defined on $\ov{V}$, and we have $f_A(\ov{V})\subset V$. We will also consider the action of $A$ on the unit sphere, given by $$\varphi_A(x):=A\cdot x =\frac{Ax}{\|x\|},$$ for a unit vector $x\in \Sp^{d}$. Consider also $\ov{U}$, the intersection of $\Sig$ with the upper hemisphere. We have $\varphi_A(\ov{U}) \subset U$. Lines through the origin provide a 1-to-1 correspondence between $\ov{U}$ and $\ov{V}$, which is bi-Lipschitz in view of the assumption $\Sig {\smallsetminus}\{0\} \subset \{x\in {{\mathbb R}}^{d+1}: x_{d+1}>0\}$. It is well-known [@Birk1957] (see also [@BV2012 Section 9]) that strictly preserving a cone implies that $\varphi_A$ is a strict contraction in the Hilbert metric on $\ov{U}$, which is by-Lipschitz with the round metric. We thus obtain the following: \[lem-Hilbert\] Suppose that the finite family $\Ak = \{A_i\}_{i\in \Lam} \subset GL_d({{\mathbb R}})$ strictly preserves a simplicial cone $\Sig =\Sig_{v_1,\ldots,v_{d+1}}\subset \{x\in {{\mathbb R}}^{d+1}: x_{d+1}>0\}\cup \{0\}$. Then the associated IFS ${{\mathcal F}}_\Ak = \{f_A\}_{A\in \Ak}$ is real-analytic and uniformly hyperbolic on $\ov{V}\subset {{\mathbb R}}^{d}$, in the sense that there exist $C>0$ and $\gam\in (0,1)$ such that $$\max_{x\in \ov{V}} \|f'_\bi(x)\| {\leqslant}C\gam^n,\ \ \mbox{for all}\ \bi\in \Lam^n,$$ where $f_\bi = f_{A_{i_1}}\circ \cdots f_{A_{i_n}}$ and $\|f_\bi'(x)\|$ is the operator norm of the differential at the point $x$. From exponential separation to the Diophantine property ------------------------------------------------------- Recall the strong exponential separation condition (Definition \[def-sep\]). \[prop-Dioph2\] Let $\Ak$ be a finite family of $GL_{d+1}({{\mathbb R}})$ matrices, and let $\Phi_\Ak$ be the induced IFS on $\Sp^d$. If $\Phi_\Ak$ satisfies the strong exponential separation on a nonempty set, then $\Ak$ is strongly Diophantine. Let $C_1 = \max_{i\in \Lam}\{1,\|A_i\|\}$ and $C_2 = \max_{i\in \Lam} \{1,\|A_i^{-1}\|\}$. Suppose that $\bi\ne \bj$ in $\Lam^n$. Let us write $$\bi = (\bi\wedge \bj) \bu,\ \ \bj = (\bi\wedge \bj) \bv,$$ where $\bi\wedge \bj$ is the common initial segment of $\bi$ and $\bj$, so that $\bu = u_1\ldots u_k,\ \bv = v_1\ldots v_k$ for some $k{\leqslant}n$, with $u_1 \ne v_1$. We have $$\label{tiu1} \|A_\bi- A_\bj\| {\geqslant}\|A_{\bi\wedge\bj}^{-1}\|^{-1} \|A_\bu - A_\bv\| {\geqslant}C_2^{-n} \|A_\bu - A_\bv\|.$$ \[lem-claim\] For any $A, B \in GL_{d+1}(\mathbb{R})$ and any unit vector $x\in {{\mathbb R}}^{d+1}$, we have $$\| A \cdot x - B \cdot x \| {\leqslant}\| A^{-1} \| \bigl( 1 + \| B \| \|B^{-1}\|\bigr)\cdot \| A - B \|.$$ We have $$\begin{aligned} \| A \cdot x - B \cdot x \| &= \left\| \frac{Ax}{\| Ax \|} - \frac{Bx}{\| Bx \|} \right\| \\ &{\leqslant}\left\| \frac{Ax}{ \| Ax \|} - \frac{ Bx }{ \| Ax \| } \right\| + \left\| \frac{Bx}{ \| Ax \| } - \frac{ Bx }{ \| Bx \| } \right\| =: R_1 + R_2. \end{aligned}$$ Since $$1 = \| A^{-1} (Ax) \| {\leqslant}\| A^{-1} \| \| Ax \|,$$ we have $\| Ax \|^{-1} {\leqslant}\| A ^{-1}\|$. Therefore, $$R_1 {\leqslant}\| A - B \| \cdot \| A^{-1} \|.$$ Similarly, $$\begin{aligned} R_2 &{\leqslant}\| B \| \cdot \big| \| Ax \| - \| Bx \| \big| \cdot \| Ax \|^{-1} \| Bx \|^{-1} \\ &{\leqslant}\| B \| \cdot \| A - B \| \cdot \| A^{-1} \| \| B^{-1} \|, \\ \end{aligned}$$ and the desired estimate follows. Applying the lemma to $A_\bu$ and $A_\bv$ yields, in view of $\|A_\bw\|{\leqslant}C_1^n$, $\|A_\bw^{-1}\|{\leqslant}C_2^n$ for any $\bw \in \Lam^k$, $k{\leqslant}n$: $$\label{eq-tiu2} \|A_\bu - A_\bv\| {\geqslant}2^{-n} C_1^{-n} C_2^{-2n} \|A_\bu\cdot x- A_\bv \cdot x\|.$$ Now we continue with the proof of the lemma. By assumption, $\Phi_\Ak$ satisfies the exponential separation condition on a nonempty set. Let $c\in (0,1)$ be the constant from the definition (\[exp\_sep\]). It follows that there exists $x\in \Sp^d$ such that $$\|A_\bu\cdot x - A_\bv\cdot x\|{\geqslant}c^k{\geqslant}c^n.$$ Combining this inequality with (\[eq-tiu2\]) and (\[tiu1\]) yields $$\|A_\bi - A_\bj\|{\geqslant}2^{-n} C_1^{-n} C_2^{-3n}c^n,$$ confirming the strong Diophantine property. Dimension of exceptions for one-parameter real-analytic families ---------------------------------------------------------------- We consider a one-parameter family of real-analytic IFS on a compact subset of ${{\mathbb R}}^d$, and show that under some mild assumptions it satisfies the exponential separation condition outside of a Hausdorff dimension zero set. This section is based on [@Hochman2014 Section 5.4] and [@Hochman2015 Section 6.6], but we had to make a substantial number of modifications in the definitions and proofs. Let $\mathcal{J}$ be a compact interval in ${{\mathbb R}}$ and $V$ a bounded open set in ${{\mathbb R}}^d$. Suppose that for each $i \in \Lambda$ we are given a real-analytic function $$f_i:\,\ov{V}\times \Jk\to V.$$ This means that it is real-analytic on some neighborhood of $\ov{V}\times \Jk$. We will sometimes write this function as $$f_{i,t}(x) = f_i(x,t),\ \ x\in \ov{V},\ \ t\in \Jk.$$ Denote $\mathcal{F}_t = \{ f_{i, t} \}_{ i \in \Lambda }$. This is a real-analytic IFS on $\ov{V}$, depending on the parameter $t\in \Jk$ real-analytically. For $\bi = i_1\ldots i_n$ we write $f_{\bi,t} = f_{i_1,t} \circ \cdots \circ f_{i_n,t}$. Further, assume that this family of IFS is uniformly hyperbolic in the following sense: there exist $C>0$ and $0 < \gam<1$, such that \[gluk1\] f’\_[,t]{}(x) C\^n,     \^n,  x, t. Here in the left-hand side is the norm of differential with respect to $x\in {{\mathbb R}}^d$. Fix $x_0 \in V$. For any finite sequence $\bi \in\Lambda^n$ we define $$F_\bi(t) = f_{\bi,t}(x_0).$$ Of course, this depends on $x_0$, but we suppress it from notation. For $\bi\in \Lam^\N$ we have $$\label{conver} \Pi_t(\bi) = F_\bi(t) := \lim_{n\to \infty} F_{\bi|_n}(t),$$ where $\Pi_t:\Lam^\N\to {{\mathbb R}}^d$ is the natural projection corresponding to the IFS ${{\mathcal F}}_t$ and $\bi|_n = i_1\ldots i_n$. Notice that this limit is well-defined, independent of $x_0$, and is uniform in $t\in \Jk$, by uniform hyperbolicity (\[gluk1\]). \[lem-analytic\] The function $F_\bi(\cdot)$ is real-analytic on $\Jk$, for any $\bi\in \Lam^\N$. Moreover, $F_{\bi|_n}(\cdot) \to F_\bi(\cdot)$ uniformly on $\Jk$ for all $\bi\in \Lam^\N$, together with derivatives of all orders. By assumption, for every $\bi \in \Lam^n$, the function $F_\bi$ extends to a holomorphic function in a complex neighborhood of $\Jk$, and we are going to prove that for all $\bi \in \Lam^\N$ the sequence $F_{\bi|_n}$ converges to $F_\bi$ on a sufficiently small neighborhood uniformly. In order to achieve this, note that since $f_i(x,t):\,\ov{V}\times \Jk \to V$ is real-analytic, it can be extended to a holomorphic (complex-analytic) function $\wtil{f}_i(z,\tau)$, defined on a neighborhood of $\ov{V}\times \Jk$ in ${{\mathbb C}}^d\times {{\mathbb C}}$. Denote by $[\ov{V}]^\delta$ the $\delta$-neighborhood of $\ov{V}$ in ${{\mathbb C}}^d$ and let $\wtil{f}_{i,\tau} = \wtil{f}_i(\cdot,\tau)$. Choose $\ell \in\N$ so that $C\gam^\ell < 1/2$. Then $\|f'_{\bi,t}(x)\|<1/2$ for $\bi\in \Lam^\ell$, and $x\in \ov{V}$. By continuity, there exists $\delta>0$ such that $\wtil{f}_{\bi,t}$, with $\bi\in \Lam^\ell$, is holomorphic on $[\ov{V}]^\delta$ and $$\label{niki1} \|{\wtil{f}_{\bi,t}}'(z)\|< 1/2,\ \mbox{for all}\ \bi\in \Lam^\ell,\ z\in [\ov{V}]^\delta,\ t\in \Jk.$$ Here in the left-hand side is the norm of the differential with respect to $z\in {{\mathbb C}}^d$. Thus, each $\wtil{f}_{\bi,t}$, with $\bi\in \Lam^\ell$, is a strict contraction on $[\ov{V}]^\delta$, and since $\wtil{f}_{\bi,t}({\ov{V}}) = f_{\bi,t}(\ov{V})\subset V$, we obtain that $[\ov{V}]^\delta$ is mapped into its interior by $\wtil{f}_{\bi,t}$, for $t\in \Jk$. Then the same must be true for all $\tau$ in a sufficiently small complex neighborhood of $\Jk$, which we denote by $\Ok$. We can find a constant $L>0$ such that $$\label{niki2} \|\wtil{f}_{\bj,\tau}'(z)\|{\leqslant}L,\ \mbox{for all}\ \bj \ \mbox{such that}\ \ |\bj| {\leqslant}\ell-1,\ z\in [\ov{V}]^\delta,\ \tau\in \Ok,$$ since there are finitely many holomorphic functions involved. Now, it follows from (\[niki1\]) and (\[niki2\]) that the function $\wtil{f}_{\bj,t}$, for [*all*]{} $\bj\in \bigcup_{n=1}^\infty \Lam^n$ and $t\in \Jk$, is well-defined and holomorphic in ${{\mathcal W}}:= [\ov{V}]^{\delta/L}$, and moreover, it maps ${{\mathcal W}}$ into $[V]^\delta$. In addition, $\wtil{{{\mathcal F}}}^\ell_\tau = \{\wtil{f}_{\bi,\tau}\}_{\bi \in \Lam^\ell}$ is a strictly contracting IFS on ${{\mathcal W}}$, depending on $\tau\in \Ok$ holomorphically. It follows that the finite iterates $\wtil{f}_{\bi|_{n},\tau}$ converge to $\Pi_\tau(\bi)$, the natural projection for $\wtil{{{\mathcal F}}}_\tau$, as $n\to \infty$, uniformly for $\tau\in \Ok$. The uniform limit of holomorphic functions in an open set in ${{\mathbb C}}$ is holomorphic, and since $F_\bi(t) = \Pi_t(\bi)$ is the restriction of a holomorphic map to an interval on the real line, it is real-analytic. The uniform convergence of holomorphic functions implies uniform convergence of their derivatives as well. Next, for $\bi,\bj \in \bigcup_{n=1}^\infty \Lam^n \cup \Lam^{\N}$, let $$\Delta_{\bi, \bj}(t) = F_\bi(t) - F_\bj(t) \in {{\mathbb R}}^d.$$ For any ${{\varepsilon}}> 0$, let $$E_{{{\varepsilon}}} = \bigcap_{N=1}^{\infty} \bigcup_{n > N} \Big( \bigcup_{\bi, \bj \in \Lambda^n, i_1 \neq j_1} \, ( \Delta_{\bi, \bj} )^{-1} B_{{{\varepsilon}}^n} \Big)$$ and $$\label{def-E} E = \bigcap_{{{\varepsilon}}> 0} E_{{{\varepsilon}}},$$ where $B_{{{\varepsilon}}^n} = \{x\in {{\mathbb R}}^d:\ \|x\|{\leqslant}{{\varepsilon}}^n\}$. It is easy to see that if $t \notin E$ then $\mathcal{F}_t$ satisfies the strong exponential separation condition. [*In [@Hochman2014; @Hochman2015] Hochman considered the case where $\mathcal{F}_t$ is an affine IFS. He defined the sets $E'_{{{\varepsilon}}}$ and $E'$ as follows: $$E'_{{{\varepsilon}}} = \bigcup_{N=1}^{\infty} \bigcap_{n > N} \Big( \bigcup_{\bi, \bj \in \Lambda^n, \bi \neq \bj} ( \Delta_{\bi, \bj} )^{-1} B_{{{\varepsilon}}^n} \Big)$$ and $$\label{def-Etag} E' = \bigcap_{{{\varepsilon}}> 0} E'_{{{\varepsilon}}}.$$ If $t \notin E'$ then $\mathcal{F}_t$ satisfies the strong exponential separation condition [*along a subsequence*]{}.* ]{} For a family of IFS ${{\mathcal F}}_t,\ t\in \Jk$, as above, and for $\bi,\bj\in \Lam^\N$ let $\Delta_{\bi,\bj}(t) = F_\bi(t) - F_\bj(t)$. We say that the family is [*non-degenerate*]{} if $$\label{nondegen} \Delta_{\bi, \bj}(\cdot) \equiv 0 \iff \bi = \bj \ \ \mbox{for all $\bi, \bj \in \Lambda^{\mathbb{N}}$.}$$ We next prove the following: \[thm\_important\] Suppose that the family of IFS ${{\mathcal F}}_t,\ t\in \Jk$, is non-degenerate. Then the set $E$ from (\[def-E\]) has Hausdorff dimension zero, and therefore, ${{\mathcal F}}_t$ satisfies the strong exponential separation condition on $\Jk$, outside of a set of zero Hausdorff dimension. \[cor\_sep\] For a family of IFS ${{\mathcal F}}_t,\ t\in \Jk$, as above, assume that there exists $t_0 \in \mathcal{J}$ such that the sets $\{ f_{i, t_0}( \overline{V} ) \}_{i \in \Lambda}$ are pairwise disjoint. Then (\[nondegen\]) holds, and hence the set $E$ from (\[def-E\]) has Hausdorff dimension zero. Hochman [@Hochman2014; @Hochman2015] proved, for a non-degenerate family of affine IFS, with a real-analytic dependence on parameter, that the set $E'$ from (\[def-Etag\]) has packing dimension zero. For any smooth function $F : \mathcal{J} \to {{\mathbb R}}^d$, denote $F^{(p)}(t) = \frac{d^p}{dt^p} F(t)$. The family $\{ \mathcal{F}_t \}_{t \in \mathcal{J} }$ is said to be *transverse of order $k$* if there exists $c > 0$ such that for all $n \in \mathbb{N}$ and $\bi, \bj \in \Lambda^{n}$, with $i_1 \neq j_1$, we have $$\forall t \in \mathcal{J} \ \ \exists p \in \{ 0, \cdots, k \} \text{\, s.t. } \| \Delta^{(p)}_{\bi, \bj}(t) \| > c.$$ Here the norm $\|\cdot\|$ is simply the Euclidean norm in ${{\mathbb R}}^d$. [*The above definition is different from [@Hochman2014] and it simplifies the proof of Theorem \[thm\_important\].* ]{} \[delta\_prop\] Suppose that the non-degeneracy condition (\[nondegen\]) holds. Then $\{ \mathcal{F}_t \}_{t \in \mathcal{J}}$ is transverse of order $k$ for some $k \in \mathbb{N}$. Suppose that for all $k\in \N$ the family $\{ \mathcal{F}_t \}_{t \in \mathcal{J}}$ is not transverse of order $k$. Then by assumption, for $\{c_k\}$ with $c_k < 1/k$, we can choose $n(k)$, $\bi^{(k)}, \bj^{(k)} \in \Lambda^{n(k)}$ with $i^{(k)}_1 \neq j^{(k)}_1$ and a point $t_k \in \mathcal{J}$ such that $$\| \Delta^{(p)}_{ \bi^{(k)}, \bj^{(k)} } (t_k) \| < c_k$$ for $0 {\leqslant}p {\leqslant}k$. Passing to a subsequence $\{ k_{l} \}$, we can assume that $t_{k_{l}} \to t_0\in \Jk$, $\bi^{( k_{l} )} \to \bi \in \Lambda^{\mathbb{N}}$ and $\bj^{( k_{l} )} \to \bj \in \Lambda^{\mathbb{N}}$, with $i_1\ne j_1$. Arguing as in the proof of Lemma \[lem-analytic\], the complex extension of $\Delta_{\bi^{( k_{l} )}, \bj^{( k_{l} )}}$ converges to the complex extension of $\Delta_{\bi, \bj}$ uniformly on a complex neighborhood of $\Jk$, and hence the same holds for $p$-th derivatives. Thus for all $p {\geqslant}0$, we have $$\| \Delta^{(p)}_{\bi, \bj} (t_0) \| = \lim_{l \to \infty} \| \Delta^{(p)}_{ \bi^{(k_{l}),} \bj^{(k_{l})} } (t_{k_{l}}) \| = 0.$$ Since $\Delta_{\bi, \bj}$ is real-analytic, the vanishing of its derivatives implies $\Delta_{\bi, \bj} \equiv 0$ on $\Jk$, contradicting (\[nondegen\]), since $\bi\ne \bj$ by construction. For a $C^k$-smooth function $F : \mathcal{V} \to \mathbb{C}$, write $$\| F \|_{ \mathcal{V}, k } = \max_{ p \in \{0, \cdots, k\} } \sup_{t \in \mathcal{V}} | F^{(p)}(t) |,\ \ \|F\|_{\Vk} = \|F\|_{\Vk,0},$$ and similarly for vector-functions. \[key\_lem\] Let $k \in \mathbb{N}$ and let $F : \mathcal{J} \to \mathbb{R}$ be a $k$ times continuously differentiable function. Let $M = \| F \|_{\mathcal{J}, k}$, and let $0 < c < 1$ be such that for every $t \in \mathcal{J}$ there is $p \in \{ 0, \cdots, k\}$ with $| F^{(p)}(t) | > c$. Then there exists $C=C_{c,M,|J|}{\geqslant}1$ such that for every $0 < \rho < (c/2)^{2^k}$, the set $F^{-1} (-\rho, \rho) \cap \mathcal{J}$ can be covered by $C^k$ intervals of length ${\leqslant}2( \rho/c )^{1/2^k} $ each. \[lem-conc\] If $\{ \mathcal{F}_t \}_{t \in \mathcal{J} }$ is transverse of order $k {\geqslant}1$ on the compact interval $\mathcal{J}$, then the set $E$ from (\[def-E\]) has Hausdorff dimension zero. Extending the real-analytic functions to the complex plane, as in Lemma \[lem-analytic\], since $$\sup_n \sup_{\bi, \bj \in \Lambda^n, i_1 \neq j_1} \| \Delta_{\bi, \bj} \|_{\Ok} < \infty$$ on a neighborhood $\Ok$ of $\Jk$, and $\Delta_{\bi, \bj}(\cdot)$ is holomorphic on $\Ok$ for all $\bi, \bj \in \Lambda^n$, we have $$\label{normal} M:=\sup_{n} \sup_{\bi, \bj \in \Lambda^n, i_1 \neq j_1} \| \Delta_{\bi, \bj} \|_{ \mathcal{J} , k} < \infty.$$ Let $$\label{En} E_{{{\varepsilon}}, n} = \bigcup_{\bi, \bj \in \Lambda^n, i_1 \neq j_1} ( \Delta_{\bi, \bj} )^{-1}( B_{{{\varepsilon}}^n} ).$$ Then $$\label{E} E_{{{\varepsilon}}} = \bigcap_{N = 1}^{\infty} \bigcup_{n > N} E_{{{\varepsilon}}, n}.$$ Let $\bi, \bj \in \Lambda^n$, with $i_1\ne j_1$, and assume that $\|\Delta_{\bi,\bj}(t)\|< {{\varepsilon}}^n$. By Lemma \[key\_lem\] applied to a component of $\Delta_{\bi,\bj}$, for ${{\varepsilon}}$ sufficiently small, the set $$( \Delta_{\bi, \bj} )^{-1} ( B_{{{\varepsilon}}^n} )$$ may be covered by $C^k$ intervals of length ${\leqslant}2( {{\varepsilon}}^n \cdot c^{-1} )^{1/2^k}$. It follows that the set $E_{{{\varepsilon}},n}$ from (\[En\]) may be covered by $O( | \Lambda |^{2n} \cdot C^k )$ intervals of length ${\leqslant}( {{\varepsilon}}^n \cdot c^{-1} )^{1/2^k}$. Fix $s>0$ and write $\Hk^s$ for the $s$-dimensional Hausdorff measure. We obtain from (\[E\]) that $$\Hk^s(E_{{\varepsilon}}) {\leqslant}O(1)\cdot \sum_{n{\geqslant}1} |\Lam|^{2n} C^k {\bigl({{\varepsilon}}^n \cdot c^{-1}\bigr)}^{s/2^k} < \infty$$ for ${{\varepsilon}}$ sufficiently small. It follows that $\Hk^s(E)=0$. This is now immediate from Lemmas \[delta\_prop\] and \[lem-conc\]. Proof of Theorem \[main\_thm\] ------------------------------ The next lemma follows by an application of Fubini’s Theorem. \[Fubini\] Let $F \subset \mathbb{R}^n$ and let $v \in \mathbb{R}^n$ be a nonzero vector. Assume that for every $x_0 \in \mathbb{R}^n$, the set $\{ x_0 + t v : t \in \mathbb{R} \} \cap F$ has 1-dimensional Lebesgue measure $0$. Then the set $F$ has $n$-dimensional Lebesgue measure $0$. \(i) Let $\Sig=\Sig_{v_1,\ldots,v_{d+1}}$ be a simplicial cone in ${{\mathbb R}}^{d+1}$. Let $\Uk \subset \Xk_{\Sig,m}$ be a small open set in $(GL_{d+1}({{\mathbb R}}))^m$ of $m$-tuples of matrices for which $\Sig$ is strictly invariant. Choose vectors $w_i \in {{\mathbb R}}^{d+1} \ (i \in \Lambda)$, with distinct directions, in such a way that \[w-cond\] w\_i \_[ A\_i v\_1, …,A\_i v\_[d+1]{} ]{}     (A\_i)\_[i ]{} . This is possible when $\Uk$ is sufficiently small. Let $(A_i)_{i \in \Lambda} \in \Uk$, and for each $t {\geqslant}0$ and $i\in \Lam$ let $A_{i,t}$ be such that $$A_{i, t} v_j = A_i v_j + t w_i,\ j=1,\ldots,d+1.$$ Condition (\[w-cond\]) guarantees that $\{A_{i,t}v_j\}_{i=1}^{d+1}$ is linearly independent, and hence $A_{i,t}\in GL_{d+1}({{\mathbb R}})$ for all $t>0$. This is a consequence of the following elementary claim. [**Claim.**]{} [*Let $y_1,\ldots,y_{d+1}\in {{\mathbb R}}^{d+1}$ be linearly independent, and suppose that $w = \sum_{k=1}^{d+1} a_k y_k$ for some $a_k{\geqslant}0$. Then the family $\{y_1 + w,\ldots, y_{d+1} + w\}$ is linearly independent as well.*]{} We have $$\sum_{j=1}^{d+1} c_j \Bigl(y_j + \sum_{k=1}^{d+1} a_k y_k\Bigr) = 0\ \ \Longrightarrow \ \ \sum_{j=1}^{d+1} \Bigl(c_j + a_j \sum_{k=1}^{d+1} c_k \Bigr) y_j = 0,$$ hence $c_j + a_j \sum_{k=1}^{d+1} c_k = 0$ for all $j$. If $\sum_{k=1}^{d+1} c_k\ne 0$, we obtain a contradiction, in view of $a_j{\geqslant}0,\ j=1,\ldots, d+1$; thus $c_j=0,\ j=1,\ldots, d+1$, as claimed. Let $\Ak_t = \{A_{i,t}\}_{i\in \Lam}$ be the family of matrices defined above, for $t{\geqslant}0$, and let ${{\mathcal F}}_t={{\mathcal F}}_{\Ak_t}$ be the corresponding one-parameter family of IFS on the set $\ov{V} \subset {{\mathbb R}}^d$ obtained by projection of $\Sig \cap \{x\in {{\mathbb R}}^{d+1}: x_d=1\}$ onto ${{\mathbb R}}^d$. Notice that the cone $\Sig$ is strictly preserved by all $\Ak_t,\ t{\geqslant}0$, by construction, hence by Lemma \[lem-Hilbert\], these IFS are all uniformly hyperbolic. It is easy to see that the IFS and their dependence on $t$ is real-analytic, since the IFS are given by rational functions. Condition (\[gluk1\]) holds for $t\in [0,M]$, for any $M<\infty$, by uniform hyperbolicity and compactness. Finally, observe that, given ${{\varepsilon}}>0$, for $t$ sufficiently large, we have $$f_{i,t}(\ov{V}) \subset {\rm P}_d\bigl(\Sig_{{\varepsilon}}(w_i)\cap \{x_{d+1}=1\}\bigr),$$ where $\Sig_{{\varepsilon}}(w_i)$ is the cone of vectors ${{\varepsilon}}$-close to $w_i$ in direction. By construction, $w_i$ are all distinct, hence Corollary \[cor\_sep\] applies. We obtain that for all $t\in [0,\infty)$ outside a set of Hausdorff dimension zero, the IFS ${{\mathcal F}}_t$ satisfies the exponential separation condition, and then Proposition \[prop-Dioph2\] implies that the $m$-tuple of matrices $(A_{i,t})_{i\in \Lam}$ is Diophantine for all $t$ outside of a zero-dimensional set, so certainly for Lebesgue-a.e. $t$. Now Lemma \[Fubini\] yields the desired claim. \(ii) We consider $(SL_{d+1}({{\mathbb R}}))^m$ as a codimension-$m$ submanifold of $(GL_{d+1}({{\mathbb R}}))^m \subset {{\mathbb R}}^{(d+1)^2m}$. In the proof of part (i) we showed that for a.e. $(A_i)_{i\in \Lam} \in \mathcal{X}_{\Sig, m}$, the induced IFS on a subset of ${{\mathbb R}}^d$ satisfies the strong exponential separation condition. Suppose that there is a positive measure subset ${{\mathcal E}}\subset \Yk_{\Sig,m}$ for which the strong Diophantine condition is violated. Then for every $(A_i)_{i\in \Lam}\in {{\mathcal E}}$, the induced IFS $\Phi$ does not have strong exponential separation, by another application of Proposition \[prop-Dioph2\]. However, $(A_i)_{i\in \Lam}\in \Yk_{\Sig,m}$ and $(c_i A_i)_{i\in \Lam}\in \Xk_{\Sig,m}$, for any $c_i>0$, induce the same IFS on the projective space, and we get a set of positive measure in $\Xk_{\Sig,m}$ for which the strong Diophantine condition does not hold. This is a contradiction, and the theorem is proved completely. Dimension of the attractor ========================== Let $A \in SL_2(\mathbb{R})$. It is easy to see that $A^{*} A$ has eigenvalues $\| A \|^2$, $\| A \|^{-2}$. Let $(\cos t_A, \sin t_A)^{\mathrm{t}}$ be the unit eigenvector corresponding to the eigenvalue $\| A \|^{-2}$, where $t_A \in [0, \pi)$. We recall some basic properties of the map $\varphi_A$. For more details see sections 2.2, 2.3 and 2.4 in [@HS2017]. The following simple lemma is [@HS2017 Section 2.4]. \[trivial0\] Let $A \in SL_2(\mathbb{R})$. Then the induced map $\varphi_{A}$ expands by at most $\| A \|^2$ and contracts by at most $\| A \|^{-2}$. Furthermore, for any ${{\varepsilon}}> 0$ there exists $C_{{\varepsilon}}> 1$ such that $\| A \|^{-2} {\leqslant}| \varphi_{A}'(x) | < C_{{\varepsilon}}\| A \|^{-2}$ for all $x \in [0, \pi) {\smallsetminus}(t_A - {{\varepsilon}}, t_A + {{\varepsilon}})$. The following lemma is now immediate. \[trivial\] Let $U \subsetneq (0, \pi)$ be an open set. Then, for every ${{\varepsilon}}> 0$ there exists $C_{{\varepsilon}}> 1$ such that for any $A \in SL_2( \mathbb{R} )$ with $(t_A-{{\varepsilon}},t_A+{{\varepsilon}}) \subset U$, we have $$\pi - C_{{\varepsilon}}\| A \|^{-2} < | \varphi_A( U ) | < \pi. $$ Lemma \[trivial\] implies the following: \[trivial3\] Let $U \subsetneq (0, \pi)$ be an open set. Then, for every ${{\varepsilon}}> 0$ there exists $M =M({{\varepsilon}})> 0$ such that the following holds: for any $A \in SL_2( \mathbb{R} )$ that satisfies $\varphi_A( U ) \subset U$ and $\| A \| > M$, we have $(t_A-{{\varepsilon}},t_A+{{\varepsilon}}) \not\subset U$. Let $\mathcal{A} = \{A_i \}_{i \in \Lambda}$ be a finite collection of $SL_2(\mathbb{R})$ matrices and let $\Phi = \{\varphi_A\}_{A\in \Ak}$ be the corresponding IFS on $[0,\pi)\cong\RP^1$. Recall the notation: $$\Phi(E) = \bigcup_{A\in \Ak} \varphi_A(E).$$ Assume that there is a strictly invariant multicone $U\subset [0,\pi)$, that is, a nonempty open set having finitely many connected components with disjoint closures, such that $\ov{U}\ne \RP^1$ and $\Phi(\ov{U})\subset U$. By Theorem \[th-unihyp\], the associated cocycle is uniformly hyperbolic, which implies that there exist $c>0$ and $\lam>1$ such that $$\label{eq-hyp} \|A_\bi\| {\geqslant}c\lam^n\ \ \mbox{for all}\ \bi\in \Lam^n,\ n\in \N,$$ see [@Yoccoz2004] and [@ABY2010 Theorem 2.2]. Fix ${{\varepsilon}}>0$ such that the $(2{{\varepsilon}})$-neighborhood of $\Phi(\ov{U})$ is contained in $U$, and let $M=M({{\varepsilon}})$ from Lemma \[trivial3\]. By (\[eq-hyp\]), there exists $n_0\in \N$ such that $\|A_\bi\|>M$ for $\bi\in \Lam^n$, $n{\geqslant}n_0$. Lemma \[trivial3\] implies that $(t_{A_\bi}-{{\varepsilon}}, t_{A_\bi}+{{\varepsilon}}) \not \subset U$, hence $$(t_{A_\bi}-{{\varepsilon}}, t_{A_\bi}+{{\varepsilon}}) \cap \Phi(\ov{U}) = \emptyset,\ \ \mbox{for all}\ \bi\in \Lam^n,\ n{\geqslant}n_0.$$ Hence, by Lemmas \[trivial0\] and \[trivial\] we obtain $$\label{eq-BDP} \|A_\bi\|^{-2} {\leqslant}|\varphi'_{\bi}(x)| {\leqslant}C_{{\varepsilon}}\|A_\bi\|^{-2},\ \ \mbox{for all}\ \ x\in \ov{U},\ \bi\in \Lam^n,\ n{\geqslant}n_0.$$ Thus we obtain \[lem-contract\] [(i)]{} The Bounded Distortion Property holds for $\Phi$ on $U$: there exists $C'>1$ such that $$\label{eq-bdp} \frac{1}{C'} {\leqslant}\frac{|\varphi_{\bi}'(x)|}{|\varphi_{\bi}'(y)|} {\leqslant}C'\ \ \mbox{for all}\ \ x,y\in \ov{U}, \ \bi\in \Lam^n,\ n\in \N.$$ [(ii)]{} The IFS $\Phi^k$ is contractive on $U$ in the metric $d_{\Prob}$ for sufficiently large $k$. More precisely, there exists $C''>0$ such that $$\label{eq-HYP} \|\varphi'_\bi\|_{\ov{U}} {\leqslant}C'' \lam^{-2n},$$ where $\lam>1$ is from (\[eq-hyp\]). [(iii)]{} We have $$s = s_\Ak/2,$$ where $s$ is the solution of the Bowen’s equation $P_\Phi(s)=0$, with the pressure given by (\[Bowen0\]) and $s_\Ak$ is the critical exponent, given by (\[critic\]). Let $p = ( p_i )_{i \in \Lambda}$ be a probability vector, and let $x_0 \in U$. Let $\chi_{\Phi, p}$ be the almost sure value of the limit $$\label{lim} \lim_{n \to \infty} - \frac{1}{n} \log | ( \varphi_{ i_1 \cdots i_n } )' (x_0) |,$$ where $i_1, i_2, \cdots \in \Lambda$ is a sequence chosen randomly according to the probability vector $p = ( p_i )_{ i \in \Lambda}$. The equations (\[eq-BDP\]) and (\[Lyap1\]) imply \[lem-Lyap\] We have $\chi_{ \Phi, p } = 2 \chi_{ \mathcal{A}, p }$. By the Birkhoff Ergodic Theorem, applied to the shift transformation on $\Lam^\N$ with the measure $\mu=p^\N$, in view of the bounded distortion (\[eq-bdp\]), we have $$\label{eq-Lyap3} \chi_{\Phi,p} = \lim_{n \to \infty} - \frac{1}{n} \log \left| ( \varphi_{ i_1 \cdots i_n } )^{'} ( \Pi (\bi)) \right|=-\int_{\Lam^\N} \log\bigl|\varphi_{i_1}'(\Pi(\bi))\bigr|\,d\mu(\bi),$$ where $\Pi:\Lam^\N\to \RP^1$ is the natural projection corresponding to $\Phi$. \[lem-Dioph\] [(i)]{} Let $\Ak$ be a finite set of matrices in $GL_2({{\mathbb R}})$, and let $\Phi$ be the IFS induced by $\Ak$ on the projective line $\RP^1$. If $\Phi$ satisfies the strong exponential separation condition on a nonempty set, then $\mathcal{A}$ is strongly Diophantine. [(ii)]{} Let $\Ak$ be a finite set of matrices in $SL_2({{\mathbb R}})$, and let $\Phi$ be the IFS induced by $\Ak$ on the projective line $\RP^1$. Then $\Phi$ satisfies the strong exponential separation condition on a set containing at least three points if and only if $\mathcal{A}$ is strongly Diophantine. \(i) This is a special case of Proposition \[prop-Dioph2\], since exponential separation for the induced action on a subset of $\RP^1$ is equivalent to that for the induced action on a subset of the circle. \(ii) One direction, that the strong exponential separation for $\Phi$ implies the strong Diophantine property for $\Ak$, follows from (i). For the converse, we refer to [@HS2017 Lemma 2.5], which says that $SL_2({{\mathbb R}})$ is quantitatively separated by the action on three points of $\RP^1$. Recall that $\A$ is a finite set of $SL_2({{\mathbb R}})$ matrices satisfying the strong Diophantine condition and having a strictly invariant multicone $U$, and $\Phi=\Phi_\Ak$ is the associated IFS on $U$. Then $\Phi$ has a compact attractor $K$, and our goal is to show that $\dim_H(K) =s$, where $P_\Phi(s)=0$ and $P_\Phi$ is given by (\[Bowen0\]). It is known that $$\label{dim_ineq} \dim_H (K) {\leqslant}s,$$ see the appendix for a short proof. Let us show the opposite inequality. Let $d_n > 0$ be the solution of the equation $$\sum_{\bi\in \Lam^n} | U_\bi |^{d_n} = 1,$$ where $U_\bi =\varphi_\bi(U)$ and $|\cdot|$ denotes the Lebesgue measure on $[0,\pi)\cong\RP^1$. It is not hard to see that $$\label{dim_lim} \lim_{n \to \infty} d_n = s.$$ For convenience of the reader, we include the proof in the appendix, following [@SiSo1999]. Let $p^{(n)} = ( p^{(n)}_\bi )_{\bi \in \Lam^n}$ be the probability vector such that $p^{(n)}_\bi = | U_\bi |^{d_n}$. Let $\eta^{(n)}$ be the invariant probability measure for the IFS $\Phi^n$ on $U$, corresponding to $p^{(n)}$. Since $\eta^{(n)}$ is supported on $K$, we have $\dim \eta^{(n)} {\leqslant}\dim_{H} (K)$. We claim that $\Ak$ satisfies the assumptions of Theorem \[thmHS\]. Indeed, the existence of a strictly invariant multicone is known to imply that all the matrices in $\Ak$ are hyperbolic, hence the group generated by $\Ak$ is unbounded. Further, we assumed that not all attracting fixed points of $\Ak$ are the same, hence this group is totally irreducible. Thus the Furstenberg measure for $(\Ak^n,p^{(n)})$ is unique, and it coincides with $\eta^{(n)}$. Since $\Ak$ is Diophantine, we have that $\Ak^n$ is Diophantine as well. Now, by Theorem \[thmHS\] and Lemma \[lem-Lyap\] we have $$\frac{ H( p^{(n)} ) }{ \chi_{ \Phi^n, p^{(n)} } } {\leqslant}\dim_{H} (K).$$ We claim that there exists $C > 0$ such that $$\label{claim2} \chi_{\Phi^n, p^{(n)} } {\leqslant}-\sum_{ \bi \in \Lambda^n } | U_\bi |^{d_n} \log | U_\bi | + C\ \ \mbox{for all}\ \ n\in \N.$$ Indeed, by (\[eq-Lyap3\]), we have $$\chi_{\Phi^n, p^{(n)} } {\leqslant}\sum_{\bi \in\Lam^n} \mu([\bi])\cdot \log\bigl(\min_{x\in \ov{U}}|\varphi_\bi'(x)|\bigr)^{-1},$$ where $[\bi]$ is the cylinder set of sequences starting with $\bi$. Now $\mu([\bi])=|U|^{d_n}$, and $$\min_{x\in \ov{U}}|\varphi_\bi'(x)|{\geqslant}\frac{|U_\bi|}{C'|U|},$$ by the Bounded Distortion Property (\[eq-bdp\]). Therefore, $$\begin{aligned} \chi_{ \Phi^n, p^{(n)} } & {\leqslant}\sum_{ \bi \in \Lambda^n } | U_\bi |^{d_n} \log \frac{ C' | U | }{ | U_\bi | } \\ &= -\sum_{\bi \in \Lambda^n} | U_\bi |^{d_n} \log | U_\bi | + \log C' | U |. \end{aligned}$$ By (\[claim2\]), we have $$\begin{aligned} \frac{ H( p^{(n)} ) }{ \chi_{ \Phi^n, p^{(n)} } } &> \frac{ -\sum_{ \bi \in \Lambda^n } | U_\bi |^{d_n} \log | U_\bi |^{d_n} } { -\sum_{ \bi \in \Lambda^n } | U_\bi |^{d_n} \log | U_\bi | + C } \\ &= d_n \left( 1 + \frac{C}{ -\sum_{ \bi \in \Lambda^n } | U_\bi |^{d_n} \log | U_\bi | } \right)^{-1}. \end{aligned}$$ Since $\lim_{n \to \infty} d_n = s$ and $\lim_{n \to \infty} -\sum_{ \bi \in \Lambda^n } | U_\bi |^{d_n} \log | U_\bi | = \infty$, we obtain $s {\leqslant}\dim_{H} (K)$, as desired. Finally, $s=s_\Ak$ by Lemma \[lem-contract\](iii). Appendix: the proof of (\[dim\_ineq\]) and (\[dim\_lim\]) {#appendix} ========================================================= Proof of (\[dim\_lim\]) [@SiSo1999] ----------------------------------- We have a projective IFS $\Phi = \{\varphi_i\}_{i\in \Lam}$ on a strictly invariant multicone $U$. Observe that $$P_\Phi(t) = \lim_{n\to \infty} \frac{1}{n} \log \sum_{\bi \in \Lam^n} \|\varphi_\bi'\|^t = \lim_{n \to \infty} \frac{1}{n} \log \sum_{\bi \in \Lambda^n} \left| U_{\bi} \right|^t,$$ by the Bounded Distortion Property (\[eq-bdp\]). Let $$Q_n = \frac{1}{n} \log \sum_{\bi \in \Lambda^n} \left| U_{\bi} \right|^s.$$ Since $P_\Phi(s ) = 0$, we have $\lim_{n \to \infty} Q_n = 0$. Let $r_1>0$ be such that $r_1 {\leqslant}|\varphi_i'(x)|$ for all $i \in \Lambda$ and $x \in \ov{U}$. Recall (\[eq-HYP\]), which says that $${\|\varphi'_\bi\|}_{\ov{U}} {\leqslant}C'' \lam^{-2n},$$ for $C''>0$ and $\lam>1$. Then $r^n_1 |U| {\leqslant}|U_\bi| < C''\lam^{-2n} |U|$ for $\bi\in\Lam^n$, and hence we have $$( r^n_1 |U| )^{s-d_n} \cdot |U_\bi|^{d_n} < |U_\bi|^s < C'' ( \lam^{-2n} |U| )^{s-d_n} \cdot |U_\bi|^{d_n}.$$ In view of $\sum_{\bi\in \Lambda^n} |U_\bi|^{d_n} = 1$, we have $$\frac{1}{n} \log ( r^n_1 |U| )^{s-d_n} < Q_n < \frac{1}{n} \bigl[\log C'' + (s-d_n)(\log|U| - 2n\log\lam)\bigr],$$ and it follows that $$\frac{Q_n -(\log C'')/n}{ -2\log \lam+ ( \log |U| ) / n } < s - d_n < \frac{Q_n}{ \log r_1 + ( \log |U| ) / n },$$ which implies $d_n\to s$, as desired. Proof of (\[dim\_ineq\]) ------------------------ Fix ${{\varepsilon}}> 0$. Then for sufficiently large $n$ we have $d_n < s + {{\varepsilon}}/2$. Thus $$\sum_{\bi \in \Lambda^n} | U_\bi |^{s + {{\varepsilon}}} < \sum_{\bi \in \Lambda^n} | U_\bi |^{d_n + {{\varepsilon}}/2} < ( r_2^n |U| )^{{{\varepsilon}}/2} \to 0, \text{\ as } n \to \infty.$$ Therefore, the $(s + {{\varepsilon}})$-dimensional Hausdorff measure of $K$ is zero. By the definition of the Hausdorff dimension, this proves (\[dim\_ineq\]). Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to acknowledge Balazs Barańy for helpful comments and for telling us about the papers [@ABY2010; @BG2009].

--- abstract: 'Public observatory project is playing more and more important role in science popularization education and scientific research, and many amateur astronomers also have began to build their own observatories in remote areas. As a result of the limitation of technical condition and construction funds for amateur astronomers, their system often breaks down, and then a stable remote unattended operation system becomes very critical. Hardware connection and control is the basic and core part in observatory design. Here we propose a conception of engineering hardware design for public observatory operation as a bridge between observatory equipment and observation software. It can not only satisfy multiple observation mode requirement, but also save cost.' author: - 'Jun Han,$^1$ Dongwei Fan,$^1$ Chenzhou Cui,$^1$ Chuanzhong Wang,$^2$ Shanshan Li,$^1$ Linying Mi,$^1$ Zheng Li,$^1$ Yunfei Xu,$^1$ Boliang He,$^1$ Changhua Li,$^1$ Yihan Tao,$^1$ and Sisi Yang$^1$' bibliography: - 'P1-4.bib' title: A Conception of Engineering Design for Remote Unattended Operation Public Observatory --- Introduction ============ Since the first microprocessor emerges, people have tried to make the telescope operation smarter and started numerous engineering attempts. Until the mid of 1960s, some began operating stably and the 8” reflector telescope at the University of Wisconsin is one of earliest examples( @1992ASPC...34....3C). They could carry out astronomical photometry according to the scheduled list, so named as Automated Scheduled Telescope and also is the beginning of robotic telescope. Then Remotely Operated Telescope is proposed to meet the needs of observation, which could be controlled by remote users and observe astronomical objects automatically. Today observatory are highly integrated, more complex and more advanced. Telescope observatory not only needs automatic unattended operation, but also can be connected remotely. It can run without human’s help and can adjust itself according to weather, equipment status and so on. This is the Robotic Autonomous Observatory, and some observatories have achieved this goal. Robotic has two main important advantages, autonomous and remote. Autonomous means that a better use of telescope time by telescope’s real-time follow-up, saving manpower in unattended operation mode, operation mistakes reduced, higher observation efficiency, and focusing science more but not operation logic. Remote means that users can be located anywhere to save time, share same telescope in different time to save cost and build observatory at high altitude and distant location to obtain the best observation environment. Robotic observatory has made significant effect in student education, for example the Bradford Robotic telescope, the original Micro-Observatory telescopes and so on(e.g. @2017AstRv..13...28G, @1996ASPC..101..380D). These learning and observation experience could motivate students to continue higher level courses and scientific career. Even some high school students continued deeper research and produced papers( @2011PASA...28...83F). As a result of robotic telescope’s huge advantages, more amateur astronomers have began to build their own observatory and have made many important scientific outputs, for example Xingming Observatory built in 2007 by an amateur astronomer. It is located in Xinjiang, China, and releases the figures by Popular Supernova Project.[^1] This project platform is managed by Chinese Virtual Observatory, and any people can participant. Until now, 17 supernova and nova candidates have been reported, and 12 of them have been confirmed by optical spectrum. Public observatory and amateur astronomers, whether in education or scientific research, have been an important astronomy strength. With economic progress, light pollution becomes worse and worse,[^2] so that building observatory in remote areas becomes inevitable. There are some solutions by using modern robotic observatory mode, but usually too expensive and complex, and also not necessary to amateur astronomers. They usually organize their own system by themselves, especially for hardware equipment connection and related control. As a result of the limitation of technical condition and construction funds for amateur astronomers, their system usually consists of multiple sub-systems made by different people. This kind of combination is very rough and less compatible, so as to break down often. A stable remote unattended operation system suitable for amateur astronomer’s observatory becomes very critical. In fact, remote unattended operation observatory is not a new conception, and it is the so called robotic telescope above, but a little difference and mainly faces the requirements of amateur astronomers’ observatory. Here we propose a conception of engineering design for public observatory. It can not only satisfy observation requirement, but also save cost. The Conception of Engineering Design ==================================== Nowadays public observatory also is not a single telescope, but an integration system with telescope, various sensors and so on. When design a common observatory hardware system, there are two key problems - the connection and intelligent control to equipment. We propose a hardware system as a bridge between observatory equipment and software system, and some criterions should be followed. - Connect and control every equipment easily without any dependence on operation system and software platform. - Multiple connection modes to meet various users. - Hardware system itself should be clever to open or close some resource according to system status. Based on these, we design a Remote Observatory System (ROS for short) to meet the criterions above, and make it to have the ability to connect and control the equipment in observatory. The framework for public observatory is showed in Figure \[P1-4\_f1\]. We define this system as a closed-loop system and make it has the capability to evaluate its operation through redundant inputs to detect errors. This system is made of multiple single chips, tiny internet chips and logic circuits, and could be reprogrammed. The ROS system consists of three control modules and five functional modules. As a result of light pollution and actual demand, observatory could be located anywhere. Different control modules should be designed to meet different users. They are mainly used to transfer and analyse control commands. Three control modules are designed and listed next. - Local Control Module - This module is the most basic part. Users can operate all the resource in the observatory, and it has the highest control priority. - Network Control Module - It has the lowest control priority and mainly used by remote users. Network protocol is independent of platform, and so it can be connected by any network equipment, for example computer, pad and so on. - Phone Control Module - This module is a special part and most useful for emergency control, for example network interruption. It connects each other by phone tower and the control is by message or voice. The following is the five functional modules. Communication interface to observation equipment is mainly by internet or serial bus. They are used to connect and control observation equipment in observatory, for example telescope, dome, all-sky camera, sky brightness, weather station, security camera, other auxiliary equipment, etc. - ARM/PC Module - This module is used to deploy related observation software for computing, data transfer, backup and telescope control, including equatorial mount, filter, focus and so on. - Dome Module - Dome open, follow-up and close. - Power Module - The power supply and control logic for every equipment. - Network Module - The network entrance and export. All network equipment should access and connect it. - Monitor Module - It not only monitor observatory equipment and its status, but also push status code to users and adjust resource’s operation by predefined algorithm. Summary ======= We propose a closed-loop hardware system as a bridge between observatory and users. It supports multiple control modules to meet different users, and provides internet and serial bus interface as the communication interface to connect observation equipment. The interface also could be extended according to factual requirements. It is a kind of open source hardware platform, and people could define control and transfer logic by themselves and then reprogramme it. Based on this hardware platform, we will develop software driver environment so as to access RTS2, ASCOM easily, and also make our own observation control software system special for public observatory in the future. This work is supported by National Natural Science Foundation of China (NSFC)(11503051, 61402325) and the Joint Research Fund in Astronomy (U1531111, U1531115, U1531246, U1731125, U1731243) under cooperative agreement between the NSFC and Chinese Academy of Sciences (CAS) and the Young Researcher Grant of National Astronomical Observatories, Chinese Academy of Sciences. We would like to thank the National R&D Infrastructure and Facility Development Program of China, “Earth System Science Data Sharing Platform” and “Fundamental Science Data Sharing Platform” (DKA2017-12-02-XX). Data resources are supported by Chinese Astronomical Data Center (CAsDC) and Chinese Virtual Observatory (China-VO). [^1]: Populsar Supernova Project reference web link <http://psp.china-vo.org/>. [^2]: The light pollution map reference web link <https://www.lightpollutionmap.info>.

--- abstract: 'This paper presents a compilation procedure which determines internal and external indices for signs in a unification based grammar to be used in improving the computational efficiency of lexicalist chart generation. The procedure takes as input a grammar and a set of feature paths indicating the position of semantic indices in a sign, and calculates the fixed-point of a set of equations derived from the grammar. The result is a set of independent constraints stating which indices in a sign can be bound to other signs within a complete sentence. Based on these constraints, two tests are formulated which reduce the search space during generation.' author: - Arturo Trujillo bibliography: - 'ref.bib' title: Determining Internal and External Indices for Chart Generation --- Introduction ============ One problem with the classical transfer approach to machine translation (MT) is that it involves complex transformations of syntactic and semantic structures from the source to the target language. These transformations can have intricate interactions with each other, making transfer modules difficult to reverse, debug and maintain. They can also make monolingual components more heavily dependent on the language pair at hand. Much of the complexity in transfer stems from the recursive nature of the syntactic and semantic frameworks normally used. However, recent work in formal semantics has found it expedient to minimise the recursive structure of semantic representations to efficiently encode certain types of ambiguity [@reyle95]. Naturally, flat semantics mitigate many structural differences between natural languages and their application to MT readily follows (see [@copestakeetal95b]). Unfortunately, simplicity in the transfer component comes at the cost of generation complexity for such representations since their lack of structure increases the non-determinism of most generation algorithms, just as lexical-only transfer increases the complexity of bag generation in Shake-and-Bake MT [@whitelock94]. For this reason, several researchers have investigated the efficiency of generators whose input has a flat structure, be this in the form of lists of semantic predicates or of lexical elements. Such generators, of the chart, bag and lexicalist varieties, differ in many ways but the source of their complexity is the same: the search space grows factorially on the size of the input for many algorithms, since they are based on a modified chart parser which essentially attempts all permutations of the input. This is the issue addressed by the paper, taking chart generation as an instance of the problem. Chart Generation ================ A chart generator [@kay96] takes as input a flat semantic representation and, using a chart data structure, outputs the string corresponding to it. The unordered character of the semantic input permits such generators to be viewed as parsers for languages with completely free word order: an active edge combines with an inactive edge only if the two edges have no semantic predicates in common; no other restrictions apply. This regime leads to the combinatorial explosion mentioned above. Example ------- Consider the following flat semantic representation corresponding to the string [*John ran fast*]{}: > r : run(r), past(r), fast(r), arg1(r,j), name(j,John) Here, $r$ is the distinguished index for the expression. These predicates will unify with the semantic component of suitably defined lexical entries resulting in the agenda entries shown below: Word Cat Semantics ------ --------- -------------------------------- John np(j) j : name(j,John) ran vp(r,j) r : run(r), arg1(r,j), past(r) fast adv(r) r : fast(r) Items are then moved into the chart and their interactions considered. Moving [*John*]{} results in no interactions, since the chart is empty. Moving [*ran*]{} results in [*John ran*]{} assuming the rule: > s(x) $\rightarrow$ np(x), vp(x,y) This is a complete sentence, but it does not subsume all the semantic material from the input; it therefore remains in the chart but cannot constitute an output sentence. Next, [*fast*]{} is moved, adding in [*ran fast*]{} to the agenda and then to the chart, at which point [*John ran fast*]{} is built. Generation thus terminates. One of the main sources of inefficiency in chart generation is that a multitude of edges are constructed which either do not subsume the entire semantics of the input or which can never be part of the solution because they omit semantic material which only they could have subsumed. In the example, [*John runs*]{} is one such edge. The problem is that these edges, if left in the chart, will interact with other edges to form yet further edges which can never be part of the final result, but which cause the search space to explode. Internal and External Indices {#inn-out-sec} ----------------------------- To overcome this inefficiency, it is necessary to discard edges which would make it impossible to incorporate all the input into an output sentence. Achieving this involves exploiting the fact that after an edge is constructed, only certain indices in its semantic predicates are accessible by other rules in the grammar. For example, treatments of English VPs (e.g. [*chased the cat*]{}) typically disallow modification of the object NP once the VP has been analysed; thus if [*cat*]{} received index [*c*]{}, it is not possible to bind into this index. Intuitively, this means that modifying the VP cannot lead to modification of the object NP. Following Kay, indices not available outside a category (i.e. outside an inactive edge) are called [*internal*]{} indices, while those which are accessible are called [*external*]{} indices. When an inactive edge is constructed, all indices in predicates not subsumed by the edge must be i) different from the indices the edge subsumes, or ii) be external to it. This ensures that inactive edges subsume all predicates indexed by their internal indices. The objective of this paper is to present a general algorithm for determining which indices are internal and external to a category without requiring the explicit identification of such information by the grammar writer. Overview of the Algorithm ========================= Ideally, internal indices should be determined directly from the rules of a grammar. However, different grammar writers adopt different index binding strategies, making such identification by automatic on-line inspection of rules very difficult at best. The algorithm proposed here therefore automatically extracts information from a grammar off-line and uses it to determine whether an index is internal or external to a category. Based on this information, it is possible to identify those edges which are incomplete with respect to the input and which may consequently be eliminated from further consideration. The algorithm has been implemented and tested on a lexicalist generator operating on a small unification-based grammar; a description of the test and further discussion of the issues involved is given in [@trujilloetal96]. The algorithm takes as input a unification-based phrase structure (PS) grammar and a set of paths and outputs a set of constraints on pairs of signs indicating which indices in the two signs can be bound for some possible derivation tree. Principal among the techniques used are those for predictive parser compilation [@ahoetal86] adapted to unification based grammars [@trujillo94]. In addition, following standard practice in data flow analysis [@kennedy81], a data structure is maintained tracing how variables (or in this case, indices) are modified (or in this case, bound) in a valid derivation. Inner and Outer Domains ----------------------- Two main phases, themselves analogous to the calculation of FIRST and FOLLOW sets for predictive parsers, constitute the bulk of the algorithm. The first phase determines the indices at the root of a tree which are bound to items at the leaves; this phase will be called the calculation of [*inner domains*]{}. The second phase uses inner domains to calculate [*outer domains*]{}, which indicate the indices in a sign which are bound to the indices of signs outside the sign’s subtree. Thus, inner domains express the relationship between phrases with related semantic material within subtrees for which they are roots, while outer domains express the relationship between a phrase and outside phrases with which the phrase shares semantic material. Calculating both inner and outer domains requires the computation of the fixed-point of a set of equations derived from the grammar. The fixed-point of a function is the value of [**X**]{} which satisfies\ f([**X**]{}) = [**X**]{} Grammar ------- We adopt the following definition of a unification grammar: A grammar is a tuple (N,T,P,S), where P is a set of productions $\alpha \Rightarrow \beta$, where $\alpha$ is a sign, $\beta$ is a list of signs, N is the set of all $\alpha$, T is the set of all signs appearing in $\beta$ such that they unify with lexical entries, and S is the start sign. The grammar must generate sequences of coherent predicates (i.e. the graph with arcs for predicates sharing indices is connected). The Triple Data Structure ------------------------- A basic data structure in the algorithm will be triples of the form [*(Left Sign, Right Sign, Bindings)*]{}, where [*Bindings*]{} is a set of pairs consisting of a path in [*Left Sign*]{} and a path in [ *Right Sign*]{} such that the values at the end of each path are assumed to be token identical; [*Left Sign*]{} and [*Right Sign*]{} are phrasal or lexical signs. The following triple for example represents part of the inner domain of an NP: > \(1) (NP\[sem:arg1:X\],Det\[sem:arg1:Y\], {$<$sem:arg1,sem:arg1$>$}) It indicates that in a complete parse, it is possible that index X be bound to index Y for these two signs. Triples of this form are used uniformly throughout to encode inner and outer domains and to represent the functions and equations for which a fixed-point needs to be found. The algorithm proper consists of three main components: general operations, inner domain compilation and outer domain compilation. General Operations ================== Fixed-Point Iteration --------------------- This is the key function in the compilation process and it is used to solve systems of equations derived from the grammar. Each such system can be interpreted as a vector function [@rayward-smith83] with one side of the equations used to calculate a value which is then assigned to the other side. The fixed-point algorithm takes a function and an initial argument value (both expressed as sets) and returns, also as a set, the result of iteratively applying the function to successive values:\ [*Fixed-point(Function, Argument) $\rightarrow$ Set of triples\ Result := {}\ A-new := Argument\ Repeat\ Temp := Function X A-new\ Result$'$ := Result $\cup_{\leq}$ Temp\ A-new := Result$' -$ Result (i.e. set difference)\ Result := Result’\ Until A-new := {}\ Return Result* ]{} Two (overloaded) operators are used in this algorithm: 1. The crossproduct operator, [*X*]{}, takes two sets, [*A*]{} and [*B*]{} and constructs the set $\{ a \times b \mid a \in A, b \in B \}$, where $\times$ is a type dependent operation, defined as follows: - If $a$ and $b$ are triples of the form [*(La,Ra,Ba)*]{} and [*(Lb,Rb,Bb)*]{}, then\ $a \times b = (La,Rb, Ba X Bb)$ if $Ra \sqcap Lb$ (i.e. they unify) and $Ba \times Bb \neq \{\}$ - If $a$ and $b$ are pairs of paths of the form $<Lpa,Rpa>$ and $<Lpb,Rpb>$, then\ $a \times b = <Lpa,Rpb>$ if $Rpa$ is equal to $Lpb$. For example, the following operations indicates that if a PP is in the outer domain of a VP, so is a preposition: > { (VP\[sem:arg1:W\],PP\[sem:arg1:X\],{$<$sem:arg1,sem:arg1$>$}) }\ > X { (PP\[sem:arg1:Y\],P\[sem:arg1:Z\],{$<$sem:arg1,sem:arg1$>$}) }\ > = { (VP\[sem:arg1:W\],P\[sem:arg1:X\],{$<$sem:arg1,sem:arg1$>$}) }\ 2. The subsume-union operator, $\cup_{\leq}$, takes two sets, $A$ and $B$ and calculates the set $\bigcup\{ a \sqcup_t b \mid a \in A, b \in B\}$, where $\sqcup_t$ is a type dependent generalisation operator, defined as follows: - If $a$ and $b$ are triples of the form [*(La,Ra,Ba)*]{} and [*(Lb,Rb,Bb)*]{} then $a \sqcup_t b$ is - [*{ (La,Ra,Bab) }*]{}, where $Bab = Ba \cup_{\leq} Bb$, if $Lb \sqsubseteq La$ (i.e. $La$ subsumes $Lb$) and $Rb \sqsubseteq Ra$. - [*{ (Lb,Rb,Bab) }*]{}, where $Bab = Ba \cup_{\leq} Bb$, if $La \sqsubseteq Lb$ and $Ra \sqsubseteq Rb$. - [*{ (La,Ra,Bab), (Lb,Rb,Bab) }*]{} otherwise. - If $a$ and $b$ are pairs of paths of the form $<Lpa,Rpa>$ and $<Lpb,Rpb>$, then $a \sqcup_t b$ is - $\{<Lpa,Rpa>\}$ if $Lpa$ is a prefix of path $Lpb$ and $Rpa$ is a prefix of path $Rpb$. - $\{<Lpb,Rpb>\}$ if $Lpb$ is a prefix of path $Lpa$ and $Rpb$ is a prefix of path $Rpa$. - $\{<Lpa,Rpa>, <Lpb,Rpb>\}$ otherwise. For example, the fact that prepositions can modify the event of a VP and also its subject, leads to the following union: > { (VP\[sem:\[arg1:U,arg2:V\]\],P\[sem:arg1:W\],{$<$sem:arg1,sem:arg1$>$}) }\ > $\cup_{\leq}$ { (VP\[sem:\[arg1:X,arg2:Y\]\],P\[sem:arg1:Z\],{$<$sem:arg2,sem:arg1$>$}) } = {(VP\[sem:\[arg1:U,arg2:V\]\],P\[sem:arg1:W\],{$<$sem:arg2,sem:arg1$>$,$<$sem:arg1,sem:arg1$>$})} Shared Indices -------------- When constructing the equations for inner and outer domains from grammar rules, the index paths shared between categories in a rule need to be identified. A dedicated function achieves this:\ [*Shared-Indices(Sign1,Sign2,Paths) $\rightarrow$\ { $<$p1,p2$> \mid$ Sign1:p1 is token identical with Sign2:p2, and p1 and p2\ are both prefixes of elements in Paths}* ]{}\ For example:\ [*Shared-Indices*]{}(NP\[sem:arg1:X\],Det\[sem:arg1:X\],{sem:arg1, sem:arg2})\ = {$<$sem:arg1,sem:arg1$>$} Compiling Inner Domain ====================== Computing inner domains proceeds by finding the fixed-point of inner domain equations derived from the grammar, represented as triples. For instance, triple 1) above would be one of the equations for the rule: > ${{\setlength{\arraycolsep}{0.4mm} > \renewcommand{\arraystretch}{0.7} > \left[ > \begin{array}{l} > \\[-2mm] > {{\mbox{\scriptsize{\sc cat}}\! =\! {{{\mbox{\scriptsize {\bf NP}}}}}}}\\ > {{\mbox{\scriptsize {\sc sem}}\! =\!\! {{{{\setlength{\fboxsep}{0.5mm} \fbox{{\scriptsize 0}} \!}}}}}} \\[-2mm] \\ > \end{array} > \right] > }} > \Longrightarrow > {{\setlength{\arraycolsep}{0.4mm} > \renewcommand{\arraystretch}{0.7} > \left[ > \begin{array}{l} > \\[-2mm] > {{\mbox{\scriptsize{\sc cat}}\! =\! {{{\mbox{\scriptsize {\bf Det}}}}}}}\\ > {{\mbox{\scriptsize {\sc sem:arg1}}\! =\!\! {{{{\setlength{\fboxsep}{0.5mm} \fbox{{\scriptsize 1}} \!}}}}}} \\[-2mm] \\ > \end{array} > \right] > }} > \hspace{1em} > {{\setlength{\arraycolsep}{0.4mm} > \renewcommand{\arraystretch}{0.7} > \left[ > \begin{array}{l} > \\[-2mm] > {{\mbox{\scriptsize{\sc cat}}\! =\! {{{\mbox{\scriptsize {\bf N1}}}}}}}\\ > {{\mbox{\scriptsize {\sc sem}}\! =\!\! {{{{\setlength{\fboxsep}{0.5mm} \fbox{{\scriptsize 0}} \!}}}}}}{{\setlength{\arraycolsep}{0.4mm} > \renewcommand{\arraystretch}{0.7} > \left[ > \begin{array}{l} > \\[-2mm] > {{\mbox{\scriptsize {\sc arg1}}\! =\!\! {{{{\setlength{\fboxsep}{0.5mm} \fbox{{\scriptsize 1}} \!}}}}}} \\[-2mm] \\ > \end{array} > \right] > }} \\[-2mm] \\ > \end{array} > \right] > }}$ The triple would therefore indicate that the inner domain of an [ *NP*]{} includes the inner domain of a [*Det*]{}. In the case of inner domains, the equations can also be interpreted as the initial value to be fed to [*Fixed-point*]{}; that is, triple 1) can also be interpreted as saying that the inner domain of [*NP*]{} includes [*Det*]{}. Inner domain equations are built as follows: > [*Inner-Equations(Grammar) $\rightarrow$ Set of triples\ > Inner-Eq := {}\ > For each rule $A \rightarrow B_1 ... B_n \in$ Grammar\ > Inner-Eq := Inner-Eq $\cup {\displaystyle \bigcup_{k=1..n}}\{(A,B_k,$Shared-Indices$(A,B_k,Paths))\}$\ > Return Inner-Eq* ]{} Here, [*Paths*]{} is a theory specific set of index paths. The set of inner domains can now be defined as: > [*Fixed-point(Inner-Equations(Grammar),Inner-Equations(Grammar))*]{} Compiling Outer Domains ======================= Outer domains encode the bindings that may exist between a sign and any signs external to it in a valid derivation. The tree:\ (.S. ‘[**NP\[sem:arg1:X\]**]{}’ (.VP\[sem:arg2:X\]. ‘[**Vtra\[sem:arg2:X\]**]{}’ ‘NP’)) would give rise to the following triple in the outer domain set: > \(2) (NP\[sem:arg1:X\],Vtra\[sem:arg2:Y\], {$<$sem:arg1,sem:arg2$>$}) This states that Vtra is in the outer domain of NP because the X in the NP can be bound to Y in Vtra. Outer domain equations are calculated as follows: > [*Outer-Equations(Grammar) $\rightarrow$ Set of triples\ > Outer-Eq := {}\ > For each rule $A \rightarrow B_1 ... B_n \in$ Grammar\ > Outer-Eq := Outer-Eq $\cup > {\displaystyle \bigcup_{k=1..n}}\{(B_k,A,$Shared-Indices$(B_k,A,Paths))\}$\ > Return Outer-Eq* ]{} Each triple in these equations states that the outer domain of a sign in a rule is the outer domain of its mother. The initial value for outer domains can be calculated from the grammar and the set of inner domains: > [*Initialise-Outer-Domains(Grammar) $\rightarrow$ Set of triples\ > Outer-Dom := {}\ > For each rule $A \rightarrow B_1 ... B_n \in$ Grammar\ > Outer-Dom := Outer-Dom $\cup {\displaystyle \bigcup_{1\leq j,k\leq n, j \neq k}}${ (B$_j$,B$_k$, Shared-Indices(B$_j$,B$_k$) }\ > Outer-Dom :=\ > Outer-Dom X Fixed-point(Inner-Equations(Grammar),Inner-Equations(Grammar))\ > Return Outer-Dom* ]{} I.e. the outer domain of a category is the inner domain of all its sisters within a rule. Once initialised, the outer domains can be computed with: > [*Fixed-point(Outer-Equations(Grammar),Initialise-Outer-Domains(Grammar))* ]{} Using Outer Domains =================== Once calculated, outer domains can be used in at least two ways for chart generation. Firstly, they can be used to determine the internal indices of an edge and thus identify predicates which may have been ‘left out’ (see Section \[inn-out-sec\]). To compute the internal indices for each inactive edge, the edge’s external indices are subtracted from all the indices subsumed by it. External indices can be determined via outer domain triples through the use of the left sign and the bindings set. The following predicate implements the relevant test; it assumes that as edges are built, a record of the indices found amongst its predicates is kept and accumulated as more complex edges are built. The function returns true if an edge includes all the predicates indexed by its internal indices. > [*Internal-Validation(Inact-Edge,Remaining-Preds,Outer-Domain) $\rightarrow$ Boolean\ > External-Indices := {}\ > For each triple (Inact-Edge, \_ ,Binds) $\in$ Outer-Domain\ > For each pair $<p,\_> \in$ Binds\ > Add the index at the end of Inact-Edge:p to External-Indices\ > Internal-Indices := Inactive-Edge.indices $-$ External-Indices\ > Return false if there is a predicate in Remaining-Preds indexed by\ > an element of Internal-Indices\ > Return true otherwise*]{} For example, given the logical form for [*the dog saw the white cat*]{}: > \(3) s : def(d), dog(d), see(s,d,c), past(s), def(c), cat(c), white(c) the edge [*saw the cat*]{} would be discarded because there would be no triple in which the third argument index in a VP is bound to an index in any category. Thus, index $c$ would be deemed internal to the VP, and, since predicate [*white(c)*]{} includes this index, the VP could not be part of an output sentence. One disadvantage of this test is that it takes no account of the category of the signs outside the inactive edge and therefore allows too many unnecessary edges. Thus, while [*saw the cat*]{} is eliminated, the edge [*the cat*]{} is still constructed because $c$ will be an external index to the NP. The second test is designed to exploit the information in the outer domains to detect such edges. The following function returns true if the external indices in an inactive edge can indeed be bound to indices on external signs. The main idea is that any lexical items indexed by external indices must be licenced by at least one triple in the outer domain set. > [*External-Validation(Inact-Edge,Remaining-Preds,Outer-Domain) $\rightarrow$ Boolean\ > For each LexSign $\in$ Remaining-Preds which includes an external index from\ > Inact-Edge\ > If there is no triple (Inact-Edge,LexSign,Binds) $\in$ Outer-Domain for\ > which Inact-Edge:a = LexSign:b for at least one $<a,b> \in$ Binds for all\ > external indices in LexSign, then\ > Terminate the loop and return False\ > Return True*]{} To disallow [*the cat*]{} when generating from (3), the outer domain set is scanned for a triple with NP as its left sign, Adj as its right sign and a pair of paths binding index $c$ in Adj to the index in the NP; since no such triple would be present, Adj could not be incorporated into the semantics of a sentence including this NP. Therefore, the NP is discarded. During generation, a conjunction of both predicates needs to be satisfied before a newly constructed inactive edge can be added to the chart: > Internal-Validation(Inact-Edge,Remaining-Preds,Outer-Domain) $\wedge$\ > External-Validation(Inact-Edge,Remaining-Preds,Outer-Domain) Evaluation and Conclusion ========================= The algorithm was implemented in Sicstus Prolog and used to compile the outer domains for a small unification-based grammar; the resulting outer domains were tested on a lexicalist chart generator [@trujilloetal96]. The grammar handles adjectival and prepositional modification, both of which are common in real text. Relative clauses and gapping in general are not handled fully since they cause a larger number of indices to become external. On a small corpus of 10 sentences (average length = 9.8 words), use of internal and external validation reduced average generation time and final number of edges in the chart by 32% and 27% respectively. The best improvements were for the sentence [*the big brown dog with the fancy collar chased a little cat in the middle of the afternoon*]{} (221 secs to 39 secs; 852 edges to 272); the worst performance came from [*the man employed the woman*]{} (29 secs to 30 secs; 50 edges to 50 edges). While the algorithm is generally applicable to grammars with a strong PS component, further work is required to extract inner and outer domains from purely lexicalized grammars such as UCG or hybrids like HPSG. The fixed-point algorithm is general and by modifying the equations constructed, it can be used for compiling parsing tables for unification-based grammars [@trujillo94]. Acknowledgements {#acknowledgements .unnumbered} ================ Thanks to two anonymous reviewers and to John Carroll, Nicolas Nicolov and the staff and students at ITRI, University of Brighton, for useful comments.

--- abstract: 'The family of left-to-right algorithms reduces input numbers by repeatedly subtracting the smaller number, or multiple of the smaller number, from the larger number. This paper describes how to extend any such algorithm to compute the Jacobi symbol, using a single table lookup per reduction. For both quadratic time algorithms (Euclid, Lehmer) and subquadratic algorithms (Knuth, Schönhage, Möller), the additional cost is linear, roughly one table lookup per quotient in the quotient sequence. This method was used for the 2010 rewrite of the Jacobi symbol computation in .' author: - Niels Möller bibliography: - 'ref.bib' date: 2019 title: Efficient computation of the Jacobi symbol --- Introduction ============ The Legendre symbol and its generalizations, the Jacobi symbol and the Kronecker symbol, are important functions in number theory. For simplicity, in this paper we focus on computation of the Jacobi symbol, since the Kronecker symbol can be computed by the same function with a little preprocessing of the inputs. Jacobi and GCD -------------- Two quadratic algorithms for computing the Kronecker symbol (and hence also the Jacobi symbol) are described as Algorithm 1.4.10 and 1.4.12 in [@cohen]. These algorithms run in quadratic time, and consists of a series of reduction steps, related to Euclid’s algorithm and the binary algorithm, respectively. Both Kronecker algorithms share one property with the binary algorithm: The reduction steps examine the current pair of numbers in both ends. They examine the least significant end to cast out powers of two, and they examine the most significant end to determine a quotient (like in Euclid’s algorithm) or to determine which number is largest (like in the binary algorithm). Fast, subquadratic, algorithms work by divide-and-conquer, where a substantial piece of the work is done by examining only one half of the input numbers. Fast left-to-right is related to fast algorithms for computing the continued fraction expansion [@schoenhage:1971; @knuth:algorithms]. These are left-to-right algorithms, in that they process the input from the most significant end. The binary recursive algorithm [@stehle] is a right-to-left algorithm, in that it processes inputs from the least significant end. The asymptotic running times of these algorithms are $O(M(n) \log n)$, where $M(n)$ denotes the time needed to multiply two $n$-bit numbers. The algorithm used in recent versions of the library [@gmp] is a variant of Schönhage’s algorithm [@moller-sgcd]. It is possible to compute the Jacobi symbol in subquadratic time, with the same asymptotic complexity as . One algorithm is described in [@bach-shallit] (solution to exercise 5.52), which says: > This complexity bound is part of the “folklore” and has apparently never appeared in print. The basic idea can be found in Gauss \[1876\]. Our presentation is based on that in Bachmann \[1902\]. H. W. Lenstra, Jr. also informed us of this idea; he attributes it to A. Schönhage. Since the quadratic algorithms for the Jacobi symbol examines the data at both ends, some reorganization is necessary to construct a divide-and-conquer algorithm that processes data from one end. The binary algorithm has the same problem. In the binary recursive algorithm, this is handled by using a slightly different reduction step using 2-adic division. Recently, the binary recursive algorithm has been extended to compute the Jacobi symbol [@brent:jacobi]. The main difference to the corresponding algorithm is that it needs the intermediate reduced values to be non-negative, and to ensure this the binary quotients must be chosen in the range $1 \leq q < 2^{k+1}$ rather than $|q| < 2^k$. As a result, the algorithm is slower than the algorithm by a small constant factor. Main contribution ----------------- This paper describes a fairly simple extension to a wide class of left-to-right algorithms, including Lehmer’s algorithm and the subquadratic algorithm in [@moller-sgcd], which computes the Jacobi symbol using only $O(n)$ extra time and $O(1)$ extra space[^1]. This indicates that also for the fastest algorithms for large inputs, the cost is essentially the same for computing the and computing the Jacobi symbol.[^2] Like the algorithm described in [@bach-shallit], the computation is related to the quotient sequence. The updates of the Jacobi symbol are somewhat different, instead following an unpublished algorithm by Schönhage [@schoenhage-brent-communication] for computing the Jacobi symbol from the quotient sequence modulo four. In the algorithms in , the quotients are not always applied in a single step; instead, there is a series of reductions of the form $a {\leftarrow}a - m b$, where $m$ is a positive number equal to or less than the correct quotient ${\lfloor a/b \rfloor}$. In the corresponding Jacobi algorithms, the Jacobi sign is updated for each such partial quotient. Most of the partial quotients are determined from truncated inputs where the least significant parts of the numbers are ignored. The least significant two bits, needed for the Jacobi computation, must therefore be maintained separately. Notation -------- The time needed to multiply two $n$-bit numbers is denoted $M(n)$, where $M(n) = O(n \log n)$ for the fastest known algorithms. [^3] The Jacobi symbol is denoted $(a | b)$. We use the convention that $[\text{condition}]$ means the function that is one when the condition is true, otherwise 0, e.g., $(0 | b) = [b = 1]$. Left-to-right GCD ================= In this paper, we will not describe the details of fast algorithms. Instead we will consider Algorithm \[alg:gcd\], which is a generic left-to-right algorithm, with a basic reduction step where a multiple of the smaller number is subtracted from the larger number. We also describe the main idea of fast instantiations of this algorithm. In: $a, b > 0$ $a \geq b$ $a {\leftarrow}a - m b$, with $1 \leq m \leq {\lfloor a/b \rfloor}$ $a = 0$ $b$ $b {\leftarrow}b - m a$, with $1 \leq m \leq {\lfloor b/a \rfloor}$ $b = 0$ $a$ This algorithm terminates after a finite number of steps, since in each iteration $\max(a,b)$ is reduced, until $a = b$ and the algorithm terminates. It returns the correct value, since $\GCD(a,b)$ is unchanged by each reduction step. The running time of an instantiation of this algorithm depends on the choice of $m$ in each step, and on the amount of computation done in each step. E.g., if $m = 1$, the worst case number of iterations in exponential. Euclid’s algorithm is a special case where, in each step, $m$ is the correct quotient of the current numbers. The faster algorithms implements an iteration that depends only on some of the most significant bits of $a$ and $b$: These bits determine which of $a$ and $b$ is largest, and they also suffice for computing an $m$ which is close to the quotient ${\lfloor a/b \rfloor}$ or ${\lfloor b/a \rfloor}$. Furthermore, one can compute an initial part of the sequence of reductions based on the most significant parts of $a$ and $b$, collect the reductions into a transformation matrix, and apply all the reductions at once to the least significant parts of $a$ and $b$ later on. This saves a lot of time, since it omits computing all the intermediate $a$ and $b$ to full precision. If one repeatedly chops off one or two of the most significant words, one gets Lehmer’s algorithm, and by chopping numbers in half, one can construct a divide-and-conquer algorithm with subquadratic complexity. We will extend this generic algorithm to also compute the Jacobi symbol. To do that, we need to investigate how the basic reduction $a - m b$ affects the Jacobi symbol. When we have sorted this out, in the next section, the result is easily applied to all variants of Algorithm \[alg:gcd\]. Left-to-right Jacobi ==================== In this section, we summarize the properties of the Jacobi symbol we use, derive the update rules needed for our left-to-right algorithm. Finally, we give the resulting algorithm and prove its correctness. Jacobi symbol properties ------------------------ The Jacobi symbol $(a | b)$ is defined for $b$ odd and positive, and arbitrary $a$. We work primarily with non-negative $a$, and make use of the following properties of the Jacobi symbol. Assume that $a$ is positive and that $b$ is odd and positive. Then (i) \[it:zero\] $(0 | b) = [b = 1]$. (ii) \[it:negation\] $(a | b) = (-1)^{(b-1)/2} (-a | b)$ (iii) \[it:reciprocity\] If both $a$ and $b$ are odd, then $$(a | b) = (-1)^{(a-1)(b-1)/4} (b | a)$$ (iv) \[it:odd-reduction\] $(a | b) = (a - m b | b)$ for any $m$. (v) \[it:even-reduction-4\] If $a = 0 \pmod 4$ and $1 \leq m \leq {\lfloor b/a \rfloor}$, then $$(a | b) = (a | b - ma)$$ (vi) \[it:even-reduction-2\] If $a = 2 \pmod 4$ and $1 \leq m \leq {\lfloor b/a \rfloor}$, then $$(a | b) = (-1)^{m(b-1)/2 + m(m-1)/2} (a | b - ma)$$ For  to  we refer to standard textbooks. The final two are not so well-known, and their use for Jacobi computation is suggested by Schönhage [@schoenhage-brent-communication]. To prove them, assume that $a$ is even and $a < b$. Then $$\begin{aligned} (a | b) &= (a - b | b) && \text{By \eqref{it:odd-reduction}}\\ &= (-1)^{(b-1)/2} (b - a | b) && \text{By \eqref{it:negation}} \\ &= (-1)^{(b-1)/2 + (b-1)(b-a-1)/4} (b | b-a) &&\text{By \eqref{it:reciprocity}}\\ &= (-1)^{(b-1)/2 + (b-1)(b-a-1)/4}(a | b-a) && \text{By \eqref{it:odd-reduction}} \end{aligned}$$ Since $b^2 - 1$ is divisible by 8 for any odd $b$, we get a resulting exponent, modulo two, of $$(b-1)/2 + (b-1)(b-a-1)/4 = a (b-1)/4$$ If $a = 0 \pmod 4$, this exponent is even and hence there is no sign change. And this continues to hold if the subtraction is repeated, which proves . Next, consider the case $a = 2 \pmod 4$. Then $a/2 = 1 \pmod 2$, and repeating the subtraction $m$ times gives the exponent $$\begin{gathered} a \{(b-1) + (b-a-1) + \cdots + (b - (m-1) a - 1)\} / 4 \\ = m(b-1)/2 + m(m-1)/2 \pmod 2 \end{gathered}$$ which proves . Finally, note that in these formulas, all the signs are determined by the least significant two bits of $a$, $b$ and $m$. The new algorithm ----------------- The algorithm works with two non-negative integers $a$ and $b$, where multiples of the smaller one is subtracted from the larger. To compute the Jacobi symbol we maintain these additional state variables: $$\begin{aligned} e &\in {\mathbb{Z}}_2 && \text{Current sign is $(-1)^e$} \\ \alpha &\in {\mathbb{Z}}_4 && \text{Least significant bits of $a$} \\ \beta &\in {\mathbb{Z}}_4 && \text{Least significant bits of $b$} \\ d &\in {\mathbb{Z}}_2 && \text{Index of denominator}\end{aligned}$$ The value of $d$ is one if the most recent reduction subtracted $b$ from $a$, and zero if it subtracted $a$ from $b$. We collect these four variables as the state $S = (e, \alpha, \beta, d)$. The state is updated by the function , Algorithm \[alg:jupdate\]. In: $d' \in {\mathbb{Z}}_2$, $m \in {\mathbb{Z}}_4$, $S = (e, \alpha, \beta, d)$ $d \neq d'$ and both $\alpha$ and $\beta$ are odd $e {\leftarrow}e + (\alpha - 1)(\beta - 1)/4$ Reciprocity $d {\leftarrow}d'$ $d = 1$ $\beta = 2$ $e {\leftarrow}e + m (\alpha - 1)/2 + m (m-1)/2$ $\alpha {\leftarrow}\alpha - m \beta$ $\alpha = 2$ $e {\leftarrow}e + m (\beta - 1)/2 + m (m-1)/2$ $\beta {\leftarrow}\beta - m \alpha$ $S' = (e, \alpha, \beta, d)$ Since the inputs of this function are nine bits, and the outputs are six bits, it’s clear it can be implemented using a lookup table consisting of $2^9$ six-bit entries, which fits in 512 bytes if entries are padded to byte boundaries.[^4] Algorithm \[alg:jacobi\] extends the generic left-to-right algorithm to compute the Jacobi symbol. The main loop of this algorithm differs from Algorithm \[alg:gcd\] only by the calls to for each reduction step. In: $a, b > 0$, $b$ odd Out: The Jacobi symbol $(a | b)$ State: $S = (e, \alpha, \beta, d)$ \[li:jacobi-init\] $S {\leftarrow}(0, a \bmod 4, b \bmod 4, 1)$ $a \geq b$ \[li:update-a\] $a {\leftarrow}a - m b$, with $1 \leq m \leq {\lfloor a/b \rfloor}$ \[li:jacobi-update-a\] $S {\leftarrow}\proc{jupdate}(S, 1, m \bmod 4)$ $a = 0$ $[b = 1] (-1)^e$ $b {\leftarrow}b - m a$, with $1 \leq m \leq {\lfloor b/a \rfloor}$ \[li:jacobi-update-b\] $S {\leftarrow}\proc{jupdate}(S, 0, m \bmod 4)$ $b = 0$ $[a = 1] (-1)^e$ Correctness ----------- Let $a_0$ and $b_0$ denote the original inputs to Algorithm \[alg:jacobi\]. Since the reduction steps and the stop condition are the same as in Algorithm \[alg:gcd\], it terminates after a finite number of steps. We now prove that it returns $(a_0 | b_0)$. Algorithm \[alg:jacobi\] clearly maintains $\alpha = a \bmod 4$ and $\beta = b \bmod 4$. We next prove that the following holds at the start of each iteration: If $d = 0$ we have $$(a_0 | b_0) = (-1)^e \times \begin{cases} (b | a) & \text{$\alpha$ odd} \\ (a | b) & \text{$\alpha$ even} \end{cases}$$ and if $d = 1$ we have $$\label{eq:invariant-1} (a_0 | b_0) = (-1)^e \times \begin{cases} (a | b) & \text{$\beta$ odd} \\ (b | a) & \text{$\beta$ even} \end{cases}$$ This clearly holds at the start of the loop, to prove that it is maintained, consider the case $a \geq b$ (the case $a < b$ is analogous). Let $a$, $b$ (unchanged) and $S = (e, \alpha, \beta, d)$ denote the values of the variables before line \[li:update-a\]. There are a couple of different cases, depending on the state: - If $\beta$ is odd and either $\alpha$ is even or $d = 1$, then $(a_0 | b_0) = (-1)^e (a | b) = (-1)^e (a - m b | b)$. - If $\alpha$ and $\beta$ are both odd and $d = 0$, then $(a_0 | b_0) = (-1)^e (b | a) = (-1)^{e + (a-1)(b-1)/4} (a - m b | b)$. - If $\beta = 0 \pmod 4$, then $(a_0 | b_0) = (-1)^e (b | a) = (-1)^e (b | a - m b)$. - If $\beta = 2 \pmod 4$, then $(a_0 | b_0) = (-1)^e (b | a) = (-1)^{e + m(a-1)/2 + m(m-1)/2} (b | a - m b)$. In each case, the call to makes the appropriate change to $e$, and Eq.  holds after the iteration. Results ======= The algorithm was implemented in -5.1.0, released 2012. In benchmarks at the time, comparing the old binary algorithm to the new Jacobi extension of Lehmer’s algorithm (both $O(n^2)$), the new algorithm computed Jacobi symbols about twice as fast for moderate size numbers (around 2000 bits), and 10 times faster for numbers of size of 500000 bits. For even larger numbers, the Jacobi extension of subquadratic brought even greater speedups. Acknowledgments {#acknowledgments .unnumbered} =============== The author wishes to thank Richard Brent for providing valuable background material and for encouraging the writing of this paper. [^1]: The size of the additional state to be maintained is $O(1)$. But in a practical implementation, which does not store this state in a global variable, either the state or a pointer to it will be copied into each activation record, which for a subquadratic recursive divide-and-conquer algorithm costs $O(\log n)$ extra space rather than $O(1)$ [^2]: Even though we cannot rule out the existence of a left-to-right algorithm which is a constant factor faster than Jacobi. Such an algorithm would lie outside the class of “generic left-to-right algorithms” we describe in this paper, e.g., it might use intermediate reduced values of varying signs and quotients that are rounded towards the nearest integer rather than towards $-\infty$. [^3]: Multiplication in is based on the more practical Schönhage-Strassen algorithm, with asymptotic complexity $O(n \log n \log \log n)$. [^4]: One quarter of the entries in this table corresponds to invalid inputs, since at least one of $\alpha$ and $\beta$ is always odd. If we also note that the value of $d$ is needed only when $\alpha = \beta = 3$, the state can be encoded into only 26 values, and then the table can be compacted to only 208 entries.

--- abstract: 'It is shown that the radial Schroedinger equation for a power law potential and a particular angular momentum may be transformed using a change of variable into another Schroedinger equation for a different power law potential and a different angular momentum. It is shown that this leads to a mapping of the spectra of the two related power law potentials. It is shown that a similar correspondence between the classical orbits in the two related power law potentials exists. The well known correspondence of the Coulomb and oscillator spectra is a special case of a more general correspondence between power law potentials.' author: - | C. V. Sukumar\ [*Wadham College,*]{}\ [*University of Oxford, Oxford OX1 3PN, U.K.* ]{} title: '**Equivalent power law potentials**' --- Introduction ============ In this study we investigate the circumstances under which Classical dynamics and Quantum Mechanics induce relationships between classical orbits, eigenvalue spectra and phaseshifts of different dynamical systems. We study in particular power law potentials which belong to a special category in the sense that they permit a unique scaling of length and energy which lead to dimensionless equations in Classical and Quantum Mechanics. The dimensionless equations allow certain transformations which reveal a connection between the orbits and spectra of two related power law potentials. It has been noted in earlier literature that the solutions to the Schroedinger equation for a Simple Harmonic Oscillator (SHO) may be related to the boundstate solutions in a Coulomb potential, a relationship that is also evident in the connection between Hermite polynomials and Laguerre polynomials in the mathematical literature. In this report we examine the possibility of more general connections between power law potentials. In section 2 of this paper we study the relation between the Schroedinger equations of two related power law potentials. This issue was discussed by Quigg and Rossner in Physics Reports [**56**]{} (1979) in their study of quarkonium states using non-relativistic Quantum Mechanics. We follow their method of analysis and extract additional features which could prove useful. In sections 3 and 4 we study the same issue from the point of view of semi-classical and Classical Mechanics and attempt to establish which of the features that appear in Classical Mechanics are preseved in the passage to Quantum Mechanics. Power law potentials in Quantum Mechanics ========================================= We start from the radial Schroedinger equation for a power law potential with power exponent $\nu_1$ and angular momentum $l_1$ $$\frac{\hbar^2}{2\mu_1} \ \frac{\partial^2u_1}{\partial r^2}\ +\ \Big(E_1\ -\ \lambda_1\ r^\nu\ -\ \frac{l_1(l_1+1}{2\mu_1 r^2}\ \Big) \ u_1\ =\ 0$$ For power law potentials a scaling length and a scaled energy may be identified $$a_1\ =\ \Big(\frac{\hbar^2}{2\mu_1|\lambda_1|}\Big)^{\frac{1}{\nu_1+2}}\ ,\ r\ =\ a_1\ \rho_1\ ,\ E_1\ =\ \frac{\hbar^2}{2\mu_1 a_1^2}\ \epsilon_1$$ in terms of which the Schroedinger equation in dimensionless form becomes $$\frac{\partial^2 u_1}{\partial\rho_1^2}\ +\ \Big(\epsilon_1\ -\ \rho_1^{\nu_1}\ -\ \frac{l_1(l_1+1)}{\rho_1^2} \Big)\ u_1\ =\ 0 \label{eq:I0}$$ We now introduce a new variable and a new function through the relations $$\rho_1^{\nu_1}\ =\ z^{-\nu_2}\ ,\ u_1(\rho_1)\ =\ z^{-\frac{\nu_1+\nu_2}{2\nu_1}}\ v(z)\label{eq:I1}$$ to tranform the radial equation to the form $$\Big(\ \Big(\frac{\nu_1}{\nu_2}\Big)^2 z^{2\big(1+\frac{\nu_1}{\nu_2}\big)} \Big[\frac{\partial^2v}{\partial z^2} + \Big(1 - \frac{\nu_2^2}{\nu_1^2}\Big) \frac{v}{4z^2}\Big] + \Big[\epsilon_1 - z^{-\nu_2} - \frac{l_1(l_1+1)}{z^2} z^{2\big(1+\frac{\nu_1}{\nu_2}\big)}\Big] v\Big)\ z^{-\frac{\nu_1+\nu_2}{2\nu_1}} \ =\ 0$$ If we now impose the conditions $$\begin{aligned} &2\Big(1 +\ \frac{\nu_2}{\nu_1}\Big)\ +\ \nu_2\ =\ 0\ \rightarrow\ \frac{1}{\nu_1}\ +\ \frac{1}{\nu_2}\ +\ \frac{1}{2}\ =\ 0 \label{eq:I2}\\ &l_1(l_1+1) + \frac{1}{4}\Big(1 - \frac{\nu_1^2}{\nu_2^2}\Big) = l_2( l_2+1)\ \frac{\nu_1^2}{\nu_2^2}\ \rightarrow\ \big(l_1 + \frac {1}{2}\Big)^2 \nu_2^2\ =\ \Big(l_2 + \frac{1}{2}\Big)^2 \nu_1^2 \label{eq:I3}\end{aligned}$$ the resulting equation is $$\frac{\partial^2v}{\partial z^2}\ +\ \Big(-\frac{\nu_2^2}{\nu_1^2}\ +\ \epsilon_1 \frac{\nu_2^2}{\nu_1^2}\ z^{\nu_2}\ -\ \frac{l_2(l_2+1)}{\rho_1^2}\Big) v\ =\ 0$$ A new scaling length and scaled energy defined by $$v(z) = u_2( \rho_2)\ ,\ z = a_2\ \rho_2\ \ ,\ a_2^{\nu_2+2}\ \epsilon_1 \Big(\frac{\nu_2}{\nu_1}\Big)^2\ =\ 1\ ,\ \epsilon_2 = - a_2^2\ \frac{\nu_2^2}{\nu_1^2}\ =\ - \epsilon_1^{\frac{\nu_1}{\nu_2}}\Big(-\frac{\nu_1}{\nu_2}\Big)^{\nu_1} \label{eq:I4}$$ may now be identified which leads to the transformed radial equation $$\frac{\partial^2 u_2}{\partial \rho_2^2}\ +\ \Big( \epsilon_2\ +\ \rho_2^{\nu_2}\ -\ \frac{l_2(l_2+1)}{\rho_2^2} \Big)\ u_2\ =\ 0 \label{eq:I5}$$ Comparison of (\[eq:I0\]) and (\[eq:I5\]) shows that the radial equation for a confining potential with $0\le \nu_1 \le \infty$ and $\epsilon_1$ positive can be transformed to a new radial equation for an attractive singular potential with $-2 \le \nu_2\le 0$ , $ \epsilon_2$ negative and the eigenvalue spectra of the confining potential and the singular potential are related by (\[eq:I4\]). If the exponents are restricted to lie in the range $[-2,\infty]$ (\[eq:I2\]) guarantees that $\nu_1$ and $\nu_2$ are always opposite in sign and the choice $$\Big(l_1 + \frac {1}{2}\Big)\ \nu_2\ =\ -\Big(l_2 + \frac{1}{2}\Big)\ \nu_1 \label{eq:I6}$$ which is consistent with (\[eq:I3\]) guarantees that a positive value of $l_1$ leads to a positive value of $l_2$ and ensures that the solution of the transformed equation vanishes as $\rho_2\rightarrow 0$. Upto this point our derivation parallels the derivation given by Quigg and Rossner in Physics Reports [**56**]{} 1979, pages 191-2. It is possible to take this discussion further which we now proceed to do. If in addition (\[eq:I6\]) transforms an integer value of $l_1$ to an integer value of $l_2$ then the mapping we have discussed relates the eigenvalue spectrum of a potential representing a possible physical system to the eigen value spectrum of another potential representing another possible physical system. If $l_1$ and $l_2$ are positive integers with $l_1 > l_2$ then the spectrum of a confining power law potential with positive exponent $\nu_1$ and angular momentum $l_1$ is related to the spectrum of an attractive potential with negative exponent $\nu_2$ and angular momentum $l_2$ if the exponents are such that $$\nu_1 = \frac{4(l_1-l_2)}{2l_2+1}\ ,\ \nu_2 = -\frac{4(l_1-l_2)}{2l_1+1}\ ,\ \rightarrow\ \epsilon_2\ =\ -\Big(\frac{2l_1+1}{2l_2+1}\Big)^{\nu_1}\ \Big(\frac{1}{\epsilon_1}\Big)^{\frac{2l_1+1}{2l_2+1}} \label{eq:I7}$$ A simple example of this relationship is realised by the choice $l_1=3l_2+1$ which yields $\nu_1=4,\ \nu_2 =-\frac{4}{3}$ and gives rise to the identification that the eigenvalues for $l_1=1,4,7,..$ in the potential $\rho_1^4$ are related to the eigenvalues for $l_2=0,1,2,..$ of the attractive potential $-(\rho_2)^{-\frac{4}{3}}$ by the mapping $\epsilon_2 = - \frac{3^4}{{\epsilon_1}^3}$. Another example of this relationship is realised by the choice $l_1=5l_2+2$ which yields $\nu_1=8,\ \nu_2 =-\frac{8}{5}$ and leads to the identification that the eigenvalues for $l_1=2,7,12,..$ in the potential $\rho_1^8$ are related to the eigenvalues for $l_2=0,1,2,..$ of the attractive potential $-(\rho_2)^{-\frac{8}{5}}$ by the mapping $ \epsilon_2 = -\frac{5^8}{{\epsilon_1}^5}$. The two examples we have given illustrate the equivalence of two sets of power law potentials but it is evident that (\[eq:I7\]) provides an entire family of pairs of potentials whose eigenvalue spectra are related. We have shown that the power law potentials $V_1=\rho_1^{\nu_1}$, $0\le\nu_1\le\infty$ and $V_2=-\rho_2^{\nu_2}$, $-2\le\nu_2\le0$ have related eigenvalue spectra if the conditions $$(\nu_1 + 2)\ (\nu_2 + 2)\ =\ 4\ \ {\hbox {and}}\ \ \frac{2l_1+1}{\nu_1}\ +\ \frac{2l_2+1}{\nu_2}\ =\ 0 \label{eq:G8}$$ are satisfied and the spectral relation may be given in the form $${\sqrt{\nu_1+2}}\ \big(\epsilon_1\big)^{\frac{1}{\nu_2}}\ =\ {\sqrt{\nu_2+2}}\ \big(-\epsilon_2\big)^{\frac{1}{\nu_1}} \label{eq:I8}$$ which may also be given in the form $$\nu_1\ \log|\epsilon_1|\ -\ \Big(\frac{\nu_1^2}{\nu_1+2}\Big)\ \log(\nu_1+2)\ =\ \nu_2\ \log|\epsilon_2|\ -\ \Big(\frac{\nu_2^2}{\nu_2+2}\Big)\ \log(\nu_2+2)$$ exhibiting a symmetrical structure of the mapping. Relation between the spectra of the Oscillator and the Coulomb potentials ------------------------------------------------------------------------- The choice $\nu_1=2$ corresponds to the radial equation for the oscillator potential and it may be shown that for any values of angular momenta $l_1$, whether integer or not, the solutions of (\[eq:I0\]) may be given as $$u_1\ =\ \rho_1^{l_1+1}\ \Big(\exp \frac{-\rho_1^2}{2}\Big)\ M\Big(\frac{2l_1+3-\epsilon_1}{4},l_1+\frac{3}{2},\rho_1^2\Big) \label{eq:I9}$$ where $M(a,b,z)$ is one of the solutions of Kummer’s equation (Abramowitz and Stegun 1965) in the form $$M(a,b,z)\ =\ 1\ +\ \frac{a}{b} z\ +\ \frac{a(a+1)}{b(b+1)} z^2\ +\ ...$$ with polynomial structure when $a=-n$, where $n$ is an integer $\ge 0$. Hence for energies $\epsilon_1=4n+2l_1+3$ the solution for $u_1$ given in (\[eq:I9\]) is normalisable and it satisfies boundstate boundary conditions at the origin and at $\infty$, irrespective of whether $l_1$ is an integer or not, but it does not correpond to a physical state unless the angular momentum is an integer. However (\[eq:I2\]) and (\[eq:I6\]) show that the radial equation for $l_1=2l_2+\frac{1}{2},\ \nu_1=2$ is related to the radial equation for $\nu_2=-1$ and integer $l_2$. This gives rise to the identification that the radial equation for the oscillator potential for unphysical angular momenta $l_1=\frac{1}{2},\ \frac{5}{2}, \frac{9}{2},..$ at energies $\epsilon_1=4(n+l_2+1)$ can be transformed to a new radial equation using the coordinate transformation $\rho_1^2 = \frac{\rho_2}{n+l_2+1}$, leading to (\[eq:I5\]) with $\nu_2=-1$, which has the solutions $$u_2\ = \rho_2^{l_2+1}\ \Big(\exp \frac{-\rho_2}{2(n+ l_2+1)}\Big)\ M\Big(-n,2l_2+2, \frac{\rho_2}{n+l_2+1}\Big)$$ which correspond to true bounstate solutions in an attractive Coulomb potential for physical values of angular momenta $l_2=0,1,2,...$ with boundstate energies $ \epsilon_2 = -\frac{4}{\epsilon_1^2}= -\frac{1}{4}\ \frac{1}{(n+l_2+1)^2}$. Using eqs.(\[eq:I0\]),(\[eq:I1\]) and (\[eq:I4\])-(\[eq:I6\]) the relationship between the solutions $u_1$ and $u_2$ can be established to be $$u_2(\rho_2)\ \sim \rho_2^{\frac{1}{4}}\ u_1(\rho_1),\ \ \rho_2\ =\ \frac{\epsilon_1}{4} \rho_1^2,\ \ l_1\ =\ 2 l_2 + \frac{1}{2}$$ This relationship is the well known SHO-Coulomb correspondence which establishes a relation between the oscillator solutions expressed in terms of polynomials and the Coulomb boundstate solutions expressed in terms of polynomials. It is to be emphasised that unlike the examples given earlier, the oscillator-Coulomb correspondence is not a relation between two physical systems in Quantum Mechanics but is a relation between two solutions of two different radial equations. However in Classical Mechanics the restriction of the angular momentum to values which are integer units of $\hbar$ does not apply and there is an exact equivalence of the SHO as a physical system to the negative energy states of a physical system correcponding to bounded motion in a Coulomb potential. The exact correspondence of the classical orbits of the SHO and the bound orbits in a Coulomb potential will be further examined in section 4. The potential $V_1=\rho_1^{\nu_1}$ with positive values of the exponent is a confining potential which has only bound states and no scattering states. The potential $V_2=-\rho_2^{\nu_2}$ with a negative exponent is a singular potential which has both bound states and scattering states. The transformation we have studied in this section establishes a correspondence between the bound states of $V_1$ and $V_2$ when the exponents and the angular momenta fulfill certain conditions. However the significance of the transformation of the scattering states of $V_2$ when there is no corresponding scattering states of $V_1$ remains to be understood. Semiclassical analysis of power law potentials ============================================== The correspondence between the radial Schroedinger equations for two related power law potentials suggests that there must be a similar correspondence between the classical action integrals too which woluld induce a relation between the WKB quantisation conditions for the two power law potentials. In semi-classical analysis the centrifugal potential in Quantum Mechanics is replaced using the Langer modification $l(l+1)\ \rightarrow\ (l+\frac{1}{2})^2$ is implemented. In Classical Mechanice the centrifugal potential is $\frac{l^2}{\rho^2}$. It is interesting to note that the Langer modification of the angular momentum of the classical value of the angular momentum $l$ through the replacement $l\rightarrow (l+\frac{1}{2})$ in the semi-classical analysis naturally arises in the study of the tranformations linking the radial Schroedinger equations for two power law potentials. The action integral associated with the attractive singular potential $v_2=-\rho_2^{\nu_2}$, $-2\le \nu_2 \le 0$, for the negative energy $\epsilon_2$ and angular momentum $l_2$ is $$S\ = \int_{t_{1,2}}^{t_{2,2}} {\sqrt {\epsilon_2\ +\ \rho_2^{\nu_2}\ -\ \frac{(2l_2+1)^2}{4\rho_2^2}}}\ \ d\rho_2 \label{eq:I12}$$ where $t_{1,2}$ and $t_{2,2}$ are the classical turning points where the integrand vanishes. Under the tranformation to a new variable $$\rho_2\ =\ {\sqrt {-\frac{1}{\epsilon_2}}}\ \Big(\frac{2+\nu_2}{2}\Big)\ \rho_1^{\frac{2}{2+\nu_2}}\ \ \ ,\ \ \ d\rho_2\ =\ {\sqrt {-\frac{1}{\epsilon_2}}}\ \rho_1^{\frac{-\nu_2}{2+\nu_2}}\ d\rho_1 \label{eq:I13}$$ the action integral transforms to $$S\ = \int_{t_{1,1}}^{t_{2,1}} {\sqrt{-\rho_1^{\frac{-2\nu_2}{2+\nu_2}} + \Big(\frac{2+\nu_2}{2}\Big)^{\nu_2} \Big(-\frac{1}{\epsilon_2}\Big)^{\frac{2+\nu_2}{2}}\ -\ \frac{(2l_2+1)^2}{4\rho_1^2} \Big(\frac{2}{2+\nu_2}\Big)^2 }} \ d\rho_1$$ where $t_{1,1}$ and $t_{2,1}$ are the turning points in the transformed coordinate $\rho_1$. If we now identify a new exponent, a new energy and a new angular momentum through the definitions $$\nu_1 = -\frac{2\nu_2}{2+\nu_2}\ ,\ \epsilon_1 = \Big(-\frac{1}{\epsilon_2}\Big)^{\frac{2+\nu_2}{2}}\ \Big(\frac{2+\nu_2}{2}\Big)^{\nu_2}\ \ ,\ 2 l_1 + 1 = \Big(\frac{2}{2+\nu_2}\Big) (2 l_2 + 1) \label{eq:I14}$$ which are exactly the same transformations as identified in eqs. (\[eq:I2\]),(\[eq:I4\]) and (\[eq:I6\]) from the study of the radial Schroedinger equation, then $S$ can be brought to the form $$S\ = \int_{t_{1,1}}^{t_{2,1}} {\sqrt {\epsilon_1\ -\ \rho_1^{\nu_1}\ -\ \frac{(2l_1+1)^2}{4\rho_1^2}}}\ \ d\rho_1 \label{eq:I15}$$ which can now be interpreted as the action integral associated with the potential $v_1=\rho_1^{\nu_1}$ for the energy $\epsilon_1$ and angular momentum $l_1$ and $t_{1,1}$ and $t_{2,1}$ are the classical turning points of the new potential. It is also possible to start from (\[eq:I15\]) and use the inverse transformation $$\begin{aligned} \rho_1\ &=\ {\sqrt {\frac{1}{\epsilon_1}}}\ \Big(\frac{2+\nu_1}{2}\Big)\ \rho_2^{\frac{2}{2+\nu_1}}\ \ \ ,\ \ \ d\rho_1\ =\ {\sqrt {\frac{1}{\epsilon_1}}}\ \rho_2^{\frac{-\nu_1}{2+\nu_1}}\ d\rho_2 \label{eq:J13}\\ \nu_2 &= -\frac{2\nu_1}{2+\nu_1}\ ,\ \epsilon_2 = - \Big(\frac{1}{\epsilon_1}\Big)^{\frac{2+\nu_1}{2}}\ \Big(\frac{2+\nu_1}{2}\Big)^{\nu_1}\ \ ,\ 2 l_2 + 1 = \Big(\frac{2}{2+\nu_1}\Big) (2 l_1 + 1) \label{eq:J14}\end{aligned}$$ and recover (\[eq:I12\]). The exact identity of the action integrals for the two related power law potentials under the mapping defined by (\[eq:I13\]) and (\[eq:I14\]) or (\[eq:J13\]) and (\[eq:J14\]) then leads to a mapping of the eigenvalues through the WKB quantisation condition appropriate to the radial equation. For non-singular potentials the quantisation condition can be shown to be $S= (n-\frac{1}{4}) \pi,\ n=1,2,3,..$ . However the quantisation condition for singular potentials requires subtle handling. The action integral for singular potentials given by (\[eq:I12\]) can be performed when $l_2=0$. The semi-classical quantisation condition for singular potentials and positive integer values of $l_2$ has been considered carefully by Quigg and Rosner (p203) and using their analysis it can be shown that the semi-classical spectrum of the singular potential $v_2$ is given by $$\epsilon_2(n,l_2)\ =\ -\Big[A_2\ \Big(n - \frac{1}{2}\ \frac{1+\nu_2-2l_2}{2+\nu_2}\Big)\Big]^{\frac{2\nu_2}{2+\nu_2}}\ \ ,\ \ A_2\ =\ 2\ {\sqrt{\pi}}\ \frac{\Gamma\Big(-\frac{1}{\nu_2}\Big)}{\Gamma\Big(-\frac{1}{2}-\frac{1}{\nu_2}\Big)}$$ The equality of the action integrals in (\[eq:I12\]) and (\[eq:I15\]) under the mapping given in (\[eq:I14\]) enables the determination of the WKB eigenvalues of $v_1$ also. The WKB eigenvalues for the potential $v_1$ for integer values of $l_1$ can be given in the form (Quigg and Rosner p205) $$\epsilon_1(n,l_1)\ =\ \Big[A_1\ \Big(n + \frac{l_1}{2} - \frac{1}{4}\Big)\Big]^{\frac{2\nu_1}{2+\nu_1}} \ \ ,\ \ A_1\ =\ {\sqrt{\pi}}\ (2+\nu_1)\ \frac{\Gamma\Big(\frac{1}{2} + \frac{1}{\nu_1}\Big)}{\Gamma\Big(\frac{1}{\nu_1}\Big)} \label{eq:G9}$$ It is simple to to verify that when the conditions (\[eq:G8\]) are satisfied then the semi-classical eigenvalues for the potentials $v_1$ and $v_2$ given above satisfy the same condition (\[eq:I8\]) as that satisfied by the exact eigen values from Quantum Mechanics. The approximate degeneracy in the spectra of $v_1=\rho_1^{\nu_1}, \nu_1>0$ -------------------------------------------------------------------------- The WKB spectrum of $v_1$ implies that the spectrum of a confining power law potential $v_1$ depends only on the combination $(2n+l_1)$ and exhibits the same degeneracy as the 3-d oscillator. This WKB result implies that for fixed $l, n>>l$, the exact Quantum Mechanical results for the spectra of power law potentials with positive exponents should also exhibit the same degeneracy. This feature can be understood from the point of view of Super Symmetric Quantum Mechanics (SUSY) which establishes a relation between a deep singular potential and a phase equivalent shallow potential which is constructed by removing a certain number $N$ of the lowest boundstates of the deep potential so that the spectrum of the shallow potential constructed by this procedure is identical to that of the deep potential except for missing the lowest $N$ eigenvalues of $V_{deep}$. It has been shown that (Baye 1987, Sukumar 1985,2005) $$\begin{aligned} V_{shallow}^{(N)} \ &=\ V_{deep}\ -2 \frac{\partial^2}{\partial\rho^2} \Big(\ln Det M\Big)\ \ ,\ \ M_{jk}\ =\ \int_0^\rho R_j^{(d)} R_k^{(d)} \ d\rho,\ \ j,k = 1,2,..,N \\ Lt_{\rho\rightarrow 0}\ \ V_{shallow}^{(N)} &\sim \frac{(l + 2N) (l + 2N + 1)}{\rho^2}\ \ {\hbox {if}}\ \ Lt_{\rho\rightarrow 0}\ \ V_{deep} \sim \frac{l(l + 1)}{\rho^2} \label{eq:S1}\\ Lt_{\rho\rightarrow \infty}\ \ V_{shallow}^{(N)} &= \ Lt_{\rho\rightarrow \infty}\ \ V_{deep} \end{aligned}$$ where $R_j^{(d)}$ are the radial eigenfunctions in the deep potential. The enhanced centrifugal barrier of the shallow potential with a $(2N+l)$ dependence seen in (\[eq:S1\]) is a short range feature valid only for $\rho<<1$. However for power law potentials with positive exponents the centrifugal potential is not significant in the long range as the $\rho^\nu$ part will dominate in the limit $\rho\rightarrow\infty$. Hence a shallow potential constructed from a power law potential by removing the lowest bound states by the SUSY procedure may, to a good approximation, be viewed as a power law potential plus a centrifugal potential corresponding to an enhanced angular momentum barrier. These results together with (\[eq:G9\]) imply that the $n^{th}$ semi-classical eigenvalue of the potential $v_1^{(eff)}=(\rho_1^{\nu_1} +$ the centrifugal potential for angular momentum $l=2l_1)$ depends on the quantum number $(n+l_1),\ n=1,2,..,$ and is degenerate with the $n^{th}$ semi-classical eigenvalue of the ’shallow’ potential ${\bar v}_1^{(l_1)}$ constructed by eliminating the lowest lying $l_1$ eigenstates of the ’deep’ potential $v_1=\rho_1^{\nu_1}$ whose eigenvalues depend on the quantum number $n=1,2,..\ $. Similarly the $n^{th}$ semi-classical eigenvalue of the potential $v_1^{(eff)}=(\rho_1^{\nu_1} +$ the centrifugal potential for angular momentum $l=2l_1+1)$ depends on the quantum number $(n+l_1+\frac{1}{2}),\ n=1,2,..,$, and is degenerate with the $n^{th}$ semi-classical eigenvalue of the ’shallow’ potential ${\bar v}_1^{(l_1)}$ constructed by eliminating the lowest lying $l_1$ eigenstates of the ’deep’ potential $\rho_1^{\nu_1} +\frac{2}{\rho_1^2}$ whose eigenvalues depend on the quantum number $ (n+\frac{1}{2}),\ n=1,2,..\ $. The potentials identified above as having degenerate spectra are not identical but the semi-classical quantisation formulae suggest that the difference $\Delta v$ between the potentials with the degenerate spectra does not play a significant role when determining the eigenvalues of states with $n>>l$. Thus the SUSY construction elucidates the feature that in the semi-classical limit, $l>>1, n>>l,$ the eigenvalues of the power law potentials with positive exponents will exhibit a $(2n+l)$ dependence. This result is also in agreement with the exactly solvable problem of the limit $\nu_1\rightarrow \infty$ which corresponds to a particle confined to move inside a unit sphere, ([*[i.e]{}*]{}) a vanishing potential inside a sphere radius $\rho_1=1$ and an infinie potential at $\rho_1=1$ representing an impenetrable wall. The radial equation for a particle inside a spherical box may be solved and the radial solution which is regular at the origin is $$R\ =\ {\sqrt{\rho_1}}\ J_{l+\frac{1}{2}}({\sqrt{\epsilon_1}}\rho_1)\ \ ,\ \ Lt_{k\rho\rightarrow \infty} \ J_{l+\frac{1}{2}}(k\rho) \sim \sin\Big(k\rho - l \frac{\pi}{2}\Big)$$ The eigenvalues arise from the requirement that the radial solution should vanish at the infinite wall, ([*[i.e]{}*]{}) when $\rho_1\rightarrow 1$. Hence the eigenvalues are related to the zeros of Spherical Bessel functions. When the argument of the Bessel function is large the asymptotic limit of the zeros can be estimated using McMahon’s expansion (Abramowitz and Stegun p371) which locates the $n^{th}$ zero $Z_n$ at $$Z_n\ \sim \ \beta\ -\ \frac{\sigma-1}{8\beta}\ -\ O\Big(\beta^{-3}\Big)\ \ ,\ \ \beta\ =\ \Big(n +\frac{l}{2}\Big) \pi\ \ , \ \ \sigma\ =\ (2l+1)^2$$ The semi-classical energy eigenvalues for a particle in a spherical box are given by $$Lt_{n\rightarrow\infty}\ \epsilon_1(n,l)\ \sim\ \Big((2n\ +\ l)\ \frac{\pi}{2}\Big)^2$$ which clearly exhibits a (2n+l) degeneracy. The (2n+l) degeneracy is an exact result for the 3-d oscillator but it is also a very good approximation for all power law potentials with positive exponents in the semi-classical limit of large quantum numbers $n >> l >> 1$. Gaussian potentials are often used to describe short range potentials in Nuclear Physics. Numerical calculation of the eigenvalues of a Gaussian potential shows that the eigenvalue spectrum exhibits a (2n+l) degeneracy to a good approximation. It is therefore possible to conjecture, without explicit proof, that the (2n+l) degeneracy is a good approximation for a large class of potentials in the limit of large quantum numbers. Classical Mechanics of power law potentials =========================================== In classical Mechanics the energy and angular momentum conservation laws lead to an equation for the orbit. For a particle in a potential $V_1=\rho_1^{\nu_1}$, $0\le\nu_1\le\infty$, with angular momentum $l_1$ and positive energy $\epsilon_1$, $$\begin{aligned} \epsilon_1\ &=\ \Big(\frac{\partial\rho_1}{\partial t}\Big)^2\ +\ \frac{l_1^2}{\rho_1^2}\ +\ \rho_1^{\nu_1}\ ,\ \ \Big(\frac{\partial\rho_1}{\partial t}\Big)^2\ \frac{\partial\theta_1}{\partial t}\ =\ l_1\ =\ {\hbox {constant}} \\ \theta_1\ &=\ l_1\ \int \frac{d\rho_1}{\rho_1^2}\ \Big(\epsilon_1 - \frac{l_1^2}{\rho_1^2} - \rho_1^{\nu_1}\Big)^{-\frac{1}{2}} \label{eq:I16}\end{aligned}$$ Similarly for a particle in an attractive singular potential $V_2=-\rho_2^{\nu_2}$, $-2\le\nu_2\le0$, with angular momentum $l_2$ and negative energy $\epsilon_2$ moving in a bound orbit $$\begin{aligned} -|\epsilon_2|\ &=\ \Big(\frac{\partial\rho_2}{\partial t}\Big)^2\ +\ \frac{l_2^2}{\rho_2^2}\ -\ \rho_2^{\nu_2}\ ,\ \ \Big(\frac{\partial\rho_2}{\partial t}\Big)^2\ \frac{\partial\theta_2}{\partial t}\ =\ l_2\ =\ {\hbox {constant}} \\ \theta_2\ &=\ l_2\ \int \frac{d\rho_2}{\rho_2^2}\ \Big(-|\epsilon_2| - \frac{l_2^2}{\rho_2^2} + \rho_2^{\nu_2}\Big)^{-\frac{1}{2}} \label{eq:I17}\end{aligned}$$ It is to be noted that in Classical Mechanice the centrifugal term is $\frac{l^2}{{\rho}^2}$ without the Langer modification used in the Semi-classical analysis. The equation of the orbit can also be found directly from the classical action integral by differentiation $$\begin{aligned} \theta_1\ &=\ -\frac{\partial S_1}{\partial l_1}\ \ ,\ S_1\ =\ \int_{t_{1,1}}^{t_{2,1}} {\sqrt {\epsilon_1\ -\ \rho_1^{\nu_1}\ -\ \frac{l_1^2}{\rho_1^2}}}\ \ d\rho_1 \\ \theta_2\ &=\ -\frac{\partial S_2}{\partial l_2}\ \ ,\ S_2\ =\ \int_{t_{1,2}}^{t_{2,2}} {\sqrt {\epsilon_2\ +\ \rho_2^{\nu_2}\ -\ \frac{l_2^2}{\rho_2^2}}}\ \ d\rho_2\end{aligned}$$ where $(t_{1,1},t_{1,2})$ and $(t_{2,1},t_{2,2})$ are the classical turning points in the two potentials. It may be shown by starting from the expression for $S_2$ and using the transformation of variables in (\[eq:I13\]) and the mapping $$\nu_1 = -\frac{2\nu_2}{2+\nu_2}\ ,\ \epsilon_1 = \Big(-\frac{1}{\epsilon_2}\Big)^{\frac{2+\nu_2}{2}}\ \Big(\frac{2+\nu_2}{2}\Big)^{\nu_2}\ \ ,\ l_1 = \Big(\frac{2}{2+\nu_2}\Big) l_2$$ that $S_2=S_1$. The mapping given above differs slightly from the transformation in (\[eq:I14\]) due to the fact that in the absence of Langer modification the relation between the angular momenta becomes $\nu_1 l_2= -\nu_2 l_1$. The equality of the classical actions in the two related power law potentials induces the relation $$\theta_2\ =\ -\frac{\partial S_2}{\partial l_2}\ =\ -\frac{\partial l_1}{\partial l_2} \frac{\partial S_1}{\partial l_1}\ =\ -\frac{\nu_1}{\nu_2}\ \theta_1$$ This relation may also be established by staring from (\[eq:I17\]) and using a transformation of coordinates to relate it to (\[eq:I16\]). The general correspondence between the orbital equations in the two related potentials may be given in the form $$\begin{aligned} \rho_1\ &=\ F(\theta_1,\ l_1,\ \epsilon_1) \ ,\ \rho_2\ =\ \frac{2}{2+\nu_1} \frac{1}{{\sqrt{-\epsilon_2}}}\ \rho_1^{\frac{2+\nu_1}{2}}\\ \theta_1\ &=\ \frac{2\theta_2}{2+\nu_1}\ ,\ l_1 = \frac{2+\nu_1}{2} \ l_2\ ,\ \epsilon_1 =\ \Big(\frac{2+\nu_1}{2}\Big)^{\frac{2\nu_1}{2+\nu_1}}\ \Big(-\epsilon_2\Big)^{-\frac{2}{2+\nu_1}} \end{aligned}$$ where the orbital function $F(\theta_1,l_1,\epsilon_1)$ is found by integration of (\[eq:I16\]). The mapping between the parameters of the two orbits has a symmetrical structure and so it is also possible to start from the orbital equation for $V_2$ and find the equivalent orbit of the tranformed potential $V_1$ using the mapping $$\begin{aligned} \rho_2\ &=\ F(\theta_2,\ l_2,\ \epsilon_2) \ ,\ \rho_1\ =\ \frac{2}{2+\nu_2} \frac{1}{{\sqrt{\epsilon_1}}}\ \rho_2^{\frac{2+\nu_2}{2}}\\ \theta_2\ &=\ \frac{2\theta_1}{2+\nu_2}\ ,\ l_2 = \frac{2+\nu_2}{2} \ l_1\ ,\ \epsilon_2 =\ -\Big(\frac{2+\nu_2}{2}\Big)^{\frac{2\nu_2}{2+\nu_2}}\ \Big(\epsilon_1\Big)^{-\frac{2}{2+\nu_2}} \end{aligned}$$ As an example of the orbital relation we consider the orbit in a three dimensional oscillator potential $V_1=\rho_1^2$ for which the equation of the orbit may be found by direct integration of (\[eq:I16\]) and may be shown to be of the form $$\frac{1}{\rho_1^2}\ =\ \frac{\epsilon_1}{2 l_1^2}\ \Big(1\ -\ {\sqrt{1 - \frac{4 l_1^2}{\epsilon_1^2}}}\ \cos 2\theta_1\Big) \label{eq:I18}$$ Only bound orbits exist with $\epsilon_1\ge0$. Under the transformation $$\rho_2 = \frac{1}{2{\sqrt{-\epsilon_2}}}\ \rho_1^2\ ,\ \ \theta_1 = \frac{\theta_2}{2}\ ,\ l_1 = 2l_2\ ,\ \epsilon_1 = 2 (-\epsilon_2)^{-\frac{1}{2}} \label{eq:J18}$$ we obtain $$\frac{1}{\rho_2}\ =\ \frac{1}{2 l_2^2}\ \Big(1\ -\ {\sqrt{1 + 4\epsilon_2 l_2^2}}\ \cos\theta_2\Big) \label{eq:I19}$$ which is a bound orbit in the Coulomb potential $V_2=-\rho_2^{-1}$ with $\epsilon_2\le0$. The inverse transformation in this case is $$\rho_1 = \frac{2}{{\sqrt{\epsilon_1}}}\ \rho_2^{\frac{1}{2}}\ ,\ \theta_2\ =\ 2\theta_1\ ,\ l_2 = \frac{l_1}{2}\ ,\ \epsilon_2 =\ -\Big(\frac{1}{2}\Big)^{-2}\ \Big(\epsilon_1\Big)^{-2} \label{eq:J19}$$ which tranforms the orbit in (\[eq:I19\]) back to the orbit given in (\[eq:I18\]). The orbit in an attractive Coulomb potential given in (\[eq:I19\]) is also the correct solution to (\[eq:I17\]) for positive values of $\epsilon_2$ and may be identified as a hyperbola. The mapping given in (\[eq:J18\]) indicates that a positive value for $\epsilon_2$ corresponds to imaginary values of $\epsilon_1$ and the solution for the oscillator given in (\[eq:I18\]) becomes complex and this complex solution for the orbit would require an interpretation. This difficulty is perhaps also related to the question raised at the end of section 2.1 regarding the transformation of the radial Schroedinger equation for positive energy scattering states of singular potentials $V_2$ to the radial Schroedinger equation for imaginary $\epsilon_1$ states of confining potentials $V_1$. References ========== \[1\] C.Quigg and J.L.Rosner, 1979 [*[Physics Reports]{}*]{} [**56**]{} 167-235. \[2\] M.Abramowitz and I.Stegun, 1965 [*[Handbook of Mathematical Functions]{}*]{} Dover: New York 504 \[3\] C.V.Sukumar, 1985 [*[J.Phys. A: Math. Gen.]{}*]{} [**18**]{} 2937-56 \[4\] D.Baye, 1987 [*[Physical Review Letters]{}*]{} [**58**]{} 2738-41 \[5\] C.V.Sukumar, 2005 [*[Latin American School of Physics XXXV, ELAF, Supersymmetries in Physics and its applications, edited bt R.Bijker, et al.]{}*]{} AIP conference proceedings [**744**]{} 166-235

--- abstract: 'Recently we have studied in great detail a model of Hybrid Natural Inflation (HNI) by constructing two simple effective field theories. These two versions of the model allow inflationary energy scales as small as the electroweak scale in one of them or as large as the Grand Unification scale in the other, therefore covering the whole range of possible energy scales. In any case the inflationary sector of the model is of the form $V(\phi)=V_0 \left(1+a \cos(\phi/f)\right)$ where $0\leq a<1$ and the end of inflation is triggered by an independent waterfall field. One interesting characteristic of this model is that the slow-roll parameter $\epsilon(\phi)$ is a non-monotonic function of $\phi$ presenting a [*maximum*]{} close to the inflection point of the potential. Because the scalar spectrum $\mathcal{P}_s(k)$ of density fluctuations when written in terms of the potential is inversely proportional to $\epsilon(\phi)$ we find that $\mathcal{P}_s(k)$ presents a [*minimum*]{} at $\phi_{min}$. The origin of the HNI potential can be traced to a symmetry breaking phenomenon occurring at some energy scale $f$ which gives rise to a (massless) Goldstone boson. Non-perturbative physics at some temperature $T<f$ might occur which provides a potential (and a small mass) to the originally massless boson to become the inflaton (a pseudo-Nambu-Goldstone boson). Thus the inflaton energy scale $\Delta$ is bounded by the symmetry breaking scale, $\Delta\equiv V_H^{1/4} <f.$ To have such a well defined origin and hierarchy of scales in inflationary models is not common. We use this property of HNI to determine bounds for the inflationary energy scale $\Delta$ and for the tensor-to-scalar ratio $r$.' author: - | Gabriel Germán$^{a} \footnote{Corresponding author: gabriel@fis.unam.mx}$, Alfredo Herrera-Aguilar$^{b,c}$, Juan Carlos Hidalgo$^{a}$,\ Roberto A. Sussman$^{d}$, José Tapia$^{a,e}$\ \ [*$^a$Instituto de Ciencias Físicas,* ]{}\ [*Universidad Nacional Autónoma de México,*]{}\ [*Apdo. Postal 48-3, C.P. 62251 Cuernavaca, Morelos, México.*]{}\ \ [*$^b$Instituto de Física,* ]{}\ [*Benemérita Universidad Autónoma de Puebla,*]{}\ [*Apdo. Postal J-48, C.P. 72570 Puebla, Puebla, México.*]{}\ \ [*$^c$Instituto de Física y Matemáticas,*]{}\ [*Universidad Michoacana de San Nicolás de Hidalgo,*]{}\ [*Edificio C–3, Ciudad Universitaria, C.P. 58040 Morelia, Michoacán, México.*]{}\ \ [*$^d$Instituto de Ciencias Nucleares,* ]{}\ [*Universidad Nacional Autónoma de México,*]{}\ [*Apdo. Postal 70Ð543, 04510 México D. F., México.*]{}\ \ [*$^e$Centro de Investigación en Ciencias,* ]{}\ [*Universidad Autónoma del Estado de Morelos,*]{}\ [*Avenida Universidad 1001, Cuernavaca, Morelos 62209, México.*]{} title: General bounds in Hybrid Natural Inflation --- Introduction {#Intro} ============ In a recent article [@Ross:2016hyb] a model of inflation [@Guth:1980zm], [@Linde:1981mu], [@Albrecht:1982wi], [@Lyth:1998xn] of the hybrid type [@Linde:1994] has been studied with great detail. To show that it is posible within Hybrid Natural Inflation (HNI) [@Ross:2016hyb] to account for inflationary energy scales as small as the electroweak scale, or as large as the Grand Unification scale, two versions of the model have been constructed based on simple effective field theories. The resulting inflationary sector in any case is described by the following potential for the inflaton field $\phi$ $$V(\phi) = V_0\left(1+a\cos \left(\frac{\phi}{f} \right) \right), \label{pot}$$ where $a$ is a positive constant less than one and $f$ is the scale of (Nambu-Goldstone) symmetry breaking. Here the end of inflation is triggered by an independent sector waterfall field in a rapid phase transition. The potential in Eq.(\[pot\]) is reminiscent of Natural Inflation [@Freese:1990rb], [@Adams:1992bn], [@Freese:2014nla] where $a=1$ sets a vanishing cosmological constant. Here, however, $a$ can take any positive value less than one and as a result the scale $f$ can be sub-Planckian. Once the waterfall field triggers the end of inflation the inflaton fast rolls to a global minimum with vanishing energy. Hybrid Natural Inflation has also the interesting property that the slow-roll parameter $\epsilon(\phi)$ turns out to be a non-monotonic function of the inflaton [@German:2015qjq]. As a consequence the scalar spectrum of density perturbations develops a [*minimum*]{} (Fig.\[Espectro\]) for a value $\phi_{min}$ close to the inflection point of the potential. We know that inflation in HNI should start before $\phi$ reaches the minimum of the spectrum at $\phi_{min}$ because the spectrum has been observed to be decreasing during at least 8 e-folds of observable inflation. Thus, there should be at least 8 e-folds of inflation from $\phi_H$, at which observable perturbations are produced[^1], to $\phi_{min}$. This minimum amount of inflation with decreasing spectrum should give an upper bound for the tensor-to-scalar ratio $r$ and for the scale of inflation $\Delta$. The remaining $42-52$ e-folds of inflation would occur with an steepening spectrum thus care should be taken to not over-produce primordial black holes (PBH) [@Kohri:2007qn], [@Josan:2009qn], [@Carr:2009jm]. Also the fact that the inflationary energy scale $\Delta$ is bounded by the symmetry breaking scale $f$ imposes [*lower*]{} bounds to these quantities, whenever the minimum of the spectrum is reached after $N_{min}\leq 60$ e«folds of inflation. If all of inflation occurs without $\phi$ reaching $\phi_{min}$ no lower bounds are found. Our paper is organised as follows: in Section \[slow\] we briefly recall expressions for the slow-roll parameters and observables. We also give an effective field theory derivation of the model we study and the hierarchy of energy scales is discussed. As a warming up exercise we initially study in Section \[NI\] this hierarchy of scales in Natural Inflation (where $a=1$) and Section \[ENI\] deals with “extended” Natural Inflation (ENI) where $a$ is not set to unity from the beginning. This allows us to study the fine tuning of $a$ (to have a vanishing cosmological constant) in terms of the parameters of the model. From here we proceed in Section \[restricted\] to HNI where the hierarchy of energy scales together with the observation that the scalar spectrum is decreasing during $8<N_{min}<60$ e-folds of observable inflation determine bounds for the inflationary energy scale $\Delta$ and for the tensor-to-scalar ratio $r$, this we call the restricted case. In Section \[general\] we obtain general bounds in HNI dropping the previous requirement that $N_{min}$ e-folds of inflation occur with decreasing spectrum. We are able to find general bounds for all the parameters (and observables) of the model and to clearly understand how the scale of inflation in HNI is able to sweep all range of values, from vanishingly small to GUT scales. A brief discussion of constraints coming from Primordial Black Hole (PBH) abundances and considerations regarding low scales of inflation can be found in Section \[PBH\]. Finally Section \[conclusions\] contains our conclusions and a discussion of the main results. Slow-roll parameters, observables and model construction {#slow} ========================================================= In slow-roll inflation, the spectral indices are given in terms of the slow-roll parameters of the model, which involve the potential $V(\phi)$ and its derivatives (see e.g. [@Liddle:94], [@Liddle:2000cg]) $$\epsilon \equiv \frac{M^{2}}{2}\left( \frac{V^{\prime }}{V }\right) ^{2},\quad \eta \equiv M^{2}\frac{V^{\prime \prime }}{V}, \quad \xi_2 \equiv M^{4}\frac{V^{\prime }V^{\prime \prime \prime }}{V^{2}},\quad \xi_3 \equiv M^{6}\frac{V^{\prime 2 }V^{\prime \prime \prime \prime }}{V^{3}}, \label{Slowparameters}$$primes denote derivatives with respect to the inflaton $\phi$ and $M$ is the reduced Planck mass $M=2.44\times 10^{18} \,\mathrm{GeV}$. In what follows we set $M=1$. In the slow-roll approximation observables are given by (see e.g. [@Liddle:2000cg]) $$\begin{aligned} n_{t} &=&-2\epsilon =-\frac{r}{8} , \label{Int} \\ n_{s} &=&1+2\eta -6\epsilon , \label{Ins} \\ n_{sk} &=&\frac{d n_{s}}{d \ln k}=16\epsilon \eta -24\epsilon ^{2}-2\xi_2, \label{Insk} \\ n_{skk} &=&\frac{d^{2} n_{s}}{d \ln k^{2}}=-192\epsilon ^{3}+192\epsilon ^{2}\eta- 32\epsilon \eta^{2} -24\epsilon\xi_2 +2\eta\xi_2 +2\xi_3, \label{Inskk} \\ \mathcal{P}_s(k)&=&\frac{1}{24\pi ^{2}}\frac{V}{\epsilon }=A_s \left( \frac{k}{k_H}\right)^{n_s-1} . \label{IA} \end{aligned}$$ Here $n_{sk}$ denotes the running of the scalar index and $n_{skk}$ the running of the running, in a self-explanatory notation. All the quantities with a subindex ${}_H$ are evaluated at the scale $\phi_{H}$, at which observable perturbations are produced, some $50-60$ e-folds before the end of inflation. The density perturbation at wave number $k$ is $\mathcal{P}_s(k)$ with amplitude at horizon crossing given by $\mathcal{P}_s(k_H)\approx 2.2\times 10^{-9}$ [@Ade:2015xua], the scale of inflation is $\Delta$ with $\Delta \equiv V_{H}^{1/4}$. The tensor power spectrum parameterised at first order in the SR parameters is $$\mathcal{P}_t(k)=A_t \left( \frac{k}{k_H}\right)^{n_t} , \label{PotHNI}$$ it allows to define the tensor-to-scalar ratio as $r\equiv \mathcal{P}_t(k)/\mathcal{P}_s(k)$. The construction of the model can proceed by initially considering a potential of the form $$V\left(\Phi\right) = V_1\left(\Phi\right)+V_2\left(\Phi\right), \label {V}$$ where $$V_1\left(\Phi\right)= -m^2|\Phi|^2+\lambda |\Phi|^4+\bar{\Delta}^4. \label {Phi}$$ is the potential invariant under the $U(1)$ symmetry, $\Phi \rightarrow e^{i\alpha}\Phi$. The origin of the constant term $\bar{\Delta}^4$ above can be traced to terms in the higher energy sector of the theory. For positive mass-square $m^2$, $\Phi$ triggers spontaneous breaking of the $U(1)$ symmetry and the field $\Phi$ gets a vev given by $\tilde{f}$ $$<\Phi_0>\, = \tilde{f}= \frac{m}{\sqrt{2\lambda}}. \label {Phi}$$ Thus, the potential can be better parameterised by $$\Phi= \frac{1}{\sqrt{2}}\left(\rho+\tilde{f}\right)\, e^{i\frac{\phi}{\tilde{f}}}, \label {Phi}$$ where $\rho$ is a radial field around the minimum of the potential and $\phi$ a Goldstone boson associated with the $U(1)$ symmetry breaking. The term $V_2\left(\Phi\right)$ in Eq. (\[V\]) explicitly breaks the $U(1)$ symmetry and generates a mass for the Goldstone boson becoming $\phi$ a Pseudo Nambu-Goldstone boson. We can write a very simple form for $V_2$ $$V_2\left(\Phi\right)=\mu^2\left(\Phi^2+ \Phi^{*2}\right) \sim \frac{1}{2}\mu^2\tilde{f}^2\left(e^{i\frac{2\phi}{\tilde{f}}} + e^{-i\frac{2\phi}{\tilde{f}}}\right) \sim \mu^2\tilde{f}^2\cos\left(\frac{2\phi}{\tilde{f}} \right)\ , \label {explicit}$$ where $\mu$ is a constant with mass dimensions. It follows that the axion potential is $$V(\phi) =V_0\left[1+a\cos\left(\frac{\phi}{f} \right) \right], \label {pot1}$$ with $V_0$ defined by $V_0\equiv \bar{\Delta}^4-\frac{m^4}{4\lambda}$, $\tilde{f}=2 f$ and $$a\equiv \frac{4\mu^2 f^2}{V_0}. \label {apar}$$ The parameter $a$ is bounded as $0\leq a\leq1$. The limiting value $a=0$ reduces the potential to a cosmological constant while $a=1$ gives Natural Inflation. The origin of the two scales occurring in the potential Eq.(\[pot1\]) is, in principle, well understood and we also know that these scales satisfy a hierarchy: a symmetry breaking phenomenon at some energy scale $f$ gives rise to a (massless) Goldstone boson while non-perturbative physics at temperature $T<f$ provides a potential (and a small mass) to the originally massless boson becoming a pseudo-Nambu-Goldstone boson, the inflaton. In the case of the $QCD$ axion, for example, non-perturbative effects are due to instantons. Thus the inflationary energy scale $\Delta$ is bounded by the symmetry breaking scale $f$ as $\Delta\equiv V_H^{1/4} \approx V_0^{1/4} < f.$ In the following sections we use this hierarchy of scales to extract bounds for the observables. Natural Inflation {#NI} ================== As a warming up exercise we beguin with Natural Inflation (NI) [@Freese:1990rb]. The potential for the NI model is given by fixing $a=1$ in Eq.(\[pot1\]) above $$V(\phi) = V_0\left(1+\cos \left(\frac{\phi}{f} \right) \right). \label{Npot}$$ The coefficient of the $\cos \left(\frac{\phi}{f} \right)$-term has been fine-tuned to one so that the potential vanishes at its minimum. We will see that in NI the hierarchy $\Delta < f$ arises in a very natural way. Defining $c_{\phi}\equiv \cos \left( \frac{\phi }{f}\right)$ the slow-roll parameters relevant for what follows are $$\begin{aligned} \epsilon &=&\frac{1}{2f^2}\frac{1-c_{\mathrm{\phi}} }{ 1+c_{\mathrm{\phi}} } , \label{NIeps}% \\ \eta &=&-\frac{1}{f^2}\, \frac{c_{\mathrm{\phi}}}{1+c_{\mathrm{\phi}}} , \label{NIeta}\end{aligned}$$Defining $\delta_{n_s}\equiv 1-n_{s_H}$, $c_{H}\equiv \cos \left( \frac{\phi _{H}}{f}\right)$ and $c_{e}\equiv \cos \left( \frac{\phi _{e}}{f}\right)$, the functions at the observable scale and at the end of inflation, the equation for the spectral index Eq.(\[Ins\]) at $\phi _H$, can be written as $$\delta_{n_s}= \frac{3 -c_H}{f^2(1+c_{H})}, \label{NIspectral}$$ solving for $c_H$ we get $$c_{H_{NI}}\equiv\frac{3-f^2\delta_{n_s}} {1+f^2\delta_{n_s}}, \label{Nch}$$ where the subindices $NI$ means that $c_H$ has been determined from the NI potential Eq.(\[Npot\]). In the following sections dealing with the Hybrid Natural Inflation (HNI) potential instead of writing $HNI$ subindices, they will be simply dropped. The end of inflation is given by the saturation of the condition $\epsilon = 1$ $$c_{e_{NI}}\equiv\frac{1-2f^2} {1+2f^2}. \label{Nce}$$ The number of e-folds from $\phi_H$ to the end of inflation at $\phi _e$ is given by $$N\equiv -\int_{\phi _H}^{\phi_e}\frac{V({\phi })}{V^{\prime }({\phi })}{d}{\phi }=f^2\ln \left( \frac{1-c_{e_{NI}}}{1-c_{H_{NI}}}\right)=f^2\ln\left[\frac{2f^2(1+f^2 \delta_{n_s})}{(1+2f^2) (-1+f^2 \delta_{n_s}) } \right], \label{NN}$$ from where it follows that in NI consistency demands that $$f > \frac{1}{\sqrt{\delta_{n_s}}}. \label{Nfbound}$$ Assuming the spectral index $n_s$ is known, the number of e-folds $N$ fixes the parameter $f$ through Eq.(\[NN\]) which then fixes $\phi_H$ and $\Delta$. For numerical results we take $n_s=0.965$ [@Ade:2015xua] thus $\delta_{n_s}=0.035$ and $f \gtrsim 5.3$. From Eq.(\[IA\]) the scale of inflation can also be expressed in terms of $f$ $$\Delta=\left(24\pi ^{2}\epsilon(\phi_H)\mathcal{P}_s(k_H)\right)^{1/4} =\left(12\pi^2 \mathcal{P}_s(k_H) \frac{1-c_{H_{NI}} }{f^2\left(1+\,c_{H_{NI}}\right)}\right)^{1/4}=\left(6\,\pi^2 \delta_{n_s}\mathcal{P}_s(k_H)\left(1-\frac{1}{f^2 \delta_{n_s}}\right) \right)^{1/4}, \label{NDelta}$$ from where it follows $$\Delta < \left(6\pi^2\, \delta_{n_s}\mathcal{P}_s(k_H)\right)^{1/4}, \label{NDeltabound}$$ or $\Delta \lesssim 8.3\times 10^{-3} M\approx 2\times 10^{16}\, GeV$. From Eqs.(\[Nfbound\]) and (\[NDeltabound\]) we find that $$\Delta < \left(6\pi^2\delta_{n_s}\mathcal{P}_s(k_H)\right)^{1/4}< \left(6\pi^2\delta_{n_s}^3\mathcal{P}_s(k_H)\right)^{1/4} f\approx 1.6 \times 10^{-3} f < f, \label{NDeltalessf}$$ thus $\Delta < f$ always. We will see that for this to be the case in HNI the parameter $a$ in the potential has to be bounded from above. The resulting bound for $r$ coming from Eqs.(\[IA\]) and (\[NDeltabound\]) is $$r= \frac{2 \Delta^4}{3 \pi^2 \mathcal{P}_s(k_H)} < 4\delta_{n_s}\approx 0.14. \label{NDeltalessf}$$ Of course, in NI, once we have determined $f$ by fixing the number of e-folds $r$ follows: for $\delta_{n_s}=0.035$, Eq.(\[NN\]) with $N=60$ requires $f\approx 8.45$ giving a value $r\approx 0.084$. From Eqs.(\[NDelta\]) and Eq.(\[apar\]) we find that tuning $a=1$ is equivalent to tuning the $\mu$ parameter to the value $$\mu^2=\frac{3\,\pi^2}{2 f^2} \left(1-\frac{1}{f^2 \delta_{n_s}}\right) \delta_{n_s}\mathcal{P}_s(k_H), \label{lambda2}$$ using the numerical values above, Eq.(\[lambda2\]) gives $\mu\approx 3.1 \times 10^{-6} M\approx 7.6 \times 10^{12}GeV$. Extended Natural Inflation {#ENI} =========================== We now make a simple extension of the NI model discussed above leaving the parameter $a$ as coefficient of the $\cos \left(\frac{\phi}{f} \right)$-term in the potential Eq.(\[pot1\]) . This parameter is now restricted to the interval $0<a<1$. The potential is thus $$V(\phi) = V_0\left(1+a \cos \left(\frac{\phi}{f} \right) \right). \label{Hpot}$$ A potential like Eq.(\[Hpot\]) has been studied in the context of Hybrid Natural Inflation (HNI) [@Ross:2016hyb], [@Ross:2009hg], [@Ross:2010fg], where the symmetry breaking scale is sub-Planckian and the end of inflation is triggered by an additional waterfall field. Here we would like to study a simple extension of NI (ENI) allowing (as in NI) super-Planckian values for $f$. This would allow us to study the fine tuning of $a$ in terms of the parameters of the model and see what requirements (if any) the inflationary dynamics impose on them. We now attempt a similar analysis as in Section\[NI\]. The slow-roll parameters are now given by $$\begin{aligned} \epsilon &=&\frac{1}{2}\left(\frac{a}{f}\right)^2\frac{1-c_{\mathrm{\phi}} ^{2}}{\left( 1+a\, c_{\mathrm{\phi}} \right)^2} , \label{HNIeps}% \\ \eta &=&-\left( \frac{a}{f^2}\right)\, \frac{c_{\mathrm{\phi}}}{1+a c_{\mathrm{\phi}}} , \label{HNIeta} \\ \xi_2 &=&-\left( \frac{a}{f^2}\right)^2\,\frac{1-c_{\mathrm{\phi}} ^{2}}{\left( 1+a\, c_{\mathrm{\phi}} \right)^2} ,\\ \xi_3 &=& +\left( \frac{a}{f^2}\right)^3\,\frac{1-c_{\mathrm{\phi}} ^{2}}{\left(1+a c_{\mathrm{\phi}}\right)^3} c_{\mathrm{\phi}} .\end{aligned}$$From the expression for the scalar spectral index Eq.(\[Ins\]) we now get $$\delta_{n_s}=\frac{a}{f^{2}}\, \frac{2 c_H+a(3-c_H^2)}{(1+ac_{H})^{2}}. \label{spectral2}$$ At $\phi _{H}$ Eq.(\[spectral2\]) can be solved for $c_H$ [@Ross:2010fg] obtaining the following $$c_{1H}\equiv\frac{1-f^2\delta_{n_s}+\sqrt{1+3a^2-3(1-a^2)f^2\delta_{n_s}}}{a(1+f^2\delta_{n_s})},\text{\ }a\geqslant \frac{1}{3} , \label{solution1}$$ and $$c_{2H}\equiv\frac{1-f^2\delta_{n_s}-\sqrt{1+3a^2-3(1-a^2)f^2\delta_{n_s}}}{a(1+f^2\delta_{n_s})},\; a<1. \label{solution2}$$ The first solution in the limit $a=1$ corresponds to NI. We thus study here the solution $c_{1H}$ only and leave $c_{2H}$ for the following sections. From the constraints $-1<c_{1H}<1$ we get restrictions on the $f$ and $a$ parameters $$\begin{aligned} \frac{1}{\sqrt{2 \delta_{n_s}}} & < f < &\frac{1}{\sqrt{\delta_{n_s}}}\,,\quad\quad\quad \frac{1}{\sqrt{3}}\left(\frac{3f^2\delta_{n_s}-1} {f^2\delta_{n_s}+1}\right)^{1/2}< a \leq \frac{f^2\delta_{n_s}} {2-f^2\delta_{n_s}} , \label{c1Ha}% \\ \frac{1}{\sqrt{\delta_{n_s}}} & < f\,, &\quad\quad\quad\quad\quad\quad \frac{1}{\sqrt{3}}\left(\frac{3f^2\delta_{n_s}-1} {f^2\delta_{n_s}+1}\right)^{1/2}< a \leq 1, \label{c1Hb}\end{aligned}$$or using $\delta_{n_s}\approx 0.035$ $$\begin{aligned} 3.78 & < f < &5.35\,,\quad\quad\quad \frac{1}{3}< a \leq 1, \label{nc1Ha}% \\ 5.35 & < f\,, &\quad\quad\quad\quad\quad\quad \frac{1}{\sqrt{3}} < a \leq 1. \label{nc1Hb}\end{aligned}$$Thus, we see that the extension of NI corresponds to Eq.(\[solution1\]) together with Eq.(\[c1Hb\]). In NI the end of inflation is given by $\epsilon=1$, here from Eq.(\[HNIeps\]) we find $$\begin{aligned} c_{e1} & = &-\frac{2f^2-\sqrt{ a^2-2(1-a^2)f^2 }}{a(1+2f^2)}, \label{ce1}% \\ c_{e2} & = &-\frac{2f^2+\sqrt{ a^2-2(1-a^2)f^2 }}{a(1+2f^2)}. \label{ce2}\end{aligned}$$Thus, there are two values of $\phi_e$ which can meet the condition $\epsilon=1$ and end inflation. The number of e-folds from $\phi_{{\rm H}}$ to the end of inflation at $\phi _{\mathrm{e}}$ is $$N\equiv -\int_{\phi _H}^{\phi_e}\frac{V({\phi })}{V^{\prime }({\phi })}{d}{\phi }=\frac{f^2}{2a}\left((1+a)\ln \left( \frac{1-c_e}{1-c_H}\right)+(1-a)\ln \left( \frac{1+c_H}{1+c_e}\right)\right), \label{N}$$ where $c_{e}\equiv \cos \left( \frac{\phi _{e}}{f}\right)$. In ENI we thus have $$N_i=\frac{f^2}{2a}\left((1+a)\ln \left( \frac{1-c_{ei}}{1-c_{1H}}\right)+(1-a)\ln \left( \frac{1+c_{1H}}{1+c_{ei}}\right)\right),\quad\quad i=1,2. \label{Ni}$$ The function $N_i(f,a)$ defines a two-dimensional sheet tightly folded (see Fig.\[folded\]) as a consequence the number of e-folds $N_1$ is not very different from $N_2$. Thus in what follows we simply talk about the number of e-folds $N$ meaning any of them. In any case the requirement of $N=60$ e-folds of inflation begs for the parameter $a$ to be very close to 1. This is so because the ENI potential esentially differs from NI by a constant term. The inflationary epoch is controled by the derivative of the potential. Thus, a constant term in the potential is not very relevant in the determination of the number of e-folds and hence of the parameters $f$ and $a$; these are very close to the NI values. Not contributing to the cosmological constant problem leads us to impose $a=1$. Thus, a modification of NI to avoid the fine tuning problem requires a more drastic solution. In the following sections we continue our discussion in the context of HNI [@Ross:2016hyb], [@Ross:2009hg], [@Ross:2010fg] where the end of inflation is not due to the saturation of the condition $\epsilon = 1$ but to the destabilisation of the inflaton direction by an extra[ *waterfall*]{} field. This approach liberates the inflationary sector from also ending inflation. Hybrid Natural Inflation, bounds in the restricted case {#restricted} ======================================================= The first solution Eq.(\[solution1\]) in the limit $a=1$ corresponds to *Natural Inflation*. However to avoid the possibility of large gravitational corrections to the potential we will concentrate on the case $f<1.$[^2] Thus, the relevant solution is the second one denoted by $c_{2H}$, hereafter $c_H$, given by $$c_H=\frac{1-f^2\delta_{n_s}-\sqrt{1+3a^2-3(1-a^2)f^2\delta_{n_s}}}{a(1+f^2\delta_{n_s})}, \label{ch}$$ corresponds to $c_{2H}$ of Eq.(\[solution2\]). From Eq.(\[ch\]) note that $c_H=1$ when $a=\frac{f^2\delta_{n_s}}{2-f^2\delta_{n_s}}$. Thus, in principle one can have very small $r$ (see Eq.(\[HNIeps\])) and very low-scale of inflation (low $\Delta$, see Eq.(\[IA\])) when $a$ gets close to $\frac{f^2\delta_{n_s}}{2-f^2\delta_{n_s}}$ for any value of $f<1$. The density perturbation for the HNI potential can be written as $$\mathcal{P}_s(k) =\frac{1}{24\pi ^{2}}\frac{V(\phi)}{\epsilon(\phi)}=\frac{f^2 V_0\left(1+a\cos(\frac{\phi}{f})\right)^3}{12\pi^2 a^2\left(1-\cos(\frac{\phi}{f})^2\right)}, \label{contraste}$$ which presents a minimum (see Fig.\[Espectro\]) for $\phi_{min}$ given by $$c_{{min}}=\cos\left(\frac{\phi_{min}}{f}\right)=\frac{1-\sqrt{1+3a^2}}{a} \approx -\frac{3}{2}a. \label{cmin}$$ This together with some other properties related with the non-monotonicity of the tensor-to-scalar ratio are studied in [@German:2015qjq]. From the equation for the number of e-folds Eq.(\[N\]) we provisionally take (in what we call the restricted case) the end of the first $8<N_{min}<60$ e-folds of inflation as given by $c_{min}$, this will determine $a$ for a given $f$. The formula for $N_{min}$ e-folds adapted from Eq.(\[N\]) is $$N_{min}=\frac{f^2}{2a}\left((1+a)\ln \left( \frac{1-c_{min}}{1-c_{H}}\right)+(1-a)\ln \left( \frac{1+c_H}{1+c_{min}}\right)\right). \label{Nmin}$$ We first study this formula analytically for small $a$. The expansion of $c_H$ for small $a$ is given by $$c_H= \frac{1}{2}\left(\frac{f^2 \delta_{n_s}}{a}\right)-\frac{3}{2}\left(1-\frac{5}{12}\left(\frac{f^2 \delta_{n_s}}{a}\right)^2\right)a+\cdot\cdot\cdot \approx \frac{1}{2}\left(\frac{f^2 \delta_{n_s}}{a}\right) +\mathcal{O}(a). \label{chapp}$$ For small $a$ the term $c_{min}$ is also small and negligible inside the $\log$ of Eq.(\[Nmin\]) thus, $N_{min}$ can be approximated by $$N_{min}\approx\frac{f^2}{2a}\ln \left( \frac{1+c_H}{1-c_{H}}\right)\approx \frac{1}{2 \delta_{n_s}}\left(\frac{f^2 \delta_{n_s}}{a}\right)^2+\frac{1}{24 \delta_{n_s}}\left(\frac{f^2 \delta_{n_s}}{a}\right)^4+\cdot\cdot\cdot, \label{N8app}$$ from where it follows that $\frac{f^2}{a}$ is approximately constant $$\frac{f^2}{a}\approx \frac{ \sqrt{6}}{\delta_{n_s}} \left(N_s-1\right)^{1/2}. \label{constant}$$ Here we have defined $N_s\equiv \sqrt{ 1+\frac{2}{3}N_{min}\delta_{n_s} }$ to simplify notation. From $$\Delta=\left(24\pi ^{2}\epsilon(\phi_H)\mathcal{P}_s(k_H)\right)^{1/4} =\left(12\pi^2 \mathcal{P}_s(k_H) \left(\frac{a}{f}\right)^2 \frac{1-c_H^2 }{\left(1+a\,c_H\right)^2}\right)^{1/4}, \label{Delta}$$ we get $$\Delta\approx\left(\pi^2 \delta_{n_s}^2\mathcal{P}_s(k_H)\left(\frac{5-3N_s } {N_s -1}\right)+\mathcal{O}(a) \right)^{1/4} f^{1/2}. \label{Deltaap}$$ The behavior $\Delta \sim f^{1/2}$ means that $\Delta$ decreases more slowly than $f$. Thus, there is a point where $\Delta=f$ signaling the minimum value of $f$ consistent with $\Delta< f$. Solving $\Delta=f$ to lowest order in $a$ $$f_{min}\approx \pi \delta_{n_s}\mathcal{P}^{1/2}_s(k_H)\left(\frac{5-3N_s} {N_s-1}\right)^{1/2}, \label{fmin}$$ from where it follows that $N_{min}<76$, sufficient for our purpose. From Eqs.(\[constant\]) and (\[Deltaap\]) with $f=f_{min}$ we get the lower limit for $\Delta$ while setting $f=1/\pi$ (for consistency with $\Delta\phi<1$) gives the upper bound $$\Delta_{min}\equiv \pi\delta_{n_s}\,\mathcal{P}^{1/2}_s(k_H)\left(\frac{5-3N_s} {N_s-1}\right)^{1/2} <\Delta< \left(\delta_{n_s}^2\mathcal{P}_s(k_H)\left(\frac{5-3N_s } {N_s -1}\right) \right)^{1/4}\equiv \Delta_{max}. \label{Deltabounded}$$ The upper limit follows simply from the requirement that $\Delta\phi<1$ and it is not derived from any stronger condition. When there is a fixed number of e-folds $N_{min}<60$ from $\phi_H$ to the minimum at $\phi_{min}$ the scale of inflation as a function of $N_{min}$ is bounded as $\Delta_{min} < \Delta < \Delta_{max}$, see Fig.\[DB\]a. For small $a$ we can also see that the tensor-to-scalar ratio scales with $a$ as follows $$r\approx \frac{2\sqrt{6}}{3}\,\delta_{n_s}\left(N_s -1\right)^{1/2} \left(\frac{5-3N_s}{N_s-1}\right)a+ \mathcal{O}(a^2), \label{rapp}$$ and so becomes small for small $a$. From Eqs.(\[IA\]) and (\[Deltabounded\]) $r$ is bounded as follows, (see Fig.\[DB\]b), $$r_{min}\equiv \frac{2}{3}\pi^2\,\delta_{n_s}^4\,\mathcal{P}_s(k_H)\left(\frac{5-3N_s}{N_s-1}\right)^2 < r < \frac{2}{3\pi^2}\delta_{n_s}^2\left(\frac{5-3N_s}{N_s-1}\right)\equiv r_{max}. \label{rbounded}$$ From Eq.(\[IA\]) we see that the scale of inflation decreases much more slowly than the tensor-to-scalar ratio, $\Delta\equiv V_H^{1/4} \sim a^{1/4}$. In Ref.[@Hebecker:2013zda], HNI was already given a detailed analysis for its feasibility to reach large values of $r$, for sub-Planckian axion decay constant and sub-Planckian field range. There, a constraint on $f$ coming from an embedding of the effective potential of Eq.  into a string theory guide the authors to choose the fiducial bound $f\lesssim \frac{\sqrt{3}}{4\pi}$ giving an upper bound $r\simeq 7.6 \times 10^{-4}$ [@Hebecker:2013zda]. Using the same bound for $f$ our upper bound on $r$ changes by a factor of 3/16, from $r\simeq 1.6 \times 10^{-3}$ to $r\simeq 3 \times 10^{-4}$, for $N_{min}=8$ and $n_s=0.965$ as can be seen from Eq. . The expressions for the spectral indices are $$\begin{aligned} n_{sk} &\approx&\frac{\delta_{n_s}^2}{6} \left(\frac{5-3N_s} {N_s-1}\right)+ \mathcal{O}(a), \\ n_{skk} &\approx&\frac{\delta_{n_s}^3}{6} \left(\frac{5-3N_s} {N_s-1}\right)+ \mathcal{O}(a), \label{nskkapp}\end{aligned}$$ both are practically constant for small $a$. For numerical values see at the end of Section \[general\]. Hybrid Natural Inflation, bounds in the general case {#general} ==================================================== In the restricted case discussed in section \[restricted\] the number of e-folds $8<N_{min}<60$ is counted from $\phi_H$ to $\phi_{min}$ with the remaining e-folds occurring with increasing spectrum. If all $N=60$ e-folds occur before $\phi$ reaches $\phi_{min}$ we can not use $\phi_{min}$ to count the number of e-folds and we are in the general case where the end of inflation is dictated by the waterfall sector of the theory. From Eq.(\[Delta\]), the equation $\Delta=f$ can actually be solved exactly for $a$ as a function of $f$, we denote this solution by $a_{max}$ $$a_{max}=f^2\left(\frac{\sqrt{3f^8+16f^2\pi^2 \mathcal{P}_s(k_H)-24 f^4\pi^2 \mathcal{P}_s(k_H)\delta_{n_s} +48\pi^4\mathcal{P}_s(k_H)^2\delta_{n_s}^2}}{\sqrt{3}\left(f^6+8\pi^2 \mathcal{P}_s(k_H)-4 f^2\pi^2 \mathcal{P}_s(k_H)\delta_{n_s}\right)} \right), \label{amax}$$ but contrary to the $N_{min}$ case $a/f^2$ is not approximately constant but clearly depends on $f$. The condition $\Delta < f$ restricts the value of the parameter $a$ to be less than $a_{max}$ for a given $f$. On the other hand the slow-roll condition $a/f^2<1$ (coming from $\eta<1$) restricts $f$ such that the formula for $a_{max}$ is only valid up to the value of $f$, denoted by $f_1$, such that the term in parenthesis in the r.h.s. of Eq.(\[amax\]) equals one, this occurs for $f_{1}\approx 2\pi\sqrt{3 \mathcal{P}_s(k_H)}\approx 5.21 \times 10^{-4}$ (left panel in Fig.\[B2\]). Thus $a_{max}$ is well approximated by $$a_{max}\approx \frac{f^2\delta_{n_s}}{2} \sqrt{1+\frac{f^2}{3\pi^2\delta^2_{n_s}\,\mathcal{P}_s(k_H)}}, \quad\quad f < f_1. \label{amaxapp}$$ For $f$ larger than $f_{1}$ the condition $\Delta < f$ is always satisfied whenever $a$ is restricted by the stronger condition $a<f^2$. Thus for a given $f < f_{1}$ we get the bounds for $a$ $$a_{min}\equiv \frac{f^2\delta_{n_s}}{2-f^2\delta_{n_s}} < a < a_{max}\,, \quad\quad f < f_1, \label{abound1}$$ while for $f > f_{1}$ the bounds are $$\frac{f^2\delta_{n_s}}{2-f^2\delta_{n_s}} < a < f^2\,, \quad\quad f > f_1. \label{abound2}$$ the lower bound in Eqs.(\[abound1\]) and (\[abound2\]) comes from requiring $c_H < 1$ in Eq.(\[ch\]). Thus as we lower the value of $f$ the range of possible $a$ values reduces according to the bounds above. In Fig.\[B2\] (right panel) for any value of $f$ all possible $a$-values define a vertical line in the shaded region. As a consequence $\Delta$ will be bounded by $f(<f_1)$ whenever $a<a_{max}$ , in this case $r$ is bounded as follows $$0 < r < \frac{2 f^4}{3\pi^2\, \mathcal{P}_s(k_H)}\,, \quad\quad\quad\quad f < f_1, \label{rGeneralBounded}$$ while for $f > f_{1}$ the scale of inflation can have any value in the interval $0<\Delta <f $ whenever $a<f^2$. Thus, in the general case $\Delta$ is unbounded from below since this was a consequence of fixing the number of e-folds from $\phi_H$ to $\phi_{min}$ to a certain number $N_{min}\leq 60$. In the general case the end of inflation can occur before $\phi$ reaches the minimum of the spectrum at $\phi_{min}$ and no relation beetwen $a$ and $f$ for a fixed $N$ can be found because $\phi_e$ is undetermined. The scale of inflation $\Delta$ is vanishingly small for $a$ approaching $\frac{f^2\delta_{n_s}}{2-f^2\delta_{n_s}}$ because $c_H$ tends to one in that limit. As $a$ goes from $ \frac{f^2\delta_{n_s}}{2-f^2\delta_{n_s}}$ to its upper limit $c_H$ diminishes from 1 and the scale $\Delta$ grows from very small values, this is how the potential is able to cover the whole range of inflationary scales (see Fig.\[B3\]). Finally the reader would have noticed that we can apply Eq.(\[amax\]) directly to the restricted case, substituting in Eq.(\[Nmin\]) for the number of e-folds from $\phi_H$ up to $\phi_{min}$, extract the values of $f$ which accommodate from 60 to 8 e-folds and evaluate all other quantities of interest. While this is certainly possible and the numerical ranges are given below in Section \[restricted\] we wanted to obtain an approximated [*analytical*]{} expression which can teach us more than a few simple numbers. The corresponding ranges for the [*lower*]{} bounds when the number of e-folds $N_{min}$ go from 8 to 60 are given below (see also Fig. \[B2\]), $$\begin{aligned} 5.75 \times 10^{-6}&< f < & 2.32\times 10^{-5}, \\ 6.84\times 10^{-13}&< a_{max} < &2.58\times 10^{-11}, \\ 60&>N_{min}>&8, \\ 1.40 \times 10^{13}GeV&<\Delta<&5.67\times 10^{13}GeV, \\ 3.22 \times 10^{-14}&<r<&8.57 \times 10^{-12}, \\ 2.43 \times 10^{-4}&<n_{sk}<&3.97\times 10^{-3}, \\ 8.52 \times 10^{-6}&<n_{skk}<&1.39\times 10^{-4}, \label{ranges}\end{aligned}$$ which compare very well with values obtained from the analytical approximations of Section\[restricted\]. Clearly, in the general case we cannot do this because the formula for the number of e-folds Eq.(\[N\]) involves $c_{e}$ which can depend on parameters different from $a$ and $f$. PBH constraints and low scales of inflation {#PBH} =========================================== The steepening of the scalar spectrum in HNI (Fig.\[Espectro\]) gives rise to a positive running allowing for the possibility of primordial black hole production during inflation [@Kohri:2007qn], [@Ross:2016hyb]. In terms of the wave number $k$ the scalar power spectrum at first order in the SR parameters is given by $$\mathcal{P}_s(k)=A_s\left( \frac{k}{k_H}\right)^{(n_s-1) + \frac{1}{2}n_{sk} \ln\left(\frac{k}{k_H}\right) +\, \cdot \, \cdot \, \cdot }.\label{power}$$ Due to the constraint coming from the possible over-production of primordial black holes (PBHs) at the end of inflation the Taylor expansion of the power spectrum around its value at horizon crossing, $N_{H}\approx 60$ is bounded by [^3], $$\label{ps:expansion} C_{PBH}\equiv \ln \left[\frac{\mathcal{P}_s(0)}{\mathcal{P}_s(N_H)}\right] = (n_s- 1) N_H + \frac{1}{2} n_{sk} N_H^2\leq 14,$$ where $\mathcal{P}_s(N= 0)\simeq 10^{-3}$ (see also Refs. [@Josan:2009qn; @Carr:2009jm]) evolves from the initial value $\mathcal{P}_s(N_H) \approx 10^{-9}$. This gives the bound $n_{sk}< 10^{-2}$. For the HNI potential this constraint can be written as $$C_{PBH}=\ln\left[\frac{(1-c_H^2)(1+a\, c_e)^3}{(1-c_e^2)(1+a\, c_H)^3} \right], \label{pbh}$$ and can be easily satisfied for all cases discussed since here $C_{PBH}<3$. A more stringent bound may be set by PBHs produced during reheating [@Hidalgo:2017dfp], [@Carr:2017edp]. However, this depends on the specific reheating model and its associated equation of state. Such restrictions will be explored elsewhere. Low scales of inflation $\Delta$ can be obtained when $r$ is very small since, from Eq.(\[IA\]), $\Delta\sim r^{1/4}$. On the other hand, from Eq.(\[HNIeps\]) we see that $r$ is small when $c_H$ is very close to 1. This occurs for $a$ approaching the lower bound in Eqs.(\[abound1\]) and (\[abound2\]). Notice that the parameters $a$ and the scale of symmetry breaking $f$ need not be very small to give small inflationary scales, instead they should be closely related by $a\approx \frac{f^2 \delta_{ns}}{2-f^2 \delta_{ns}}$. In the absence of a mechanism which sets $a$ close to $\frac{f^2 \delta_{ns}}{2-f^2 \delta_{ns}}$ this is fine tuning which is equivalent to starting inflation with $\phi_H$ very close to the origin. In any case the value of $\phi_{H}$ should exceed the quantum fluctuations of the field $\delta\phi\approx\frac{H}{2\pi}\approx\frac{\Delta^2}{2\pi\sqrt{3}}$. From Eqs.(\[HNIeps\]) and (\[IA\]) we get the small $\phi$ behaviour $$r \approx \frac{\delta_{ns}^2}{2} \phi_{H}^2=\frac{2}{3 \pi^2A_s}\Delta^4, \label{rsmall}$$ from where it follows that $$\phi_{H}\approx\left(\frac{4}{3 \pi^2A_s \delta_{ns}^2}\right)^{1/2}\Delta^2\approx 2.2\times10^5\Delta^2>>\frac{\Delta^2}{2\pi\sqrt{3}}=\delta\phi. \label{rsmall}$$ Summary and conclusions {#conclusions} ======================= An interesting characteristic of Hybrid Natural Inflation is that the tensor-to-scalar ratio is a non-monotonic function of $\phi$ with the parameter $\epsilon(\phi)$ developing a [*maximum*]{}. A consequence of this is that the scalar spectrum of density fluctuations develops a [*minimum*]{} for some value $\phi_{min}$ of the inflaton. The value of $\phi_{min}$ is always slightly larger than the value of $\phi$ at the inflection point of the potential at $\phi_{I}=\pi/2$. Since the scalar spectrum has been observed to be decreasing during some 8 e-folds of observable inflation we can determine upper bounds for the scale of inflation and for the tensor-to-scalar ratio. In the [*restricted*]{} case considered in section \[restricted\] we do this by requiring a minimum of $8<N_{min}<60$ e-folds of inflation from the scale $\phi_{H}$, at which observable perturbations are produced, to $\phi_{min}$ where the spectrum stops decreasing. The remaining $0-52$ e-folds of inflation would occur with an steepening spectrum thus care is taken to not over-produce primordial black holes. The condition of having $N_{min}$ e-folds of inflation with a decreasing spectrum fixes the parameter $a$ once the symmetry breaking scale $f$ is determined. A minimum value for $f$ can be obtained by the requirement that the inflationary energy scale $\Delta$ is bounded by $f$. This allows to determine [*lower*]{} bounds for the inflationary energy scale and the tensor-to-scalar ratio. The general case is discussed in section \[general\] where we find upper bounds for $\Delta$ as well as $r$. In the general case the inflationary scale $\Delta$ is unbounded from below and can go all the way to vanishing values. The lower bound of section \[restricted\] is a consequence of fixing the number of e-folds from $\phi_H$ to $\phi_{min}$ to a certain number $N_{min}\leq 60$. In the general case the end of inflation can occur before $\phi$ reaches the minimum of the spectrum at $\phi_{min}$ and no relation beetwen $a$ and $f$ for a fixed $N$ can be found because $\phi_e$ is undetermined. The scale of inflation $\Delta$ is vanishingly small for $a$ approaching $\frac{f^2\delta_{n_s}}{2-f^2\delta_{n_s}}$ because $c_H$ tends to one in that limit. As $a$ goes from $ \frac{f^2\delta_{n_s}}{2-f^2\delta_{n_s}}$ to its upper limit $c_H$ diminishes from 1 and the scale $\Delta$ grows from very small values, this is how the potential is able to sweep the whole range of inflationary scales. By finding lower as well as upper bounds for the parameters $a$ and $f$ we can clearly understand how the scale of inflation in HNI is able to cover the complete range of values, from vanishingly small up to the GUT scale. Acknowledgements ================ We are grateful to SNI for partial financial support. AHA also acknowledges a VIEP-BUAP-HEAA-EXC17-I research grant. We acknowledge financial support from PAPIIT-UNAM grant IA-103616 [*Observables en cosmología relativista*]{} as well as CONACyT grants 269639 and 269652. 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A. Hebecker, S. C. Kraus and A. Westphal, Phys. Rev. D [**88**]{} (2013) 123506 doi:10.1103/PhysRevD.88.123506 \[arXiv:1305.1947 \[hep-th\]\]. J. C. Hidalgo, J. De-Santiago, G. German, N. Barbosa-Cendejas and W. Ruiz-Luna, arXiv:1705.02308 \[astro-ph.CO\]. B. Carr, T. Tenkanen and V. Vaskonen, arXiv:1706.03746 \[astro-ph.CO\]. [^1]: All quantities with a subindex ${}_H$ are evaluated at the scale $\phi_{H}$, at which observable perturbations are produced, some $50-60$ e-folds before the end of inflation. [^2]: Recall that $f>\frac{1}{\sqrt{\delta_{n_s}}} \approx 5.3$ in NI. [^3]: To lowest order in slow-roll $d/dN =- d / d\ln k$. The next order term in the expansion of Eq. , involving the parameter $n_{skk}$, is subdominant.

--- abstract: 'A computational method based on a first-principles multiscale simulation has been used for calculating the optical response and the ablation threshold of an optical material irradiated with an ultrashort intense laser pulse. The method employs Maxwell’s equations to describe laser pulse propagation and time-dependent density functional theory to describe the generation of conduction band electrons in an optical medium. Optical properties, such as reflectance and absorption, were investigated for laser intensities in the range $10^{10} \, \mathrm{W/cm^{2}}$ to $2 \times 10^{15} \, \mathrm{W/cm^{2}}$ based on the theory of generation and spatial distribution of the conduction band electrons. The method was applied to investigate the changes in the optical reflectance of $\alpha$-quartz bulk, half-wavelength thin-film and quarter-wavelength thin-film and to estimate their ablation thresholds. Despite the adiabatic local density approximation used in calculating the exchange–correlation potential, the reflectance and the ablation threshold obtained from our method agree well with the previous theoretical and experimental results. The method can be applied to estimate the ablation thresholds for optical materials in general. The ablation threshold data can be used to design ultra-broadband high-damage-threshold coating structures.' author: - 'Kyung-Min Lee' - Chul Min Kim - 'Shunsuke A. Sato' - Tomohito Otobe - Yasushi Shinohara - Kazuhiro Yabana - Tae Moon Jeong title: 'First-principles simulation of the optical response of bulk and thin-film $\alpha$-quartz irradiated with an ultrashort intense laser pulse' --- \[sec:intro\] Introduction ========================== The advances made in femtosecond (fs) high-power laser technology in the last decade have made it possible to achieve laser intensities as high as $10^{22} \, \mathrm{W/cm^{2}}$ [@Bahk:2004]. With such a wide range of laser intensities available for investigations, the optical response of a material can be expected to show fairly different characteristics with varying intensity. For example, at very low intensities below $10^{10} \, \mathrm{W/cm^{2}}$, the optical properties of a medium follow a linear response to laser intensity variation [@Hecht:2001; @Born:1999], but start showing a nonlinear response as the laser intensity increases beyond a certain level [@Boyd:2008]. However, at still higher laser intensities of greater than $10^{14} \, \mathrm{W/cm^{2}}$, the optical medium suddenly starts behaving like a plasma medium, and its optical properties follow the properties of a plasma medium [@Kruer:1989]. In the intermediate intensity range ($10^{11} \, \mathrm{W/cm^{2}}$ to $10^{14} \, \mathrm{W/cm^{2}}$), the physical behavior of an optical medium is very complicated and many interesting phenomena, e.g., generation and heating of conduction band (CB) electrons and energy transfer to the lattice, followed by melting, boiling and ablation of the material, can be observed. These behaviors are related to the transition mechanism from solid to plasma and have been intensively studied in previous reports. A theoretical understanding of the laser–matter interactions in the intermediate intensity range is, therefore, of great interest. In addition, it can also provide important insights into laser-induced damage and ablation of optical materials in general. Studies on laser-induced damage date back to as far as the late 1960s. The dependence of the damage on laser characteristics such as the wavelength, pulse duration and energy fluence as well on material type was investigated by Wood using nanosecond (ns) laser pulses [@Wood:2003]. Later, investigations of laser-induced damage in the picosecond (ps) and fs regime gained significance when the advent of the chirped-pulse amplification (CPA) technique [@Strickland:1985] made it feasible to develop fs and petawatt-class laser systems [@Sung:2010; @Yu:2012]. In particular, laser ablation occurring on the fs time scale became critical because a laser pulse duration of few tens of fs is much shorter than the time scale for electron energy transfer to the lattice and subsequent lattice heating. In 1995, Stuart $\mathit{et \, al.}$, investigated the laser-induced damage threshold at $1053 \, \mathrm{nm}$ and $526 \, \mathrm{nm}$ for pulse durations ranging from $270 \, \mathrm{fs}$ to $1 \, \mathrm{ns}$, through a theoretical model based on CB electron production via multiphoton ionization, Joule heating and collisional ionization [@Stuart:1995]. Subsequent studies by other groups were conducted for a more accurate analysis of the damage and ablation threshold by including energy dependence of the CB electrons [@Rethfeld:2002] and nonlinear pulse propagation effect in a medium [@Penano:2005; @Petrov:2008; @Gulley:2010; @Apalkov:2012] in the fs regime. However, all these studies were based on theoretical models that used experimental and/or empirical values of the material parameters such as ionization rate, refractive index, relaxation rate, and band structure. Hence, the need for developing a method that uses non-empirical values of the material parameters grew continuously in the search for a comprehensive and reliable method of investigating laser–matter interactions in the intermediate laser intensity range. In this paper, we employ an alternative method to compute the optical response and the ablation threshold of an optical medium. In contrast to the previous studies, our method is based on first-principles simulations computed from fundamental equations. A multiscale approach using the wave equation and the time-dependent density functional theory (TDDFT) is applied to calculate directly the density of the CB electrons generated in the optical medium. The first report on the use of such a multiscale approach for investigation of the interaction between a laser pulse and an optical medium was made for crystalline silicon, where it was said to yield reliable results [@Yabana:2012]. In this approach, no empirical parameters and approximations were used except for information on the crystal structure and on the exchange–correlation potential. As far as these parameters and approximations are valid for a given set of conditions, our first-principles simulations can produce the most reliable and comprehensive results. We applied the method to calculate the reflectance, the CB electron density and the absorbed energy for investigating the changes in the optical properties of bulk and thin-film $\alpha$-quartz (having different thicknesses) on being irradiated by fs laser pulses in the intensity range of $10^{10} \, \mathrm{W/cm^{2}}$ to $2 \times 10^{15} \, \mathrm{W/cm^{2}}$. By comparing the absorbed energy based on some criterion for laser-induced ablation, the ablation threshold can be computationally determined without the help of empirical values. The proposed approach can be easily applied to other optical materials and structures to design high-performance optical coatings, such as a high-damage-threshold broadband optical coating. The organization of the paper is as follows. Section \[sec:method\] describes in brief the theoretical methods and the simulation details. The calculated results and discussion are presented in Section \[sec:results\]. Finally, the conclusion of the paper is given in Section \[sec:conclusion\]. \[sec:method\] Theoretical methods ================================== \[ssec:multiscale\] Multiscale description of laser-matter interaction ---------------------------------------------------------------------- We employ a theoretical method and a computational code developed by some of the present authors [@Yabana:2012]. In the following, we briefly describe the formalism. The interaction between a laser pulse and matter involves two characteristic lengths: the wavelength of the laser pulse and the electronic structure size of the atoms constituting the matter. In the case of fs laser pulses, the former lies on the macroscopic scale comprising the $\mu\mathrm{m}$ range, while the latter lies on the microscopic scale comprising the $\mathrm{nm}$ range. Any first-principles description of the interaction should incorporate these two different scales simultaneously. Let **R** denote the macroscopic scale in which the laser pulse evolves and **r** the microscopic scale in which the electrons move. To describe the dynamics of electrons in a unit cell under an external electromagnetic field, the time-dependent Kohn–Sham (TDKS) equation is used [@Runge:1984]: $$\begin{aligned} \label{eq:tdks} \mathrm{i}\hbar\frac{\partial}{\partial t}\psi_{i,\mathbf{R}}(\vec{r},t)&=&\biggl\{ \frac{1}{2m_{\mathrm{e}}}\left(-\mathrm{i}\hbar\nabla_{\mathbf{r}}+\frac{e}{c}\vec{A}_{\mathbf{R}}(t)\right)^{2}+V_{\mathrm{ion},\mathbf{R}}(\vec{r}) \nonumber \\ &&+V_{\mathrm{h},\mathbf{R}}(\vec{r},t)+V_{\mathrm{xc},\mathbf{R}}(\vec{r},t)\biggr\}\psi_{i,\mathbf{R}}(\vec{r},t),\end{aligned}$$ where $\psi_{i,\mathbf{R}}$ is the $i$th Kohn–Sham (KS) orbital, $\vec{A}_{\mathbf{R}}$ the vector potential of the laser pulse in the Coulomb gauge, $V_{\mathrm{ion},\mathbf{R}}$ the ionic potential, $V_{\mathrm{h},\mathbf{R}}$ the Hartree potential and $V_{\mathrm{xc},\mathbf{R}}$ the exchange–correlation potential. Since the laser pulse we considered slowly varies over the electronic length scale, it can be assumed that $\vec{A}_{\mathbf{R}}$ does not depend on $\vec{r}$. Once the TDKS equation is solved with a given vector potential, the electron density ($n_{\mathbf{R}}$) and current ($\vec{j}_{\mathbf{R}}$) can be calculated from the KS orbitals: $$\label{eq:density} n_{\mathbf{R}}(\vec{r},t)=\sum_{i}\left|\psi_{i,\mathbf{R}}(\vec{r},t)\right|^{2},$$ $$\begin{aligned} \label{eq:current} \vec{j}_{\mathbf{R}}(\vec{r},t)&=&\frac{1}{2m_{\mathrm{e}}}\sum_{i}\biggl\{\psi_{i,\mathbf{R}}^{\ast}(\vec{r},t)\left(-\mathrm{i}\hbar\nabla_{\mathbf{r}}+\frac{e}{c}\vec{A}_{\mathbf{R}}(t)\right)\psi_{i,\mathbf{R}}(\vec{r},t) \nonumber \\ && -\psi_{i,\mathbf{R}}(\vec{r},t)\left(-\mathrm{i}\hbar\nabla_{\mathbf{r}}-\frac{e}{c}\vec{A}_{\mathbf{R}}(t)\right)\psi_{i,\mathbf{R}}^{\ast}(\vec{r},t)\biggr\}.\end{aligned}$$ This microscopic current is averaged over a unit cell to define the macroscopic current ($\vec{J}_{\mathbf{R}}$) as: $$\label{eq:maccurrent} \vec{J}_{\mathbf{R}}(t)=\frac{1}{\Omega}\int_{\Omega} \vec{j}_{\mathbf{R}}(\vec{r},t) \, \mathrm{d}\vec{r},$$ where $\Omega$ is the unit cell volume. It should be noted that there is also a contribution to the current from a nonlocal pseudopotential. The propagation of the laser pulse is described by the wave equation with the macroscopic current as its source term: $$\label{eq:we} \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\vec{A}_{\mathbf{R}}(t)-\nabla^2_{\mathbf{R}}\vec{A}_{\mathbf{R}}(t)=-\frac{4\pi e}{c}\vec{J}_{\mathbf{R}}(t).$$ The vector potential obtained by solving Eq. (\[eq:we\]) is used to solve Eq. (\[eq:tdks\]) at the next time step. The interaction between a laser pulse and matter can be fully described by solving Eqs. (\[eq:tdks\]) and (\[eq:we\]) self-consistently via the macroscopic current and the vector potential [@Yabana:2012]. It should be noted that the electron motion is restricted to be within a unit cell; non-local processes such as electron transport among unit cells cannot be accounted for in the present method. Moreover, the ionic motion is neglected since the motion of ions is slow enough in comparison with electrons due to their large mass. However, these are beyond the scope of our interest since we consider laser intensities smaller than $10^{17} \, \mathrm{W/cm^{2}}$ and wavelength of the pulse in the near-visible region. \[ssec:details\] Simulation details ----------------------------------- For the sake of simplicity of the laser–matter interaction geometry, the case of normal incidence of the pulse was considered in the simulation. The $\left(\bar{2}10\right)$ surface of $\alpha$-quartz was taken to be the transverse plane. The laser pulse was assumed to be linearly polarized along the $z$-axis and propagating along the $x$-axis. It had a wavelength of $\lambda_{0}=800 \, \mathrm{nm}$, corresponding to a photon energy of $\hbar\omega=1.55 \, \mathrm{eV}$ and a pulse duration of $T=20 \, \mathrm{fs}$. In the simulation, a uniform laser intensity was assumed in the transverse plane. The vector potential of the incident pulse was given as $$A_{X}(t)=-\frac{E_{0}}{\omega}\sin^{2}\left\{ \frac{\pi\left(X-ct\right)}{cT}\right\} \cos\left\{ \frac{\omega\left(X-ct\right)}{c}\right\}$$ for $0<X-ct<cT$ and $A_{X}(t)=0$ otherwise. Here, $X$ denotes the macroscopic coordinate in the laser propagation direction and $E_{0}$ is the maximum electric field strength, which is related to the laser intensity ($I_{0}$) as $I_{0}=cE_{0}^{2}/8\pi$. The spatial step size along the $z$-axis had a value of $\Delta X=12.67 \, \mathrm{nm}$. The incident electric field was related to the vector potential by $E_{X}(t)=-\left(1/c\right)\cdot\left(dA_{X}(t)/dt\right)$. The thickness of the $\alpha$-quartz sample was appropriately chosen so as to simulate bulk and thin-film structures. The $\alpha$-quartz bulk sample had a thickness of $d_{\mathrm{bulk}}=3.548 \, \mu\mathrm{m}$, which was considered large enough for the assumption that there was no reflection from the rear surface during pulse propagation. The thin-film samples of $\alpha$-quartz had thicknesses of $d_{\mathrm{HWTF}}=\lambda_{0}/2n_{0}=253.28 \, \mathrm{nm}$ for the half-wavelength thin film (HWTF) and $d_{\mathrm{QWTF}}=\lambda_{0}/4n_{0}=126.64 \, \mathrm{nm}$ for the quarter-wavelength thin film (QWTF). Here, $n_{0}$ is the refractive index of $\alpha$-quartz at $\lambda_{0}=800 \, \mathrm{nm}$. A value of $n_{0}=1.578$ obtained from the TDDFT calculations [@Otobe:2009] was used rather than an experimental value of $n_{0}=1.538$ [@Ghosh:1999]. An orthogonal unit cell containing six $\mathrm{SiO_2}$ molecular units was used as a unit cell for $\alpha$-quartz. Three sides of the unit cell had lengths of $\mathrm{a}=9.28 \, \mathrm{a.u.}$, $\mathrm{b}=16.08 \, \mathrm{a.u.}$ and $\mathrm{c}=10.21 \, \mathrm{a.u.}$, respectively [@Schober:1993; @Adeagbo:2008]. The calculation results numerically converged well when the sides were discretized into $26$, $36$ and $50$ points, respectively. The number of $k$-points in the reciprocal space was $4^{3}$ in the simulation. The TDKS equations were solved by applying the explicit time evolution operator containing up to the fourth-order term in the Taylor expansion of the complete time evolution operator [@Yabana:1996; @Yabana:2006]: $$e^{-\mathrm{i}H_{\mathrm{KS}}\Delta t/\hbar}\approx\sum_{n=0}^{4}\frac{\left(-\mathrm{i} H_{\mathrm{KS}} \Delta t / \hbar \right)^{n}}{n!},$$ where $\Delta t=0.2 \, \mathrm{a.u.}$ ($1 \, \mathrm{a.u.}$ of time corresponds to $0.024189 \, \mathrm{fs}$) is the time step and $H_{\mathrm{KS}}$ is the Kohn–Sham Hamiltonian in Eq. (\[eq:tdks\]). In this paper, we employed a norm-conserving pseudopotential [@Troullier:1991] with the separable approximation [@Kleinman:1982]. In this approximation, only the valence band electrons were explicitly treated in the simulation, while the effect of core electrons was included in the pseudopotential. To calculate the exchange–correlation potential in Eq. (\[eq:tdks\]), the adiabatic local density approximation (LDA) was used [@Perdew:1981]. The calculated band gap energy was $6.5 \, \mathrm{eV}$ in our simulation, while the experimental one was $9.0 \, \mathrm{eV}$ [@Arnold:1994]. This underestimation of the band gap energy is a well-known characteristic of the LDA. Consequently, the number of photons responsible for interband transitions is reduced, which indicates that more CB electrons are generated at the same laser intensity. This discrepancy in the band gap energy should be kept in mind when the calculated quantities are compared to the experimental ones. A more quantitative evaluation of the band gap energy can be systematically achieved by using an elaborate functional, e.g., meta-GGA [@Tran:2009; @Rasanen:2010], which is being implanted in our simulation code, and will be presented in a further study. \[sec:results\] Results and discussion ====================================== This section consists of three major parts. The first part describes the laser intensity dependence of the reflectance for bulk and thin-film samples of $\alpha$-quartz irradiated by an ultrashort intense laser pulse (see \[ssec:reflectance\]). The second part describes explicitly the generation and spatial distribution of the CB electrons, which are responsible for the change in the reflectance, in the $\alpha$-quartz medium (see \[ssec:CBE\] and \[ssec:thickness\]). In the last part, the extent of laser-induced ablation is estimated based on the energy absorbed by the CB electrons in the medium (see \[ssec:absorption\]). \[ssec:reflectance\] Reflectance as a function of laser intensity ----------------------------------------------------------------- ![\[fig:reflectance\] Reflectance of $\alpha$-quartz as a function of laser intensity: bulk (circles and solid line), HWTF (squares and dashed line) and QWTF (triangles and dotted line).](reflectance.eps){width="80mm"} The optical response of the $\alpha$-quartz materials under investigation was described by the reflectance curve for various laser intensity conditions. The reflectance in our simulation was calculated as the fraction of power of the incident laser pulse that is reflected at the surface when the reflected and the transmitted pulses are well separated. In this study, only static results, which mean the results after pulse propagation is over, were considered, although our simulation could intrinsically deal with time-dependent processes. Figure \[fig:reflectance\] shows the calculated reflectances of $\alpha$-quartz materials under laser intensities of $10^{10} \, \mathrm{W/cm^{2}}$ to $2 \times 10^{15} \, \mathrm{W/cm^{2}}$. As shown in Fig. \[fig:reflectance\], at low intensities below $2.5 \times 10^{13} \, \mathrm{W/cm^{2}}$, all the reflectances have different but constant values. The constant values of the reflectances at low intensities can be attributed to the linear response of lossless dielectric materials. According to Fresnel’s equation [@Born:1999], the reflectance of a bulk material is given by $$\label{eq:Rbulk} R_{\mathrm{bulk}}=\left(\frac{n_0-1}{n_0+1}\right)^2,$$ where $n_{0}$ refers to the refractive index of the material at a given wavelength. Eq. (\[eq:Rbulk\]) yields $R_{\mathrm{bulk}}=0.050$ with $n_{0}=1.578$, which is almost the same as $R_{\mathrm{bulk}}=0.051$ obtained from the simulation (see Fig. \[fig:reflectance\]). For a thin film, interference between the secondary waves reflected from the front and rear surfaces should be considered as well. The formula is thus modified to [@Dressel:2002] $$\label{eq:Rfilm} R_{\mathrm{film}}(\beta)=\frac{4R_{\mathrm{bulk}}\sin^{2}\beta}{(1-R_{\mathrm{bulk}})^{2}+4R_{\mathrm{bulk}}\sin^{2}\beta},$$ where $\beta=2\pi n_{0}d/\lambda_{0}$ and $d$ is the thickness of the film. Eq. (\[eq:Rfilm\]) gives $R_{\mathrm{HWTF}}(\pi)=0$ and $R_{\mathrm{QWTF}}(\pi/2) =0.181$, which are almost identical to the results of $R_{\mathrm{HWTF}}(\pi)=0.013$ and $R_{\mathrm{QWTF}}(\pi/2)=0.184$ obtained from Fig. \[fig:reflectance\]. The slight differences might have been caused by material dispersion because we used a $20 \, \mathrm{fs}$ pulse. Material dispersion cannot be described by a single Fresnel’s equation at a fixed frequency. In the case of thin films, interference between the secondary waves generated at the front and rear surfaces plays a dominant role in determining the reflectances. It should be noted that secondary waves can also be generated inside the medium, but they get summed to zero along the reflection direction as long as the medium has a high uniform density [@Hecht:2001], according to the Ewald–Oseen extinction theorem [@Fearn:1996]. However, as the laser intensity increases, the CB electrons, which behave like free electrons, can be non-uniformly generated inside the medium. By absorbing and reflecting the laser pulse, the non-uniform distribution of the CB electrons can change the interference pattern among the secondary waves and may result in a change in reflectance. As shown in Fig. \[fig:reflectance\], as the laser intensity is increased beyond $2.5 \times 10^{13} \, \mathrm{W/cm^{2}}$, the reflectance for QWTF and HWTF first decreases–though this decrease is only $8 \, \%$ for HWTF at an intensity of $5 \times 10^{13} \, \mathrm{W/cm^{2}}$–and thereafter starts increasing. In constrast, the reflectance for bulk $\alpha$-quartz is almost constant up to $5 \times 10^{13} \, \mathrm{W/cm^{2}}$ and then increases monotonously. To understand the role of CB electrons in changing the reflectance, the generation and spatial distribution of the CB electrons should be analyzed in detail, as further discussed in Sect. \[ssec:CBE\] and Sect. \[ssec:thickness\]. At high intensities, $I_{0}>5 \times 10^{14} \, \mathrm{W/cm^{2}}$, the reflectances for all the structures rapidly increased and converged to the same value. Convergence in the reflectances implies that the CB electrons, which are mostly generated at the front surface, play a leading role in the bringing about changes in reflectance. To investigate this effect quantitatively, we define a parameter called the skin depth [@Gibbon:2005], which expresses light penetration into the medium as $$\label{eq:skindepth} l_{\mathrm{s}}=\frac{c}{\sqrt{\omega_{\mathrm{p}}^{2}-\omega^{2}}},$$ where $c$ is the velocity of light, $\omega_{\mathrm{p}}=\sqrt{4\pi e^{2} N_{\mathrm{CB}}/m_{\mathrm{e}}^{\ast}}$ the plasma frequency and $\omega$ the laser frequency. With the CB electron density ($N_{\mathrm{CB}}$) calculated at an intensity of $5 \times 10^{14} \, \mathrm{W/cm^{2}}$, we estimated the skin depth ($l_{\mathrm{s}}$) as $28 \, \mathrm{nm}$ for HWTF, $30 \, \mathrm{nm}$ for QWTF and $29 \, \mathrm{nm}$ for bulk, depths that are much smaller than the thickness of even QWTF. At intensities higher than $5 \times 10^{14} \, \mathrm{W/cm^{2}}$, the skin depths became much smaller due to the high $N_{\mathrm{CB}}$. This small skin depth indicates that the transmitted waves barely reached the rear surface and the reflection mainly occurred at the front surface by the many CB electrons present there. It should be noted that the optical Kerr effect can also contribute to the change in reflectance. The third-order nonlinear susceptibility of $\alpha$-quartz is given as $\chi^{(3)}=3.81 \times 10^{-14} \, \mathrm{esu}$ [@Buchalter:1982] and the corresponding nonlinear refractive index is $n_{2}=6.04 \times 10^{-16} \, \mathrm{cm^{2}/W}$ with $n_{0}=1.578$. Substituting this value of $n_{0}$ into $n=n_{0}+n_{2}I_{0}$ in Eq. (\[eq:Rbulk\]) at an intensity of $I_{0}=10^{15} \, \mathrm{W/cm^{2}}$, the reflectance for bulk $\alpha$-quartz is calculated to be $0.14$, which is less than $0.34$ obtained from the simulation. Therefore, the optical Kerr effect would be less important for the change of the bulk reflectance, nor is it valid for the case of thin-film reflectance because interference among the secondary waves has been ignored. \[ssec:CBE\] Generation of CB electrons --------------------------------------- ![\[fig:drude\] Change in the bulk reflectance calculated from Eq. (\[eq:drude\]) as a function of the calculated CB electron density ($N_{\mathrm{CB}}$) and the collision time ($\tau$). The open circles and the solid line represent our simulation results and the open rectangles and the dashed line represent the best fit (from Eq. (\[eq:drude\])) to our simulation results, keeping the effective electron mass fixed at $m_{\mathrm{e}}^{\ast}=0.5 \, m_{\mathrm{e}}$. The open triangles and the dotted line represent the fit from the experimental value of $\tau=0.2 \, \mathrm{fs}$.](ref_fit_Ncb.eps){width="80mm"} In Section \[ssec:reflectance\], changes in the reflectance were attributed to the generation of CB electrons. For large band gap materials under a weak and infrared laser pulse, the valence band (VB) electrons cannot be directly excited into the CB since the photon energy is smaller than the band gap energy. As the laser intensity increases, material begins to absorb multiple photons and then the VB electrons can be excited to the CB. In our simulation, the increase in the CB electrons was proportional to the laser intensity ($I_{0}$) in the form of $I_{0}^{4}$. This indicates that the excitation occurred by means of a four-photon absorption process that overcame the calculated band gap energy of $6.3 \, \mathrm{eV}$. The generated CB electrons absorbed laser energy through the inverse Bremsstrahlung process, which resulted in the change in optical response of material such as permittivity. To confirm this scenario, we calculated the reflectance by using a modified Drude model as an empirical model, which includes free electron generation and effect on decrease of VB electrons by the electron excitation to CB, and compared the calculated results with our simulation results. According to the model, the permittivity of $\alpha$-quartz can be written as $$\begin{aligned} \label{eq:drude} \epsilon&=&1+\frac{N_{\mathrm{VB}}}{N_{\mathrm{VB}}^{0}}\left(\epsilon_{0}-1\right)-\frac{\omega_{\mathrm{p}}^{2}}{\omega^{2}\left(1+i\tilde{\nu}\right)} \nonumber \\ &=&\epsilon_{0}-N_{\mathrm{CB}}\cdot\left(\frac{1}{N_{\mathrm{cr}}\cdot\left(1+\mathrm{i}\tilde{\nu}\right)}+\frac{\epsilon_{0}-1}{N_{\mathrm{VB}}^{0}}\right),\end{aligned}$$ where $N_{\mathrm{VB}}^{0}$ is the number of the initial VB electrons, $N_{\mathrm{VB}}^{0}=N_{\mathrm{VB}}+N_{\mathrm{CB}}=4.25 \times 10^{23} \, \mathrm{cm}^{-3}$, $\tilde{\nu}$ is the relative collision frequency given by $\tilde{\nu}=\nu/\omega$ and $N_{\mathrm{cr}}$ is the critical density by $N_{\mathrm{cr}}=\omega^2 m_{\mathrm{e}}^{\ast}/4\pi e^{2}=8.7 \times 10^{20} \, \mathrm{cm}^{-3}$. A collision time ($\tau$) and an effective mass of $\alpha$-quartz ($m_{\mathrm{e}}^{\ast}$) were defined as $\tau=1/\nu$ and $m_{\mathrm{e}}^{\ast}=0.5 m_{\mathrm{e}}$ [@Vexler:2005], respectively. In the simulation, the CB electron density was calculated at the macroscopic points, i.e., the unit cells, in the medium as follows: $$\label{eq:NCB} N_{\mathrm{CB},\mathbf{R}}(t)=\sum_{i,j}\left(\delta_{ij}-\left| \braket{\psi_{i,\mathbf{R}}(t=0)|\psi_{j,\mathbf{R}}(t)} \right|^2\right),$$ where $i$ and $j$ are indices for the Kohn–Sham orbitals and $\psi_{i,\mathrm{R}}(t=0)$ is the ground-state Kohn–Sham orbital. Figure \[fig:drude\] shows the change in the reflectance of bulk $\alpha$-quartz calculated from Eq. (\[eq:drude\]) as a function of the calculated $N_{\mathrm{CB}}$ and $\tau$. It should be noted that the calculated $N_{\mathrm{CB}}$ was taken as the value of Eq. (\[eq:NCB\]) after the pulse passed the medium. The calculation based on the model could qualitatively reproduce our simulation results; the bulk reflectance increased with increase in the CB electrons. For more quantitative comparison, the best fit to the simulation result was achieved when $\tau=0.05 \, \mathrm{fs}$, which was much smaller than the experimental value, $\tau=0.2 \, \mathrm{fs}$ [@Mao:2003]. The discrepancy could be understood by the fact that our simulation intrinsically considers the time-dependent CB electrons, which means that when the pulse reaches to medium, CB electrons are initially zero and increase with propagation of the pulse in the medium. However, in Eq. (\[eq:drude\]), only static values of CB electrons, which are the values after the pulse passes the medium, are considered. Therefore the quantitative comparison between the model and our simulation could be difficult. It should also be mentioned that the underestimation of the band gap energy in our simulation might affect the results. Four-photon absorption rather than six-photon absorption, which is the correct one for an experimental band gap energy of about $9 \, \mathrm{eV}$, can generate more CB electrons at the same laser intensity. This might cause an increase in the reflectance at a lower intensity than that expected from six-photon absorption. When more elaborate exchange–correlation functionals reproducing six-photon absorption are used, the increase in reflectance would occur at higher intensities. \[ssec:thickness\] Dependence of CB electron generation on material thickness ----------------------------------------------------------------------------- The empirical model described by Eq. (\[eq:drude\]) could qualitatively explain the changes in reflectance of bulk $\alpha$-quartz based on generation of CB electrons. However, it may not be a suitable explanation for the case of thin films because the interference effect was not considered in the model. As mentioned earlier, the CB electrons generated inside the medium absorb and reflect the laser pulse, resulting in a change in the condition for interference, which plays a dominant role in determining the reflectance of thin films. In this regard, it is important to know the spatial distribution of the CB electrons for understanding how it changes the interference condition, and hence the reflectance. {width="120mm"} Figure \[fig:CBEdensity\] shows the spatial distributions of CB electrons at various laser intensities after the pulse passes the medium. Since there is no interference in the bulk, the spatial distribution of the CB electrons in the bulk $\alpha$-quartz was relatively uniform inside the medium at low intensities (See Fig. \[fig:CBEdensity\](a)). As the intensity increases, the CB electrons were accumulated around the front surface and exceeded the critical density when $I_{0}=6 \times 10^{13} \, \mathrm{W/cm^{2}}$, at which the bulk reflectance increased only by $5 \, \%$. A non-uniform spatial distribution of the CB electrons was observed for thin-film cases. For HWTF, more CB electrons were generated around the two surfaces than in the middle of HWTF, due to interference between the secondary waves generated from the two surfaces. With increases in intensity, the CB electrons exceeded the critical density at the two surfaces for a given intensity. The CB electrons generated around the front and rear surfaces change the interference condition in a way that destructively contributes to the reflectance. The change of the interference condition increases the reflectance because the destructive interference effect is reduced. Since we consider a laser pulse with a finite pulse duration, the broad spectrum of the laser pulse can affect the interference condition and thus the reflectance, for example, in the case of HWTF it can give rise to non-zero values of the CB electrons at the center of the film and also of reflectance at low intensities. For QWTF, the CB electrons were dominantly generated around the rear surface and exceeded the critical density on increasing the laser intensity. As in the case of HWTF, the CB electrons above the critical density modify the interference condition. However, in the case of QWTF, the interference constructively contributes to the reflectance. This indicates that the change from an initial constructive interference condition can reduce the reflectance. As seen in Fig. \[fig:reflectance\], the reflectance for QWTF started to decrease at an intensity of $5 \times 10^{13} \, \mathrm{W/cm^{2}}$, at which the CB electrons reached the critical density on the rear surface, and had the minimum reflectance at an intensity of $1.7 \times 10^{14} \, \mathrm{W/cm^{2}}$, at which the CB electrons density reached up to $9 \times 10^{21} \, \mathrm{cm^{-3}}$. This value is much higher than the critical density. Therefore, it can be inferred that significant change in reflectance for QWTF occurs by the CB electrons above the critical density. At higher intensities above $5 \times 10^{14} \, \mathrm{W/cm^{2}}$, more CB electrons were populated at the front surface in all the structures. Since a major part of the pulse was reflected from the front surface, interference between the secondary waves from the front and rear surfaces does not play an important role anymore and all the reflectances converged and rapidly increased to the reflectance of the over-dense plasma. It should be noted that, at a given laser intensity, the films had a larger population of CB electrons than the bulk at the front and/or the rear surfaces. This implies that the optical response of the films might be more sensitive to changes in the laser intensity than the bulk. \[ssec:absorption\] Estimation of ablation threshold ---------------------------------------------------- ![\[fig:absenergy\] Absorbed energy at the surfaces as a function of laser intensity: (a) bulk, (b) HWTF and (c) QWTF. The horizontal dotted line represents the cohesive energy calculated by the LDA functional, $9.5 \times 10^{4} \, \mathrm{J/cm^{3}}$ [@Liu:1994].](absenergy.eps){width="80mm"} The CB electrons can cause a permanent structural damage of materials by transferring their kinetic energy to the lattice, in addition to changing the optical response. A knowledge of the damage threshold is thus equally important in designing an optical device such as a high-damage-threshold mirror. A criterion indicating the occurrence of damage is needed to establish a damage threshold. In this study, an energy criterion was used rather than the critical density as a criterion [@Penano:2005; @Jia:2003; @Chimier:2011]. Since we used a $20 \, \mathrm{fs}$ laser pulse, the typical damage type was laser-induced ablation; so the cohesive energy was adopted as a criterion for laser-induced ablation [@Chimier:2011]. For a consistent description, a cohesive energy value of $9.5 \times 10^{4} \, \mathrm{J/cm^{3}}$ obtained by the LDA functional rather than the experimental value of $8.2 \times 10^{4} \, \mathrm{J/cm^{3}}$ was used in the simulation. Note that the $15 \, \%$ difference in the cohesive energy did not severely influence our interpretations. The absorbed energy in the simulation was defined as the difference in total energies between before and after the laser pulse passes through a unit cell that was closest to the front surface and the rear surface. Figure \[fig:absenergy\] shows the energies absorbed by the bulk and thin film samples at various laser intensities. For the bulk case, at laser intensities below $I_{0}=5 \times 10^{13} \, \mathrm{W/cm^{2}}$, the increase in the absorbed energy was proportional to $I_{0}^{4}$, as seen in Fig. \[fig:absenergy\](a) (see the dashed line), which is an evidence for four-photon absorption. The rate of increase in the absorbed energy became less as the intensity increased. This could be related to an increase in the reflectance, since a high reflectance implies a relatively low fraction of the transmitted wave, which in turn implies reduced availability of the wave for absorption, and hence a decrease in the energy absorbed by the medium. The ablation threshold was determined from a cross point at which the absorbed energy and the cohesive energy intersected. The calculated ablation threshold at the bulk surface was $2.2 \times 10^{14} \, \mathrm{W/cm^{2}}$ and the corresponding fluence was $1.7 \, \mathrm{J/cm^{2}}$. This value is slightly lower than the experimental value of $2 \, \mathrm{J/cm^{2}}$ reported in Ref. . Although our simulation underestimated the band gap energy and considered only the multiphoton ionization for CB electron generation, the estimated ablation threshold showed a good agreement with the experimental value. The minor difference might have come from the underestimation in the band gap energy, which can be systematically modified by using a more elaborate functional that reproduces the experimental band gap energy. It should be mentioned that our estimated ablation threshold can be considered as a maximum operational intensity, which means the maximum intensity that the material can be exposed to without producing any ablation. This is because we assumed that the kinetic energy of the CB electrons was completely transferred to the lattice, while in reality some losses may actually be occurring, pushing the threshold higher up. It is worth comparing the estimated threshold value with the one obtained by using another threshold criterion, i.e., the critical density. When the critical density was used, the threshold fluence for laser-induced ablation was $0.5 \, \mathrm{J/cm^{2}}$, which is significantly lower than that obtained from experiments. This may again be coming from the underestimated band gap energy used in the simulation. If the band gap energy becomes larger and closer to the experimental value, the electrons will not be easily excited from the VB to the CB. Therefore, a higher laser intensity would be needed to reach the critical density, resulting in an increase in ablation threshold fluence. However, the exact relation between the band gap energy and the generation of CB electrons needs to be investigated for an accurate evaluation of the critical density criterion. The absorbed energies for the case of thin films showed interesting characteristics, depending on the thickness and the surface, whether front or rear. For HWTF, the absorbed energies at the front and rear surfaces had similar values at low laser intensities. However, the absorbed energy at the front surface was more than the rear surface at higher laser intensities. This can be understood from the CB electrons distributed around the front surface. The threshold for HWTF was $1.8 \times 10^{14} \, \mathrm{W/cm^{2}}$, which was slightly lower than that for the bulk, indicating that the HWTF is weaker than the bulk when it comes to the intense laser pulses. On the other hand, for QWTF, the absorbed energies at the front and rear surfaces showed a huge difference in the low laser intensity range, but the difference gradually decreased in the high intensity region. The calculated ablation thresholds for QWTF were $2.6 \times 10^{14} \, \mathrm{W/cm^{2}}$ for the front surface and $2.9 \times 10^{14} \, \mathrm{W/cm^{2}}$ for the rear surface. These results for the absorbed energy in the case of QWTF can be explained by the fact that at low laser intensities the CB electrons are mainly generated near the rear surface, but at high intensities the CB electrons near the front surface become dominant in the absorption process. Consequently, the ablation threshold is closely related to CB electrons generation and energy absorption by the CB electrons generated inside the medium. \[sec:conclusion\] Conclusion ============================= Through first-principles simulations, we have investigated the changes in the optical response of bulk and thin-film $\alpha$-quartz when irradiated with an intense ultrashort laser pulse. The generation of CB electrons in the medium has also been investigated in detail. The change in the reflectance with laser intensity was mainly attributed to the generation and spatial distribution of the CB electrons in the medium. The simulation studies performed for laser intensities in the range $10^{10} \, \mathrm{W/cm^{2}}$ to $2 \times 10^{15} \, \mathrm{W/cm^{2}}$ successfully described the transition (from a dielectric to plasma) property of the medium as well as the laser intensity required for this change in the optical properties. At low laser intensities, the interference effect between the secondary waves from the front and rear surfaces was the dominant process in the reflectance behavior of thin films. However, at high laser intensities, the CB electrons generated on the front surface played a dominant role and interference a minor role in changing the reflectance. The energy absorbed by the CB electrons in the medium was used to estimate the laser-induced ablation threshold of $\alpha$-quartz materials. The theoretical estimation showed a good agreement with the experimental value, despite some limitations in the simulations, particularly, underestimation of the band gap energy. This limitation can be easily overcome by using more elaborate functionals; these will be discussed in a further study. 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--- abstract: 'The perimeter and area generating functions of exactly solvable polygon models satisfy $q$-functional equations, where $q$ is the area variable. The behaviour in the vicinity of the point where the perimeter generating function diverges can often be described by a scaling function. We develop the method of $q$-linear approximants in order to extract the approximate scaling behaviour of polygon models when an exact solution is not known. We test the validity of our method by approximating exactly solvable $q$-linear polygon models. This leads to scaling functions for a number of $q$-linear polygon models, notably generalized rectangles, Ferrers diagrams, and stacks.' author: - | C. Richard and A. J. Guttmann\ Department of Mathematics and Statistics\ The University of Melbourne, Parkville, Victoria 3052, Australia date: title: | **$q$-linear approximants:\ Scaling functions for polygon models** --- Introduction ============ Models of polygons and related combinatorial objects have received considerable attention in recent years (for a recent monograph, see [@J00]). They are of interest in physics as models of vesicles or polymer molecules in solution. The interplay between bulk energy and surface energy in these models gives rise to a phase transition from an extended phase to a compact, ball-shaped phase [@BOP93]. There have been many studies of combinatorial aspects of these models, including a general method for deriving the perimeter and area generating function of column-convex models [@B96]. Less is known about analytic aspects of the solutions, which are needed to understand the phase transitions of these models. Scaling functions which describe the crossover behaviour at critical points have been computed for a number of polygon models, mostly by indirect methods such as from a semi-continuous version of the models [@PO95; @PB95]. The only direct derivation of scaling behaviour has been for staircase polygons [@P94] by methods of uniform asymptotic expansions. There is, however, no known general method to obtain scaling functions directly from functional equations. This paper presents such a method in the simplest case of a $q$-linear functional equation in the perimeter variable. This class of functional equations is satisfied by rectangles, Ferrers diagrams, and stacks [@PO95]. As a step towards the analysis of more complicated classes, we introduce $q$-linear approximants of first order so as to analyze models which do not obey a first-order $q$-linear equation, but which can be well approximated by one. We will test our method by approximating exactly solvable $q$-linear polygon models of generalized rectangles, Ferrers diagrams and stacks. In particular, we will analyze the model of Ferrers diagrams with a hole and obtain a differential equation for the scaling function by analysis of the approximants. We discuss the connection between this new type of approximant and the method of partial differential approximants [@FC82; @SF82; @RF88; @S90]. Finally we indicate how our methods can be extended to more general classes of polygon models. In a subsequent publication we will consider $q$-quadratic and other non-linear approximants, which we expect will give good approximations to the scaling function of as yet unsolved models, such as self-avoiding polygons. Phase diagrams and scaling functions ==================================== Let us briefly review phase diagrams of $q$-linear polygon models in order to fix our notation. (We follow [@PO95; @PB95; @Ow00]). The perimeter and area generating function of a polygon model is given by f(x,y,q) = \_[r,s,n=1]{}\^f\_[r,s,n]{} x\^r y\^s q\^n = \_[n=1]{}\^f\_n(x,y) q\^n, where $f_{r,s,n}$ denotes the number of configurations of area $n$, horizontal perimeter $r$ and vertical perimeter $s$. We introduce the area activity $q$, the horizontal perimeter activity $x$, and the vertical perimeter activity $y$. The perimeter generating function of the polygon model is given by $f(x,y,1)$. A phase diagram is the graph of the radius of convergence of $f(x,y,q)$ in the parameter space $x,y,q$. Let us consider the isotropic version $f(t,q) := f(t,t,q)$ of the model, with $t$ denoting the total perimeter, so that the phase diagram is two-dimensional. For a typical $q$-linear polygon model such as Ferrers diagrams or stacks, defined below, the phase diagram is as depicted in Figure \[fig:stacphase\]. Let us interpret the phase diagram using the grand-canonical ensemble in which we count, for fixed area, all polygons by perimeter. The curve $q_c(t)$ where the polygon generating function diverges is related to the [*free energy*]{} per unit area of the ensemble, -q\_c(t) = \_[n ]{} f\_n(t). The phase $q_c(t)<1$ consists of [*inflated*]{} polygons, whose perimeter grows like their area. The phase $q_c(t)=1$ consists of [*ball-shaped*]{} polygons, their perimeter growing as the square-root of their area. This results in a vanishing free energy for the ensemble. This behaviour is characteristic of a first order phase transition at the point $t_f$ where both phases meet. In the ball-shaped phase $q_c(t)=1$, there are contributions to the [*boundary*]{} free energy, however[^1]. Let us denote by $t_c$ the point where the perimeter generating function diverges. At this point a phase transition in the boundary free energy occurs: For $t<t_c$, the contributions to the boundary free energy are given by polygons of finite size, whereas for $t>t_c$ the contributions to the boundary free energy derive from polygons of infinite size. In the remainder of this paper we will concentrate on the critical behaviour about the point $t_c$ where the perimeter generating function diverges. This point is the natural one to look at from the perspective of power series approximations, which we employ. Moreover, for rectangles and more complicated models such as self-avoiding polygons, the distinction between the two phase transitions is irrelevant since $t_c$ and $t_f$ coincide for these models. To describe the singular behaviour about $t_c$ in more detail, consider $f(t,q)$ for $t$ fixed, as $q$ approaches unity. For $q$-linear polygon models, $q=1$ is a point of an essential singularity in the generating function: For $t<t_c$, $f$ converges to a finite limit. If $t=t_c$, $f$ has a power-law divergence with an exponent generally different from that of the perimeter generating function. If $t>t_c$, $f$ diverges with an essential singularity. In many cases, the crossover between these types of critical behaviour can be described by a scaling function $\bar{P}(\bar{s})$ of combined argument $\bar{s}= (t_c-t)(1-q)^{-\phi}$, f(t,q) \~ |[P]{}( ) ( ( t,q) (t\_c\^-,1\^-) ). \[eqn:scalingfctn\] The asymptotic behaviour of the scaling function at infinity is related to the behaviour of $f(t,q)$ for $t<t_c$. To see this, assume that $f(t,q)$ admits an asymptotic expansion of the form f(t,q) = \_[n=0]{}\^f\_n(t) (1-q)\^n (t&lt;t\_c) about $q=1$, where the leading contributions of the coefficients $f_n(t)$ are given by f\_n(t) = + [O]{} ( (t\_c - t)\^[-\_n+1]{}), as $t$ approaches $t_c$[^2]. For $q$-linear polygon models, the asymptotic expansion can be computed recursively from the defining functional equation. It can be inferred from (\[eqn:scalingfctn\]) that the existence of a scaling function implies the restriction \_n = + \[eq:gamn\] on the exponents $\gamma_n$. Moreover, it can be seen that the numbers $p_n$ are the coefficients in the asymptotic expansion of the scaling function |[P]{}(|[s]{}) = \_[n=0]{}\^p\_n |[s]{}\^[-\_n]{}. We assume that the scaling function $\bar{P}(\bar{s})$ is regular at the origin. (This assumption is not always fulfilled. The simplest counterexample is the model of rectangles in its isotropic version.) In this situation, the behaviour of $f$ at $t=t_c$ is given by f(t\_c,q) \~ (q 1\^-). The exponents $\theta$ and $\gamma_0$ are called [*critical exponents*]{} of the model. $\theta$ describes the behaviour of $f(t_c,q)$ about $q=1$, whereas $\gamma_0$ describes the power-law behaviour of the perimeter generating function about $t_c$. The exponent $\phi$ is called the [*crossover exponent*]{} and relates the two critical exponents, see (\[eq:gamn\]). $q$-linear polygon models ========================= We call a polygon model $q$-linear of $N$th order if its generating function satisfies a $q$-linear functional equation[^3] G(t,q) = \_[k=1]{}\^N a\_k(t,q) G(q\^kt,q) + b(t,q), \[form:leq\] where $a_k(t,q)$ and $b(t,q)$ are rational functions in $t$ and $q$. Here, $t$ may denote the total or horizontal or vertical perimeter. Explicit realizations include rectangles, Ferrers diagrams, and stacks [@PO95]. They all satisfy a $q$-linear functional equation of first order in the horizontal perimeter activity $x$. We may construct new polygon models from given ones by allowing for decorations of the polygons. In this way we may obtain models of polygons with holes, of coloured polygons, or the like. Decorations may be interpreted in physical terms as allowing for a refined structure of polygons, which may be a better approximation to vesicles than undecorated polygons. These models have a natural interpretation in terms of random tilings of polygonal shape [@RHHB98]. The question arises as to which decorations lead to models which continue to satisfy $q$-linear functional equations. This is the case for models whose generating function can be obtained by application of a linear differential operator (w.r.t. $x$, $y$, and $q$) to the generating function of the undecorated model. For example, the model of Ferrers diagrams with a 1-hole, defined in the appendix, satisfies a $q$-linear functional equation of order 3. Let us now focus on the simplest class of decorated polygon models which satisfy a $q$-linear functional equation of first order. To this end, consider first decorated rectangles of unit height. We denote the generating function of this model by $b_1(x,y,q)$. For example, the generating function of a rectangle with exactly $k$ black unit squares is given by b\_1(x,y,q) = , while the generating function of undecorated rectangles is given by $b_1(x,y,q)=yqx/(1-qx)$. Let us now consider the polygon model of rectangles with a decorated top layer. As indicated in Fig. \[fig:functional\], we can consider all rectangles of height $m+1$ as being obtained from rectangles of height $m$ by adding a row of squares below the bottom layer. This construction misses out all decorated rectangles of height $1$. A similar construction can be applied to Ferrers diagrams and stacks with a decorated top layer. The figure indicates that the models defined above satisfy a $q$-linear functional equation of first order G\_s(x,y,q) = G\_s(qx, y, q) + b\_1(x,y,q), \[form:funceq\] where $s=0,1,2$ denotes rectangles, Ferrers diagrams, and stacks respectively, and $b_1(x,y,q)$ is the generating function of decorated rectangles of unit height. Equation (\[form:funceq\]) can be iterated to give a closed form for the area and perimeter generating function G\_s(x,y,q) = \_[n=1]{}\^, \[form:linchain\] where $(t;q)_n = \prod_{k=0}^{n-1}(1-q^k t)$ denotes the $q$-product. The perimeter generating function can be obtained from the functional equation by setting $q=1$ and solving for $G_s(x,y,1)$, G\_s(x,y,1) = b\_1(x,y,1). The models of rectangles, Ferrers diagrams and stacks are recovered as special cases where the trivial decoration is chosen. Scaling functions via dominant balance ====================================== We now discuss how to derive scaling exponents and scaling functions from $q$-linear functional equations about the critical point where the perimeter generating function diverges. Our technique relies on the method of dominant balance [@BO78], which has previously been applied to the derivation of scaling equations for a number of semi-continuous models [@PB95; @PO95]. Our approach lies in deriving scaling functions for the discrete models by manipulating the $q$-functional equation directly. It consists of three steps: Firstly, the critical point is shifted to the origin by a change of variables. Then a scaling variable is introduced, and a consistent set of scaling exponents is sought. Finally, the resulting differential or difference equation is solved. Consider the $q$-linear functional equation of first order a\_0(x,q) f(x,q) - a\_1(x,q)f(qx,q) - b(x,q)=0. \[form:dbeq\] Bearing later applications in mind, we restrict $a_0(x,q), a_1(x,q), b(x,q)$ to polynomials in $x$ and $q$. For readability, we suppress all subsequent dependencies on $q$. As a first step, assume that the critical point is at $x=x_c$ and $q=1$. We expand the functional equation about the critical point. To this end we introduce small variables $\epsilon=1-q$ and $s=x_c-x$, and define $P(s)=f(x_c-s)$. In these variables, the functional equation reads ( a\_0 - a\_1)(x\_c-s) P(s) - a\_1(x\_c-s) \_[n=1]{}\^ P(s) - b(x\_c-s) =0. \[form:shifted\] Second, we introduce scaled quantities s = \^|[s]{}, P = \^[-]{} |[P]{}, and write equation (\[form:shifted\]) in terms of $\bar{s}$ and $\bar{P}$. These quantities are just the scaling variable of combined argument and the scaling function (\[eqn:scalingfctn\]). Here, $\phi$ is assumed to be positive. This leads to additional factors of $\epsilon$ in each summand. The scaling equation results from taking only the terms with [*smallest*]{} exponents in $\epsilon$, for suitable choices of the exponents $\theta$ and $\phi$. Let us analyze the contribution from the $n$-th order in the expansion of $f(qx)$. Scaling leads to exponents of the form $n(1-\phi)$. This implies the constraint $\phi \le 1$, since other values lead to arbitrarily small exponents with increasing $n$. If $\phi < 1$, only the first order contributes, leading to $x_c \frac{d}{d\bar{s}} \bar{P}(\bar{s}) $. If $\phi=1$, higher orders cannot be ignored, and the dominant part of the sum equals $\bar{P}(\bar{s} + x_c) - \bar{P}(\bar{s})$. This results in a differential or difference equation for the scaling function. We will present typical examples of both kinds below. We concentrate on critical points given by the smallest pole of the perimeter generating function, that is at the smallest positive $x_c$ satisfying a\_1(x\_c,1) = a\_0(x\_c,1). For polygon models, the coefficients of the perimeter generating function are all positive. This generating function is obtained by setting $q = 1$ in (\[form:shifted\]), and hence implies that $b(x)/(a_0(x)-a_1(x))$ has nonnegative Taylor-coefficients. This implies in particular that $b(x_c,1)$ is non-zero. Therefore, the leading contribution of $b(x)$ in the scaling limit is of order $\epsilon^0$. The contributions from $a_0(x)-a_1(x)$ and from $a_1(x)$ have to be analyzed for each model separately. To obtain a nontrivial scaling equation, we demand that we get contributions from each of the three terms in (\[form:shifted\]). This means that all three exponents in $\epsilon$ have to be equal. We thus arrive at a set of equations determining $\theta$ and $\phi$, …- = …- + (1-) = 0, where $\ldots$ denotes contributions from $a_0(x)-a_1(x)$ and $a_1(x)$, respectively. We discuss particular examples below. We finally mention that a scaling analysis of the $q$-linear equation of order $N$ can be carried out by the same method. If $\phi<1$, this results in a differential equation, as found above for the case $N = 1.$ Ferrers diagrams ---------------- Ferrers diagrams satisfy a $q$-linear functional equation with $$\begin{aligned} a_0(x,q)&=& 1-qx,\\ a_1(x,q)&=& y, \nonumber \\ b(x,q)&=& y q x. \nonumber\end{aligned}$$ The perimeter generating function diverges at $x_c=1-y$. $a_1(x,q)$ and $b(x,q)$ are nonzero at the critical point, whereas $a_0(x,q)-a_1(x,q)$ vanishes linearly in $s=x_c-x$. The leading contributions in the three terms of equation (\[form:shifted\]) give - = -+ (1-) = 0. This leads to exponents = , =, and to the differential equation ([|s]{}) = ([|s]{}) + 1. The solution, which is uniquely determined if we demand power-law behaviour as $\bar{s} \to \infty$, is given by a complementary error-function |[P]{}(|[s]{}) = ( ) ( ), where $a=y(1-y)$. The asymptotic behaviour is given by $$\begin{aligned} \bar{P}(\bar{s}) &\sim& \frac{a}{\bar{s}} \qquad (\bar{s} \to +\infty) \\ \bar{P}(\bar{s}) &\sim& \sqrt{2\pi a} \, \exp\left(\frac{\bar{s}^2}{2a} \right) \qquad (\bar{s} \to -\infty) \nonumber \\ \bar{P}(0) &=& \sqrt{\frac{\pi a}{2}}. \nonumber\end{aligned}$$ This is in agreement with both the scaling function found for the semi-continuous model and with the asymptotic behaviour computed previously [@PO95]. Decorated rectangles -------------------- We next consider the model of rectangles with a decorated top layer. The decoration consists of exactly $k$ black squares. The generating function of the top layer is b\_1(x,y,q) = , As shown above, the model satisfies a $q$-linear functional equation with $$\begin{aligned} a_0(x,q)&=& (1-qx)^{k+1},\label{eq:rectchain}\\ a_1(x,q)&=& y(1-q x)^{k+1}, \nonumber \\ b(x,q)&=& y (q x)^k. \nonumber\end{aligned}$$ The perimeter generating function has a pole of order $k+1$ at $x_c=1$. In contrast with the model of Ferrers diagrams, this point coincides with the point $x_f$ where the first order phase transition in the free energy occurs. The case $k=0$ is closely related to rectangles[^4]: Comparison of (\[eq:rectchain\]) with the functional equation for rectangles shows that both models obey the same equations about the critical point and hence have the same scaling functions. The leading terms in the three summands of equation (\[form:shifted\]) give (k+1)- = (k+1)- + (1-) = 0. This leads to exponents = k+1, =1, and we get the difference equation (1+ |[s]{})\^[k+1]{} ( |[P]{}\_k(|[s]{}) - y |[P]{}\_k(|[s]{}+1) ) -y =0. In order to obtain the correct scaling function from this recursion, we have to force the asymptotic behaviour as $\bar{s} \to \infty$ to be of power-law type. This fixes the constant term, and for $y<1$ we arrive at the Lerch functions |[P]{}\_k(|[s]{}) = \_[n=1]{}\^. The asymptotic behaviour is given by $$\begin{aligned} \bar{P}_k(\bar{s}) &\sim& \frac{y}{1-y}\frac{1}{\bar{s}^{k+1}} \qquad (\bar{s} \to +\infty) \\ \bar{P}_k(0) &=& \sum_{n=1}^\infty \frac{y^n}{n^{k+1}} . \nonumber\end{aligned}$$ The case $k=0$ (rectangles) is in agreement with both the scaling function of the semi-continuous model and the asymptotics computed in [@PO95]. The case $y=1$ is different. The perimeter generating function diverges for all values of $x$, but it is possible to obtain scaling behaviour about $x_c=1$, as $q$ approaches unity. For $k>0$, the solution of the difference equation is given by |[P]{}\_k(|[s]{}) = \_k(1+|[s]{}) where $\Psi_k(x)$ denotes the $k$-th derivative of the $\Psi$-function $\Psi(x)=\partial_x \log\Gamma(x)$. The asymptotic behaviour is given by $$\begin{aligned} \bar{P}_k(\bar{s}) &\sim& \frac{1}{k}\frac{1}{\bar{s}^k} \qquad (\bar{s} \to +\infty) \\ \bar{P}_k(0) &=& \zeta(k+1), \nonumber\end{aligned}$$ where $\zeta(k)=\sum_{n=1}^\infty n^{-k}$ denotes the Riemann zeta function. If $k=0$, the scaling function diverges logarithmically as $\bar{s} \to \infty$. In order to obtain a well-defined crossover behaviour, we compensate for this by the addition of a logarithmic term, |[P]{}\_0(|[s]{}) = -(1+|[s]{}) -. \[eq:rect0\] The asymptotic behaviour then follows as $$\begin{aligned} \bar{P}_0(\bar{s}) &\sim& -\log (\bar{s}\epsilon) = - \log s \qquad (\bar{s} \to +\infty) \\ \bar{P}_0(0) &=& -\log \epsilon + \gamma, \nonumber\end{aligned}$$ where $\gamma=-\int_0^\infty e^{-t} \ln t \, dt \approx 0.57721$ denotes Euler’s constant. This gives the (non-uniform) asymptotic behaviour first computed in [@PO95]. The scaling function describing this behaviour was not previously known, however. $q$-linear approximants ======================= We now consider the more general situation where we cannot obtain the generating function, but only a finite number of terms thereof. The method ---------- The basic idea of $q$-linear approximants (of first order) is to fit a $q$-linear functional equation to a function $f(t,q)$, a\_0(t,q) f(t,q) = a\_1(t,q) f(qt,q) + b(t,q), \[form:appeq\] such that (\[form:appeq\]) is [*exact*]{} up to a given degree in the variables. Here, we restrict $a_0(t,q),$ $a_1(t,q),$ and $b(t,q)$ to be polynomials in $t$ and $q$ of degree $n_t$ and $n_q$, say. The coefficients of the polynomials can be found by solving the system of linear equations deriving from the expansion of (\[form:appeq\]) in its two variables. This process is not unique, since there are many choices of sets of linear equations. Moreover, (\[form:appeq\]) could be expanded about points other than the origin, resulting in approximants accurate about these points. Our approach is to demand that $q$-linear approximants shall reduce to [*linear approximants*]{} with polynomials of order $n_t$, as $q$ approaches unity, which corresponds to an approximation of the perimeter generating function by rational functions. Thus such approximants are only likely to be good if the perimeter generating function is dominated by a pole. We therefore expand (\[form:appeq\]) about $t=0$ and $q=1$, taking into account only terms up to a fixed order $N_t(n_t)$ in $t$. In order to obtain the desired limit, $N_t(n_t)$ has to be chosen large enough. We order the resulting terms by increasing powers in $1-q$ and, for each power, by increasing powers in $t$. For given $N_t(n_t)$, we compute $q$-linear approximants with polynomials of order $n_t$ in $t$ and of order $n_q=0,1,\ldots$ For given $N_t(n_t)$, the highest obtainable order $n_q$ depends on the number of equations deriving from (\[form:appeq\]). To fix the multiplicative constant, we require $a_0(0,0)=1+a_1(0,0)$. Information about the scaling function is obtained by applying the method of dominant balance to the approximants, as described above. In order to test the method of $q$-approximants, we will apply the method to a number of exactly solvable $q$-linear polygon models, which are mainly isotropic versions of the models defined above. These models have a rational perimeter generating function and can be shown to obey a $q$-linear equation of higher than first order. Therefore, $q$-linear approximants (of first order) should give the correct differential equation for the scaling function. This can be checked by computing the leading coefficients in the asymptotic expansion of the model, using the anisotropic functional equation directly. If the functional equation of the isotropic model is known, the scaling function can alternatively be obtained by applying the method of dominant balance. We will illustrate the method of $q$-approximants by deriving the scaling behaviour of the $q$-linear models, defined above in the isotropic case, which do not obey a $q$-linear functional equation of first order. Stacks ------ The generating function $f(t,q)$ of stacks with equal horizontal and vertical perimeter activity $x=y=t$ satisfies the $q$-linear functional equation of order 2 a\_0(t,q) f(t,q) = a\_1(t,q) f(qt,q) +a\_2(t,q) f(q\^2t,q) + b(t,q), where the polynomials $a_0(t,q)$, $a_1(t,q)$, $a_2(t,q)$ and $b(t,q)$ are given by $$\begin{aligned} a_0(t,q)&=& (1-q^2t)^2 (1-qt)^3, \label{eq:stacks}\\ a_1(t,q)&=& t^2(1+q)(1-q^2t)^2, \nonumber \\ a_2(t,q)&=& -q^3t^4, \nonumber \\ b(t,q)&=& -qt^2(1-q^2t)(q^4t^3-q^3t^2+q^2t+qt-1). \nonumber\end{aligned}$$ We found this relation by computing $q$-linear approximants of second order to stacks. It is possible to interpret this relation combinatorially [@BM01]. The perimeter generating function $f(t,1)$ diverges at $t_c=(3-\sqrt5)/2$ with a simple pole. The method of dominant balance can be applied to compute the scaling function about $t_c$. It is of the same form as the scaling function for Ferrers diagrams, which corresponds to the observation made for the semi-continuous models [@PO95]. We used stacks in order to test the method of $q$-linear approximants. Their rational perimeter generating function is obtained by approximants of cubic order ($n_t$=3). A scaling analysis of cubic approximants at $q=1$ and $t_c=(3-\sqrt5)/2$ yields the correct type of differential equation for the scaling functions for each approximant, with generally incorrect coefficients. The accuracy of approximation increases with the degree $n_q$ in $(1-q)$ of the polynomials. For $N_t(3) > 6$ and $n_q\ge1$, the approximants yield the [*correct*]{} scaling equation. This result is robust against increasing the order $n_t$. Decorated Ferrers diagrams -------------------------- We introduce Ferrers diagrams with a decorated top layer: Consider a decoration consisting of exactly $k$ black squares. Approximants indicate that the scaling function is the same as for (pure) Ferrers diagrams. We checked this for $k=0,1,\ldots,5$. Instead of considering more complicated decorations of the top layer, we will now consider Ferrers diagrams with a decorated bottom layer. Let us approximate the model with a decoration consisting of exactly two black squares which may be placed anywhere in the bottom layer. The model has the generating function G(x,y,q) = \_[n=2]{}\^ , \[form:ferrch\] where $(t;q)_n = \prod_{k=0}^{n-1}(1-q^k t)$ denotes the $q$-product. The perimeter generating function of the model is G(x,y,1) = . We consider the isotropic case where $x=y=t$. The critical point is $t_c=1/2$. Since the perimeter generating function is rational with numerator and denominator polynomials of order 4 and 3, we use $q$-linear approximants with $n_t=4$ and $N_t=20$. For $n_q>2$, the dominant terms of the approximants give the differential equation ( [|s]{}\^3 + ) [|P]{}([|s]{}) - ( + ) ([|s]{}) - =0 with exponents =, =. This result is robust against varying the values of $n_q$ and $n_t$ in the approximation. We checked this up to $n_q=8$ and also for increasing values of $n_q$ at $n_t=5$ and $n_t=6$. The above equation leads to the scaling function ([|s]{}) = ( \_[|s]{}\^ d t ) 2 e\^[8 [|s]{}\^2]{} (1+16 [|s]{}\^2). The asymptotic behaviour is given by $$\begin{aligned} \bar{P}(\bar{s}) &\sim& \frac{2^{-7}}{\bar{s}^3} \qquad (\bar{s} \to +\infty) \\ \bar{P}(\bar{s}) &\sim& 2 \bar{P}(0) \, e^{8 {\bar s}^2} {\bar s}^2 \qquad (\bar{s} \to -\infty) \nonumber \\ \bar{P}(0) &=& \frac{\sqrt{2\pi}}{8} \nonumber.\end{aligned}$$ Using the methods described at the end of the appendix, it can be shown that the isotropic model satisfies a $q$-linear equation of third order which can be used to test that the scaling function obtained by $q$-linear approximants is correct. Ferrers diagrams with a 1-hole ------------------------------ Ferrers diagrams with a 1-hole are defined in the Appendix. We again consider the isotropic model where $t=x=y$. Since the perimeter generating function is rational with numerator and denominator polynomials of order 6 and 5, we use $q$-linear approximants with $n_t=6$ and $N_t = 20$. For $n_q>2$, the dominant terms of the approximants give the differential equation ( [|s]{}\^3 + ) [|P]{}([|s]{}) - ( + ) ([|s]{}) - =0 with exponents =, =. This is the same equation as that for decorated Ferrers diagrams. This result is robust against varying the values of $n_q$ and $n_t$ in the approximation. We checked this up to $n_q=8$ and also by increasing values of $n_q$ at $n_t=7$ and $n_t=8$. Comparison with partial differential approximants ================================================= There is an existing approximating method designed to compute critical exponents and scaling functions about multi-singular points, known as the method of partial differential approximants (p.d.a.), due to Fisher and co-workers [@FC82; @SF82; @RF88; @S90]. We will briefly explain the method and compare it to our approach. The basic idea of p.d.a. derives from the observation that a scaling function f(t,q) = |[P]{} ( ) + f\_0 \[form:pda\] obeys the first-order partial differential equation f(t,q) + f\_0 = (t\_c-t) f(t,q) + (q\_c-q) f(t,q). \[form:pde\] Therefore, p.d.a. of the form a(t,q) f(t,q) + b(t,q) = c(t,q) f(t,q) + d(t,q) f(t,q), \[form:pdeq\] where $a(t,q)$, $b(t,q)$, $c(t,q)$ and $d(t,q)$ are polynomials in $t$ and $q$, may serve to detect possible scaling behaviour about [*multi-critical points*]{} $(t_c,q_c)$ defined by the simultaneous vanishing of $c$ and $d$, c(t\_c,q\_c) = d(t\_c,q\_c) = 0. The critical exponents $\theta$ and $\phi$ can then be read off as lowest order coefficients in the expansion of the approximating polynomials about the multi-critical point. Numerical methods can be used to determine subsequent terms in the Taylor-expansion of the scaling function. It has been shown [@RF88] that numerical integration works if the crossover exponent $\phi$ is restricted to $1/2<\phi<2$. In contrast to this general setup, $q$-linear approximants are only suited to the detection of critical behaviour for models whose scaling function obeys a linear differential or difference equation of first order. Equivalently, $q$-linear approximants can be used to test whether a scaling function obeys an equation of the above type: If it does not, the approximants are likely to fail to converge. If it does, the approximants will converge and give the underlying differential or difference equation. This is then more specific information than can be gained from the p.d.a. approach, though the p.d.a. approach currently approximates a broader range of scaling behaviour. In subsequent work we will extend to non-linear approximants, which should combine the generality of the p.d.a. approach with the specificity of information obtainable by the $q$-approximants. We conclude with remarks about Ferrers diagrams and rectangles. Since Ferrers diagrams have a crossover exponent $\phi=1/2$, numerical integration using p.d.a. [@RF88] to obtain the coefficients of the scaling function is unlikely to converge. The model of rectangles (\[eq:rect0\]) does not obey a scaling law of type (\[form:pda\]). It can be shown that the scaling function gives [*quadratic*]{} prefactors for the derivatives in (\[form:pde\]). Therefore, the p.d.a. method cannot be successfully applied here in its standard form. For rectangles with the random chain considered above, p.d.a. may provide good estimates. In summary, we would expect that the method reported here is likely to be better than p.d.a. for those systems whose scaling function is described by, or well approximated by, a linear differential or difference equation of first order, while the p.d.a. method might be expected to better approximate those systems that do not. As we extend the method reported here to higher order $q$-functional equations, and possibly other types of functional equation as well, we would expect these new $q$-approximants to more appropriately represent a correspondingly larger class of systems. We emphasise that these are remarks of a general nature, and not the result of rigorous numerical comparisons, which we have not carried out. Conclusion ========== We have developed techniques to obtain scaling functions for $q$-linear polygon models about the point where the perimeter generating function diverges, using the method of dominant balance. This led to scaling functions for a number of $q$-linear polygons models generalizing rectangles, Ferrers diagrams and stacks. The question arises as to what extent can the scaling behaviour for these models be obtained by direct methods? The most direct approach is to approximate the perimeter and area generating function by their Euler-Maclaurin sum and to estimate the resulting integral by methods of uniform asymptotic expansions [@W89], in the spirit of [@P94]. The authors are, however, not aware of standard methods to do this, apart from Bleistein’s method [@O74], which can be used to analyze Ferrers diagrams. We introduced the method of $q$-linear approximants to obtain scaling functions of models where an exact $q$-linear functional equation of first order does not exist. The method yields the correct scaling functions for exactly solvable models which can be described by $q$-linear functional equations of higher than first order, such as for decorated $q$-linear models. We claim that $q$-approximants will be appropriate for the analysis of the scaling behaviour of statistical models whose scaling function may be well described by a difference or differential equation of first order. The method of dominant balance can also be applied to obtain differential equations for scaling functions from $q$-functional equations different from $q$-linear. For example, it is possible to derive the differential equation for the scaling function of the model of staircase polygons [@PB95], which belongs to the $q$-quadratic class. This indicates that the idea of $q$-linear approximants can be generalized to more complex classes of polygon models satisfying $q$-algebraic functional equations. This leads to $q$-algebraic approximants. Even the $q$-quadratic class is interesting to analyze, since there are solvable models where the scaling behaviour is not known (such as staircase polygons with a hole; see also [@J001] for an example), and more interestingly it can be used to approximate the generating function for self-avoiding polygons, which displays a square-root divergence in its perimeter generating function. We are developing our method in that direction. Acknowledgements {#acknowledgements .unnumbered} ================ We thank M. Bousquet-Mélou for a number of comments on the manuscript and A. Owzcarek for stimulating discussions. Financial support from the German Science Foundation (DFG) and from the Australian Research Council (ARC) is gratefully acknowledged. Appendix: Ferrers diagrams with a 1-hole {#appendix-ferrers-diagrams-with-a-1-hole .unnumbered} ======================================== We define the polygon model of Ferrers diagrams with a 1-hole and give a closed form for the perimeter generating function and for the perimeter and area generating function. The model of Ferrers diagrams with a 1-hole consists of all Ferrers diagrams where a unit square is removed from the interior. For a typical Ferrers diagram, this is graphically depicted in Fig. \[fig:Ferrerhole\]. If we denote the last column height by $m$, the last-but-one column height by $m_1$, the bottom row length by $n$ and the second row length by $n_1$, the number of possible sites for a hole $h(n,m,n_1,m_1)$ is given by h(n,m,n\_1,m\_1) = - n -m -n\_1-m\_1 +4. \[form:hole\] This translates into a formula for the generating function $G(x,y,q)$ of Ferrers diagrams with a 1-hole G(x,y,q) = ( F - x (\_x F) - y (\_y F) + q(\_q F) )(qx,qy,q), \[eq:ferrhole\] where $F(x,y,q)$ is the generating function of Ferrers diagrams (without holes) of height and width greater than one. In particular, we have G(x,y,1) = ( )\^3 . The perimeter generating function is rational with the same critical point as the Ferrers diagrams (without holes). The perimeter and area generating function is given by G(x,y,q) = \_[n=2]{}\^ \_[k=3]{}\^[n+1]{}, where $(t;q)_n = \prod_{k=0}^{n-1}(1-q^k t)$ denotes the $q$-product. $G(x,y,q)$ satisfies a $q$-linear equation of third order. This equation can be derived by expressing $G(x,y,q)$, $G(qx,qy,q)$ and $G(q^2x,q^2y,q)$ in terms of $F$, $(\partial_x F)$, $(\partial_y F)$, $(\partial_q F)$ at argument $(q^3x,q^3y,q)$. This is done by using (\[eq:ferrhole\]) and the symmetrized version of the functional equation for $F(x,y,q)$. The resulting system of linear equations can be solved for $F$ in terms of $G$. Insertion of the result into the functional equation for $F$ gives the functional equation for $G$. 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--- abstract: 'In this paper, we develop a fully discrete Galerkin method for solving initial value fractional integro-differential equations(FIDEs). We consider Generalized Jacobi polynomials(GJPs) with indexes corresponding to the number of homogeneous initial conditions as natural basis functions for the approximate solution. The fractional derivatives are used in the Caputo sense. The numerical solvability of algebraic system obtained from implementation of proposed method for a special case of FIDEs is investigated. We also provide a suitable convergence analysis to approximate solutions under a more general regularity assumption on the exact solution.' author: - | P. Mokhtary\ \ \ title: 'Discrete Galerkin Method for Fractional Integro-Differential Equations' --- [**Subject Classification:**]{}[34A08; 65L60]{} [**Keywords:**]{} Fractional integro-differential equation(FIDE), Galerkin Method, Generalized Jacobi Polynomials(GJPs), Caputo derivative. Introduction ============ In this paper, we provide a convergent numerical scheme for solving FIDE $$\label{1} \left\{\begin{array}{l} \mathcal D^q u(x)=p(x) u(x)+f(x)+\lambda \int\limits_0^x{K(x,t) u(t) dt},~~~ x \in \Omega=[0,1],\\ \\ u(0)=0, \end{array}\right.$$ where $q\in \mathbb R^+ \bigcap (0,1)$. The symbol $\mathbb R^+$ is the collection of all positive real numbers. $p(x)$ and $f(x)$ are given continuous functions and $K(x,t)$ is a given sufficiently smooth kernel function, $u(x)$ is the unknown function. Note that the condition $u(0)=0$ is not restrictive, due to the fact that (\[1\]) with nonhomogeneous initial condition $u(0)=d,~~d \neq 0$ can be converted to the following homogeneous FIDE $$\left\{\begin{array}{l} \mathcal D^q \tilde u(x)=p(x) \tilde u(x)+\tilde f(x)+\lambda \int\limits_0^x{K(x,t) \tilde u(t) dt},~~~ x \in \Omega=[0,1],\\ \\ \tilde u(0)=0, \end{array}\right.$$ by the simple transformation $\tilde u(x)=u(x)-d$, where $\tilde f(x)=f(x)+d\bigg(p(x)+\lambda \int_{0}^{x}{K(x,t)dt}\bigg)$. Such kind of equations arising in the mathematical modeling of various physical phenomena, such as heat conduction, materials with memory, combined conduction, convection and radiation problems([@r2], [@r5], [@r20], [@r21]). $\mathcal D^q u(x)$ denotes the fractional Caputo differential operator of order $q$ and defines as([@r8], [@r13], [@r22]) $$\label{2} \mathcal D^q u(x) = \mathcal I^{1-q} u'(x),$$ where $$\label{3} \mathcal I^\mu u(x)=\frac{1}{\Gamma{(\mu)}} \int\limits_0^x{(x-s)^{\mu-1} u(s) ds},$$ is the fractional integral operator from order $\mu$. $\Gamma{(\mu)}$ is the well known Gamma function. The following relation holds[@r8] $$\label{20}\mathcal I^q(\mathcal D^q u(x))=u(x)-u(0).$$ From the relation above, it is easy to check that (\[1\]) is equivalent with the following weakly singular Volterra integral equation $$\label{5} u(x)=g(x)+\lambda \int\limits_0^x{{\bar K}(x,t) u(t) dt}.$$ Here $g(x)=\mathcal I^q f(x)$ and ${\bar K}(x,t)=\frac{(x-t)^{q-1}}{\Gamma{(q)}}p(t)+\int\limits_t^x{\frac{(x-s)^{q-1}}{\Gamma{(q)}} K(s,t)ds}.$ From the well known existence and uniqueness Theorems([@r3], [@r7]), it can be concluded that if the following conditions are fulfilled - $f(x) \in C^l(\Omega),~~l \ge 1$ - $p(x) \in C^l(\Omega),~~l \ge 1$ - $K(x,t) \in C^l(D),~~ D=\{(x,t);0 \le t \le x\le 1\},~~l \ge1$ - $K(x,x)\neq 0$, the regularity of the unique solution $u(x)$ of (\[5\]) and also (\[1\]) is described by $$\label{6} u(x)=\sum\limits_{(j,k)}{\gamma_{j,k} x^{j+kq}}+U_l(x;q) \in C^l(0,1]\bigcap C(\Omega),\hspace{.5 cm} \text{with} \hspace{.5 cm} |u'(x)| \le C_q x^{q-1},$$ where the coefficients $\gamma_{j,k}$ are some constants, $U_l(.;q) \in C^l(\Omega)$ and $(j,k):=\{(j,k):~~j,k \in \mathbb N_0,~j+kq<l\}$. Here $\mathbb N_0=\mathbb N \bigcup \{0\}$, where the symbol $\mathbb N$ denotes the collection of all natural numbers. Thus, we must expect the first derivative of the solution to has a discontinuity at the origin. More precisely, if the given functions $g(x), p(x)$ and $K(x,t)$ are real analytic in their domains then it can be concluded that there is a function $U=U(z_1,z_2)$ real and analytic at $(0,0)$, so that solutions of (\[5\])and also (\[1\]) can be written as $u(x)=U(x,x^q)$([@r3], [@r7]). Recently, several numerical methods for the numerical solution of FIDE’s have been proposed. In [@r19], fractional differential transform method was developed to solve FIDE’s with nonlocal boundary conditions. In [@r23], Rawashdeh studied the numerical solution of FIDE’s by polynomial spline functions. In [@r1], an analytical solution for a class of FIDE’s was proposed. Adomian decomposition method to solve nonlinear FIDE’s was proposed in [@r17]. In [@r25], authors solved fractional nonlinear Volterra integro differential equations using the second kind Chebyshev wavelets. In [@r11], Taylor expansion approach was presented for solving a class of linear FIDE’s including those of Fredholm and Volterra types. In [@r16], authors were solved FIDE’s by adopting Hybrid Collocation method to an equivalent integral equation of convolution type. In [@r12], Chebyshev Pseudospectral method was implemented to solve linear and nonlinear system of FIDE’s. In [@r15], authors proposed an analyzed spectral Jacobi Collocation method for the numerical solution of general linear FIDE’s. In [@r9], authors applied Collocation method to solve the nonlinear FIDE’s. In [@r18], Mokhtary and Ghoreishi, proved the $L^2$ convergence of Legendre Tau method for the numerical solution of nonlinear FIDE’s. Many of the techniques mentioned above or have not proper convergence analysis or if any, very restrictive conditions including smoothness of the exact solution are assumed. In this paper we will consider non smooth solutions of (\[1\]). In this case although the discrete Galerkin method can be implemented directly but this method leads to very poor numerical results. Thus it is necessary to introduce a regularization procedure that allows us to improve the smoothness of the given functions and then to approximate the solution with a satisfactory order of convergence. To this end, we propose a regularization process which the original equation (\[1\]) will be changed into a new equation which possesses a more regularity properties by taking a suitable coordinate transformation. Our logic in choosing proper transformation is based upon the formal asymptotic expansion of the exact solution in (\[6\]). Consider (\[1\]), using the variable transformation $$\label{6xx} x=v^{\frac{1}{q}},\;\; v=x^{q},\;\; t=w^{\frac{1}{q}},\;\; w=t^q,$$ we can change (\[1\]) to the following equation $$\label{6x} \mathcal M^q \bar u(v)=\bar p(v) \bar u(v)+\bar f(v)+\lambda\int\limits_0^v{\tilde{K}(v,w) \bar{u}(w)dw},$$ where $$\begin{aligned} \label{rv4} \nonumber\bar p(v)&=&p(v^{\frac{1}{q}}),\;\; \bar f(v)=f(v^{\frac{1}{q}}),~~{\tilde K}(v,w)=\frac{w^{{\frac{1}{q}}-1}}{q} K(v^{\frac{1}{q}},w^{\frac{1}{q}}).\\ \mathcal M^q \bar u(v)&=&\frac{1}{\Gamma{(1-q)}}\int\limits_0^v{(v^{\frac{1}{q}}-w^{\frac{1}{q}})^{-q} {\bar u}'(w)dw}.\end{aligned}$$ From (\[6\]), the exact solution $\bar u(v)$ can be written as $\bar{u}(v)=u(v^{\frac{1}{q}})=\sum\limits_{(j,k)}{\gamma_{j,k} v^{\frac{j}{q}+k}}+U_l(v^{\frac{1}{q}};q)$. It can be easily seen that $\bar u'(v) \in C(\Omega)$. It is trivial that for $q=\frac{1}{n},~n \in \mathbb N$, the unknown function $\bar u(v)$ will be in the form $$\bar{u}(v)=u(v^n)=\sum\limits_{(j,k)}{\gamma_{j,k} v^{nj+k}}+U_l(v^n;q), \quad n \in \mathbb N,$$ which is infinitely smooth. Then we can deduce that the solution $\bar u(v)$ of the new equation (\[6x\]) possesses better regularity and discrete Galerkin theory can be applied conveniently to obtain high order accuracy. In the sequel, we introduce the discrete Galerkin solution $\bar u_N(v)$ based upon GJPs to (\[6x\]). Since the exact solutions of (\[1\]) can be written as $u(x)=\bar u(v)$ then we define $u_N(x)=\bar u_N(v),\; x, v \in \Omega$ as the approximate solution of (\[1\]). Spectral Galerkin method is one of the weighted residual methods(WRM), in which approximations are defined in terms of truncated series expansions, such that residual which should be exactly equal to zero, is forced to be zero only in an approximate sense. It is well known that, in this method, the expansion functions must satisfy in the supplementary conditions. The two main characteristics behind the approach are that, first it reduces the given problems to those of solving a system of algebraic equations, and in general converges exponentially and almost always supplies the most terse representation of a smooth solution([@a13], [@a14], [@aa26]). In this article, we use shifted GJPs on $\Omega$, which are mutually orthogonal with respect to the shifted weight function $\delta^{\alpha,\beta}(v)=(2-2v)^\alpha(2v)^\beta$ on $\Omega$ where $\alpha, \beta$ belong to one of the following index sets $$\begin{aligned} {\mathcal N_1}&=&\{(\alpha,\beta); \alpha, \beta \le -1, ~\alpha, \beta \in \mathbb Z\},\quad~~~~~~~~~~~~~~ {\mathcal N_2}=\{(\alpha,\beta); \alpha \le -1, \beta > -1,~~\alpha \in \mathbb Z, \beta \in \mathbb R\}, \\ {\mathcal N_3}&=&\{(\alpha,\beta); \alpha>-1, \beta \le -1,~\alpha \in \mathbb R, \beta \in \mathbb Z\},\quad {\mathcal N_4}=\{(\alpha,\beta); \alpha, \beta > -1,~\alpha, \beta \in \mathbb R\},\end{aligned}$$ where the symbol $\mathbb Z$ is the collection of all integer numbers. The main advantage of GJPs is that these polynomials, with indexes corresponding to the number of homogeneous initial conditions in a given FIDE, are the natural basis functions to the Galerkin approximation of this problem([@a15], [@a16]). The organization of this paper is as follows: we begin by reviewing some preliminaries which are required for establishing our results in Section 2. In Section 3, we introduce the discrete Galerkin method based on the GJPs and its application to (\[6x\]). Numerical solvability of the algebraic system obtained from discrete Galerkin discretization of a special case of (\[6x\]) with $0<q<\frac{1}{2}$ and $\bar p(v)=1$ based on GJPs is given in Section 4. Convergence analysis of the proposed scheme is provided in Section 5. Numerical experiments are carried out in Section 6. Preliminaries and Notations =========================== In this section, we review the basic definitions and properties that are required in the sequel. Defining weighted inner product $$\Big(u_1,u_2\Big)_{\alpha, \beta}=\int_{\Omega}{u_1(v) u_2(v) \delta^{\alpha, \beta}(v) dv},$$ and discrete Jacobi-Gauss inner product $$\bigg(u_1,u_2\bigg)_{N,\alpha,\beta}=\sum\limits_{k=0}^N{u_1(v_k^{\alpha,\beta}) u_2(v_k^{\alpha,\beta}) \delta_k^{\alpha,\beta}},$$ we recall the following norms over $\Omega$ $$\|u\|_{\alpha,\beta}^2=\Big(u,u\Big)_{\alpha,\beta}, \quad \|u\|_{N,\alpha,\beta}^2=\Big(u,u\Big)_{N,\alpha,\beta},\quad \|u\|_{\infty}=\sup_{v \in \Omega} |u(v)|.$$ Here, $v_k^{\alpha,\beta}$ and $\delta_k^{\alpha,\beta}$ are the shifted Jacobi Gauss quadrature nodal points on $\Omega$ and corresponding weights respectively. The non-uniformly Jacobi-weighted Sobolev space denotes by $B_{\alpha,\beta}^{k}(\Omega)$ and defines as follows $$B_{\alpha , \beta}^{k}(\Omega)=\{ u: \|u^{(s)}\|_{\alpha+s,\beta+s} < \infty;~~ 0 \le s\le k\},$$ equipped with the norm and semi-norm $$||u||_{\alpha,\beta,k}^2=\sum\limits_{s = 0}^k ||u^{(s)}||_{\alpha+s,\beta+s}^2, \quad |u|_{\alpha,\beta,k}=||u^{(k)}||_{\alpha+k,\beta+k}.$$ The space $B_{\alpha,\beta}^{k}(\Omega)$ distinguishes itself from the usual weighted Sobolev space $H_{\alpha,\beta}^{k}(\Omega)$ by involving different weight functions for derivatives of different orders. The usual weighted Sobolev space $H_{\alpha,\beta}^{k}(\Omega)$ is defined as $$H_{\alpha , \beta}^{k}(\Omega)=\{ u: \|u^{(s)}\|_{\alpha,\beta} < \infty;~~ 0 \le s\le k\},$$ equipped with the norm $$||u||_{H_{\alpha,\beta}^{k}(\Omega)}^2=\sum\limits_{s = 0}^k ||u^{(s)}||_{\alpha,\beta}^2.$$ We denote the shifted GJPs on $\Omega$ by $G_n^{\alpha,\beta}(v)$ and define as $$\label{7} G_n^{\alpha,\beta}(v) = \left\{ \begin{array}{l} {(2 -2v)^{-\alpha}}{(2v)^{-\beta}}J_{n-n_0}^{-\alpha,-\beta}(v),~ (\alpha,\beta) \in {\mathcal N_1}, ~~n_0 = -(\alpha+\beta),\\ \\ {(2 -2v)^{-\alpha}}J_{n-n_0}^{-\alpha,\beta}(v), \quad \quad \quad ~~~\;\;(\alpha,\beta) \in {\mathcal N_2},~~ n_0 = -\alpha, \\ \\ {(2v)^{-\beta}}J_{n-n_0}^{\alpha,-\beta}(v), \quad \quad \quad ~~~\;\;(\alpha,\beta) \in {\mathcal N_3},~~ n_0 = -\beta,\\ \\ J_{n-n_0}^{\alpha,\beta}(v),\quad \quad \quad \;\;\quad \quad \quad~~~~~~~ \;(\alpha,\beta) \in {\mathcal N_4},~~ n_0=0,\\ \end{array} \right.$$ where $J_n^{\alpha,\beta}(v)$ is the classical shifted Jacobi polynomials on $\Omega$; see [@aa26]. An important fact is that the shifted GJPs $\{G_n^{\alpha,\beta}(v); n \ge 1 \}$ form a complete orthogonal system in $L_{\alpha,\beta}^2(\Omega)$; see([@a15], [@a16]). To present a Galerkin solution for (\[6x\]) it is fundamental that the basis functions in the approximate solution satisfy in the homogeneous initial condition. To this end, since $G_n^{0,-1}(0) = 0,\;\; n \ge 1$, then we can consider $\{G_n^{0,-1}(v),~~n\geq 1\}$ as suitable basis functions to the Galerkin solution of (\[6x\]). From (\[7\]) and the following formula [@aa16] $$J_i^{\alpha,\beta}(v)=\sum\limits_{k=0}^{i}{(-1)^{i-k}\frac{\Gamma{(i+\beta+1)}\Gamma{(i+k+\alpha+\beta+1)}} {\Gamma{(k+\beta+1)}\Gamma{(i+\alpha+\beta+1)}(i-k)!k!}v^k},\quad \alpha,\beta\in \mathcal N_4,$$ we can obtain the following explicit formula for $G_i^{0,-1}(v)$ $$\label{8} G_i^{0,-1}(v)= (2v)J_{i-1}^{0,1}(v)=2 \sum_{k=0}^{i-1}(-1)^{i-1-k} \frac{(i+k)!}{(k+1)! (i-1-k)! k!}v^{k+1},~~~~~i \ge 1.$$ For any continuous function $Z(v)$ on $\Omega$, we define the Legendre Gauss interpolation operator $\mathcal I_N$, as $$\label{9} \mathcal I_N Z(v) =\sum\limits_{s=0}^N {\frac{\bigg(Z,J_s^{0,0}\bigg)_{N,0,0}}{\|J_s^{0,0}\|_{N,0,0}^2} J_s^{0,0}(v)}.$$ Let $\mathcal P_N$ be the space of all algebraic polynomials of degree up to $N$. We introduce Legendre projection $\Pi_N: L^2(\Omega) \to \mathcal P_N$ which is a mapping such that for any $Z(v) \in L^2(\Omega)$, $$\label{cc5} \bigg(Z-\Pi_N Z, \phi\bigg)_{0,0}=0,\quad \forall \phi \in \mathcal P_N.$$ Discrete Galerkin Approach ========================== In this section, we present the numerical solution of (\[6x\]) by using the discrete Galerkin method based on GJPs. Let $$\label{10c} \tilde u_N(v)=\sum\limits_{i=1}^N{b_i G_{i}^{0,-1}(v)},$$ be the Galerkin solution of (\[6x\]). It is trivial that $\tilde u_N(0)=0.$ Galerkin formulation of (\[6x\]) is to find $\tilde u_N(v)$, such that $$\label{rv17} \bigg(\mathcal M^q \tilde u_N,G_i^{0,-1}\bigg)_{0,-1}=\bigg(\bar p(v) \tilde u_N(v),G_i^{0,-1}\bigg)_{0,-1}+\bigg(\bar f(v),G_i^{0,-1}\bigg)_{0,-1}+\lambda\bigg(\mathcal K(\tilde u_N),G_i^{0,-1}\bigg)_{0,-1}, ~~i=1,2,...,N$$ where $\mathcal K(\tilde u_N)=\int\limits_0^v{\tilde K(v,w)\tilde u_N(w)dw}.$ Applying transformation $w(\theta)=v \theta,~~\theta \in \Omega$  we get $$\label{rv5} \mathcal K(\tilde u_N)=\mathcal K_\theta(\tilde u_N)=v \int\limits_0^1{\tilde K(v,w(\theta)) \tilde u_N(w(\theta))d\theta}.$$ Substituting (\[rv5\]) in (\[rv17\]) yields $$\begin{gathered} \label{12} \bigg(\mathcal M^q \tilde u_N,G_i^{0,-1}\bigg)_{0,-1}=\bigg(\bar p(v) \tilde u_N(v),G_i^{0,-1}\bigg)_{0,-1}+\bigg(\bar f(v),G_i^{0,-1}\bigg)_{0,-1}+\lambda\bigg(\mathcal K_\theta(\tilde u_N),G_i^{0,-1}\bigg)_{0,-1},\\ i=1,2,...,N.\end{gathered}$$ Inserting (\[10c\]) in (\[12\]) we get $$\begin{gathered} \label{13} \sum\limits_{j=1}^{N}{b_j \bigg\{\bigg(\mathcal M^q G_j^{0,-1}(v),G_i^{0,-1}\bigg)_{0,-1}-\bigg(\bar p(v) G_j^{0,-1}(v),G_i^{0,-1}\bigg)_{0,-1}-\lambda\bigg(\mathcal K_{\theta}(G_j^{0,-1}),G_i^{0,-1}\bigg)_{0,-1}\bigg\}} \\ =\bigg(\bar f(v),G_i^{0,-1}\bigg)_{0,-1}, ~~i=1,2,...,N.\hspace{.01 cm}\end{gathered}$$ Following the relation $G_i^{0,-1}(v) \delta^{0,-1}(v)=G_{i-1}^{0,1}(v)$, we can rewrite (\[13\]) as $$\begin{gathered} \label{14} \sum\limits_{j=1}^{N}{b_j \bigg\{\bigg(\mathcal M^q G_j^{0,-1}(v),G_{i-1}^{0,1}\bigg)_{0,0}-\bigg(\bar p(v) G_j^{0,-1}(v),G_{i-1}^{0,1}\bigg)_{0,0}-\lambda\bigg(\mathcal K_{\theta}(G_j^{0,-1}),G_{i-1}^{0,1}\bigg)_{0,0}\bigg\}} \\=\bigg(\bar f(v),G_{i-1}^{0,1}\bigg)_{0,0},~~ i=1,2,...,N.\end{gathered}$$ Now, we try to find an explicit form for $\mathcal M^q G_j^{0,-1}$. To this end, using (\[8\]) we have $$\begin{aligned} \label{15} \mathcal M^q G_j^{0,-1}(v)&=& \frac{2}{\Gamma{(1-q)}}\sum\limits_{k=0}^{j-1}{(-1)^{j-1-k}\frac{(j+k)!}{(k+1)!k! (j-1-k)!} \int\limits_0^v{\big(v^{\frac{1}{q}}-w^{\frac{1}{q}}\big)^{-q}(w^{k+1})'dw}}\\ \nonumber &=&\frac{2}{\Gamma{(1-q)}}\sum\limits_{k=0}^{j-1}{(-1)^{j-1-k}\frac{(j+k)!}{(k!)^2 (j-1-k)!} \int\limits_0^v{\big(v^{\frac{1}{q}}-w^{\frac{1}{q}}\big)^{-q} w^{k} dw}}.\end{aligned}$$ Applying the relation[@ax15] $$\int\limits_0^v{\bigg(v^{\frac{1}{q}}-w^{\frac{1}{q}}\bigg)^{-q} w^k dw}=\bigg(\frac{q \pi \csc{(\pi q)}\Gamma{(q+qk)}}{\Gamma{(q)} \Gamma{(1+k q)}}\bigg)v^k,~~ k \ge 0,$$ in (\[15\]) we can obtain the following explicit formula for $\mathcal M^q G_j^{0,-1}$: $$\label{15x} \mathcal M^q G_j^{0,-1}(v) =\frac{2}{\Gamma{(1-q)}}\sum\limits_{k=0}^{j-1}{(-1)^{j-1-k}\frac{(j+k)!}{(k!)^2 (j-1-k)!}\bigg(\frac{q \pi \csc{(\pi q)}\Gamma{(q+qk)}}{\Gamma{(q)} \Gamma{(1+k q)}}\bigg)v^k}=:\Psi_{j,q}(v),$$ Substituting (\[15x\]) in (\[14\]) we obtain $$\begin{gathered} \label{16} \sum\limits_{j=1}^{N}{b_j \bigg\{\bigg(\Psi_{j,q}(v),G_{i-1}^{0,1}\bigg)_{0,0}-\bigg(\bar p(v) G_j^{0,-1}(v),G_{i-1}^{0,1}\bigg)_{0,0}-\lambda\bigg(\mathcal K_{\theta}(G_j^{0,-1}),G_{i-1}^{0,1}\bigg)_{0,0}\bigg\}}\\=\bigg(\bar f(v),G_{i-1}^{0,1}\bigg)_{0,0},~~i=1,2,...,N.\end{gathered}$$ In this position, we approximate the integral terms of (\[16\]) using $(N+1)-$point Legendre Gauss quadrature formula. Our discrete Galerkin method is to seek $$\label{10}\bar u_N(v)=\sum\limits_{i=1}^N{a_i G_{i}^{0,-1}(v)},$$ such that coefficients $\{a_j\}_{j=1}^N$ satisfies in the following algebraic system of linear equations $$\begin{gathered} \label{17} \sum\limits_{j=1}^{N}{a_j \bigg\{\bigg(\Psi_{j,q}(v),G_{i-1}^{0,1}\bigg)_{0,0}-\bigg(\bar p(v) G_j^{0,-1}(v),G_{i-1}^{0,1}\bigg)_{N,0,0}-\lambda\bigg(\mathcal K_{N,\theta}(G_j^{0,-1}),G_{i-1}^{0,1}\bigg)_{N,0,0}\bigg\}} \\=\bigg(\bar f(v),G_{i-1}^{0,1}\bigg)_{N,0,0}, i=1,2,...,N,\end{gathered}$$ where $$\label{rv6} \mathcal K_{N,\theta}(G_j^{0,-1})=v \sum\limits_{k=0}^{N}{\tilde K(v,w(\theta_k)) G_j^{0,-1}(w(\theta_k))\delta_k}.$$ Here $\{\theta_k\}_{k=0}^N$ and $\{\delta_k\}_{k=0}^N$ are the shifted Legendre Gauss quadrature points on $\Omega$ and corresponding weights respectively. Note that, from (\[15x\]) we can see that $\Psi_{j,q}(v)$ is a polynomial from degree at most $N$, then we have $\bigg(\Psi_{j,q}(v),G_{i-1}^{0,1}\bigg)_{N,0,0}=\bigg(\Psi_{j,q}(v),G_{i-1}^{0,1}\bigg)_{0,0}.$ It is trivial that the solution of (\[17\]) gives us unknown coefficients $\{a_i\}_{i=1}^N$ in (\[10\]). Existence and Uniqueness Theorem for Discrete Galerkin Algebraic System ======================================================================= The main object of this section is providing an existence and uniqueness Theorem for a special case of the discrete Galerkin algebraic system of equations (\[17\]) with $\bar p(v)=1$ and $0<q<\frac{1}{2}$. Throughout the paper, $C_i$ will denote a generic positive constant that is independent on $N$. First, we give some preliminaries which will be used in the sequel. Let $\mathcal X, \mathcal Y$ be normed spaces. A linear operator $\mathcal A: \mathcal X \to \mathcal Y $ is compact if the set $\{\mathcal A {x}|~ ||x||_{\mathcal X}\leq 1\}$ has compact closure in $\mathcal Y$. \[rt1\]\[\[rvv1\], \[rvv2\], \[rvv3\]\] Assume $\mathcal X,~\mathcal Y$ be two banach spaces. Let $$\label{rv7}v=\mathcal A v+f,$$ be a linear operator equation where $\mathcal A:\mathcal X \to \mathcal Y$ is a linear continuous operator, and the operator $I-\mathcal A$ is continuously invertible. As an approximation solution of (\[rv7\]) we consider the equation $$\label{rv8c} v_N=\mathcal A_N v_N+\mathcal B_N f,$$ which can be rewritten as $$\label{rv8} v_N=\tilde{\mathcal B}_N \mathcal A v_N+\mathcal S_N v_N+\mathcal B_N f,$$ where $\mathcal A_N$ is a linear continuous operator in a closed subspace $\tilde{\mathcal Y}$ of $\mathcal Y$. $\mathcal B_N, \tilde{\mathcal B}_N:\mathcal Y\to \tilde{\mathcal Y}$ are linear continuous projection operators and $\mathcal S_N=\mathcal A_N-\mathcal{\tilde B}_N \mathcal A$ is a linear operator in $\tilde{\mathcal Y}$. If the following conditions are fulfilled - for any $Z \in \tilde{\mathcal Y}$ we have $\|\mathcal S_N Z\|\to 0$  as $N \to \infty$ - $\|\mathcal A-\tilde{\mathcal B}_N \mathcal A\| \to 0$ as  $N \to \infty$ - $\|f-\mathcal B_N f\| \to 0$  as   $N \to \infty$ then (\[rv8\]) possesses a uniquely solution $v_N \in \tilde{\mathcal Y}$, for a sufficiently large $N$. \[rl1\] [@rvv4] Let $\mathcal X, \mathcal Y$ be banach spaces and $\tilde{\mathcal Y}$ be a subspace of $\mathcal Y$. Let $\tilde{\mathcal B}_N:\mathcal Y \to \tilde{\mathcal Y}$ be a family of linear continuous projection operators with $$\tilde{\mathcal B}_N y \to y~~ \text{as}~~ N \to \infty, ~~ y \in \mathcal Y.$$ Assume that linear operator $\mathcal A:\mathcal X \to \mathcal Y$ be compact. Then $$\|\mathcal A-\tilde{\mathcal B}_N \mathcal A\| \to 0~~\text{as}~~ N \to \infty.$$ \[l1\] (Interpolation error bound[@aa26]) Let $\mathcal I_N Z$ be the interpolation polynomial approximation of the function $Z(v)$ defined in (\[9\]). For any $Z(v) \in B_{0,0}^k(\Omega)$ with $k \ge 1$, we have $$\|Z-\mathcal I_N Z\|_{0,0} \le C N^{-k} |Z|_{0,0,k}.$$ \[l3\] [@r6] For every bounded function $Z(v)$, there exists a constant $C$ independent of $Z$ such that $$\sup\limits_{N}\|\mathcal I_N Z\|_{0,0} \le C \sup\limits_{v}|Z(v)|.$$ \[l4\] (Legendre Gauss quadrature error bound[@aa26]) If $Z(v) \in B_{0,0}^k(\Omega)$ for some $k \ge 1$ and $\Phi \in \mathcal P_N$, then for the Legendre Gauss integration we have $$\bigg|(Z,\Phi)_{0,0}-(Z,\Phi)_{N,0,0}\bigg| \le C N^{-k} \|Z\|_{0,0,k} \|\Phi\|_{0,0}.$$ Now we intend to prove existence and uniqueness Theorem for a special case of the discrete Galerkin system (\[17\]) with $\bar p(v)=1$ and $0<q<\frac{1}{2}$. (Existence and Uniqueness)Let $0<q<\frac{1}{2}$ and $\bar p(v)=1$. If (\[6x\]) has a uniquely solution $\bar u(v)$ then the linear discrete Galerkin system (\[17\]) has a uniquely solution $\bar u_N(v) \in \mathcal P_N$ for sufficiently large $N$. Our strategy in proof is based on two steps. First, we try to represent (\[17\]) in the operator form (\[rv8\]). Then by applying Theorem \[rt1\] to operator form obtained in the first step the desired result have been concluded. [**Step 1:**]{} In this step, we show that the discrete Galerkin system (\[17\]) can be written in the operator form (\[rv8\]). To this end, consider (\[6x\]) and define $$\mathcal{\bar R}_N(v)=\mathcal M^q \bar u_N(v)-\bar u_N(v)-\lambda \mathcal K_{N,\theta}(\bar u_N)-\bar f(v).$$ According to the proposed method, we have $$\bigg(\mathcal{\bar R}_N(v),G_{i-1}^{0,1}(v)\bigg)_{N,0,0}=0,\quad i=1, 2,...,N.$$ From interpolation and Legendre Gauss quadrature properties, we can write $$\label{18} \bigg(\mathcal{\bar R}_N(v),G_{i-1}^{0,1}(v)\bigg)_{N,0,0}=\bigg(\mathcal I_N(\mathcal{\bar R}_N),G_{i-1}^{0,1}(v)\bigg)_{N,0,0}=\bigg(\mathcal I_N(\mathcal{\bar R}_N),G_{i-1}^{0,1}(v)\bigg)_{0,0}=0,\quad i=1, 2,...,N.$$ Since $\mathcal I_N(\mathcal{\bar R}_N(v))$ is a polynomial, it can be represented by a linear orthogonal polynomial expansion based on $\big\{G_i^{0,-1}(v)\big\}_{i=0}^N$, as $$\label{cc6} \mathcal I_N(\mathcal{\bar R}_N)=\sum\limits_{i=1}^{N}{\frac{\bigg(\mathcal I_N(\mathcal{\bar R}_N),G_{i}^{0,-1}\bigg)_{0,-1}}{\|G_i^{0,-1}\|_{0,-1}^2} G_i^{0,-1}(v)}=\sum\limits_{i=1}^{N}{\frac{\bigg(\mathcal I_N(\mathcal{\bar R}_N),G_{i-1}^{0,1}\bigg)_{0,0}}{\|G_i^{0,-1}\|_{0,-1}^2} G_i^{0,-1}(v)}.$$ Using the relations (\[18\]) and (\[cc6\]) yields $\mathcal I_N(\mathcal{\bar R}_N)=0$. Thus $$\mathcal I_N\bigg(\mathcal M^q \bar u_N-\bar u_N(v)-\lambda \mathcal \mathcal K_{N,\theta}(\bar u_N)-\bar f\bigg)=0,$$ which can be rewritten as $$\label{rv9c} \bar u_N(v)=\mathcal I_N\bigg(\mathcal M^q -\lambda \mathcal \mathcal K_{N,\theta}\bigg)\bar u_N-\mathcal I_N \bar f=\mathcal I_N \mathcal T_N \bar u_N -\mathcal I_N \bar f,$$ and thereby $$\label{rv9} \bar u_N(v)=\Pi_N \mathcal T \bar u_N+\mathcal S_N \bar u_N -\mathcal I_N \bar f,$$ where $\mathcal I_N$ and $\Pi_N$ are defined in (\[9\]) and (\[cc5\]) respectively and $$\begin{aligned} \mathcal T&=&\mathcal M^q-\lambda \mathcal K_\theta,\\ \mathcal T_N&=&\mathcal M^q -\lambda \mathcal \mathcal K_{N,\theta},\\ \mathcal S_N&=&\mathcal I_N \mathcal T_N-\Pi_N \mathcal T.\end{aligned}$$ Since (\[rv9\]) is the form in which the discrete Galerkin method is implemented, as it leads directly to the equivalent linear system (\[17\]). It can be easily check that (\[rv9\]) can be considered in the operator form (\[rv8\]) by assuming $$\begin{aligned} \label{cc1} \nonumber\mathcal X&=&H_{0,0}^1(\Omega),~~\mathcal Y=L^2(\Omega),~~\tilde{\mathcal Y}=\mathcal P_N,\\ v_N&=&\bar u_N,~~\quad ~~\tilde{\mathcal B_N}=\Pi_N,~~\quad \mathcal B_N=\mathcal I_N,\\ \nonumber\mathcal A&=&\mathcal T,~~\quad ~~ \mathcal A_N=\mathcal I_N \mathcal T_N,\end{aligned}$$ which can be completed the desired result of step 1. [**Step 2:**]{} In this step we intend to apply Theorem \[rt1\] with the assumptions (\[cc1\]) to prove the Theorem. To this end, following Theorem \[rt1\] we must show that $$\begin{aligned} \label{cc3} \nonumber&&1)~\text{for any}~ Z \in \mathcal P_N ~\text{we have}~ \|\mathcal S_N Z\|_{0,0}=\|\mathcal I_N \mathcal T_N Z-\Pi_N \mathcal T Z\|_{0,0} \to 0 ~\text{as}~ N \to \infty,\\ &&2) \|\mathcal T-\Pi_N \mathcal T\|_{0,0} \to 0~ \text{as}~ N \to \infty,\\ \nonumber&&3) \|\bar f-\mathcal I_N \bar f\|_{0,0} \to 0 ~\text{as}~ N \to \infty.\end{aligned}$$ First, we prove the first condition in (\[cc3\]). For this, we can write $$\label{rv13} \|\mathcal S_N Z\|_{0,0}=\|\bigg(\mathcal I_N \mathcal T_N-\Pi_N \mathcal T\bigg)Z\|_{0,0}\le \|\mathcal I_N\bigg(\mathcal T_N-\mathcal T\bigg)Z\|_{0,0}+\|\bigg(\mathcal I_N-\Pi_N\bigg)\mathcal T Z\|_{0,0}.$$ Since $\mathcal M^q Z \in \mathcal P_N$ for any $Z \in \mathcal P_N$ then $$\label{rv14} \|\mathcal S_N Z\|_{0,0}\le \lambda\Bigg(\|\mathcal I_N\bigg(\mathcal K_{N,\theta}-\mathcal K_{\theta}\bigg)Z\|_{0,0}+\|\bigg(\mathcal I_N-\Pi_N\bigg)\mathcal K_\theta Z\|_{0,0}\Bigg).$$ Using Lemmas \[l3\] and \[l4\] and the relations (\[rv5\]) and (\[rv6\]) we can obtain $$\label{25rv} \|\mathcal I_N\bigg(\mathcal K_{N,\theta}-\mathcal K_{\theta}\bigg)z\|_{0,0} \le \sup\limits_{v \in \Omega}{|\mathcal K_\theta(z(v))-\mathcal K_{N,\theta}(z(v))|} \le C_1 N^{-k_1} \sup\limits_{v \in \Omega}{\bigg(\|\tilde K(v,w(\theta))\|_{0,0,k_1}\|v Z(w(\theta))\|_{0,0}\bigg)},$$ where norms $\|\tilde K(v,w(\theta))\|_{0,0,k_1}$ and $\|v Z(w(\theta))\|_{0,0}$ are applied with respect to the variable $\theta$. According to Lemma \[l4\] we have $$\label{cc7} \Bigg|\bigg(\mathcal K_\theta,J_s^{0,0}\bigg)_{0,0}-\bigg(\mathcal K_\theta,J_s^{0,0}\bigg)_{N,0,0}\Bigg| \le CN^{-k_2} \|\mathcal K_\theta\|_{0,0,k_2} \|J_s^{0,0}\|_{0,0},~~s=0,1,...,N,$$ where norm $\|\mathcal K_\theta\|_{0,0,k_2}$ is applied with respect to the variable $v$. Using (\[cc7\]) and the relations (\[9\]), (\[cc5\]) we can yields $$\label{rv15} \|\bigg(\mathcal I_N-\Pi_N\bigg)\mathcal K_\theta Z\|_{0,0} \to 0~~\text{as}~~N \to \infty.$$ Substituting (\[25rv\]) and (\[rv15\]) in (\[rv14\]) we can conclude the first condition in (\[cc3\]). Applying Lemma \[l4\] gives us the third condition in (\[cc3\]). To complete the proof, it is sufficient that we prove the second condition of (\[cc3\]). To this end, we apply Lemma \[rl1\] with the assumptions (\[cc1\]). Since $\|y-\Pi_N y\|_{0,0} \to 0~\text{as}~N \to \infty$ for $y \in L^2(\Omega)$(see\[\]), the second condition in (\[cc3\]) can be achieved by proving compactness of the operator $\mathcal T$. Since $\tilde k(v,w)$ is a continuous kernel, then the operator $\mathcal T$ will be compact if $\mathcal M^q(\bar u):H_{0,0}^1(\Omega) \to L^2(\Omega)$ be a compact operator. For this, define $M=\big[\int_{0}^{1}{\int_{0}^{1}{\bigg(v^{\frac{1}{q}}-w^{\frac{1}{q}}\bigg)^{-2q} dw}dv}\big]^{\frac{1}{2}}< \infty$. For $\bar u \in H_{0,0}^1(\Omega)$, using Cauchy-Schwartz inequality we have $$\begin{aligned} \|\mathcal M^q \bar u\|_{0,0}^2&=& \int_{0}^{1}{|\int_{0}^{v}{\bigg(v^{\frac{1}{q}}-w^{\frac{1}{q}}\bigg)^{-q} \bar u'(w) dw}|^2 dv} \le \int_{0}^{1}\bigg({\int_{0}^{v}{|\bigg(v^{\frac{1}{q}}-w^{\frac{1}{q}}\bigg)^{-q} \bar u'(w)| dw}\bigg)^2 dv} \\ & \le & \int_{0}^{1}\bigg({\int_{0}^{1}{|\bigg(v^{\frac{1}{q}}-w^{\frac{1}{q}}\bigg)^{-q} \bar u'(w)| dw}\bigg)^2 dv} \le \int_{0}^{1}{\bigg(\int_{0}^{1}{\bigg(v^{\frac{1}{q}}-w^{\frac{1}{q}}\bigg)^{-2q}dw} \int_{0}^{1}{|\bar u'(w)|^2dw}\bigg)dv} \\ & \le & M^2 \|\bar u'\|_{0,0}^2 \le M^2 \|\bar u\|_{H_{0,0}^1(\Omega)}^2.\end{aligned}$$ So $\mathcal M^q \bar u \in L^2(\Omega)$ and $\|\mathcal M^q\|=\sup \bigg\{\frac{\|\mathcal M^q \bar u\|_{0,0}}{\|\bar u\|_{H_{0,0}^1(\Omega)}}~~\Big|~~ \bar u \ne 0, \bar u \in H_{0,0}^1(\Omega)\bigg\} \le M < \infty$. Therefore $\mathcal M^q:H_{0,0}^{1}(\Omega) \to L^2(\Omega)$ defined by (\[rv4\]), is a bounded operator. If we proceed same as Theorem 3.4 in \[\[arr1\]\], we can conclude compactness of the operator $\mathcal M^q(\bar u)$. Then the operator $\mathcal T$ is a compact operator and from Lemma \[rl1\] we have $$\|\mathcal T-\Pi_N \mathcal T\|_{0,0} \to 0~\text{as}~ N\to \infty$$ In this position, all conditions in (\[cc3\]) that are required to deduce existence and uniqueness of solutions of the discrete Galerkin system (\[17\]) have been proved and then the proof is completed. Convergence analysis ==================== In this section, we will try to provide a reliable convergence analysis which theoretically justify convergence of the proposed discrete Galerkin method for the numerical solution of a special case of (\[6x\]) with $\bar p(v)=1$. In the sequel, our discussion is based on these Lemmas: [@aa26]\[l2\]For any $Z \in B_{0,0}^1(\Omega)$, we have $$\|\mathcal I_{N}Z\|_{0,0} \le C\bigg(\|Z\|_{0,0}+N^{-1}\|Z'\|_{1,1}\bigg).$$ \[l5\] The fractional integral operator $\mathcal I^\mu$ is bounded in $L^2(\Omega)$ and $$\|\mathcal I^\mu Z\|_{0,0} \le C \|Z\|_{0,0},~~ Z \in L^2(\Omega).$$ \[l6\] [@r6](Gronwall inequality)Assume that $Z(v)$ is a non-negative, locally integrable function defined on $\Omega$ which satisfies $$Z(v) \le b(v)+B \int_{0}^{v}{(v-w)^\alpha w^\beta Z(w) dw}, \quad w \in \Omega,$$ where $\alpha, \beta>-1$, $b(v) \ge 0$ and $B \ge 0$. Then, there exist a constant $C$ such that $$Z(v) \le b(v)+ C \int_{0}^{v}{(v-w)^\alpha w^\beta b(w) dw}, \quad w \in \Omega.$$ Now, we state and prove the main result of this section regarding the error analysis of the proposed method for the numerical solution of a special case of (\[6x\]) with $\bar p(v)=1$. \[t1\] (Convergence)Let $u(x)$ and $\bar u(v)$ are the exact solutions of the equations (\[1\]) and (\[6x\]) respectively that is related by $u(x)=\bar u(v)$. Assume that $u_N(x)=\bar u_N(v)$ be the approximate solution of (\[1\]), where $\bar u_N(v)$ is the discrete Galerkin solution of the transformed equation (\[6x\]). If the following conditions are fulfilled 1. $\bar f(v) \in B_{0,0}^{k_1}(\Omega)$ for $k_1 \ge 1$, 2. $\mathcal K(\bar u) \in B_{0,0}^{k_2}(\Omega)$ for $k_2 \ge 1$, then for sufficiently large $N$ we have $$\|e_N(u)\|_{0,0} \le C N^{-\xi}\bigg(|\bar f|_{0,0,k_1}+|\mathcal K(\bar u)|_{0,0,k_2}\bigg)$$ where $\xi=\min\{k_1,k_2\}$ and $e_N(u)= u(x)-u_N(x)$ denotes the error function. Since $\mathcal M^q \bar u_N$ is a polynomial from degree of at most $N$ then we can rewrite (\[rv9c\]) as $$\label{7x} \mathcal M^q \bar u_N-\bar u_N(v)-\mathcal I_N\bigg(\lambda \mathcal \mathcal K_{N,\theta}(\bar u_N)\bigg)=\mathcal I_N \bar f.$$ Subtracting (\[6x\]) from (\[7x\]) and some simple manipulations we can obtain $$\label{19} \mathcal M^q \bar e_N=\bar e_N(v)+e_{\mathcal I_N}(\bar f)+\lambda \mathcal K(\bar e_N)+\lambda e_{\mathcal I_N}(\mathcal K(\bar u_N))+\lambda \mathcal I_N\bigg(\mathcal K_{\theta}(\bar u_N)-\mathcal K_{N,\theta}(\bar u_N)\bigg),$$ where $e_{\mathcal I_N}(g)=g-\mathcal I_N(g)$ denotes the interpolating error function and $\bar e_N(v)=\bar u(v)-\bar u_N(v)$ is the error function of approximating (\[6x\]) using discrete Galerkin solution $\bar u_N(v)$. Applying the transformation (\[6xx\]) to the operator $\mathcal I^q u$ we get $$\mathcal M_1^q \bar u=\frac{1}{\Gamma{(q)}}\int\limits_0^v{\bigg(v^{\frac{1}{q}}-w^{\frac{1}{q}}\bigg)^{q-1} \bar u(w) \frac{w^{\frac{1}{q}-1}}{q}dw}.$$ Following (\[20\]) it is easy to check that $$\label{7xx} \mathcal M_1^q \big(\mathcal M^q \bar u(v)\big)=\bar u(v)-\bar u(0).$$ Applying the operator $\mathcal M_1^q$ on the both sides of (\[19\]) and using (\[7xx\]) we can yield $$\label{rv1} |\bar e_N(v)| \le \mathcal M_1^q(|\bar e_N|+|\lambda \mathcal K(\bar e_N)|)+\bigg|\mathcal M_1^q\bigg(e_{\mathcal I_N}(\bar f)+\lambda e_{\mathcal I_N}(\mathcal K(\bar u_N))+\lambda \mathcal I_N\bigg(\mathcal K_{\theta}(\bar u_N)-\mathcal K_{N,\theta}(\bar u_N)\bigg)\bigg)\bigg|.$$ Since $$\begin{aligned} \label{rv2} \nonumber\mathcal M_1^q \int\limits_{0}^{v}{|\tilde K(v,w) \bar e_N(w)| dw}&=&\frac{1}{\Gamma{(q)}}\int\limits_{0}^{v}{\int\limits_{0}^{w}{\big(v^{\frac{1}{q}}-w^{\frac{1}{q}}\big)^{q-1} \frac{w^{\frac{1}{q}-1}}{q} |\tilde K(w,s)||\bar e_N(s)|ds}dw}\\ \nonumber&=&\frac{1}{\Gamma{(q)}}\int\limits_{0}^{v}{\int\limits_{s}^{v}{\big(v^{\frac{1}{q}}-w^{\frac{1}{q}}\big)^{q-1} \frac{w^{\frac{1}{q}-1}}{q} |\tilde K(w,s)||\bar e_N(s)|dw}ds}\\ &=&\frac{1}{\Gamma{(q)}}\int\limits_{0}^{v}{\tilde K_1(v,s) |\bar e_N(s)|ds}\le \frac{\|\tilde K_1(v,s)\|_\infty}{\Gamma{(q)}}\int\limits_{0}^{v}{|\bar e_N(s)|ds},\end{aligned}$$ then (\[rv1\]) can be rewritten as $$\begin{aligned} \label{rv3} \nonumber|\bar e_N(v)| &\le & \frac{1}{\Gamma{(q)}}\bigg(\int\limits_{0}^{v}{\big(v^{\frac{1}{q}}-w^{\frac{1}{q}}\big)^{q-1} \frac{w^{\frac{1}{q}-1}}{q}|\bar e_N(w)|dw}+|\lambda|\|\tilde K_1(v,s)\|_\infty \int\limits_{0}^{v}{|\bar e_N(w)| dw}\bigg)\\ \nonumber &+&\bigg|\mathcal M_1^q\bigg(e_{\mathcal I_N}(\bar f)+\lambda e_{\mathcal I_N}(\mathcal K(\bar u_N))+\lambda \mathcal I_N\bigg(\mathcal K_{\theta}(\bar u_N)-\mathcal K_{N,\theta}(\bar u_N)\bigg)\bigg)\bigg|\\ &\le& \tilde C \bigg(\int\limits_{0}^{v}{\big(v^{\frac{1}{q}}-w^{\frac{1}{q}}\big)^{q-1} |\bar e_N(w)|dw}\bigg)\\ \nonumber &+&\bigg|\mathcal M_1^q\bigg(e_{\mathcal I_N}(\bar f) +\lambda e_{\mathcal I_N}(\mathcal K(\bar u_N))+\lambda \mathcal I_N\bigg(\mathcal K_{\theta}(\bar u_N)-\mathcal K_{N,\theta}(\bar u_N)\bigg)\bigg)\bigg|,\end{aligned}$$ where $$\tilde K_1(v,s)=\int\limits_{s}^{v}{\big(v^{\frac{1}{q}}-w^{\frac{1}{q}}\big)^{q-1} \frac{w^{\frac{1}{q}-1}}{q} |\tilde K(w,s)|dw},\quad \tilde C=\max\bigg\{\frac{1}{q \Gamma{(q)}},\frac{|\lambda|\|\tilde K_1(v,s)\|_\infty }{\Gamma{(q)}}\bigg\} \|1+\big(v^{\frac{1}{q}}-w^{\frac{1}{q}}\big)^{1-q}\|_\infty.$$ Using Gronwall inequality(Lemma \[l6\]) in (\[rv3\]) yields $$\label{21c} \|\bar e_N\|_{0,\frac{1}{q}-1} \le C_1 \bigg(\|\mathcal M_1^q\bigg(e_{\mathcal I_N}(\bar f)+\lambda e_{\mathcal I_N}(\mathcal K(\bar u_N))+\lambda \mathcal I_N\bigg(\mathcal K_{\theta}(\bar u_N)-\mathcal K_{N,\theta}(\bar u_N)\bigg)\bigg)\|_{0,\frac{1}{q}-1}\bigg).$$ It can be checked that by applying the transformation (\[6\]) we obtain $$\label{cc8}\|\bar e_N(v)\|_{0,\frac{1}{q}-1}^2=q 2^{\frac{1}{2}-1}\|e_N(x)\|_{0,0}^2.$$ On the other hand, by applying (\[6\]) and Lemma \[l5\] we can get $$\label{cc4} \|\mathcal M_1^q \bar Z(v)\|_{0,\frac{1}{q}-1}^2=q 2^{\frac{1}{q}-1}\|\mathcal I^q Z(x)\|_{0,0}^2 \le C q 2^{\frac{1}{q}-1}\|Z(x)\|_{0,0}^2=C \|\bar Z(v)\|_{0,\frac{1}{q}-1}^2,$$ where $Z(x)$ is a given function and $\bar Z(v)=z(v^{\frac{1}{q}})$. Using the relations (\[cc8\]) and (\[cc4\]) in (\[21c\]) we can obtain $$\begin{aligned} \label{21cc} \sqrt{q 2^{\frac{1}{q}-1}}\|e_N\|_{0,0} &\le & C_2 \| e_{\mathcal I_N}(\bar f)+\lambda e_{\mathcal I_N}(\mathcal K(\bar u_N))+\lambda \mathcal I_N\bigg(\mathcal K_{\theta}(\bar u_N)-\mathcal K_{N,\theta}(\bar u_N)\bigg)\|_{0,\frac{1}{q}-1} \\ \nonumber&\le & C_2 \bigg(\| e_{\mathcal I_N}(\bar f)\|_{0,\frac{1}{q}-1}+|\lambda|\|e_{\mathcal I_N}(\mathcal K(\bar u_N))\|_{0,\frac{1}{q}-1}+|\lambda|\|\mathcal I_N\bigg(\mathcal K_{\theta}(\bar u_N)-\mathcal K_{N,\theta}(\bar u_N)\bigg)\|_{0,\frac{1}{q}-1}\bigg).\end{aligned}$$ Since $\delta^{0,\frac{1}{q}-1} \le \delta^{0,0}$ then we have $\|.\|_{0,\frac{1}{q}-1} \le \|.\|_{0,0}$ and thereby (\[21cc\]) can be written as $$\label{21} \|e_N\|_{0,0} \le C_3 \bigg(\mathcal L_1+\mathcal L_2+\mathcal L_3\bigg),$$ where $$\mathcal L_1=\|e_{\mathcal I_N}(\bar f)\|_{0,0}, \quad \mathcal L_2=\|e_{\mathcal I_N}(\mathcal K(\bar u_N))\|_{0,0}, \quad \mathcal L_3=\|\mathcal I_N\big(\mathcal K_{\theta}(\bar u_N)-\mathcal K_{N,\theta}(\bar u_N)\big)\|_{0,0}.$$ Now, it is sufficient to find suitable upper bounds to the above norms. According to Lemma \[l1\] we have $$\begin{aligned} \label{22} \mathcal L_1&=&\|e_{\mathcal I_N}(\bar f)\|_{0,0} \le C_3 N^{-k_1} |\bar f|_{0,0,k_1},\\ \nonumber \\ \nonumber\mathcal L_2&=&\|e_{\mathcal I_N}(\mathcal K(\bar u_N))\|_{0,0} \le C_4 N^{-k_2}|\mathcal K(\bar u-\bar e_N(u))|_{0,0,k_2}.\end{aligned}$$ Using Lemmas \[l3\] and \[l4\] we can obtain $$\begin{aligned} \label{25} \nonumber \mathcal L_3=\|\mathcal I_N\big(\mathcal K_\theta(\bar u_N)-\mathcal K_{N,\theta}(\bar u_N)\big)\|_{0,0} &\le & \sup\limits_{v \in \Omega}{|\mathcal K_\theta(\bar u_N)-\mathcal K_{N,\theta}(\bar u_N)|}\\ \nonumber &\le & C_7 N^{-k_3} \sup\limits_{v \in \Omega}{\bigg(\|\tilde K(v,w(\theta))\|_{0,0,k_3}\|v \bar u_N(w(\theta))\|_{0,0}\bigg)}\\ \nonumber &\le & C_7 N^{-k_3}\bigg(\sup\limits_{v \in \Omega}{\|\tilde K(v,w(\theta))\|_{0,0,k_3}\bigg)}\|\bar u_N(w)\|_{0,0}\\ &\le & C_7 N^{-k_3} \bigg(\sup\limits_{v \in \Omega}{\|\tilde K(v,w(\theta))\|_{0,0,k_3}\bigg)}\bigg(\|\bar u\|_{0,0}+\|\bar e_N\|_{0,0}\bigg),\end{aligned}$$ where norms $\|\tilde K(v,w(\theta))\|_{0,0,k_3}$ and $\|v \bar u_N(w(\theta))\|_{0,0}$ are applied with respect to the variable $\theta$. For sufficiently large $N$ the desired result can be concluded by inserting (\[22\]) and (\[25\]) in (\[21\]). Numerical Results ================= In this section we apply a program written in Mathematica to a numerical example to demonstrate the accuracy of the proposed method and effectiveness of applying GJPs. the “Numerical Error” always refers to the $L^2$-norm of the obtained error function. \[e1\]Consider the following FIDE $$\mathcal D^{\frac{1}{2}}u(x)=u(x)+f(x)+\frac{1}{2}\int\limits_0^x{\sqrt{x t} u(t) dt},\quad u(0)=0,$$ with the exact solution $u(x)=\frac{\sin{x}}{\sqrt{x}}$ and $$f(x) =\frac{-x+\sqrt{x}\bigg(\sqrt{x} \cos{x}+\sqrt{\pi}\bigg(J_0(\frac{x}{2})\cos{\frac{x}{2}}-J_1(\frac{x}{2})\sin{\frac{x}{2}}\bigg)\bigg)-2 \sin{x}}{2 \sqrt{x}}.$$ Here $J_n(z)$ gives the Bessel function of the first kind. This example has a singularity at the origin, i.e., $$u'(x) \sim \frac{1}{\sqrt{x}}.$$ In the theory presented in the previous section, our main concern is the regularity of the transformed solution. For the present problem by applying coordinate transformation $x=v^2$, the infinitely smooth solution $$\bar u(v)=u(v^2)=\frac{\sin{v^2}}{v},$$ is obtained. The main purpose is to check the convergence behavior of the numerical solutions with respect to the approximation degree $N$. Numerical results obtained are given in the Table 1 and Figure 1. As expected, the errors show an exponential decay, since in this semi-log representation(Figure 1) one observes that the error variations are essentially linear versus the degrees of approximation. Table $1$: The numerical errors of example \[e1\]. N ---- ------------------------ -- -- 2 $7.86 \times 10^{-3}$ 4 $4.71 \times 10^{-5}$ 6 $2.15 \times 10^{-6}$ 8 $1.05 \times 10^{-8}$ 10 $4.42 \times 10^{-10}$ 12 $2.05 \times 10^{-12}$ 14 $2.64 \times 10^{-14}$ 16 $1.91 \times 10^{-16}$ =2.8 in =2.8 in [9]{} \[rvv4\] K. E. Atkinson, [*The Numerical Solution of Integral Equations of the Second Kind,*]{} Cambridge, (1997). \[r1\]F. Awawdeh, E. A. Rawashdeh, H. M. Jaradat, Analytic solution of fractional integro-differential equations, [*Ann. Univ. Craiova math. Comput. Sci. Ser.*]{} [**38**]{} (2011), 1-10. R. L. Bagley, P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, [*J. Rheol.,*]{} [**27**]{}(1983), 201-210. \[r3\]H. Brunner, [*Collocation Methods for Volterra and Related Functional Equations,*]{} Cambridge University Press, Cambridge, (2004). \[r5\] M. Caputo, Linear models of dissipation whose Q is almost frequency independent II, [*Geophys. J. Roy. Astron. 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--- abstract: 'We study the convexity of mutual information along the evolution of the heat equation. We prove that if the initial distribution is log-concave, then mutual information is always a convex function of time. We also prove that if the initial distribution is either bounded, or has finite fourth moment and Fisher information, then mutual information is eventually convex, i.e., convex for all large time. Finally, we provide counterexamples to show that mutual information can be nonconvex at small time.' author: - bibliography: - 'mi\_arxiv\_v2.bib' title: | Convexity of mutual information\ along the heat flow --- Introduction ============ The heat equation plays a fundamental role in many fields. In thermodynamics, it describes the diffusion of heat in a body due to temperature differences. In probability theory, it describes the evolution of the Brownian motion. In information theory, it describes the additive white Gaussian noise channel, which is one of the most important communication channels. In general, the heat equation can be used to model the transport of any quantity in a medium via a diffusion process. It also forms the basis for more general stochastic processes, such as the Ornstein-Uhlenbeck process or the Fokker-Planck process. Therefore, the heat equation has found applications in diverse scientific disciplines—from explaining the evolution of zebra stripes [@Tur52] to modeling stock prices via the Black-Scholes formula [@BlaSch73]. We are interested in the heat flow, which is the flow of the heat equation in the space of random variables. The properties of the heat flow are closely linked to entropy. Indeed, one important interpretation of the heat flow is as the flow that increases entropy as fast as possible. More precisely, heat flow is the gradient flow (i.e., the steepest descent flow) of negative entropy in the space of probability distributions with the Wasserstein metric structure [@JKO98]. In this paper we will not need this result, but only use a certain key identity in our calculation. Nevertheless, this relation suggests an intricate connection between entropy and the heat flow. The behavior of entropy along the heat flow has been long studied. The gradient flow interpretation above shows that entropy is increasing along the heat flow. In particular, De Bruijn’s identity [@Sta59] states that the time derivative of entropy along the heat flow is given by the Fisher information, which is always positive. Moreover, entropy is a concave function of time along the heat flow. This is because the second time derivative of entropy along the heat flow is the negative of the second-order Fisher information [@McKean66; @Tos99; @Vil00]; the latter identity also implies the concavity of entropy power along the heat flow [@Cos85; @Dembo89; @Dembo91]. It is further conjectured that the higher derivatives of entropy along the heat flow have alternating signs [@McKean66; @Vil02; @Che15]. In one dimension, this has been verified up to the fourth derivative [@Che15]; in multi dimension, this is true for the third derivative when the initial distribution is log-concave [@Tos15]. On the other hand, the behavior of mutual information along the heat flow has been less explored. Clearly mutual information is decreasing along the heat flow by the data processing inequality, since the heat flow is a Markov chain. De Bruijn’s identity implies that the time derivative of mutual information along the heat flow is the negative of the mutual Fisher information; the latter is proportional to the minimum mean square error (mmse) of estimating the initial from the final distribution, thus recovering the I-MMSE relation for the additive Gaussian channel [@GuoEtAl05]. Similarly, the second time derivative of mutual information along the heat flow is the mutual version of the second-order Fisher information; unfortunately, it does not always have a definite sign. In this paper we study the convexity of mutual information along the heat flow. This amounts to determining when the mutual second-order Fisher information is positive along the heat flow. We show that in general, the mutual second-order Fisher information is positive whenever the final distribution is log-concave. Since the heat flow preserves log-concavity, this implies our first main result: If the initial distribution is log-concave, then mutual information is always convex along the heat flow. In some cases, for example when the initial distribution is bounded, the heat flow implies eventual log-concavity, which means the final distribution eventually becomes log-concave; this implies mutual information is eventually convex along the heat flow for these cases. Furthermore, we prove that in general, regardless of log-concavity, mutual information is eventually convex along the heat flow whenever the initial distribution has finite fourth moment and Fisher information. Unlike entropy, however, we show that mutual information can be nonconvex along the heat flow. We provide explicit counterexamples, namely mixtures of point masses and mixtures of Gaussians, for which mutual information along the heat flow is nonconvex at small time; furthermore, by scaling we can arrange the region of nonconvexity to engulf any finite time. We elaborate on these results below. Background and problem setup ============================ The heat flow ------------- The heat equation in ${\mathbb{R}}^n$ is the partial differential equation: $${\frac{\partial \rho}{\partial t}} = \frac{1}{2} \Delta \rho$$ where $\rho = \rho(x,t)$ for $x \in {\mathbb{R}}^n$, $t \ge 0$, and $\Delta = \sum_{i=1}^n {\frac{\partial ^2}{\partial x_i^2}}$ is the Laplacian operator. This equation conserves mass, so if $\rho_0 = \rho(\cdot,0)$ is a probability distribution, then so is $\rho_t = \rho(\cdot,t)$ for all $t > 0$. The heat equation admits a closed-form solution via convolution: $$\rho_t = \rho_0 \ast \gamma_t$$ where $\gamma_t(x) = (2\pi t)^{-\frac{n}{2}} e^{-\frac{\|x\|^2}{2t}}$ is the heat kernel at time $t$. Probabilistically, if $X_0 \sim \rho_0$ is a random variable in ${\mathbb{R}}^n$, then $X_t \sim \rho_t$ that evolves following the heat equation is given by $$X_t = X_0 + \sqrt{t} Z$$ where $Z \sim {\mathcal{N}}(0,I)$ is the standard Gaussian random variable in ${\mathbb{R}}^n$ independent of $X_0$. We call this the heat flow. (Note that the true solution to the heat equation is the Brownian motion, but at each time $t$ it has the same distribution as $X_t$ above.) Observe that even when $X_0 \sim \rho_0$ has a singular density, $X_t \sim \rho_t$ has a smooth positive density for all $t > 0$. If $X_0 \sim \delta_{a}$ is a point mass at some $a \in {\mathbb{R}}^n$, then $X_t \sim {\mathcal{N}}(a, tI)$ is Gaussian with mean $a$ and covariance $tI$. If $X_0 \sim {\mathcal{N}}(\mu,\Sigma)$ is Gaussian, then $X_t \sim {\mathcal{N}}(\mu,\Sigma+tI)$ is also Gaussian with the same mean and increasing covariance. If $X_0 \sim \sum_{i=1}^k p_i \delta_{a_i}$ is a mixture of point masses, then $X_t \sim \sum_{i=1}^k p_i {\mathcal{N}}(a_i,tI)$ is a mixture of Gaussians with the same covariance $tI$. If $X_0 \sim \sum_{i=1}^k p_i {\mathcal{N}}(a_i, \Sigma_i)$ is a mixture of Gaussians, then $X_t \sim \sum_{i=1}^k p_i {\mathcal{N}}(a_i,\Sigma_i+tI)$ is also a mixture of Gaussians with the same means and increasing covariance. Entropy and Fisher information ------------------------------ Let $X$ be a random variable in ${\mathbb{R}}^n$ with a smooth positive density $\rho$. The (differential) [*entropy*]{} of $X \sim \rho$ is $$H(X) = -\int_{{\mathbb{R}}^n} \rho(x) \log \rho(x) \, dx.$$ The [*Fisher information*]{} of $X \sim \rho$ is $$J(X) = \int_{{\mathbb{R}}^n} \rho(x) \|\nabla \log \rho(x)\|^2 \, dx.$$ The [*second-order Fisher information*]{} of $X \sim \rho$ is $$K(X) = \int_{{\mathbb{R}}^n} \rho(x) \|\nabla^2 \log \rho(x)\|_{{\mathrm{HS}}}^2 \, dx.$$ Here $\|A\|_{{\mathrm{HS}}}^2 = \sum_{i,j=1}^n A_{ij}^2 = \sum_{i=1}^n \lambda_i(A)^2$ is the Hilbert-Schmidt (or Frobenius) norm of a symmetric matrix $A = (A_{ij}) \in {\mathbb{R}}^{n \times n}$ with eigenvalues $\lambda_i(A) \in {\mathbb{R}}$. In general we have the inequality $$\begin{aligned} \label{Eq:KJ} K(X) \ge \frac{J(X)^2}{n}\end{aligned}$$ which is equivalent to the entropy power inequality [@Cos85; @Dembo89; @Dembo91; @Vil00]. If $X \sim {\mathcal{N}}(\mu,\Sigma)$ is Gaussian, then $$\begin{aligned} H(X) &= \frac{1}{2} \log \det (2 \pi e \Sigma) = \frac{1}{2} \sum_{i=1}^n \log (2 \pi e \lambda_i) \\ J(X) &= \operatorname{Tr}(\Sigma^{-1}) = \sum_{i=1}^n \frac{1}{\lambda_i} \\ K(X) &= \|\Sigma^{-1}\|^2_{{\mathrm{HS}}} = \sum_{i=1}^n \frac{1}{\lambda_i^2}\end{aligned}$$ where $\lambda_1,\dots,\lambda_n > 0$ are the eigenvalues of $\Sigma \succ 0$. Our interest in the first and second-order Fisher information is because they are the first and second derivatives of entropy along the heat flow. \[Lem:DerEnt\] Along the heat flow $X_t = X_0 + \sqrt{t} Z$, $$\begin{aligned} \frac{d}{dt} H(X_t) &= \frac{1}{2} J(X_t) \\ \frac{d^2}{dt^2} H(X_t) &= -\frac{1}{2} K(X_t).\end{aligned}$$ Note that since $J(X_t) \ge 0$, the first derivative of entropy is positive, which means entropy is increasing along the heat flow. Similarly, since $K(X_t) \ge 0$, the second derivative of entropy is negative, which means entropy is a concave function along the heat flow. Mutual information and mutual Fisher information {#Sec:Mut} ------------------------------------------------ Let $(X,Y)$ be a joint random variable in ${\mathbb{R}}^n \times {\mathbb{R}}^n$ with a joint density $\rho_{XY}$, which we can factorize into a product of marginal and conditional densities: $$\rho_{XY}(x,y) = \rho_X(x) \, \rho_{Y|X}(y\,|\,x) = \rho_Y(y) \, \rho_{X|Y}(x\,|\,y).$$ We assume $\rho_Y$ and $\rho_{Y|X}(\cdot\,|\,x)$ are smooth and positive for all $x \in {\mathbb{R}}^n$. The [*mutual information*]{} of $(X,Y)$ is $$I(X;Y) = H(Y) - H(Y\,|\,X)$$ where $H(Y\,|\,X) = \int \rho_X(x) H(\rho_{Y|X}(\cdot\,|\,x))\,dx$ is the expected entropy of the conditional densities. The [*mutual Fisher information*]{} of $(X,Y)$ is $$J(X;Y) = J(Y\,|\,X) - J(Y)$$ where $J(Y\,|\,X) = \int \rho_X(x) J(\rho_{Y|X}(\cdot\,|\,x))\,dx$ is the expected Fisher information of the conditional densities. The [*mutual second-order Fisher information*]{} of $(X,Y)$ is $$K(X;Y) = K(Y\,|\,X) - K(Y)$$ where $K(Y\,|\,X) = \int \rho_X(x) K(\rho_{Y|X}(\cdot\,|\,x))\,dx$ is the expected second-order Fisher information of the conditional densities. Mutual information is symmetric: $I(X;Y) = I(Y;X)$. However, mutual first and second-order Fisher information are not symmetric: in general, $J(X;Y) \neq J(Y;X)$ and $K(X;Y) \neq K(Y;X)$. The mutual Fisher information $J(X;Y)$ can be shown to be equal to the [*backward (statistical) Fisher information*]{} $\Phi(X\,|\,Y)$, which is manifestly positive. The mutual second-order Fisher information $K(X;Y)$, on the other hand, is not always positive, but it can be represented in terms of the [*backward (statistical) second-order Fisher information*]{} $\Psi(X\,|\,Y)$; see Appendix \[App:ProofKJMut\] for detail. Analogous to the basic (non-mutual) inequality , we have the following result. Recall that a smooth probability distribution $\rho$ in ${\mathbb{R}}^n$ is [*$\alpha$-log-semiconcave*]{} for some $\alpha \in {\mathbb{R}}$ if $$-\nabla^2 \log \rho(x) \succeq \alpha I~~~~\forall \, x \in {\mathbb{R}}^n.$$ When $\alpha \ge 0$, we say $\rho$ is log-concave. \[Lem:KJMut\] If $Y \sim \rho_Y$ is $\alpha$-log-semiconcave for some $\alpha \in {\mathbb{R}}$, then $$K(X;Y) \ge \frac{J(X;Y)^2}{n} + 2\alpha J(X;Y).$$ In particular, if $\rho_Y$ is log-concave, then $K(X;Y) \ge 0$. Mutual information along the heat flow {#Sec:MutHeat} -------------------------------------- Now consider when $Y = X_t$ is the heat flow from $X = X_0$. By the linearity of the channel, the identities for the derivatives of entropy in Lemma \[Lem:DerEnt\] imply the following identities for the derivatives of mutual information along the heat flow. \[Lem:DerMut\] Along the heat flow $X_t = X_0 + \sqrt{t} Z$, $$\begin{aligned} \frac{d}{dt} I(X_0;X_t) &= -\frac{1}{2} J(X_0;X_t) \\ \frac{d^2}{dt^2} I(X_0;X_t) &= \frac{1}{2} K(X_0;X_t).\end{aligned}$$ Since $J(X_0;X_t) = \Phi(X_0\,|\,X_t) \ge 0$, the first identity above shows that mutual information is decreasing along the heat flow. In fact along the heat flow $\Phi(X_0\,|\,X_t) = \frac{1}{t^2} \operatorname{Var}(X_0\,|\,X_t)$ is proportional to the mmse of estimating $X_0$ from $X_t$, thus recovering the I-MMSE relation for Gaussian channel [@GuoEtAl05; @WibisonoJL17]. From the second identity above, we see that the convexity of mutual information along the heat flow is equivalent to the positivity of $K(X_0;X_t)$, for which Lemma \[Lem:KJMut\] will be useful. Finally, we note that since $X_t \,|\, X_0$ is Gaussian, the various mutual quantities in Lemma \[Lem:DerMut\] are simply comparisons against a baseline Gaussian: $I(X_0;X_t) = H(X_t) - \frac{n}{2} \log (2 \pi t e)$, $$J(X_0;X_t) = \frac{n}{t} - J(X_t), ~~\text{ and }~~ K(X_0;X_t) = \frac{n}{t^2} - K(X_t).$$ In the opposite order, mutual information stays the same: $I(X_t;X_0) = I(X_0;X_t)$. On the other hand, the mutual first and second-order Fisher information can be computed explicitly and do not depend on $X_t$: $$J(X_t;X_0) = \frac{n}{t} ~~~~\text{ and }~~~~ K(X_t;X_0) = \frac{n}{t^2} + \frac{2}{t} J(X_0).$$ See Appendix \[App:DetMutHeat\] for detail. Convexity of mutual information =============================== We present our main results on the convexity of mutual information along the heat flow. Throughout, let $X_t = X_0 + \sqrt{t} Z$ denote the heat flow. Perpetual convexity when initial distribution is log-concave ------------------------------------------------------------ Recall from Lemma \[Lem:KJMut\] and \[Lem:DerMut\] that mutual information is convex whenever the final distribution is log-concave. Since the heat flow preserves log-concavity, this implies mutual information is always convex when the initial distribution is log-concave. \[Thm:PerpConv\] If $X_0 \sim \rho_0$ has a log-concave distribution, then mutual information $t \mapsto I(X_0;X_t)$ is convex for all $t \ge 0$. Eventual convexity when initial distribution is bounded ------------------------------------------------------- Next, we ask when the final distribution is eventually convex under the heat flow, which also implies the eventual convexity of mutual information. We can show that if the initial distribution is bounded, then the final distribution is eventually log-concave; this fact has also been observed in [@Lee03]. We say a probability distribution $\rho$ is [*$D$-bounded*]{} for some $D \ge 0$ if it is supported on a domain of diameter at most $D$. \[Thm:EventConv\] If $X_0 \sim \rho_0$ has a $D$-bounded distribution, then mutual information $t \mapsto I(X_0;X_t)$ is convex for all $t \ge D^2$. Since convolution with log-concave distribution preserves log-concavity, we also have the following corollary. Note that when the bounded part is a point mass (with diameter $D = 0$) this recovers Theorem \[Thm:PerpConv\] above. \[Cor:EventConv\] If $X_0 \sim \rho_0$ is a convolution of a $D$-bounded and a log-concave distribution, then mutual information $t \mapsto I(X_0;X_t)$ is convex for all $t \ge D^2$. For example, if $X_0 \sim \sum_{i=1}^k p_i {\mathcal{N}}(a_i, \Sigma)$ is a mixture of Gaussians with the same covariance, then the bounded part $\sum_{i=1}^k p_i \delta_{a_i}$ has diameter $D = \max_{i \neq j} \|a_i-a_j\|$. Eventual convexity when Fisher information is finite ---------------------------------------------------- We now investigate when mutual information is eventually convex in general, regardless of the log-concavity of the distributions. We show that if the initial distribution has finite fourth moment and Fisher information, then mutual information is eventually convex. For $p \ge 0$, let $M_p(X) = {\mathbb{E}}[\|X-\mu\|^p]$ denote the $p$-th moment of a random variable $X$ with mean ${\mathbb{E}}[X] = \mu \in {\mathbb{R}}^n$. \[Thm:EventConvFI\] If $X_0 \sim \rho_0$ has finite fourth moment $M_4(X_0) < \infty$ and Fisher information $J(X_0) < \infty$, then mutual information $t \mapsto I(X_0;X_t)$ is convex for all $t \ge \frac{1}{n^2} J(X_0)M_4(X_0)$. Thus, we see that under a wide variety of conditions, mutual information is eventually convex along the heat flow. However, it turns out mutual information is [*not*]{} always convex along the heat flow, in contrast to the concavity of entropy or entropy power along the heat flow. Nonconvexity of mutual information {#Sec:NonConv} ================================== We present some counterexamples for which mutual information along the heat flow is not convex at some small time. Concretely, we study mixtures of point masses and mixtures of Gaussians as initial distribution of the heat flow. Mixture of two point masses {#Sec:MixtPoint} --------------------------- Let $X_0 \sim \frac{1}{2} \delta_{-a} + \frac{1}{2} \delta_a$ be a uniform mixture of two point masses centered at $a$ and $-a$, for some $a \in {\mathbb{R}}^n$, $a \neq 0$. Along the heat flow, $X_t \sim \frac{1}{2} {\mathcal{N}}(-a,tI) + \frac{1}{2} {\mathcal{N}}(a,tI)$ is a uniform mixture of two Gaussians with equal covariance $tI$. For $u > 0$, let $$V_u = {\mathcal{N}}(u,u) \in {\mathbb{R}}$$ denote the one-dimensional Gaussian random variable with mean and variance both equal to $u$. Then by direct calculation: $$\begin{aligned} I(X_0;X_t) &= \frac{\|a\|^2}{t} - {\mathbb{E}}[\log \cosh(V_{\frac{\|a\|^2}{t}})] \\ J(X_0;X_t) &= \frac{\|a\|^2}{t^2} {\mathbb{E}}[\operatorname{sech}^2(V_{\frac{\|a\|^2}{t}})] \\ K(X_0;X_t) &= \frac{2\|a\|^2}{t^3} {\mathbb{E}}[\operatorname{sech}^2(V_{\frac{\|a\|^2}{t}})] \! - \! \frac{\|a\|^4}{t^4} {\mathbb{E}}[\operatorname{sech}^4(V_{\frac{\|a\|^2}{t}})].\end{aligned}$$ Note the dependence on dimension is only implicit via $\|a\|^2$. The behavior of these quantities is illustrated in Figure \[Fig:MixtPoint\]. Mutual information is not convex at small time since it starts at some finite value (in fact $\log 2$), and stays flat for a while before decreasing. Its second derivative, the mutual second-order Fisher information, starts at $0$ and becomes negative before eventually becoming positive. Thus, mutual information is concave for all small time. Furthermore, by scaling $\|a\|^2$ we can stretch the region of nonconvexity to cover any finite time interval. Mixture of two Gaussians {#Sec:MixtGaus} ------------------------ Let $X_0 \sim \frac{1}{2} {\mathcal{N}}(-a,sI) + \frac{1}{2} {\mathcal{N}}(a,sI)$ be a uniform mixture of two Gaussians with the same covariance $sI$ for some $s > 0$, centered at $-a$ and $a$ for some $a \in {\mathbb{R}}^n$, $a \neq 0$. Note, the limit $s \to 0$ recovers the mixture of two point masses above. Along the heat flow, $X_t \sim \frac{1}{2} N(-a,(s+t)I) + \frac{1}{2} {\mathcal{N}}(a,(s+t)I)$ is also a mixture of two Gaussians with increasing covariance. Then with $V_u = {\mathcal{N}}(u,u)$ as above, we have: $$\begin{aligned} I(X_0;X_t) &= \frac{n}{2} \log\left(1+\frac{s}{t}\right) + \frac{\|a\|^2}{s+t} - {\mathbb{E}}[\log \cosh(V_{\frac{\|a\|^2}{s+t}})] \\ J(X_0;X_t) &= \frac{ns}{t(s+t)} + \frac{\|a\|^2}{(s+t)^2} {\mathbb{E}}[\operatorname{sech}^2(V_{\frac{\|a\|^2}{s+t}})] \\ K(X_0;X_t) &= \frac{ns(s+2t)}{t^2(s+t)^2} + \frac{2\|a\|^2}{(s+t)^3} {\mathbb{E}}[\operatorname{sech}^2(V_{\frac{\|a\|^2}{s+t}})] \\ &~~~~ - \frac{\|a\|^4}{(s+t)^4} {\mathbb{E}}[\operatorname{sech}^4(V_{\frac{\|a\|^2}{s+t}})].\end{aligned}$$ Note the explicit dependence on the dimension $n$. The behavior of these quantities is illustrated in Figure \[Fig:MixtGaus\] for $n=1$. Mutual information initially starts at $+\infty$, but it decreases quickly and exhibits a similar pattern of nonconvexity as the mixture of point masses. Its second derivative, the mutual second-order Fisher information, also starts at $+\infty$, but decreases quickly and becomes negative for some time before eventually becoming positive. Thus, mutual information is concave at some small time, and by scaling $\|a\|^2$ we can enlarge the region of nonconvexity. [0.2311]{} ![Behavior of mutual information and its two derivatives along the heat flow. (a) Left: $X_0 \sim \frac{1}{2} \delta_{-1} + \frac{1}{2} \delta_1$. (b) Right: $X_0 \sim \frac{1}{2} {\mathcal{N}}(-1,s) + \frac{1}{2} {\mathcal{N}}(1,s)$ with $s = 10^{-3}$.[]{data-label="Fig:Mixt"}](IMixtPoint.eps "fig:"){width="\textwidth"} [0.2311]{} ![Behavior of mutual information and its two derivatives along the heat flow. (a) Left: $X_0 \sim \frac{1}{2} \delta_{-1} + \frac{1}{2} \delta_1$. (b) Right: $X_0 \sim \frac{1}{2} {\mathcal{N}}(-1,s) + \frac{1}{2} {\mathcal{N}}(1,s)$ with $s = 10^{-3}$.[]{data-label="Fig:Mixt"}](IMixtGaus.eps "fig:"){width="\textwidth"} [0.22]{} ![Behavior of mutual information and its two derivatives along the heat flow. (a) Left: $X_0 \sim \frac{1}{2} \delta_{-1} + \frac{1}{2} \delta_1$. (b) Right: $X_0 \sim \frac{1}{2} {\mathcal{N}}(-1,s) + \frac{1}{2} {\mathcal{N}}(1,s)$ with $s = 10^{-3}$.[]{data-label="Fig:Mixt"}](JMixtPoint.eps "fig:"){width="\textwidth"}    [0.22]{} ![Behavior of mutual information and its two derivatives along the heat flow. (a) Left: $X_0 \sim \frac{1}{2} \delta_{-1} + \frac{1}{2} \delta_1$. (b) Right: $X_0 \sim \frac{1}{2} {\mathcal{N}}(-1,s) + \frac{1}{2} {\mathcal{N}}(1,s)$ with $s = 10^{-3}$.[]{data-label="Fig:Mixt"}](JMixtGaus.eps "fig:"){width="\textwidth"} [0.22]{} ![Behavior of mutual information and its two derivatives along the heat flow. (a) Left: $X_0 \sim \frac{1}{2} \delta_{-1} + \frac{1}{2} \delta_1$. (b) Right: $X_0 \sim \frac{1}{2} {\mathcal{N}}(-1,s) + \frac{1}{2} {\mathcal{N}}(1,s)$ with $s = 10^{-3}$.[]{data-label="Fig:Mixt"}](KMixtPoint.eps "fig:"){width="\textwidth"}    [0.22]{} ![Behavior of mutual information and its two derivatives along the heat flow. (a) Left: $X_0 \sim \frac{1}{2} \delta_{-1} + \frac{1}{2} \delta_1$. (b) Right: $X_0 \sim \frac{1}{2} {\mathcal{N}}(-1,s) + \frac{1}{2} {\mathcal{N}}(1,s)$ with $s = 10^{-3}$.[]{data-label="Fig:Mixt"}](KMixtGaus.eps "fig:"){width="\textwidth"} General mixture of point masses ------------------------------- Let $X_0 \sim \sum_{i=1}^k p_i \delta_{a_i}$ be a mixture of point masses centered at distinct $a_i \in {\mathbb{R}}^n$, with mixture probabilities $p_i > 0$, $\sum_{i=1}^k p_i = 1$. Along the heat flow, $X_t \sim \sum_{i=1}^k p_i {\mathcal{N}}(a_i,tI)$ is a mixture of Gaussians with increasing covariance $tI$ at the same centers. We show that mutual information starts at a finite value which is equal to the discrete entropy of the mixture probability, and it is exponentially concentrated at small time. Let $\|p\|_\infty = \max_{i,j} p_i/p_j$ and $m = \min_{i \neq j} \|a_i-a_j\| > 0$. Let $h(p) = -\sum_{i=1}^k p_i \log p_i$ denote the discrete entropy. \[Thm:GenMixt\] For all $0 < t \le \frac{m^2}{676\|p\|_\infty^2}$, $$0 \,\le\, h(p) - I(X_0;X_t) \,\le\, 3(k-1)\|p\|_\infty e^{-0.085 \frac{m^2}{t}}.$$ The theorem above implies that $$\lim_{t \to 0} I(X_0;X_t) = h(p).$$ In particular, the initial value of mutual information does not depend on the locations of the centers, as long as they are distinct. This is interesting, because by moving the centers and merging them we can obtain discontinuities of the mutual information with respect to the initial random variable at the origin (moving the centers changes the mutual information curve but preserves the starting point, while merging the centers makes the starting point jump). Furthermore, if a function converges exponentially fast, then all its derivatives must converge to zero exponentially fast. Thus, we have the following corollary. \[Cor:Last\] For all $\ell \in \mathbb{N}$, $\lim_{t \to 0} \frac{d^\ell}{dt^\ell} I(X_0;X_t) = 0$. In particular, the first derivative of mutual information, which is negative mutual Fisher information, starts at $0$. Since the initial distribution is bounded, mutual information is eventually convex by Theorem \[Thm:EventConv\], which means mutual Fisher information is eventually decreasing. Since mutual Fisher information is always nonnegative, this means it must initially increase, during which mutual information is concave; this is similar to the behavior observed in $\S\ref{Sec:MixtPoint}$. Moreover, by the continuity of the second-order Fisher information, this suggests that when the initial distribution is a mixture of Gaussians, mutual information may be also be concave at some small time. Discussion and future work ========================== In this paper we have studied the convexity of mutual information along the heat flow. We have shown that under a wide variety of conditions mutual information is eventually convex, and we have shown examples where mutual information may be concave at some small time. Many questions remain. One question is how much we can extend the results to general stochastic processes. We can show most of our results still hold for the Ornstein-Uhlenbeck process [@WJOU18]. For general Fokker-Planck processes the situation is more complicated, but at least there are explicit formulae for the second derivatives [@Vil08]. Another question is whether there are other conditions that imply eventual log-concavity under the heat flow. Currently we only know it for when the initial distribution is a convolution of a bounded and a log-concave distribution. It is interesting to study what happens for a larger class of initial distributions, for example sub-Gaussian. Alternatively, for each point in space we can define the notion of a “time to log-concavity,” after which the final distribution is log-concave at that point. In general, this time is finite for each fixed point, and eventual log-concavity occurs if the supremum of this time over space is finite. There is a generic bound for this time to log-concavity in terms of the variance, and we can prove a slightly better bound under sub-Gaussian assumption, but not much is known. We are seeking a proof of the nonconvexity of mutual information for the examples presented in $\S$\[Sec:NonConv\]. The nonconvexity is clear from Figure \[Fig:Mixt\], and we have explicit formulae for the second derivatives, but it is desirable to have a formal proof. It is also interesting to study the effect of dimension in this problem, whether it makes convexity of mutual information easier or more difficult to occur. From Theorem \[Thm:EventConvFI\], and taking into account the growth of Fisher information and fourth moment with dimension, we see that the effect of dimension seems to be to delay the eventual convexity. Finally, for mixtures of point masses, we have shown that the definition of self-information under the heat flow remembers the discrete initial data. We can show this also holds for the Ornstein-Uhlenbeck process [@WJOU18]. It is interesting to study whether the self-information limit is the same under more general flows such as the Fokker-Planck process. Proofs ====== Proof of Lemma \[Lem:DerEnt\] ----------------------------- These identities follow by direct calculation and integration by parts (and Bochner’s formula for the second identity). The first derivative of entropy along the heat flow is De Bruijn’s identity [@Sta59]. The second derivative of entropy along the heat flow is by McKean [@McKean66] in one dimension, and by Toscani [@Tos99] in multi dimension; see also Villani [@Vil00] for a clean proof. Proof of Lemma \[Lem:KJMut\] {#App:ProofKJMut} ---------------------------- We first introduce some definitions. We view the joint distribution $\rho_{XY}(x,y) = \rho_Y(y) \rho_{X|Y}(x\,|\,y)$ as a family of probability distributions $\rho_{X|Y}(\cdot\,|\,y)$ parameterized by $y \in {\mathbb{R}}^n$, which has distribution $\rho_Y$. We also assume the density $\rho_{X|Y}(\cdot\,|\,y)$ is smooth with respect to $y$. The [*pointwise backward Fisher information matrix*]{} of $X$ given $Y=y$ is $$\begin{gathered} \widetilde \Phi(X\,|\,Y=y) = \\ \int_{{\mathbb{R}}^n} \rho_{X|Y}(x\,|\,y) (\nabla_y \log \rho_{X|Y}(x\,|\,y))(\nabla_y \log \rho_{X|Y}(x\,|\,y))^\top dx.\end{gathered}$$ By integration by parts (assuming boundary terms vanish), we can also write $$\widetilde \Phi(X\,|\,Y\!=\!y) = -\int_{{\mathbb{R}}^n} \rho_{X|Y}(x\,|\,y) \nabla^2_y \log \rho_{X|Y}(x\,|\,y) dx.$$ The [*pointwise backward Fisher information*]{} of $X$ given $Y=y$ is $$\begin{aligned} \Phi(X\,|\,Y\!=\!y) &= \operatorname{Tr}(\widetilde \Phi(X\,|\,Y\!=\!y)) \\ &= \int_{{\mathbb{R}}^n} \rho_{X|Y}(x\,|\,y) \|\nabla_y \log \rho_{X|Y}(x\,|\,y)\|^2 dx.\end{aligned}$$ The [*backward Fisher information matrix*]{} of $X$ given $Y$ is $$\widetilde \Phi(X\,|\,Y) = \int_{{\mathbb{R}}^n} \rho_Y(y) \, \widetilde \Phi(X\,|\,Y\!=\!y) \, dy.$$ The [*backward Fisher information*]{} of $X$ given $Y$ is $$\Phi(X\,|\,Y) = \operatorname{Tr}(\widetilde \Phi(X\,|\,Y)).$$ Note $\widetilde \Phi(X\,|\,Y\!=\!y) \succeq 0$ and $\Phi(X\,|\,Y\!=\!y) \ge 0$ for all $y \in {\mathbb{R}}^n$, so $\widetilde \Phi(X\,|\,Y) \succeq 0$ and $\Phi(X\,|\,Y) \ge 0$. Similarly, the [*pointwise backward second-order Fisher information*]{} of $X$ given $Y=y$ is $$\Psi(X\,|\,Y\!=\!y) = \int_{{\mathbb{R}}^n} \rho_{X|Y}(x\,|\,y) \|\nabla^2_y \log \rho_{X|Y}(x\,|\,y)\|^2_{{\mathrm{HS}}} \, dx.$$ The [*backward second-order Fisher information*]{} of $X$ given $Y$ is $$\Psi(X\,|\,Y) = \int_{{\mathbb{R}}^n} \rho_Y(y) \, \Psi(X\,|\,Y\!=\!y) \, dy.$$ Note that $\Psi(X\,|\,Y=y) \ge 0$ for all $y \in {\mathbb{R}}^n$, so $\Psi(X\,|\,Y) \ge 0$. Finally, the [*Fisher information matrix*]{} of $Y$ is $$\widetilde J(Y) = \int_{{\mathbb{R}}^n} \rho_Y(y) (\nabla_y \log \rho_Y(y))(\nabla_y \log \rho_Y(y))^\top dy.$$ By integration by parts (assuming boundary terms vanish), we can also write $$\widetilde J(Y) = -\int_{{\mathbb{R}}^n} \rho_Y(y) \nabla^2_y \log \rho_Y(y) dy.$$ Note that $\widetilde J(Y) \succeq 0$ and Fisher information is its trace: $J(Y) = \operatorname{Tr}(\widetilde J(Y))$. As stated in $\S\ref{Sec:Mut}$, mutual Fisher information is in fact equal to the backward Fisher information. \[Lem:JPhi\] For any joint random variable $(X,Y)$, $$J(X;Y) = \Phi(X\,|\,Y).$$ From the factorization $$\rho_X(x) \rho_{Y|X}(y\,|\,x) = \rho_Y(y) \rho_{X|Y}(x\,|\,y)$$ we have $$-\nabla^2_y \log \rho_{Y|X}(y\,|\,x) = -\nabla^2_y \log \rho_Y(y) - \nabla^2_y \log \rho_{X|Y}(x\,|\,y).$$ We integrate both sides with respect to $\rho_{XY}(x,y)$. The left-hand side gives the expected Fisher information matrix $\widetilde J(Y\,|\,X)$. The first term on the right-hand side gives $\widetilde J(Y)$, while the second term gives the $\widetilde \Phi(X\,|\,Y)$. That is, $\widetilde J(Y\,|\,X) = \widetilde J(Y) + \widetilde \Phi(X\,|\,Y)$, or equivalently, $$\widetilde J(X;Y) = \widetilde J(Y\,|\,X) - \widetilde J(Y) = \widetilde \Phi(X\,|\,Y).$$ Taking trace gives $$J(X;Y) = \operatorname{Tr}(\widetilde J(X;Y)) = \operatorname{Tr}(\widetilde \Phi(X\,|\,Y)) = \Phi(X;Y)$$ as desired. Similarly, mutual second-order Fisher information can be represented in terms of the backward second-order Fisher information, albeit in a more complicated way. \[Lem:KPsi\] For any joint random variable $(X,Y)$, $$\begin{gathered} K(X;Y) = \Psi(X\,|\,Y) \, + \\ 2 \int_{{\mathbb{R}}^n} \rho_Y(y) \langle -\nabla^2 \log \rho_Y(y), \, \widetilde \Phi(X\,|\,Y\!=\!y)\rangle_{{\mathrm{HS}}} \, dy.\end{gathered}$$ As before we have the decomposition $$-\nabla^2_y \log \rho_{Y|X}(y\,|\,x) = -\nabla^2_y \log \rho_Y(y) - \nabla^2_y \log \rho_{X|Y}(x\,|\,y).$$ Taking the squared norm on both sides and expanding, we get $$\begin{aligned} &\|\nabla^2_y \log \rho_{Y|X}(y\,|\,x)\|^2_{{\mathrm{HS}}} \\ &~~~~= \|\nabla^2_y \log \rho_Y(y)\|^2_{{\mathrm{HS}}} + \|\nabla^2_y \log \rho_{X|Y}(x\,|\,y)\|^2_{{\mathrm{HS}}} \\ &~~~~~~~~ + 2 \langle \nabla^2_y \log \rho_Y(y), \nabla^2_y \log \rho_{X|Y}(x\,|\,y) \rangle_{{\mathrm{HS}}}.\end{aligned}$$ We integrate both sides with respect to $\rho_{XY}(x,y)$. On the left-hand side we get $K(Y\,|\,X)$. The first term on the right-hand side gives $K(Y)$; the second term gives $\Psi(X\,|\,Y)$; for the third term, by first integrating over $\rho_{X|Y}(x\,|\,y)$ we obtain an inner product with $\widetilde \Phi(X\,|\,Y\!=\!y)$. That is, $$\begin{gathered} K(Y\,|\,X) = K(Y) + \Psi(X\,|\,Y) \\ + 2 \int_{{\mathbb{R}}^n} \rho_Y(y) \langle -\nabla^2_y \log \rho_Y(y), \widetilde \Phi(X\,|\,Y\!=\!y)\rangle_{{\mathrm{HS}}} \, dy.\end{gathered}$$ This implies the desired expression for $K(X;Y) = K(Y\,|\,X)-K(Y)$. We can prove a lower bound for $K(X;Y)$ under log-semiconcavity assumption on $Y$. \[Lem:KLC\] If $Y \sim \rho_Y$ is $\alpha$-log-semiconcave for some $\alpha \in {\mathbb{R}}$, then $$K(X;Y) \ge \Psi(X\,|\,Y) + 2\alpha \Phi(X\,|\,Y).$$ Since $-\nabla^2 \log \rho_Y(y) \succeq \alpha I$ and $\widetilde \Phi(X\,|\,Y=y) \succeq 0$ for all $y \in {\mathbb{R}}^n$, we have $$\begin{aligned} \langle -\nabla^2 \log \rho_Y(y), \, \widetilde \Phi(X\,|\,Y\!=\!y) \rangle_{{\mathrm{HS}}} &\ge \, \langle \alpha I, \, \widetilde \Phi(X\,|\,Y\!=\!y) \rangle_{{\mathrm{HS}}} \\ &=\, \alpha \operatorname{Tr}(\widetilde \Phi(X\,|\,Y\!=\!y)) \\ &=\, \alpha \, \Phi(X\,|\,Y\!=\!y).\end{aligned}$$ Integrating with respect to $\rho_Y(y)$ gives $$\begin{aligned} &\int_{{\mathbb{R}}^n} \rho_Y(y) \langle -\nabla^2 \log \rho_Y(y), \, \widetilde \Phi(X\,|\,Y\!=\!y) \rangle_{{\mathrm{HS}}} \, dy \\ &~~~~~~~ \ge\, \alpha \int_{{\mathbb{R}}^n} \rho(y) \Phi(X\,|\,Y\!=\!y) \, dy \,=\, \alpha \, \Phi(X\,|\,Y).\end{aligned}$$ Adding $\Psi(X\,|\,Y)$ and using Lemma \[Lem:KPsi\] gives the result. Furthermore, we have the following result which is reminiscent of the inequality  between first and second-order Fisher information. \[Lem:PsiPhi\] For any joint random variable $(X,Y)$ in ${\mathbb{R}}^n \times {\mathbb{R}}^n$, $$\Psi(X\,|\,Y) \ge \frac{\Phi(X\,|\,Y)^2}{n}.$$ Let $A_{x,y} = -\nabla^2_y \log \rho_{X|Y}(x\,|\,y)$. By Cauchy-Schwarz inequality, $$\|A_{x,y}\|^2_{{\mathrm{HS}}} = \operatorname{Tr}(A_{x,y}^2) \ge \frac{(\operatorname{Tr}(A_{x,y}))^2}{n}.$$ Taking expectation over $(X,Y) \sim \rho_{XY}$ and applying Cauchy-Schwarz again, we get the desired result $$\begin{aligned} \Psi(X\,|\,Y) &= {\mathbb{E}}[\|A_{X,Y}\|^2_{{\mathrm{HS}}}] \\ &\ge \frac{{\mathbb{E}}[(\operatorname{Tr}(A_{X,Y}))^2]}{n} \\ &\ge \frac{({\mathbb{E}}[\operatorname{Tr}(A_{X,Y})])^2}{n} \\ &= \frac{\Phi(X\,|\,Y)^2}{n}.\end{aligned}$$ Finally, we are ready to prove Lemma \[Lem:KJMut\]. By Lemma \[Lem:KLC\] and \[Lem:PsiPhi\], $$K(X;Y) \ge \frac{\Phi(X\,|\,Y)^2}{n} + 2\alpha \Phi(X\,|\,Y).$$ Since $J(X;Y) = \Phi(X\,|\,Y)$ by Lemma \[Lem:JPhi\], the result follows. Proof of Lemma \[Lem:DerMut\] ----------------------------- These identities follow from Lemma \[Lem:DerEnt\] and the linearity of the heat flow channel. Concretely, recall by Lemma \[Lem:DerEnt\] that $\frac{d}{dt} H(X_t) = \frac{1}{2} J(X_t)$. We apply this result to the conditional density $\rho_{X_t|X_0}(\cdot\,|\,x_0)$ to get $\frac{d}{dt} H(X_t\,|\,X_0=x_0) = \frac{1}{2} J(X_t\,|\,X_0=x_0)$ for each $x_0 \in {\mathbb{R}}^n$. Taking expectation over $X_0 \sim \rho_0$ and interchanging the order of expectation and time differentiation yields $\frac{d}{dt} H(X_t\,|\,X_0) = \frac{1}{2} J(X_t\,|\,X_0)$. Combining this with the earlier result above yields $\frac{d}{dt} I(X_0;X_t) = \frac{1}{2} J(X_0;X_t)$, as desired. The proof for $\frac{d^2}{dt^2} I(X_0;X_t) = -\frac{1}{2} K(X_0;X_t)$ proceeds identically using the second identity in Lemma \[Lem:DerEnt\]. Detail for $\S\ref{Sec:MutHeat}$ {#App:DetMutHeat} -------------------------------- We compute $J(X_t;X_0)$ and $K(X_t;X_0)$ along the heat flow $X_t = X_0 + \sqrt{t} Z$. Let $X_0 \sim \rho_0$, $X_t \sim \rho_t$, $(X_0,X_t) \sim \rho_{0t}$, and we write the conditionals as $$\rho_0(x) \rho_{t|0}(y\,|\,x) = \rho_{0t}(x,y) = \rho_t(y) \rho_{0|t}(x\,|\,y).$$ Then $$-\nabla_x \log \rho_{0|t}(x\,|\,y) = -\nabla_x \log \rho_0(x) - \nabla_x \log \rho_{t|0}(y\,|\,x).$$ Along the heat flow $X_t\,|\,X_0$ is Gaussian with covariance $tI$, so we have explicitly $-\nabla_x \log \rho_{t|0}(y\,|\,x) = \frac{x-y}{t}$. Therefore, $$\begin{aligned} \label{Eq:DetMutHeatCalc} -\nabla_x \log \rho_{0|t}(x\,|\,y) = -\nabla_x \log \rho_0(x) +\frac{x-y}{t}.\end{aligned}$$ Take the squared norm on both sides and expand: $$\begin{aligned} \|\nabla_x \log \rho_{0|t}(x\,|\,y)\|^2 &= \|\nabla_x \log \rho_0(x)\|^2 + \frac{\|x-y\|^2}{t^2} \\ &~~~~ + \frac{2}{t} \langle -\nabla_x \log \rho_0(x), x-y \rangle.\end{aligned}$$ Now we take expectation of both sides over $(X_0,X_t)$. The left-hand side gives $J(X_0\,|\,X_t)$. The first term on the right-hand side gives $J(X_0)$; the second term gives $\frac{1}{t^2}{\mathbb{E}}[\|X_0-X_t\|^2] = \frac{1}{t^2} {\mathbb{E}}[\|\sqrt{t}Z\|^2] = \frac{n}{t}$ where $Z \sim {\mathcal{N}}(0,I)$; while the third term gives $0$ by integrating over $y$ first for each fixed $x$. That is, $$\begin{aligned} \label{Eq:JXYX} J(X_0\,|\,X_t) = J(X_0) + \frac{n}{t}.\end{aligned}$$ Therefore, $$J(X_t;X_0) = J(X_0\,|\,X_t) - J(X_0) = \frac{n}{t}.$$ Next, we differentiate  again with respect to $x$ to get $$-\nabla^2_x \log \rho_{0|t}(x\,|\,y) = -\nabla^2_x \log \rho_0(x) +\frac{I}{t}.$$ Take the squared norm on both sides and expand: $$\begin{aligned} &\|\nabla^2_x \log \rho_{0|t}(x\,|\,y)\|_{{\mathrm{HS}}}^2 \\ &= \|\nabla^2_x \log \rho_0(x)\|_{{\mathrm{HS}}}^2 + \frac{\|I\|^2_{{\mathrm{HS}}}}{t^2} + \frac{2}{t} \langle -\nabla^2_x \log \rho_0(x), I \rangle_{{\mathrm{HS}}} \\ &= \|\nabla^2_x \log \rho_0(x)\|_{{\mathrm{HS}}}^2 + \frac{n}{t^2} - \frac{2}{t} \Delta_x \log \rho_0(x).\end{aligned}$$ Now we take expectation of both sides over $(X_0,X_t)$. The left-hand side gives $K(X_0\,|\,X_t)$. The first term on the right-hand side gives $K(X_0)$; the second term is a constant; while the third term gives $\frac{2}{t}J(X_0)$. That is, $$K(X_0\,|\,X_t) = K(X_0) + \frac{n}{t^2} + \frac{2}{t} J(X_0).$$ Therefore, $$K(X_t;X_0) = K(X_0\,|\,X_t) - K(X_0) = \frac{n}{t^2} + \frac{2}{t} J(X_0).$$ Proof of Theorem \[Thm:PerpConv\] --------------------------------- Recall that the heat flow preserves log-concavity. This is because the Gaussian density (the heat kernel) is log-concave, and convolution with a log-concave distribution preserves log-concavity by the Prékopa-Leindler inequality. By assumption $X_0 \sim \rho_0$ is log-concave, so $X_t \sim \rho_t$ is also log-concave for all $t \ge 0$. By Lemma \[Lem:KJMut\] and \[Lem:DerMut\], this implies $\frac{d^2}{dt^2} I(X_0;X_t) = K(X_0;X_t) \ge 0$, which means mutual information is always convex. Proof of Theorem \[Thm:EventConv\] {#App:ProofEventConv} ---------------------------------- Throughout, let $X_t = X_0 + \sqrt{t} Z$ denote the heat flow. Let $X_0 \sim \rho_0$, $X_t \sim \rho_t$, $(X_0,X_t) \sim \rho_{0t}$, and we write the conditionals as $$\rho_0(x) \rho_{t|0}(y\,|\,x) = \rho_{0t}(x,y) = \rho_t(y) \rho_{0|t}(x\,|\,y).$$ We first establish the following result to help us determine when we have eventual log-concavity under the heat flow; see Appendix \[App:ProofHesHeat\] for the proof. \[Lem:HesHeat\] Along the heat flow, for all $y \in {\mathbb{R}}^n$, $$\begin{aligned} \label{Eq:HesHeat} -\nabla^2_y \log \rho_{t}(y) = \frac{1}{t} \left(I - \frac{1}{t} \operatorname{Cov}(\rho_{0|t}(\cdot\,|\,y))\right).\end{aligned}$$ In particular, for bounded initial distribution we have the following eventual log-concavity. \[Lem:BddLC\] If $X_0 \sim \rho_0$ is $D$-bounded, then along the heat flow, $X_t \sim \rho_t$ is log-concave for all $t \ge D^2$. Since $X_0 \sim \rho_0$ is $D$-bounded, the conditional distributions $\rho_{0|t}(\cdot\,|\,y)$ are also $D$-bounded for all $y \in {\mathbb{R}}^n$. In particular, $\operatorname{Cov}(\rho_{0|t}(\cdot\,|\,y)) \preceq D^2 I.$ Therefore, by Lemma \[Lem:HesHeat\], $$-\nabla^2_y \log \rho_{t}(y) \succeq \frac{1}{t} \left(1 - \frac{D^2}{t}\right) I.$$ If $t \ge D^2$, then $-\nabla^2_y \log \rho_{t}(y) \succeq 0$ for all $y \in {\mathbb{R}}^n$, which means $X_t \sim \rho_t$ is log-concave. We are now ready to prove Theorem \[Thm:EventConv\]. By Lemma \[Lem:BddLC\], $X_t \sim \rho_t$ is log-concave for $t \ge D^2$. By Lemma \[Lem:KJMut\] and \[Lem:DerMut\], this implies mutual information is convex for all $t \ge D^2$. Proof of Corollary \[Cor:EventConv\] ------------------------------------ Analogous to Lemma \[Lem:BddLC\], we have the following result. \[Lem:BddConvLC\] If $X_0 \sim \rho_0$ is a convolution of a $D$-bounded and a log-concave distribution, then along the heat flow, $X_t \sim \rho_t$ is log-concave for all $t \ge D^2$. We write $X_0 = B_0+C$ where $B_0$ is a $D$-bounded random variable and $C$ is a log-concave random variable independent of $B$. Then $X_t = X_0 + \sqrt{t}Z = (B_0 + \sqrt{t}Z) + C = B_t+C$ where $B_t = B_0 + \sqrt{t}Z$ is the heat flow from $B_0$. By Lemma \[Lem:BddLC\], $B_t$ is log-concave for $t \ge D^2$. Then by the Prékopa Leindler inequality, $X_t = B_t+C$ is also log-concave for all $t \ge D^2$. We are now ready to prove Corollary \[Cor:EventConv\]. By Lemma \[Lem:BddConvLC\], $X_t \sim \rho_t$ is log-concave for $t \ge D^2$. By Lemma \[Lem:KJMut\] and \[Lem:DerMut\], this implies mutual information is convex for all $t \ge D^2$. Proof of Lemma \[Lem:HesHeat\] {#App:ProofHesHeat} ------------------------------ We use the same setting and notation as in Appendix \[App:ProofEventConv\]. We first establish the following result. \[Lem:HessCov\] Along the heat flow, for all $x,y \in {\mathbb{R}}^n$, $$\begin{aligned} \label{Eq:HessCov} -\nabla^2_y \log \rho_{0|t}(x\,|\,y) = \frac{\operatorname{Cov}(\rho_{0|t}(\cdot\,|\,y))}{t^2}.\end{aligned}$$ We observe that the conditional density $\rho_{0|t}(x\,|\,y)$ can be written as an exponential family distribution over $x$ with parameter $\eta = \frac{y}{t}$: $$\rho_{0|t}(x\,|\,y) = h(x) e^{\langle x,\eta \rangle - A(\eta)}$$ where $h(x) = \rho_0(x) e^{-\frac{\|x\|^2}{2t}}$ is the base measure, and $$A(\eta) = \log \int_{{\mathbb{R}}^n} h(x) e^{\langle x,\eta \rangle} \, dx$$ is the log-partition function, or normalizing constant. Then we have $$-\nabla^2_y \log \rho_{0|t}(x\,|\,y) = \frac{1}{t^2} \nabla^2_\eta A(\eta).$$ By a general identity for exponential family [@WJ08], or simply by differentiating, we have that $$\nabla^2_\eta A(\eta) = \operatorname{Cov}(\rho_{0|t}(\cdot\,|\,y)).$$ Combining the two expressions above yields the result. We are now ready to prove Lemma \[Lem:HesHeat\]. From the factorization $$\rho_t(y) \rho_{0|t}(x\,|\,y) = \rho_0(x) \rho_{t|0}(y\,|\,x)$$ we have, along the heat flow and by Lemma \[Lem:HessCov\], $$\begin{aligned} -\nabla^2_y \log \rho_t(y) &= -\nabla^2_y \log \rho_{t|0}(y\,|\,x) + \nabla^2_y \log \rho_{0|t}(x\,|\,y) \\ &= \frac{1}{t} I -\frac{1}{t^2}\operatorname{Cov}(\rho_{0|t}(\cdot\,|\,y)),\end{aligned}$$ as desired. Proof of Theorem \[Thm:EventConvFI\] ------------------------------------ Let $X_t = X_0 + \sqrt{t} Z$ denote the heat flow. We first establish some results. \[Lem:KHeat\] Along the heat flow, $$K(X_0;X_t) = \frac{2}{t^3} \operatorname{Var}(X_0\,|\,X_t) - \frac{1}{t^4} {\mathbb{E}}[\|\operatorname{Cov}(\rho_{0|t}(\cdot\,|\,X_t))\|^2_{{\mathrm{HS}}}].$$ Squaring and taking the expectation of the identity  in Lemma \[Lem:HesHeat\] gives $$\begin{aligned} K&(X_t) = \frac{1}{t^2} {\mathbb{E}}\Big[\Big\|I - \frac{1}{t} \operatorname{Cov}(\rho_{0|t}(\cdot\,|\,X_t))\Big\|^2_{{\mathrm{HS}}}\Big] \\ &= \frac{\|I\|^2_{{\mathrm{HS}}}}{t^2} - \frac{2}{t^3} {\mathbb{E}}[\operatorname{Var}( \rho_{0|t}(\cdot\,|\,X_t) )] \\ &~~~~ + \frac{1}{t^4} {\mathbb{E}}[\|\operatorname{Cov}(\rho_{0|t}(\cdot\,|\,X_t))\|^2_{{\mathrm{HS}}}] \\ &= \frac{n}{t^2} - \frac{2}{t^3} \operatorname{Var}(X_0\,|\,X_t) + \frac{1}{t^4} {\mathbb{E}}[\|\operatorname{Cov}(\rho_{0|t}(\cdot\,|\,X_t))\|^2_{{\mathrm{HS}}}].\end{aligned}$$ Since $K(X_t\,|\,X_0) = n/t^2$, this implies the desired result. \[Lem:VarJ\] Assume $J(X_0) < +\infty$. Along the heat flow, $$\operatorname{Var}(X_0\,|\,X_t) \ge \frac{n^2}{J(X_0) + \frac{n}{t}}.$$ For any random variable $X \sim \rho$ in ${\mathbb{R}}^n$ with a smooth density, recall the uncertainty relationship $$J(X) \operatorname{Var}(X) \ge n^2,$$ which also follows from the Cauchy-Schwarz inequality and integration by parts. Applying this to the conditional densities $\rho_{0|t}(\cdot\,|\,y)$ yields $$\operatorname{Var}(\rho_{0|t}(\cdot\,|\,y)) \ge \frac{n^2}{J(\rho_{0|t}(\cdot\,|\,y))}.$$ Taking expectation over $Y = X_t$ and noting that ${\mathbb{E}}[\frac{1}{J}] \ge \frac{1}{{\mathbb{E}}[J]}$ by Cauchy-Schwarz, we get $$\operatorname{Var}(X_0\,|\,X_t) \ge {\mathbb{E}}\left[\frac{n^2}{J(\rho_{0|t}(\cdot\,|\,X_t))}\right] \ge \frac{n^2}{J(X_0\,|\,X_t)}.$$ Finally, recall from  that $J(X_0\,|\,X_t) = J(X_0)+\frac{n}{t}$. Recall that $M_4(X_0)$ is the fourth moment of $X_0$. \[Lem:CovM4\] Assume $M_4(X_0) < +\infty$. Along the heat flow, $${\mathbb{E}}[\|\operatorname{Cov}(\rho_{0|t}(\cdot\,|\,X_t))\|^2_{{\mathrm{HS}}}] \le M_4(X_0).$$ Let $\mu_0 = {\mathbb{E}}[X_0]$. For each $y \in {\mathbb{R}}^n$, $$\begin{aligned} \|\operatorname{Cov}(\rho_{0|t}(\cdot\,|\,y))\|^2_{{\mathrm{HS}}} &\le (\operatorname{Tr}(\operatorname{Cov}(\rho_{0|t}(\cdot\,|\,y))))^2 \\ &= (\operatorname{Var}(\rho_{0|t}(\cdot\,|\,y)))^2 \\ &\le \left( {\mathbb{E}}_{\rho_{0|t}(\cdot\,|\,y)}[\|X-\mu_0\|^2]\right)^2 \\ &\le {\mathbb{E}}_{\rho_{0|t}(\cdot\,|\,y)}[\|X-\mu_0\|^4].\end{aligned}$$ Taking expectation over $Y=X_t$ and applying the tower property of expectation gives the result. We are now ready to prove Theorem \[Thm:EventConvFI\]. By Lemma \[Lem:KHeat\], \[Lem:VarJ\], and \[Lem:CovM4\], we have $$K(X_0;X_t) \ge \frac{2n^2}{t^3(J(X_0)+\frac{n}{t})} - \frac{M_4(X_0)}{t^4}.$$ The right-hand side above is nonnegative if $2n^2t^4 \ge t^3M_4(X_0)(J(X_0)+\frac{n}{t})$, or equivalently, if $$2n^2 t^2 - t J(X_0)M_4(X_0) - nM_4(X_0) \ge 0.$$ Therefore, $K(X_0;X_t) \ge 0$ if $t$ is larger than the upper root of the quadratic polynomial above, which is the case when $$t \ge \frac{J(X_0)M_4(X_0)}{4n^2}\left(1+\sqrt{1+\frac{8n}{J(X_0)^2 M_4(X_0)}}\right).$$ Furthermore, by Cauchy-Schwarz and the uncertainty relationship, $$J(X_0)^2 M_4(X_0) \ge J(X_0)^2 \operatorname{Var}(X_0)^2 \ge n^4.$$ Plugging this to the bound above and further using $n \ge 1$, we conclude that $K(X_0;X_t) \ge 0$, and hence mutual information is convex, whenever $$t \ge \frac{J(X_0)M_4(X_0)}{n^2}.$$ Proof of Theorem $\ref{Thm:GenMixt}$ ------------------------------------ At each $t > 0$, the density of $X_t \sim \sum_{i=1}^k p_i {\mathcal{N}}(a_i,tI)$ is $$\rho_t(y) = \frac{1}{(2\pi t)^{n/2}} \sum_{i=1}^k p_i e^{-\frac{\|y-a_i\|^2}{2t}}.$$ The entropy of $X_t$ is $$H(X_t) = \frac{n}{2} \log (2\pi t) - {\mathbb{E}}\left[ \log \left( \sum_{i=1}^k p_i e^{-\frac{\|X_t-a_i\|^2}{2t}} \right) \right].$$ The expectation is over the mixture $X_t \sim \sum_{i=1}^k p_i {\mathcal{N}}(a_i, tI)$, which we split into a sum over $i=1,\dots,k$ of the individual expectations over $Y \sim {\mathcal{N}}(a_i, tI)$. When $Y \sim {\mathcal{N}}(a_i, tI)$, we write $Y = a_i + \sqrt{t} Z$ where $Z \sim {\mathcal{N}}(0,I)$. Then we can write the entropy above as [$$\begin{aligned} &H(X_t) - \frac{n}{2} \log (2\pi t) \\ &= - \sum_{i=1}^k p_i {\mathbb{E}}\Big[\log\Big(p_i e^{-\frac{\|Z\|^2}{2}} + \sum_{j \neq i} p_j e^{-\frac{\|\sqrt{t}Z+a_i-a_j\|^2}{2t}}\Big)\Big] \\ &= - \sum_{i=1}^k p_i {\mathbb{E}}\Big[ \log p_i - \frac{\|Z\|^2}{2} + \log \Big(1 + \sum_{j \neq i} \frac{p_j}{p_i} e^{\frac{\|Z\|^2}{2}-\frac{\|\sqrt{t}Z+a_i-a_j\|^2}{2t}}\Big)\Big] \\ &= h(p) + \frac{n}{2} - \sum_{i=1}^k p_i {\mathbb{E}}\Big[ \log \Big(1 + \sum_{j \neq i} \frac{p_j}{p_i} e^{\frac{\|Z\|^2}{2}-\frac{\|\sqrt{t}Z+a_i-a_j\|^2}{2t}}\Big)\Big]\end{aligned}$$ ]{} where $h(p) = -\sum_{i=1}^k p_i \log p_i$ is the discrete entropy. Since $H(X_t\,|\,X_0) = \frac{n}{2} \log (2\pi t e)$, we have for mutual information [$$h(p) - I(X_0;X_t) = \sum_{i=1}^k p_i {\mathbb{E}}\Big[ \log \Big(1 + \sum_{j \neq i} \frac{p_j}{p_i} e^{\frac{\|Z\|^2}{2}-\frac{\|\sqrt{t}Z+a_i-a_j\|^2}{2t}}\Big)\Big].$$ ]{} Clearly $h(p)-I(X_0;X_t) \ge 0$ since the logarithm on the right-hand side above is positive. On the other hand, using the inequality $\log(1+\sum_j x_j) \le \sum_j \log(1+x_j)$ for $x_j > 0$, we also have the upper bound [$$h(p)-I(X_0;X_t) \le \sum_{i=1}^k p_i \sum_{j \neq i} {\mathbb{E}}\Big[ \log \Big(1 + \frac{p_j}{p_i} e^{\frac{\|Z\|^2}{2}-\frac{\|\sqrt{t}Z+a_i-a_j\|^2}{2t}}\Big)\Big].$$ ]{} For each $i \neq j$, the exponent on the right-hand side above is $$\frac{\|Z\|^2}{2}-\frac{\|\sqrt{t}Z+a_i-a_j\|^2}{2t} = -\frac{\langle Z,a_i-a_j\rangle}{\sqrt{t}} - \frac{\|a_i-a_j\|^2}{2t},$$ which has the ${\mathcal{N}}(-\frac{\|a_i-a_j\|^2}{2t},\frac{\|a_i-a_j\|^2}{t})$ distribution in ${\mathbb{R}}$, so it has the same distribution as $-\frac{\|a_i-a_j\|^2}{2t} + \frac{\|a_i-a_j\|}{\sqrt{t}} Z_1$ where $Z_1 \sim {\mathcal{N}}(0,1)$ is the standard one-dimensional Gaussian. Thus, we can write the upper bound above as $$h(p) - I(X;Y) \le \sum_{i=1}^k p_i \sum_{j \neq i} {\mathbb{E}}\left[ \log \left(1 + b_{ij} e^{c_{ij} Z_1 - \frac{c_{ij}^2}{2}}\right)\right]$$ where $b_{ij} = \frac{p_j}{p_i}$ and $c_{ij} = \frac{\|a_i-a_j\|}{\sqrt{t}}$, and $Z_1 \sim {\mathcal{N}}(0,1)$ in ${\mathbb{R}}$. By Lemma \[Lem:Log2\] below, if $c_{ij} \ge \max\{1,\frac{26}{b_{ij}}\}$, then we have $$\begin{aligned} h(p) - I(X_0;X_t) \le 3 \sum_{i=1}^k p_i \sum_{j \neq i} b_{ij} e^{-0.085c_{ij}^2}.\end{aligned}$$ Note that $b_{ij} = \frac{p_j}{p_i} \le \|p\|_\infty$ and $c_{ij}^2 = \frac{\|a_j-a_i\|^2}{t} \ge \frac{m^2}{t}$, so $$\begin{aligned} h(p) - I(X_0;X_t) &\le 3 \sum_{i=1}^k p_i \sum_{j \neq i} \|p\|_\infty e^{-0.085 \frac{m^2}{t}} \\ &= 3(k-1) \|p\|_\infty e^{-0.085 \frac{m^2}{t}}.\end{aligned}$$ Now, the condition $c_{ij} \ge \max\{1,\frac{26}{b_{ij}}\}$ is equivalent to $$t \le \frac{\|a_i-a_j\|^2}{\max\{1,\frac{26}{b_{ij}}\}^2}.$$ Since $\|a_i-a_j\|^2 \ge m^2$ and $\frac{1}{b_{ij}} = \frac{p_i}{p_j} \le \|p\|_\infty$, the condition above is satisfied when $$t \le \frac{m^2}{\max\{1,26\|p\|_\infty\}^2} = \frac{m^2}{26^2\|p\|_\infty^2}.$$ Thus, we conclude that if $t \le \frac{m^2}{676\|p\|_\infty^2}$, then $$\begin{aligned} h(p) - I(X_0;X_t) \le 3 (k-1) \|p\|_\infty e^{-0.085\frac{m^2}{t}}\end{aligned}$$ as desired. To complete the proof of Theorem \[Thm:GenMixt\], it remains to prove the following estimate. \[Lem:Log2\] Let $b > 0$, $c \ge \max\{1,\frac{26}{b}\}$, and $Z \sim {\mathcal{N}}(0,1)$. Then $${\mathbb{E}}[\log(1 + be^{cZ-\frac{c^2}{2}})\big] \le 3be^{-0.085 c^2}.$$ We use the standard tail bound $\Pr(Z \ge x) \le \frac{1}{\sqrt{2\pi}} \frac{1}{x} e^{-\frac{x^2}{2}}$ for all $x > 0$, which follows from using the inequality $1 \le \frac{z}{x}$ in the integration. In particular, for $x \ge \frac{1}{\sqrt{2\pi}}$ we have $\Pr(Z \ge x) \le e^{-\frac{x^2}{2}}$. Let $0 < \eta < 1$. We split the expectation into three parts: 1. For $Z < (1-\eta)\frac{c}{2}$: We have $cZ-\frac{c^2}{2} < -\eta\frac{c^2}{2}$, so $\log(1 + be^{cZ-\frac{c^2}{2}}) \le \log(1+be^{-\eta\frac{c^2}{2}}) \le be^{-\eta\frac{c^2}{2}}$. The contribution to the expectation from this region is at most $be^{-\eta\frac{c^2}{2}} \Pr(Z < (1-\eta)\frac{c}{2}) \le be^{-\eta\frac{c^2}{2}}$. 2. For $(1-\eta)\frac{c}{2} \le Z < \frac{c}{2}$: We have $cZ-\frac{c^2}{2} < 0$, so $\log(1 + be^{cZ-\frac{c^2}{2}}) \le \log(1+b) \le b$. The contribution to the expectation from this region is at most $b \Pr((1-\eta)\frac{c}{2} \le Z < \frac{c}{2}) \le b \Pr(Z \ge (1-\eta)\frac{c}{2}) \le b e^{-(1-\eta)^2\frac{c^2}{8}}$ where the last inequality holds for $c \ge \frac{2}{(1-\eta)\sqrt{2\pi}}$. 3. For $Z \ge \frac{c}{2}$: We have $cZ-\frac{c^2}{2} \ge 0$, so $\log(1 + be^{cZ-\frac{c^2}{2}}) \le \log((1+b)e^{cZ-\frac{c^2}{2}}) = \log(1+b) + cZ - \frac{c^2}{2} \le b + cZ$. The contribution to the expectation from this region is at most $\int_{\frac{c}{2}}^\infty (b+cz) \frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}} dz = b \Pr(Z \ge \frac{c}{2}) + \frac{c}{\sqrt{2\pi}} e^{-\frac{c^2}{8}} \le (b+c) e^{-\frac{c^2}{8}}$, where the last inequality holds for $c \ge \frac{2}{\sqrt{2\pi}}$. Combining the three parts above, we have that for $c \ge \frac{2}{1-\eta}$, $${\mathbb{E}}[\log(1 + be^{cZ-\frac{c^2}{2}})] \le be^{-\eta\frac{c^2}{2}} + b e^{-(1-\eta)^2\frac{c^2}{8}} + (b+c) e^{-\frac{c^2}{8}}.$$ The leading exponent is $\min\{\eta, \frac{(1-\eta)^2}{4}\} \frac{c^2}{2}$, which is maximized by $\eta^\ast = 3-\sqrt{8} \approx 0.1716$. Set $\eta = \eta^\ast$. Note that for $c \ge \frac{2}{(\frac{1}{4}-\eta^\ast)b}$ we have $\frac{c^2}{2}(\frac{1}{4}-\eta^\ast) \ge \frac{c}{b} \ge \log(1+\frac{c}{b})$, so $(b+c)e^{-\frac{c^2}{8}} \le b e^{-\eta^\ast \frac{c^2}{2}}$. Thus, for $c \ge \max\{\frac{2}{(1-\eta^\ast)\sqrt{2\pi}}, \frac{2}{(\frac{1}{4}-\eta^\ast)b}\}$, we have $${\mathbb{E}}\left[\log\left(1 + be^{cZ-\frac{c^2}{2}}\right)\right] \le 3be^{-\eta^\ast\frac{c^2}{2}}.$$ Since $\frac{\eta^\ast}{2} \approx 0.0858 > 0.085$, $\frac{2}{(1-\eta^\ast)\sqrt{2\pi}} \approx 0.9631 < 1$, and $\frac{2}{\frac{1}{4}-\eta^\ast} \approx 25.5014 < 26$, we can simplify this conclusion by saying that for $c \ge \max\{1,\frac{26}{b}\}$ we have $${\mathbb{E}}\left[\log\left(1 + be^{cZ-\frac{c^2}{2}}\right)\right] \le 3be^{-0.085 c^2},$$ as desired. Proof of Corollary \[Cor:Last\] ------------------------------- From Theorem \[Thm:GenMixt\], we have for small $t$, $$\left|\frac{I(X_0;X_t) - h(p)}{t^\ell} \right| \le 3(k-1) \|p\|_\infty \frac{e^{-0.085 \frac{m^2}{t}}}{t^\ell}.$$ Inductively, this implies all derivatives of $I(X_0;X_t)$ tend to $0$ exponentially fast as $t \to 0$.

--- abstract: 'This paper studies Bloch oscillations of ultracold atoms in an optical lattice, in the presence of atom-atom interactions. A new, interaction-induced Bloch period is identified. Analytical results are corroborated by realistic numerical calculations.' author: - 'Andrey R. Kolovsky' title: New Bloch period for interacting cold atoms in 1D optical lattices --- The response of a quantum system to a static field has been a longstanding problem since the early days of quantum mechanics. A topic of particular interest in this wide field is the dynamics of a quantum particle in a periodic potential induced by a static force (modelling a crystal electron in an electric field). In this system, the effect of the field manifests in a very unintuitive way. Indeed, as already emphasised by Bloch [@Bloc28] and Zener [@Zene34], according to the predictions of wave mechanics, the motion of electrons in a perfect crystal should be oscillatory rather than uniform. This phenomenon, nowadays known as Bloch oscillations (BO), has recently received renewed interest which was stimulated by experiments on cold atoms in optical lattices [@Daha96; @Wilk96; @Raiz97; @Ande98]. This system (which mimics a solid state system – with the electrons and the crystal lattice substituted by the neutral atoms and the optical potential, respectively) offers unique possibilities for the experimental study of BO and of related phenomena. In turn, these fundamentally new experiments have stimulated considerable progress in theory (see review [@PR], and references therein), and it can be safely stated that BO in diluted quasi one-dimensional gases is well understood today. Other directions of research focus on BO in the presence of relaxation processes (spontaneous emission) [@PRA2], BO in 2D optical lattices [@PRL3], and BO in the presence of atom-atom interactions (‘BEC-regime’) [@Berg98; @Choi99; @Chio00; @Mors01]. The present Letter deals with the third problem, which is approached here by an ‘ab initio’ analysis of the dynamics of a system of many atoms. This distinguishes this work from previous studies of BO in the BEC regime [@Berg98; @Choi99; @Chio00], which were based on the a mean field approach using a nonlinear Schrödinger equation. A new effect, so far unaddressed by these earlier studies, is predicted: besides the usual Bloch dynamics, the atomic oscillations may exhibit another fundamental period, entirely defined by the strength of the atom-atom interactions. Let us first recall some results on BO in the single-particle case. Using the tight-binding approximation [@Fuku73], the Hamiltonian of a single atom in an optical lattice has the form $$H = E_0\sum_l |l\rangle\langle l| -\frac{J}{2}\left(\sum_l |l+1\rangle\langle l|+h.c.\right)$$ $$\label{1} +dF\sum_l l |l\rangle\langle l| \;.$$ In Eq. (\[1\]), $|l\rangle$ denotes the $l$th Wannier state $\phi_l(x)$ corresponding to the energy level $E_0$ [@remark1], $J$ is the hopping matrix elements between neighbouring Wannier states, $d$ is the lattice period, and $F$ is the magnitude of the static force. The Hamiltonian (\[1\]) can be easily diagonalised, which yields the spectrum $E_l=E_0+dFl$ (the so-called Wannier-Stark ladder) and the eigenstates (Wannier-Stark states) $$\label{1a} |\psi_l\rangle=\sum_m {\cal J}_{m-l}(J/dF)|m\rangle \;,\quad \langle x|m\rangle=\phi_m(x) \;,$$ (here ${\cal J}_m(z)$ are the Bessel functions). As a direct consequence of the equidistant spectrum, the evolution of an arbitrary initial wave function is periodic in time, with the Bloch period $T_B=2\pi\hbar/dF$. In particular, we shall be interested in the time evolution of the Bloch states $|\psi_\kappa\rangle=\sum_l \exp(id\kappa l)|l\rangle$. Using the explicite expression for the Wannier-Stark states (\[1a\]), it is easy to show that $|\psi_\kappa(t)\rangle=\exp\{-i(J/dF)\sin(d\kappa(t))\} |\psi_{\kappa(t)}\rangle$, where $\kappa(t)=\kappa+Ft/\hbar$ (from now on $E_0=0$ for simplicity). Note that the exponential pre-factor in the last equation contains the same parameter $J/dF$ as the argument of the Bessel function in Eq. (\[1a\]). Depending on the value of this parameter, the regimes of weak ($dF\ll J$) or strong ($dF\gg J$) static fields can be distinguished. In this Letter, we shall restrict ourselves to the strong field case, which, in some sense, is easier to treat than the weak field regime. Indeed, for $J/dF\ll1$, the Wannier-Stark states practically coincide with Wannier states, and $|\psi_\kappa(t)\rangle\approx|\psi_{\kappa(t)}\rangle$. A remark concerning the characteristic values of the parameters is at place here: In the numerical simulations below, we use scaled variables, where $\hbar=1$, $d=2\pi$, and the energy is measured in units of the photon recoil energy. In typical experiments with cold atoms in an optical lattice, the amplitude $v$ of the optical potential equals few recoil energies. Then, for example, for $v$ equal to 10 recoil energies, the value of the dimensionless hopping matrix element is $J=0.0384$. The strength of the static field is restricted from below by the condition $dF> J$, and from above by the condition that Landau-Zener tunnelling events can be neglected. Since the probability of Landau-Zener tunnelling is proportional to $\exp(-\pi\delta^2/8dFJ)$ ($\delta$ is the energy gap separating the lowest Bloch band from the remaining part of the spectrum) [@Zene34; @PR], we have $F<30$ for $v=10$. ![Momentum distribution of the atoms in the optical lattice, for different amplitudes $v$ of the optical potential. (The amplitude $v$ is measured in units of the recoil energy, the momentum $k$ in units of $2\pi\hbar/d$.) The figure illustrates the transition from the SF-phase to the MI-phase as $v$ is varied ($F=0$, $L=N=7$).[]{data-label="fig1"}](fig1.eps){width="8cm"} We proceed with the multi-particle case. A natural extension of the tight-binding model (\[1\]), which accounts for the repulsive interaction of the atoms, is given by the Bose-Hubbard model [@Fish89], $$H=-\frac{J}{2}\left(\sum_{l=1}^L \hat{a}^\dag_{l+1}\hat{a}_l +h.c.\right) +\frac{W}{2}\sum_{l=1}^L \hat{n}_l(\hat{n_l}-1)$$ $$\label{2} +2\pi F\sum_{l=1}^L l\hat{n}_l \;.$$ In Eq. (\[2\]), $\hat{a}_l^\dag$ and $\hat{a}_l$ are the bosonic creation and annihilation operators, $\hat{n}_l= \hat{a}_l^\dag\hat{a}_l$ is the occupation number operator of the $l$th lattice site, and the parameter $W$ is proportional to the integral over the Wannier function raised to the fourth power. Since the Bose-Hubbard Hamiltonian conserves the total number of atoms $N$, the wave function of the system can be represented in the form $|\Psi\rangle=\sum_{\bf n} c_{\bf n}|{\bf n}\rangle$, where the vector ${\bf n}$, consisting of $L$ integer numbers $n_l$ ($\sum_l n_l=N$), labels the $N$-particle bosonic wave function constructed from $N$ Wannier functions. (In what follows, if not stated otherwise, $|\Psi\rangle$ refers to the ground state of the system.) As known, in the thermodynamic limit, and for $F=0$, the system (\[2\]) shows a quantum phase transition from a superfluid (SF) to a Mott insulator (MI) phase as the ratio $J/W$ is varied (see [@Sach01] and references therein). It is interesting to note that an indication of this transition can already be observed in a system of few atoms [@Jaks98]. As an example, Fig. \[fig1\] shows the diagonal elements of the one-particle density matrix, $$\label{4} \rho(k,k')=\langle\Psi|\hat{\Phi}^\dag(k)\hat{\Phi}(k') |\Psi\rangle \;,\quad \hat{\Phi}(k)=\sum_{l=1}^L \hat{a}_l\phi_l(k) \;,$$ for $N=L=7$, $5\le v\le 35$, and $W=0.1\int dk \phi_l^4(k)$ (here, $\phi_l(k)$ are the Wannier states in the momentum representation, and $k=p/(2\pi\hbar/d)$ is the dimensionless momentum). Physically, this quantity corresponds to the momentum distribution $P(k)=\rho(k,k)$ of the atoms, directly measured in the experiment. It is seen in Fig. \[fig1\] that, around $v=15$, there is a qualitative change in the momentum distribution, in close analogy with that observed in the experiment [@Grei02]. It should be noted, however, that this qualitative change of the momentum distribution alone does not yet prove the occurrence of a phase transition. A more reliable indication of a SF-MI phase transition are the fluctuations of the number of atoms in a single well, which drops from $\langle\Delta n^2\rangle\approx0.72$ at $v=5$ to $\langle\Delta n^2\rangle\approx0$ at $v=35$ [@remark3]. ![Bloch oscillations of the atoms, induced by the static force $F=1/2\pi$ ($v=10$). One Bloch period is shown.[]{data-label="fig2"}](fig2.eps){width="8cm"} ![Dephasing of Bloch oscillations due to the atom-atom interaction. The period $T_W=2\pi F/W$ is clearly seen. ($F=1/2\pi$, $v=10$, $W=0.1\int dx \phi_l^4(x)=0.0324$.)[]{data-label="fig3"}](fig3.eps){width="8cm"} Let us now discuss the effect of the static force. Figure \[fig2\] shows the dynamics of the momentum distribution of the atoms (which were initially in SF-phase) in presence of a force $F=1/2\pi$ [@remark2]. This numerical simulation illustrates atomic BO as observed in laboratory experiments [@Daha96; @Mors01]. It is seen that after one Bloch period the initial momentum distribution practically coincides with the final distribution. A small difference, which can be noticed by closer inspection of the figure, is obviously due to the atom-atom interaction [@remark4]. This difference becomes evident once the system evolved over several Bloch periods. In Fig. \[fig3\], the momentum distribution $P(k)$ at integer multiples of the Bloch period is shown. A periodic change of the distribution from SF to MI-like and back is clearly seen. (The use of the term ‘MI-like’ stresses the fact that the variance $\langle\Delta n^2\rangle$ does not change as time evolves.) In addition to Fig. \[fig2\] and Fig. \[fig3\], Fig. \[fig4\] depicts the mean momentum $p(t)$ of the atoms for two different values of the occupation number $\bar{n}=N/L$ (number of atoms per lattice cite) – $\bar{n}=1$ (upper panel) and $\bar{n}=2/7$ (lower panel). As to be expected, the dynamics of the system depends on the value of $\bar{n}$, and for a larger occupation number the deviations of many-particle BO from the non-interacting result $p(t)=NJ\sin(2\pi Ft)$ becomes larger. ![Normalised mean momentum ($p/NJ\rightarrow p$) as a function of time, for two different values of the occupation number $\bar{n}=N/L$: $N=L=7$ (upper panel), and $N=4$, $L=14$ (lower panel). The dashed lines show the analytical result for the envelope function in the thermodynamic limit.[]{data-label="fig4"}](fig4.eps){width="8cm"} Our explanation for the numerical results is the following. It is convenient to treat the atom-atom interaction as a perturbation. Let us denote by $U_0(t)$ the evolution operator of the system for $W=0$, by $U(t)$ the evolution operator for $W\ne0$, and by $U_W(t)$ the evolution operator defined by the decomposition $U(t)=U_W(t)U_0(t)$. Since $U_0(T_B)$ is the identity matrix, one has to find $U_W(T_B)$ to reproduce the result of Fig. \[fig3\]. In the interaction representation, the formal solution for $U_W(T_B)$ has the form $$\label{5} U_W(T_B)=\widehat{\exp}\left(-i\frac{W}{2} \int_0^{T_B} dt U_0^\dag(t) \sum_{l=1}^L \hat{n}_l(\hat{n_l}-1)U_0(t)\right) \;,$$ where the hat over the exponential denotes time ordering. Now we make use of the above strong-field condition $dF>J$. Under this premise the Wannier states are the eigenstates of the atom in the optical lattice \[see Eq. (\[1a\])\]. In the multi-particle case this means that the Fock states $|{\bf n}\rangle$ are the eigenstates of the system (\[2\]) and, thus, that the operator $U_0(t)$ is diagonal in the Fock state basis. Then the integral in Eq. (\[5\]) can be calculated explicitely, which yields $$\label{6} \langle {\bf n}|U_W(T_B)|{\bf n}\rangle= \exp\left(-i\frac{W}{2F}\sum_{l=1}^L\langle {\bf n}| \hat{n}_l(\hat{n_l}-1)|{\bf n}\rangle \right) \;.$$ Finally, by noting that the quantity $\langle {\bf n}|\hat{n}_l(\hat{n_l}-1)|{\bf n}\rangle$ is always an even integer, one comes to the conclusion that, besides the Bloch period, there is additional period, $$\label{7} T_W=2\pi F/W \;,$$ which characterises the dynamics of the system. Further analytical results can be obtained if we approximate the ground state of the system for $F=0$ by the product of $N$ Bloch waves with quasimomentum $\kappa=0$, i.e., $$\label{8} |\Psi\rangle=\frac{1}{\sqrt{N!}}\left( \frac{1}{\sqrt{L}}\sum_{l=1}^L \hat{a}^\dag_l\right)^N |0\ldots0\rangle \;.$$ Indeed, let us consider, for example, the dynamics of the mean momentum. Using the interaction representation (now with respect to the Stark energy term) the mean momentum is given by $$\label{9} p(t)=J\; {\rm Im}\left(\langle \Psi U^\dag_W(t)| \sum_{l=1}^L \hat{a}^\dag_{l+1}\hat{a}_l |U_W(t) \Psi \rangle e^{-i2\pi Ft}\right) \;,$$ where $U_W(t)$ is the continuous-time version ($T_B\rightarrow t$) of the diagonal unitary matrix (\[6\]). Substituting Eq. (\[8\]) and Eq. (\[6\]) into Eq. (\[9\]), we obtain the following exact expression, $$\label{10} \frac{p(t)}{NJ}=\frac{L}{N} {\rm Im}\left( \sum_{n,n'} n {\cal P}(n,n')e^{i(n'-n+1)Wt}e^{-i2\pi Ft}\right) \;,$$ where ${\cal P}(n,n')$ is the joint probability to find $n$ and $n'$ atoms in two neighbouring wells. In the thermodynamic limit $N,L\rightarrow\infty$, $N/L=\bar{n}$, the function ${\cal P}(n,n')$ factorises into a product of the Poisson distributions ${\cal P}(n)=\bar{n}^n\exp(-\bar{n})/n!$, and the double sum in Eq. (\[10\]) converges to the positive periodic function, $f(t)=\exp(-2\bar{n}[1-\cos(Wt)])$, indicated in Fig. \[fig4\] by the dashed line. Good agreement between the envelope of $p(t)$ and the dashed line proves that in the numerical simulation presented above the convergence was indeed achieved. In summary, Bloch oscillations of interacting cold atoms have been studied, both numerically and analytically. We have shown that in the strong field regime atom-atom interactions cause the reversible dephasing of Bloch oscillations. As a result, the momentum distribution of the atoms changes periodically from SF to MI-like distributions, with a period given by Eq. (\[7\]). Using the original (unscaled) parameters, this period reads $T_W=(dF/W)T_B=2\pi\hbar/W$, where $W$ is the strength of the atom-atom interactions. Since the momentum distribution can be measured easily in the laboratory experiment, this effect suggests an alternative method for studying atom-atom interactions by applying a static force to the system. It is worth to stress one more time that the reported result is valid only in the strong field limit, $dF> J$. If this condition is violated, the evolution operator (\[6\]) is no more a diagonal matrix, and the system dynamics get significantly more complicated. An analysis of this latter case will be presented elsewhere. Discussions with A. Buchleitner are gratefully acknowledged. [10]{} F. Bloch, Z. Phys **52**, 555 (1928). C. Zener, Proc. R. Soc. A **145**, 523 (1934). BenDahan [*et al.*]{}, Phys. Rev. Lett. **76**, 4508 (1996). S. R. Wilkinson [*et al.*]{}, Phys. Rev. Lett. **76**, 4512 (1996). M. G. Raizen, C. Salomon, and Qian Niu, Physics Today **50**(8), 30 (1997). B. P. Anderson and M. A. Kasevich, Science **282**, 1686 (1998). M. Glück, A. R. Kolovsky, and H. J. Korsch, Phys. Rep. **366**, 103 (2002). A. R. Kolovsky, A. V. Ponomarev, and H. J. Korsch, Phys. Rev. A. **66**, 053405 (2001). M. Glück, F. Keck, A. R. Kolovsky, and H. J. Korsch, Phys. Rev. Lett. **86**, 3116 (2001); A. R. Kolovsky, and H. J. Korsch, Phys. Rev. A., to appear. K. Berg-Sorensen and K. Molmer, Phys. Rev. A **58**, 1480 (1998). D. I. Choi and Q. Niu, Phys. Rev. Lett. **82**, 2022 (1999). M. L. Chiofalo, M. Polini, and M. P. Tosi, Eur. Phys. J. D **11**, 371 (2000). O. Morsch [*et al.*]{}, Phys. Rev. Lett. **87**, 140402 (2001). H. Fukuyama, R. A. Bari, and H. C. Fogedby, Phys. Rev. B **12**, 5579 (1973). The Wannier states (which should not be confused with Wannier-Stark states) are defined as the Fourier coefficients of the Bloch states over the quasimomentum. For the considered range the $v$, the $l$th Wannier state is essentially a Gaussian centred at the $l$th potential well of the optical lattice. M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Phys. Rev. B **40**, 546 (1989). S. Sachdev, Quantum phase transitions (Cambridge Univ. Press., Cambridge, 2001). D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. **81**, 3108 (1998). M. Greiner [*et al.*]{}, Nature **415**, 39 (2002). A SF-MI transition can also be tracked by calculating the entropy $S=-\sum_{\bf n} |c_{\bf n}|^2 \log |c_{\bf n}|^2$, which changes from $S\sim\log {\cal N}$ (${\cal N}$ is the dimension of the Hilbert space) in the SF-phase, to $S\approx0$ in the MI-phase. In all numerical simulations periodic boundary conditions were used, i.e., the site $l=L+1$ is identified with $l=1$. Note that in the case $F\ne0$ this can be done only after elimination of the static term, what is achieved in the interaction representation. Thus, the dynamics of the system (\[2\]) was actually calculated on the basis of the time-dependent Hamiltonian $\widetilde{H}(t)=-(J/2)\left(\exp(-i2\pi Ft)\sum \hat{a}^\dag_{l+1} \hat{a}_l +h.c.\right)+(W/2)\sum \hat{n}_l(\hat{n_l}-1)$. One gets exact coincidence between the initial and final distributions only for $W=0$. It might also be useful to mention two obvious facts concerning this case. First, for $W=0$ the momentum distribution $P(k)$ calculated for the many atoms system coincides with that obtained within the single-particle approach. Second, within the framework of our present tight-binding model, the function $P(k,t/T_B)$ does not depend on the particular value of the static force $F$.

--- abstract: | In the past years, analyzers have been introduced to detect classes of non-terminating queries for definite logic programs. Although these non-termination analyzers have shown to be rather precise, their applicability on real-life Prolog programs is limited because most Prolog programs use non-logical features. As a first step towards the analysis of Prolog programs, this paper presents a non-termination condition for Logic Programs containing integer arithmetics. The analyzer is based on our non-termination analyzer presented at ICLP 2009. The analysis starts from a class of queries and infers a subclass of non-terminating ones. In a first phase, we ignore the outcome (success or failure) of the arithmetic operations, assuming success of all arithmetic calls. In a second phase, we characterize successful arithmetic calls as a constraint problem, the solution of which determines the non-terminating queries. Keywords: non-termination analysis, numerical computation, constraint-based approach author: - | Dean Voets[^1] $~~~~~$ Danny De Schreye\ Department of Computer Science, K.U.Leuven, Belgium\ Celestijnenlaan 200A, 3001 Heverlee\ {Dean.Voets, Danny.DeSchreye }@cs.kuleuven.be bibliography: - 'prolog.bib' title: 'Non-termination Analysis of Logic Programs with integer arithmetics' --- **Note:** This article has been published in *Theory and Practice of Logic Programming, volume 11, issue 4-5, pages 521-536, 2011*. Introduction ============ The problem of proving termination has been studied extensively in Logic Programming. Since the early works on termination analysis in Logic Programming, see e.g. [@DBLP:journals/jlp/SchreyeD94], there has been a continued interest from the community for the topic. Lots of in-language and transformational tools have been developed, e.g. [@Giesl06aprove1.2] and [@DBLP:journals/corr/abs-0912-4360], and since 2004, there is an annual Termination Competition[^2] to compare the current analyzers on the basis of an extensive database of logic programs. In contrast with termination analysis, the dual problem, to detect non-terminating classes of queries, is a fairly new topic. The development of the first and most well-known non-termination analyzer, $NTI$ [@nti_06], was motivated by difficulties in obtaining precision results for termination analyzers. Since the halting problem is undecidable, one way of demonstrating the precision of a termination analyzer is with a non-termination analyzer. For $NTI$ it was already shown that for many examples one can partition queries in terminating and non-terminating. $NTI$ compares the consecutive calls in the program using binary unfoldings and proves non-termination by comparing the head and body of these binary clauses with a special more general relation. Recently, in joined work with Yi-Dong Shen, we integrated loop checking into termination analysis, yielding a very accurate technique to predict the termination behavior for classes of queries described using modes [@term_prediction]. Classes of queries are represented as *moded queries*. A moded query consists of a query and a label, input or output, for each variable in the query. These moded queries are then evaluated with a *moded SLD-tree* obtained by applying clauses to the partially instantiated query and propagating the labels. To guarantee a finite analysis, this moded SLD-tree is constructed using a complete loop check. After evaluating the moded query, the analysis predicts the termination behavior of the program for the considered queries based on the labels and substitutions in the moded SLD-tree. Motivated by the elegance of this approach and the accuracy of the predictions, our research focused on defining a non-termination condition based on these moded queries. In [@DBLP:conf/iclp/VoetsS09], we introduced a non-termination condition identifying paths in a moded SLD-tree that can be repeated infinitely often. This approach was implemented in a system called $P2P$, which proved more accurate than $NTI$ on the benchmark of the termination competition. An evaluation of the classes of queries not handled by current approaches lead to considerable improvements in our non-termination analysis. These improvements were presented in [@VDS10] and implemented in the analyzer $pTNT$. Both termination and non-termination analyzers have been rather successful in analyzing the termination behavior of definite logic programs, but only a few termination analyzers, e.g. [@DBLP:conf/lpar/SerebrenikS01a], and none of the non-termination analyzers handle non-logical features such as arithmetics or cuts, typically used in practical Prolog programs. In this paper, we introduce a technique for proving non-termination of logic programs containing a subset of the built-in predicates for integer arithmetic, commonly found in Prolog implementations. Given a program, containing integer arithmetics, and a class of queries, described using modes, we infer a subset of these queries for which we prove existential non-termination (i.e. the derivation tree for these queries contains an infinite path). The inference and proof are done in two phases. In the first phase, non-termination of the logic part of the program is proven by assuming that all comparisons between integer expressions succeed. We will show that only a minor adaption of our technique presented in [@DBLP:conf/iclp/VoetsS09] is needed to achieve this. In the second phase, given the moded query, integer arguments are identified and constraints over these arguments are formulated, such that solutions for these constraints correspond to non-terminating queries. The paper is structured as follows. In the next section, we introduce some preliminaries concerning logic programs, integer arithmetics and we present the symbolic derivation trees used to abstract the computation. In Section 3, we introduce our non-termination condition for programs containing integer arithmetics. In Section 4, we describe our prototype analyzer and some results. Finally, we conclude in Section 5. Preliminaries ============= Logic Programming ----------------- We assume the reader is familiar with standard terminology of logic programs, in particular with SLD-resolution as described in [@Lloyd_foundations]. Variables are denoted by strings beginning with a capital letter. Predicates, functions and constant symbols are denoted by strings beginning with a lower case letter. We denote the set of terms constructible from a program $P$ by $Term_P$. Two atoms are called *variants* if they are equal up to variable renaming. An atom $A$ is *more general* than an atom $B$ and $B$ is an *instance* of $A$ if there exists a substitution $\theta$ such that $A\theta = B$. We restrict our attention to definite logic programs. A logic program $P$ is a finite set of clauses of the form $H\leftarrow A_1,..., A_n$, where $H$ and each $A_i$ are atoms. A goal $G_i$ is a headless clause $\leftarrow A_1,..., A_n$. A top goal is also called the query. Without loss of generality, we assume that a query contains only one atom. Let $P$ be a logic program and $G_0$ a goal. $G_0$ is evaluated by building a *generalized SLD-tree* as defined in [@term_prediction], in which each node is represented by $N_i:G_i$ where $N_i$ is the name of the node and $G_i$ is a goal attached to the node. Throughout the paper, we choose to use the best-known *depth-first, left-most* control strategy, as is used in Prolog, to select goals and atoms. So by the *selected atom* in each node $N_i:\leftarrow A_1,..., A_n$, we refer to the left-most atom $A_1$. For any node $N_i:G_i$, we use $A_i^1$ to refer to the selected atom in $G_i$. Let $A_i^1$ and $A_j^1$ be the selected atoms at two nodes $N_i$ and $N_j$, respectively. $A_i^1$ is an *ancestor* of $A_j^1$ if the proof of $A_i^1$ goes through the proof of $A_j^1$. A derivation step is denoted by $N_i:G_i\Longrightarrow_{C} N_{i+1}:G_{i+1}$, meaning that applying a clause $C$ to $G_i$ produces $N_{i+1}:G_{i+1}$. Any path of such derivation steps starting at the root node $N_0:G_0$ is called a *generalized SLD-derivation*. Integer arithmetics ------------------- Prolog implementations contain special purpose predicates for handling integer arithmetics. Examples are $is/2, \geq/2, =:=/2,\ldots$ \[integer\_expressions\] An expression $Expr$ is an *integer expression* if it can be constructed by the following recursive definition. - $Expr = z \in {{\mathbb{Z}}}\mid -Expr \mid Expr+Expr \mid Expr-Expr \mid Expr*Expr$ $\hfill \square$ An atom `"V is Expr"`, with $V$ a free variable and $Expr$ an integer expression, is called an *integer constructor*. An atom $Expr1 \circ Expr2$ is called an *integer condition* if $Expr1$ and $Expr2$ are integer expressions and $\circ \in \lbrace$`>,>=,=<,<,=:=,=/=`$\rbrace$. Moded SLD-trees and loop checking --------------------------------- In [@DBLP:conf/iclp/VoetsS09], classes of queries are represented as *moded queries*. Moded queries are partially instantiated queries, in which variables can be labeled as *input*. Variables labeled input are called *input variables* and represent arbitrary ground terms. To indicate that a variable is labeled as input, the name of the variable is underlined. A query in which no variable is labeled as input is called a *concrete query*. The set of concrete queries represented by a moded query $Q$ is called the *denotation* of $Q$. \[def:denotation\] Let $Q$ be a query and $\lbrace \underline{I_1},\ldots,\underline{I_n} \rbrace$ its set of input variables. The *denotation* of $Q$, $Den(Q)$, is defined as: - $Den(Q) = \left\lbrace Q\lbrace \underline{I_1} \setminus t_1,\ldots,\underline{I_n} \setminus t_n \rbrace \mid t_i \in Term_P, t_i~is~ground, 1\leq i \leq n \right\rbrace $. $\hfill \square$ Note that the denotation of a concrete query is a singleton containing the query itself. Denotations of moded goals and atoms are defined similarly. A moded query $\leftarrow Q$ is evaluated by constructing a *moded SLD-tree*, representing the derivations of the queries in $Den(\leftarrow Q)$. This moded SLD-tree is constructed by applying SLD-resolution to the query and propagating the labels. An input variable $\underline{I}$ can be unified with any term $t \in Term_P$. After unifying $\underline{I}$ and $t$, all variables of $t$ will be considered input as well. \[example:moded\_sld\] Figure \[fig:eq\_plus\_symbolic\] shows the moded SLD-tree of the program $eq\_plus$ for the moded query $\leftarrow eq\_plus(\underline{I},\underline{J},\underline{P})$. This program is non-terminating for any query in $Den(\leftarrow eq\_plus(\underline{I},\underline{I},0))$ and fails for all other queries in $Den(\leftarrow eq\_plus(\underline{I},\underline{J},\underline{P}))$. A query fails if its derivation tree is finite, with no path ending with the empty goal. eq_plus(I,J,P):- eq(I,J), plus(P,I,In), eq_plus(In,J,P). eq(A,A). plus(0,B,B). plus(s(A),B,s(C)):- plus(A,B,C). ![Moded SLD-tree $eq\_plus$[]{data-label="fig:eq_plus_symbolic"}](figs/eq_plus_symbolic.pdf){width="70ex"} Substitutions on input variables express conditions for the clause to be applicable. The edge from node $N_2$ to $N_3$ shows that clause two is applicable if the concrete term denoted by $\underline{P}$ can be unified with $0$. The substitution, $I1 \setminus \underline{I}$, shows that applying this clause unifies $I1$ with the term corresponding to $\underline{I}$. Every derivation in a moded SLD-tree for a query $\leftarrow Q$ corresponds to a concrete derivation for a subclass of $Den(\leftarrow Q)$. The subclass of queries for which a derivation to node $N_i$ is applicable is obtained by applying all substitutions on input variables from $N_0$ to $N_i$. Our condition of [@DBLP:conf/iclp/VoetsS09] proves non-termination for every query for which the derivation to $N_3$ is applicable. The substitutions on input variables in the derivation to $N_3$ are $\underline{J}\setminus \underline{I}$ and $\underline{P} \setminus 0$. Applying these to the query proves non-termination for the queries in $Den(\leftarrow eq\_plus(\underline{I},\underline{I},0))$. $\hfill \square$ As in the example, moded SLD-trees are usually infinite. To obtain a finite analysis, a complete loop check is applied during the construction of the tree. As in our previous works, [@DBLP:conf/iclp/VoetsS09] [@VDS10], we use the complete loop check *LP-check*, [@shen_dynamic_approach]. Without proof, we state that this loop check can also be used for moded SLD-trees and refer to [@shen_dynamic_approach] for more information. In Figure \[fig:eq\_plus\_symbolic\], LP-check cuts clause 4 at node $N_6$ and clause 3 at node $N_7$. $\hfill \square$ Combined with the loop check, a moded SLD-tree can be considered a light-weight alternative to an abstract interpretation for mode analysis. Non-termination analysis for programs with integer arithmetics ============================================================== In this section, we introduce a non-termination condition for programs containing integer arithmetics. To abstract the computations for the considered queries, the moded SLD-tree of [@term_prediction] is used, with some modifications to handle integer constructors and integer conditions. LP-check ensures finiteness of the tree and detects paths that may correspond to infinite loops. For every such path, two analyses are combined to identify classes of non-terminating queries. In the first phase, an adaption of the non-termination condition of [@DBLP:conf/iclp/VoetsS09] detects a class of queries such that each query is non-terminating or fails due to the evaluation of an integer condition such as $>/2$. This class of queries is a moded query with an additional integer label for variables representing unknown integers. In the second phase, the class of queries is restricted to a class of non-terminating queries by formulating additional constraints on the integer variables of the moded query. To prove that this class of non-terminating queries is not empty, these constraints over unknown integers are transformed to constraints over the natural numbers and solved by applying well-known techniques from termination analysis. Then we try to solve these constraints by transforming them to constraints over the natural numbers and applying well-known techniques on them. Moded SLD-tree for programs with integer arithmetics ---------------------------------------------------- The first step of the extension is rather straightforward. The extensions to the moded SLD-tree of [@term_prediction] are limited to the introduction of the label *integer variable* and additional transitions to handle integer constructors and integer conditions. Integer variables are also input variables and will also be represented by underlining the name of the variable. An integer constructor, i.e. $is/2$, is applicable if the first argument is a free variable and the second argument is an integer expression. The application of an integer constructor labels the free variable as an integer variable. An integer condition, e.g. $\geq/2$, is applicable if both arguments are integer expressions. Since integer variables denote unknown integers, integer expressions are allowed to contain integer variables. Applications of integer constructors and integer conditions in the moded SLD-tree are denoted by derivation steps $N_i:G_i\Longrightarrow_{cons} N_{i+1}:G_{i+1}$ and $N_i:G_i\Longrightarrow_{cond} N_{i+1}:G_{i+1}$, respectively. \[example:count\_to\] The following program, $count\_to$, is a faulty implementation of a predicate generating the list starting from 0 up to a given number. The considered class of queries is represented by the moded query $\leftarrow count\_to(\underline{N},L)$ with $\underline{N}$ an integer variable. count_to(N,L):- count(0,N,L). count(N,N,[N]). count(M,N,[M|L]):- M > N, M1 is M+1, count(M1,N,L). In the last clause, the integer condition should be `M < N` instead of `M > N`. Due to this error, the program: - fails for the queries for which $\underline{N}>0$ holds, - succeeds for $\leftarrow count\_to(0,L)$, - loops for the queries for which $\underline{N} < 0$ holds. ![Moded SLD-tree $count\_to$[]{data-label="fig:count_to"}](figs/count_to.pdf){width="60ex"} Figure \[fig:count\_to\] shows the moded SLD-tree for the considered query, constructed using LP-check. LP-check cuts clause 3 at node $N_9$. $\hfill \square$ Note that by ignoring the possible values for the integer variables when constructing the tree, some derivations in it may not be applicable to any considered query. For example the refutations at nodes $N_6$ and $N_{10}$ in the previous example cannot be reached by the considered queries. Adapting the non-termination condition -------------------------------------- In [@DBLP:conf/iclp/VoetsS09], programs are shown to be non-terminating for a moded query, by proving that a path in the moded SLD-tree can be repeated infinitely often. Such a path, from a node $N_b$ to a node $N_e$, is identified based on three properties. The path should be applicable, independent from the concrete terms represented by the input variables. Therefore, the first property states that no substitutions on input variables may occur between $N_b$ and $N_e$. The second property states that the selected atom of $N_b$ – i.e. $A_b^1$ – has to be an ancestor of $A_e^1$. These two properties prove that the sequence of clauses in the path from $N_b$ to $N_e$ is applicable to any goal with a selected atom from $Den(A_b^1)$. Therefore, non-termination is proven by requiring that $Den(A_e^1)$ is a subset of $Den(A_b^1)$. This property can be relaxed by requiring that each atom in $Den(A_e^1)$ is more general than some atom in $Den(A_b^1)$. If this is the case, $A_e^1$ is called *moded more general* than $A_b^1$. For definite logic programs, these three properties imply non-termination. Let $A$ and $B$ be moded atoms. $A$ is *moded more general* than $B$ if - $ \forall I \in Den(A),~ \exists J \in Den(B): I \textit{ is more general than } J$.$\hfill \square$ In Figure \[fig:eq\_plus\_symbolic\], the path from $N_3$ to $N_6$ satisfies these properties. The ancestor relation holds. There are no substitutions on input variables in the path. Finally, the selected atoms are identical and therefore denote the same concrete atoms. $\hfill \square$ The following proposition provides a practical sufficient condition to verify whether the moded more general relation holds. \[prop:mmg\] Let $A$ and $B$ be moded atoms. Let $A_1$ and $B_1$ be renamings of these atoms such that they do not share variables. $A$ is moded more general than $B$ if $A_1$ and $B_1$ are unifiable with most general unifier $\lbrace V_1\setminus t_1,\ldots, V_n \setminus t_n \rbrace$, $t_i \in Term_P$, $1 \leq i, \leq n$, such that for each binding $V_i \setminus t_i$, either: - $V_i \in Var(B_1)$ and $V_i$ is labeled as input, or - $V_i \in Var(A_1)$, $V_i$ is not labeled as input and no variable of $Var(t_i)$ is labeled as input. $\hfill \square$ As stated, we want to prove that every query in the denotation of the considered moded query is either non-terminating or terminates due to the evaluation of an integer condition. To achieve this, we need to guarantee that integer constructors are repeatedly evaluated with a free variable and an integer expression as arguments and that integer conditions are repeatedly evaluated with integer expressions as arguments. Proposition \[prop:mmg\] already implies that the first argument of all integer constructors are free variables in the subsequent iterations of the loop. To prove the repeated behavior on integer constructors and integer expressions stated above, the *integer-similar to* relation is defined. Intuitively, given some loop in the computation, if an atom at the end of the loop is integer-similar to an atom at the start of the loop, then it will provide the required integer expressions to the first atom. First, we introduce positions to identify subterms and a function to obtain a subterm from a given position. \[def:func\_subterm\] Let $L$ be a list of natural numbers, called a *position*, and $A$ a moded atom or term. The function *subterm(L,A)* returns the subterm obtained by: - if $L = [I]$ and $A=f(A_1,\ldots,A_I,A_{I+1},\ldots,A_n)$ then $subterm(L,A) = A_I$ - else if $L=[I|T]$ and $A=f(A_1,\ldots,A_I,A_{I+1},\ldots,A_n)$ then $subterm(L,A) = subterm(T,A_I)$ $\hfill \square$ An atom $A$ is integer-similar to an atom $B$ if it has integer expressions on all positions corresponding to integer expressions in $B$. Let $A$ and $B$ be moded atoms. $A$ is *integer-similar to* $B$ if for every integer expression $t_B$ of $B$, with $subterm(L,B) = t_B$, there exists an integer expression $t_A$ of $A$, with $subterm(L,A) = t_A$. $\hfill \square$ - $count(0,\underline{N},L)$ is integer-similar to $count(\underline{M},\underline{N},L)$ - $count(\underline{M},\underline{N},L)$ is integer-similar to $count(0,\underline{N},L)$ - $count(\underline{M} + 1,\underline{N},L)$ is integer-similar to $count(\underline{M},\underline{N},L)$ - $count(\underline{M},\underline{N},L)$ is not integer-similar to $count(\underline{M}+1,\underline{N},L)$ Note that the last one is a counterexample because $count(\underline{M}+1,\underline{N},L)$ has integer expressions on $[1,1]$ and $[1,2]$, while $count(\underline{M},\underline{N},L)$ does not have any subterms on these positions. $\hfill \square$ \[th:analysis1\] Let $N_b$ and $N_e$ be nodes in a moded SLD-tree for a moded query $Q$. Let $Q'$ be the moded atom obtained by applying to $Q$ all substitutions on input variables from $N_0$ to $N_b$. Every query in $Den(Q')$ is either non-terminating or terminates due to the evaluation of an integer condition if the following properties hold: - $A_b^1$ is an ancestor of $A_e^1$ - no substitutions on input variables occur from $N_b$ to $N_e$ - $A_e^1$ is moded more general than $A_b^1$ - $A_e^1$ is integer-similar to $A_b^1$ $\hfill \square$ \[example:mmg\_adaption\] The path between nodes $N_5$ and $N_9$ in Figure \[fig:count\_to\] satisfies the conditions of Theorem \[th:analysis1\]. There are no substitutions on input variables from $N_0$ to $N_5$ and thus, every query in $Den(\leftarrow count\_to(\underline{N},L))$ is either non-terminating or fails due to the evaluation of an integer condition. Note that although $\leftarrow count\_to(0,L)$ has a succeeding derivation to $N_2$, its derivation to $N_9$ fails due to the integer condition $0 > \underline{N}$. $\hfill \square$ To verify the last property automatically, we strengthen Proposition \[prop:mmg\] to imply both the moded more general relation and the integer-similar to relation. \[prop:mmg\_int\_ins\] Let $A$ and $B$ be moded atoms. Let $A_1$ and $B_1$ be renamings of these atoms such that they do not share variables. $A$ is moded more general than $B$ and $A$ is integer-similar to $B$, if $A_1$ and $B_1$ are unifiable with most general unifier $\lbrace V_1\setminus t_1,\ldots, V_n \setminus t_n \rbrace$, such that for each binding $V_i \setminus t_i$, $1\leq i \leq n$, either: - $V_i \in Var(B_1)$ and $V_i$ is labeled as integer and $t_i$ is an integer expression, or - $V_i \in Var(B_1)$ and $V_i$ is labeled as input but not as integer variable, or - $V_i \in Var(A_1)$, $V_i$ is not labeled as input, no variable of $Var(t_i)$ is labeled as input and $t_i$ does not contain integers. $\hfill \square$ Since the selected atoms of nodes $N_5$ and $N_9$ in Figure \[fig:count\_to\] are variants, Proposition \[prop:mmg\_int\_ins\] holds. $\hfill \square$ Generating the constraints on the integers of the query ------------------------------------------------------- In this subsection, we introduce the constraints on the integer variables of the moded query, identifying values for which all integer conditions in the considered derivations succeed. These constraints consist of reachability constraints, identifying queries for which the derivation up till the last node is applicable, and an implication proving that the integer conditions will also succeed in the following iterations. \[example:count\_to\_int\_cons\] As a first example, we introduce the constraints for the path between $N_5$ and $N_9$ in the moded SLD-tree of $count\_to$ in Figure \[fig:count\_to\]. For this path, Theorem \[th:analysis1\] holds and thus every query denoted by $\leftarrow count\_to(\underline{N},L)$ is either non-terminating or terminates due to an integer condition. To restrict the class of considered queries to those for which the derivation to $N_9$ is applicable, all integer conditions in the derivation are expressed in terms of the integers of the query, yielding $0 > \underline{N}$ and $0 + 1 > \underline{N}$. For this program and considered class of queries, the condition $0 > \underline{N}$ implies that the derivation is applicable until node $N_9$. The following implication states that if the condition of node $N_7$ holds for any two values $M$ and $N$, then it also holds for the values of the next iteration. $$\forall M,N \in {{\mathbb{Z}}}: M > N \Longrightarrow M+1>N$$ This implication is correct and thus proves non-termination for the considered queries if the precondition holds in the first iteration. This is the case for all queries in $Den(\leftarrow count\_to(\underline{N},L))$ with $0 > \underline{N}$ since the value corresponding to $M$ in the first iteration is $0$ and the value corresponding to $N$ is $\underline{N}$. This proves non-termination of all considered queries for which $0 > \underline{N}$. $\hfill \square$ In the following example, applicability of the derivation does not imply non-termination. To detect a class of non-terminating queries, a domain constraint is added to the pre- and postcondition of the implication. \[example:constants\_nt\_cond\] constants(I,J):- I =:= 2, In is J*2, Jn is I-J, constants(In,Jn). The clause in *constants* is applicable to any goal with $constants(2,\underline{J})$ as selected atom, with $\underline{J}$ an integer variable. Since the first argument in the next iteration is the value corresponding to $\underline{J}*2$, only goals with the selected atom $constants(2,1)$ are non-terminating for this program. Since applicability of the derivation does not imply non-termination, a similar implication as in the previous example is false, $\forall I,J \in {{\mathbb{Z}}}: I=2 \Longrightarrow J*2 = 2$. To overcome this, a constraint is added to the pre- and post-condition of this implication, restricting the considered values of $\underline{J}$ to an unknown set of integers, called its *domain*. $$\exists Dom_j \subset {{\mathbb{Z}}}, \forall I,J \in {{\mathbb{Z}}}: I=2, J \in Dom_j \Longrightarrow J*2 = 2, I-J \in Dom_j$$ The resulting implication is true for $Dom_j = \lbrace 1 \rbrace$. By requiring that the considered moded query satisfies both the reachability constraint and the additional constraint in the pre-condition, the non-terminating query $\leftarrow constants(2,1)$ is obtained. $\hfill \square$ All information needed to construct these constraints can be obtained from the moded SLD-tree. Let $C$ be an integer condition or expression and $N_i$ and $N_j$ two nodes in a moded SLD-tree $D$. Let $Cons$ be the set of all integer constructors occurring as selected atom in a node $N_p~(i \leq p \leq j)$ in $D$. The function *$apply\_cons(C,N_i,N_j)$* returns the integer condition or expression obtained by exhaustively applying $\underline{I}\setminus Expr$ to $C$, for any $\underline{I} ~is~ Expr \in Cons$. $\hfill \square$ The constraints guaranteeing a derivation to $N_j$ to be applicable, can be obtained using $apply\_cons(Cond,N_0,N_i)$ for any integer condition $Cond$ in a node $N_i$ in the considered derivation. For a path from $N_b$ to $N_e$, the precondition of the implication is obtained using $apply\_cons(Cond,N_b,N_i)$, for each condition $Cond$ in a node $N_i$ between nodes $N_b$ to $N_e$ and universally quantifying the integer variables of $N_b$. \[example:apply\_cons\] The derivation to $N_9$ in Figure \[fig:count\_to\], contains integer conditions in nodes $N_3$ and $N_7$. These are expressed on the integer variable of the query, $\underline{N}$, using $apply\_cons$. - $apply\_cons(0>\underline{N},N_0,N_3) = 0 > \underline{N}$ - $apply\_cons(\underline{M1}>\underline{N},N_0,N_7) = 0 + 1 > \underline{N}$ To obtain the precondition of the implication, the integer condition in $N_7$ is expressed in terms of the integer variables of $N_5$. - $apply\_cons(\underline{M1}>\underline{N},N_5,N_7) = \underline{M1} > \underline{N}$ Universally quantifying these variables yields the precondition. $\hfill \square$ To obtain the consequence of the implication for a path from $N_b$ to $N_e$, one first replaces the integer variables of $N_b$ in the precondition by the corresponding integer variables of $N_e$. Then, $apply\_cons$ is used to express the consequence in terms of the values in the previous iteration. Let $LHS$ be the precondition of an implication, consisting of integer conditions and constraints of the form $I \in Dom_I$. Let $N_i$ and $N_j$ be two nodes in a moded SLD-derivation such that all integer variables in $LHS$ are in $A_i^1$ and let $\underline{I_1},\ldots,\underline{I_n}$ be all integer variables of $A_i^1$. If there exist subterms of $A_j^1$, $t_1,\ldots,t_n$, such that $\forall L: subterm(L,A_i^1)=\underline{I_p} \Longrightarrow subterm(L,A_j^1)=t_p, 1 \leq p \leq n$, then *$replace(LHS,N_i,N_j)$* is obtained by applying $\lbrace \underline{I_1} \setminus t_1, \ldots, \underline{I_n} \setminus t_n\rbrace$ to all constraints in $LHS$. $\hfill \square$ In Example \[example:apply\_cons\], we generated the precondition of the implication, $\underline{M1} > \underline{N}$. To obtain the consequence, $replace(\underline{M1} > \underline{N},N_5,N_9)$ is applied, yielding $\underline{M2} > \underline{N}$. Then, the integer variable of $N_9$, $\underline{M_2}$, is expressed in terms of the integer variables of $N_5$ using $apply\_cons(\underline{M2} > \underline{N},N_5,N_9)=\underline{M_1}+1 > \underline{N}$. Adding the domains to the pre- and postcondition yields the desired implication: $\exists Dom_N, Dom_{M1} \subset {{\mathbb{Z}}}, \forall N,M1 \in {{\mathbb{Z}}}: M1 > N,~N \in Dom_N,~M1 \in Dom_{M1} \Longrightarrow$\ $~~~~~~~~~M1+1 > N,~N \in Dom_N,~M1+1 \in Dom_M$ $\hfill \square$ Adding these constraints to the class of queries detected by Theorem \[th:analysis1\], yields a class of non-terminating queries. Proving that the constraints on integers are solvable ----------------------------------------------------- The previous subsection introduced constraints, implying that all integer conditions in a considered derivation succeed. In this subsection, we introduce a technique to check if these constraints have solutions, using a constraint-based approach. Symbolic coefficients represent values for the integers in the query and domains in the implication, for which the considered path is a loop. After these coefficients are introduced, the implication is transformed into a set of equivalent implications over natural numbers. These implications can then be solved automatically in the constraint-based approach, based on Proposition 3 of [@DBLP:journals/corr/abs-0912-4360]. \[prop:rem\_imp\] Let $prem$ be a polynomial over $n$ variables and $conc$ a polynomial over 1 variable, both with natural coefficients, where $conc$ is not a constant. Moreover, let $p_1,\ldots,p_{n+1},q_1,\ldots,q_{n+1}$ be arbitrary polynomials with integer coefficients[^3] over the variables $\overline{X}$. If $$\forall \overline{X} \in {{\mathbb{N}}}: conc(p_{n+1})-conc(q_{n+1})-prem(p_1,\ldots,p_n)+ prem(q_1,\ldots,q_n) \geq 0$$ is valid, then $\forall \overline{X} \in {{\mathbb{N}}}: p_1 \geq q_1, \ldots,p_n\geq q_n \Longrightarrow p_{n+1}\geq q_{n+1}$ is also valid. $\hfill \square$ ### Introducing the symbolic coefficients. To represent half-open domains in the implication by symbolic coefficients, the domains are described by two symbolic coefficients, one upper or lower limit and one for the direction. Constraints of the form $Exp \in Dom_I$ in the implication, are replaced by constraints of the form $d_I * Exp \geq d_I* c_I$ with $d_I$ either $1$ or $-1$, describing the domain $\lbrace c_I, c_I-1, \ldots\rbrace$ for $d_I=-1$ and $\lbrace c_I, c_I+1, \ldots\rbrace$ for $d=1$. The values to be inferred for the integers of the query should satisfy the precondition of the implication. Off course, the symbolic coefficients $c_I$ should also be consistent with the values of the integers in the query. In Example \[example:count\_to\_int\_cons\], we introduced constraints on the integer variable $\underline{N}$, $0 > \underline{N}$ and $0 + 1 > \underline{N}$, proving non-termination for queries in $Den(\leftarrow count\_to(\underline{N},L))$. By convention, we denote the symbolic coefficients as constants. For the integer variable $\underline{N}$, we introduce the symbolic coefficient $n$. The implication introduced in Example \[example:count\_to\_int\_cons\], for the path from $N_5$ to $N_9$ in Figure \[fig:count\_to\], does not contain constraints on the domains. When adding these constraints to the pre- and postcondition, we obtain the following implication. - $\forall M,N \in {{\mathbb{Z}}}: ~M > N, ~N \in Dom_N, ~M \in Dom_M \Longrightarrow $\ $~~~~~~~~~~~M+1>N, ~N \in Dom_N, ~M+1 \in Dom_M$ Representing these domains by symbolic coefficients yields the following implication. - $\forall M,N \in {{\mathbb{Z}}}: ~M > N, ~d_N * N \geq d_N * c_N, ~d_M * M \geq d_M * c_M \Longrightarrow $\ $~~~~~~~~~~~M+1>N, ~d_N * N \geq d_N * c_N, ~d_M * (M+1) \geq d_M * c_M$ To guarantee that the precondition succeeds for the considered derivation, $c_M$ and $c_N$ are required to be the values for $\underline{M}$ and $\underline{N}$ in node $N_5$. Combining these constraints implies non-termination for the query $\leftarrow count\_to(n,L)$, for which the following constraints are satisfied with some unknown integers $c_N,c_M,d_N$ and $d_M$. - $0>n,~0+1>n$ to guarantee applicability of the derivation - $c_N = n, ~c_M = 0+1$ to guarantee that the precondition holds - $d_N = 1 \lor d_N = -1, ~d_M = 1 \lor d_M = -1$, - $\forall M,N \in {{\mathbb{Z}}}: M > N, d_N * N \geq d_N * c_N, d_M * M \geq d_M * c_M \Longrightarrow $\ $~~~~~~~~~~~M+1>N, d_N * N \geq d_N * c_N, d_M * (M+1) \geq d_M * c_M$ to prove that the condition succeeds infinitely often. Due to the implication, $d_M$ has to be $1$. $d_N$ can be either $1$ or $-1$. $\hfill \square$ To be able to infer singleton domains, we allow the constant describing the direction of the interval to be $0$. If in such a constant $d_I$ is zero, the constraints on the domain are satisfied trivially because they simplify to $0 \geq 0$. To guarantee that the domain is indeed a singleton when $d_I$ is inferred to be zero, a constraint of the form $(1-d_I^2)Exp=(1-d_I^2)*c_I$ is added to the postcondition for every constraint $d_I * I \geq d_I * c_I$. This constraint is trivially satisfied for half-open domains and proves that $\lbrace c_I \rbrace$ is the domain in the case that $d_I = 0$. In Example \[example:constants\_nt\_cond\], we introduced constraints on the integer variables $\underline{I}$ and $\underline{J}$, proving non-termination for queries in $Den(\leftarrow constants(\underline{I},\underline{J}))$. Introducing symbolic coefficient $i$ and $j$ for the integers of the query and for the domains of $\underline{I}$ and $\underline{J}$, yields the following constraints. 1. $i = 2$ to guarantee applicability of the derivation 2. $c_I = i, ~c_J = j$ to guarantee that the precondition holds 3. $d_I \leq 1, ~d_I \geq -1, ~d_J \leq 1, ~d_J \geq -1$, 4. $\forall I,J \in {{\mathbb{Z}}}: I=2, ~d_I * I \geq d_I * c_I, ~d_J * J \geq d_J * c_J \Longrightarrow $\ $~~~~~J*2=2, ~d_I * (J*2) \geq d_I * c_I, (1-d_I^2)*(J*2) = (1-d_I^2)*c_I, $\ $~~~~~d_J * (I-J) \geq d_J * c_J, (1-d_J^2)*(I-J) = (1-d_J^2)*c_J$ The implication in $(4)$ can only be satisfied with $d_J$ equal to zero. $\hfill \square$ ### To implications over the natural numbers The symbolic coefficients to be inferred which represent the domains, allow to transform the implication over ${{\mathbb{Z}}}$ to an equivalent implication over ${{\mathbb{N}}}$. - for $d_I = 1$, any integer in $\lbrace c_I,~c_I+1,~\ldots\rbrace$ that satisfies the precondition is in $\lbrace c_I+d_I*N \mid N \in {{\mathbb{N}}}\rbrace$ - for $d_I = -1$, any integer in $\lbrace c_I,~c_I-1,~\ldots\rbrace$ that satisfies the precondition is in $\lbrace c_I+d_I*N \mid N \in {{\mathbb{N}}}\rbrace$ - for $d_I = 0$, any integer in $\lbrace c_I \rbrace$ that satisfies the precondition is in $\lbrace c_I+d_I*N \mid N \in {{\mathbb{N}}}\rbrace$ Therefore, we obtain an equivalent implication over the natural numbers by replacing each integer $I$ by its corresponding expression $c_I+d_I*N$ and replacing the universal quantifier over $I$ by a quantifier over $N$. ### Automation by a translation to diophantine constraints To solve the resulting constraints, we use the approach of [@DBLP:journals/corr/abs-0912-4360]. Constraints of the form $A =:= B$ in the implication, are replaced by the conjunction $A\geq B,~B\geq A$. Constraints of the form $A =/= B$, yield two disjunctive cases. One obtained by replacing the $=/=$ in the pre- and postcondition by $>$ and one obtained by replacing it by $<$. The other conditions – i.e. $>,<$ and $\leq$ – are transformed into $\geq$-constraints in the obvious way. Implications with only one consequence are obtained by creating one implication for each consequence, with the pre-condition of the original implication. The resulting implications allow to apply Proposition \[prop:rem\_imp\]. These inequalities of the form, $p\geq0$, are then transformed into a set of *diophantine constraints*, i.e. constraints without universally quantified variables, by requiring that all coefficients of $p$ are non-negative. As proposed in [@DBLP:journals/corr/abs-0912-4360], the resulting diophantine constraints are then transformed into a SAT-problem. The constraints are then proven to have solutions by a SAT solver by inferring one possible solution. Evaluation ========== We have implemented our analysis and integrated it within our existing non-termination analyzer $pTNT$. The analyzer can be downloaded from\ http://www.cs.kuleuven.be/\~dean/iclp2011.html. We tested our analysis on a benchmark of 16 programs similar to those in the paper. These programs are also available online. To solve the resulting SAT-Problem, MiniSat [@ES03] is used. $ $ linear-class, 3 bits linear-class, 4 bits max2-class, 3 bits max2-class, 4 bits ----------- ---------------------- ---------------------- -------------------- -------------------- count\_to $+$ $+$ $+$ $+$ constants $+$ $+$ $+$ $OS$ int1 $+$ $+$ $+$ $+$ int2 $+$ $+$ $+$ $+$ int3 $+$ $+$ $+$ $OS$ int4 $+$ $+$ $+$ $OS$ int5 $+$ $+$ $+$ $OS$ int6 $+$ $+$ $+$ $OS$ int7 $+$ $+$ $OS$ $OS$ int8 $+$ $+$ $OS$ $OS$ int9 $-$ $+$ $OS$ $OS$ int10 $-$ $-$ $+$ $OS$ int11 $-$ $+$ $-$ $OS$ int12 $-$ $+$ $-$ $OS$ int13 $+$ $+$ $+$ $+$ int14 $+$ $+$ $+$ $OS$ : An overview of the experiments[]{data-label="Table:evaluation"} We experimented with different bit-sizes in the translation to SAT and different classes of functions for the $prem$ functions in Proposition \[prop:rem\_imp\]. As $conc$ functions, the identity function was used. Table \[Table:evaluation\] shows the results for the considered settings, $+$ denotes that non-termination is proven successfully, $-$ denotes that non-termination could not be proven and $OS$ denotes that the computation went out of stack. The considered settings are 3 and 4 as bit-sizes and $linear$ and $max2$ as forms for the symbolic $prem$-functions. The $linear$ class is a weighted sum of each argument. The $max2$ class contains a weighted term for each multiplication of two arguments. The analysis time is between $1$ and $20$ seconds for all programs and settings. Table \[Table:evaluation\] shows non-termination can be proven for any program of the benchmark when choosing the right combination of parameters, but no setting succeeds in proving non-termination for all programs. Programs $int9$ and $int12$ require a constant that cannot be represented with bit-size 3. Linear prem-functions cannot prove non-termination for $int10$. However, the setting with 4 as a bit-size and $max2$ as class of $prem$-function usually fails, because these settings cause an exponential increase in memory use during the translation to SAT. Conclusion ========== In this paper we introduced a technique to detect classes of non-terminating queries for logic programs with integer arithmetic. The analysis starts with a given program and class queries, specified using modes, and detects subclasses of non-terminating queries. First, the derivations for the given class of queries are abstracted by building a moded SLD-tree [@term_prediction] with additional transitions to handle integer arithmetic. Then, this moded SLD-tree is used to detect subclasses of non-terminating queries in two phases. In the first phase, we ignore the conditions over integers, e.g. $>/2$, and detect paths in the moded SLD-tree that correspond to infinite derivations if all conditions on integers in those derivations succeed. For every such path, the corresponding subclass of queries is generated. In the second phase, the obtained classes of queries are restricted to classes of non-terminating queries, by formulating constraints implying that all conditions on integers will succeed. These constraints are then solved by transforming them into a SAT problem. We implemented this approach in our non-termination analyzer $pTNT$ and evaluated it on small benchmark of non-terminating Prolog programs with integer arithmetic. The evaluation shows that the proposed technique is rather powerful, but also that the parameters in the transformation to SAT must be chosen carefully to avoid excessive memory use. For future work, we plan to improve the efficiency by using SMT solvers. #### Acknowledgment We thank the referees for their useful and constructive comments. [^1]: Supported by the Fund for Scientific Research - FWO-project G0561-08 [^2]: Results are available at http://termcomp.uibk.ac.at/ [^3]: Proposition 3 in [@DBLP:journals/corr/abs-0912-4360] uses natural coefficients, but the proposition also holds for polynomials with integer coefficients.

--- abstract: 'We report on results of [*BeppoSAX*]{} Target Of Opportunity (TOO) observations of the source MXB 1730-335, also called the Rapid Burster (RB), made during its outburst of February–March 1998. We monitored the evolution of the spectral properties of the RB from the outburst decay to quiescence. During the first TOO, the X–ray light curve of the RB showed many Type II bursts and its broadband (1-100 keV) spectrum was acceptably fit with a two blackbody plus power law model. Moreover, to our knowledge, this is the first time that this source is detected beyond 30 keV.' author: - 'F. Frontera$^{1,2}$, N. Masetti$^1$, M. Orlandini$^1$, L. Amati$^1$, E. Palazzi$^1$, D. Dal Fiume$^1$, S. Del Sordo$^3$, G. Cusumano$^3$, A.N. Parmar$^4$, G. Pareschi$^5$, I. Lapidus$^6$ and L. Stella$^7$' title: | Discovery of hard X-ray emission from\ Type II bursts of the Rapid Burster --- Observations ============ Four Target Of Opportunity (TOO) observations were performed with [*BeppoSAX*]{} (Boella et al. 1997a) on the Rapid Burster (=MXB 1730–335; hereafter RB) during the activity state which started on January 28, 1998 (Fox et al. 1998). These TOOs spanned over one month (from February 18 to March 18) and caught the object in four different snapshots, from the post–maximum decay to the quiescent state. Figure 1, left panel, shows the ASM light curve of the [*Rossi-XTE*]{} satellite with superimposed the times of the four [*BeppoSAX*]{} observations. Here we report on RB data from three of the four instruments mounted on [*BeppoSAX*]{}: LECS (0.1-10 keV; Parmar et al. 1997), MECS (1.5-10 keV; Boella et al. 1997b, and PDS (15-300 keV; Frontera et al. 1997). For the PDS the default rocking collimator law was modified by offsetting the RB by 40$^{'}$ from the center of the field of view in order to reduce as much as possible the contamination from a nearby variable X–ray source, GX 354-0 (=4U 1728-34), located at about 30$^{'}$ from the RB. Unfortunately, due to failure of the rocking law setup program, the collimator did not move as requested during TOO2 and TOO3; so, we have only LECS and MECS data for these two observations. In this paper we report on preliminary results of these observations. Definitive results along with their implications will be the subject of another paper (Masetti et al. 2000). In the following, for the luminosity estimates we will assume that the RB lies at a distance $d$ = 8 kpc (Ortolani et al. 1996). Spectral analysis and temporal evolution of the RB ================================================== During TOO1, the RB was in a strong state of bursting activity. The 2-10 keV light curve obtained with MECS (see Fig. 1, right panel, part [*a*]{}) showed 113 Type II X–ray bursts during 9457 seconds of good observational data. Evidence of Type II bursts was also observed in the 0.1-2 keV data obtained with LECS. We divided the MECS TOO1 data into two subsets: persistent emission (PE; below 5 counts s$^{-1}$) and bursting emission (BE; above 5 counts s$^{-1}$). The MECS PE and BE spectra could be well fit with a photoelectrically absorbed two-component blackbody (2BB); these BB components may originate from the neutron star (NS) surface, a boundary layer between the NS and the inner edge of the accretion disk, or the inner region of the disk itself. The same model was used for the RB by Guerriero et al. (1998) who found values consistent with ours. In Table 1 we report the best-fit parameters along with their 90% confidence errors. The temperature values of the two BB components were slightly higher during the PE than during the BE, while their luminosities were much higher (by a factor 20 for the cooler BB and 60 for the hotter BB) during the BE than during the PE. We also remark that during the BE the hotter BB component was brighter (by more than a factor 3) than the cooler BB, while during the PE they had similar luminosities. This implies that the BE influences more the higher temperature component than the other one. If the BE is due to spasmodic accretion onto the compact object, the higher temperature component should be the one coming from the NS surface. The source was also visible in the hard X–ray (15-100 keV) energy range. However the statistics of the PDS light curve was much lower and did not allow distinguishing the Type II bursts. In order to construct the BE spectrum we used the time intervals in which the bursts were observed with MECS. Also, we could not derive the correct 15-100 keV flux and spectrum of both BE and PE given the residual source contamination by GX 354-0. Thus, in order to overcome this problem, we used as background level for the 1-100 keV BE spectrum the total count rate level measured during the PE time intervals. The combined LECS+MECS+PDS PE-subtracted bursting spectrum, shown in Fig. 2, was no longer fit with a 2BB model. By adding a power law component we obtained an acceptable fit (see Table 1). The further addition of a Fe K emission line at 6.5 keV slightly improved the fit, with parameter values found for this line in general agreement with the findings by Stella et al. (1988) for Type II bursts. During TOO2 the RB drastically reduced its bursting activity, and the bursts were concentrated at the beginning of this TOO (Fig. 1, right panel, part [*b*]{}). Also, the emission intensity level decreased. The best–fit model was a photoelectrically absorbed 2BB model (Table 1). No evidence of a Fe emission line was present. During TOO3 the object further reduced its bursting activity, and no Type II bursts were seen throughout the observation. The best–fit model spectrum was still an absorbed 2BB (Table 1). As in the case of TOO2, no iron emission line at 6.5 keV was found. The RB was instead no longer visible in MECS/LECS images during TOO4. Stray light from GX354-0 prevented us to get a deep observation of the source. The 3$\sigma$ upper limit to the RB X–ray emission was 1.5$\times$10$^{-12}$ erg cm$^{-2}$ s$^{-1}$ in the 2-10 keV energy band. Boella G., Butler R.C., Perola G.C. et al., 1997a, A&AS 122, 299\ Boella G., Chiappetti L., Conti G. et al., 1997b, A&AS 122, 327\ Fox D., Guerriero R., Lewin W.H.G., 1998, ATEL n. 9\ Frontera F. Costa E., Dal Fiume D. et al., 1997, A&AS 122, 357\ Guerriero R., Fox D., Kommers J. et al., 1999, MNRAS 307, 179\ Masetti N., Frontera F., et al., 2000, A&A, submitted\ Ortolani S., Bica E., Barbuy B., 1996, A&A 306, 134\ Parmar A., Martin D.D.E., Bavdaz M. et al., 1997, A&AS 122, 309\ Stella L., Haberl F., Lewin W.H.G. et al., 1988, ApJ 324, 379

--- abstract: | A closed form expression for multiplicity-free quantum 6-j symbols (MFS) was proposed in [@MFS] for symmetric representations of $U_q(sl_N)$, which are the simplest class of multiplicity-free representations. In this paper we rewrite this expression in terms of q-hypergeometric series ${}_4\Phi_3$. We claim that it is possible to express any MFS through the 6-j symbol for $U_q(sl_2)$ with a certain factor. It gives us a universal tool for the extension of various properties of the quantum 6-j symbols for $U_q(sl_2)$ to the MFS. We demonstrate this idea by deriving the asymptotics of the MFS in terms of associated tetrahedron for classical algebra $U(sl_N)$. Next we study MFS symmetries using known hypergeometric identities such as argument permutations and Sears’ transformation. We describe symmetry groups of MFS. As a result we get new symmetries, which are a generalization of the tetrahedral symmetries and the Regge symmetries for $N=2$. author: - '[**Victor Alekseev$^{a,b,c}$[^1], Andrey Morozov$^{a,b,c}$[^2], Alexey Sleptsov$^{a,b,c}$[^3]**]{}' bibliography: - 'bib.bib' date: title: | [**Multiplicity-free $U_q(sl_N)$ 6-j symbols:\ relations, asymptotics, symmetries**]{} --- ITEP-TH-25/19\ IITP-TH-17/19\ MIPT-TH-15/19 $^a$\ $^b$\ $^c$\ Introduction ============ Racah-Wigner coefficients or 6-j symbols play an important role in mathematics and theoretical physics, because they appear in many different problems. From mathematical point of view they describe the associativity data, which are still unknown for $U_q(sl_N)$. The main difficulty is in the appearance of the so-called multiplicities, which happens when the algebra rank $N$ is greater than 2. However, even for multiplicity-free representations analytical formulas for 6j-symbols are known only for a small class of representations, namely, for symmetric representations. In theoretical physics the algebra $U_q(sl_N)$ is very important especially in quantum physics. Here is an incomplete list of topics, in which 6-j symbols of quantum Lie algebra $U_q(sl_N)$ or its classical version $U(sl_N)$, appear:\ $\bullet$ quantum mechanics [@LL] and quantum computing [@qcomp],\ $\bullet$ quantum $\mathcal{R}$-matrices and integrable systems [@Rint],\ $\bullet$ WZW conformal field theory and 3d Chern-Simons theory [@WZW1; @WZW2],\ $\bullet$ lattice gauge theory [@lattice],\ $\bullet$ 3-d quantum gravity [@qgrav],\ $\bullet$ quantum $sl_N$ invariants of knots [@RT],\ $\bullet$ Turaev-Viro invariants of 3-manifolds and topological field theory [@TV1; @TV2],\ $\bullet$ Drinfeld associator and Kontsevich integral [@DA; @KI],\ $\bullet$ orthogonal polynomials [@ortpol; @ortpol2; @ortpol3]. One can see that 6-j symbols are widely used in both classical and modern works. Note that in many situations, e.g. in the quantum gravity or in statistical models, one considers partition functions, which contain a sum over all possible 6-j symbols of the given gauge group. In such problems it would be very useful to use symmetries between different 6-j symbols in order to reduce the sum and simplify the computation. Quantum 6-j symbols have a lot of symmetries, most of them are still unknown. Nowadays we have different situations for $U_q(sl_2)$ and more general $U_q(sl_N)$ 6-j symbols. All symmetries of $U_q(sl_2)$ 6-j symbols are well known and well studied, many interesting and surprising results are obtained, see e.g. [@Roberts; @Boalch; @brehamet2015regge; @sleptsov_new_sym]. In the present paper we are interested in the so-called *linear* symmetries. *Non-linear* symmetries (e.g. the pentagon relation), that are more complicated, are out of the scope of this paper. Linear symmetries of $U_q(sl_2)$ Racah coefficients include Regge symmetries, the tetrahedral symmetries and transformation $q \leftrightarrow q^{-1}$ [@klimyk]. [*Known*]{} symmetries of $U_q(sl_N)$ include complex conjugation, a $q\leftrightarrow {q^{-1}}$ and the tetrahedral symmetries [@WZW2]. Some symmetries may be obtained with the help of the eigenvalue hypothesis [@NewSymsFromEvHyp; @Mironov:2016; @cabling; @Dhara:2017ukv; @Alekseev:2019] including some generalization for Regge symmetries. It says that the Racah matrices are uniquely defined by the eigenvalues of the $\hat{\mathcal{R}}$-matrices. All studied examples says that it is true and this hypothesis becomes a useful tool to derive symmetries. Moreover, there is an exact expression for the Racah matrices through the $\hat{\mathcal{R}}$-matrix eigenvalues for the matrices of the size up to $5\times 5$ [@Ev_Hyp] and $6\times 6$ [@Universality]. The 6-j symbols calculation is a big problem for $U_q(sl_N)$ representations. There are few calculation methods and each of them is extremely tedious. Unlike the $U_q(sl_2)$ case, where the answer is known in a closed form for each representation [@KR], the analytical expression for arbitrary representations is still unknown. However, for the special case of symmetric and conjugated symmetric $U_q(sl_N)$ representations, the analytical expression was proposed recently [@MFS; @Mironov:2014]. The result gives us plenty of new questions. In particular, which properties of the expression are special for $U_q(sl_2)$ and which can be generalized to the more complex cases. For instance, in this context it was found [@racah_pol] that 6-j symbols for symmetric representations of $U_q(sl_N)$ can be expressed in terms of orthogonal q-Racah polynomials as well as their counterpart for $U_q(sl_2)$. Also note that 6j-symbols of $U_q(sl_N)$ for non-symmetric representations were studied in [@Morozov:2019haw; @Morozov:2019jqp; @Morozov:2019kgx]. In this paper we study the analytical expression from [@MFS] in order to find new symmetries. In section \[S2\] we start by introducing Racah coefficients and 6-j symbols for $U_q(sl_N)$. In this paper we consider 6-j symbols that have only symmetric and conjugate to symmetric representations. All these 6-j symbols may be transformed via tetrahedral symmetries into either type I and type II [@WZW2]. For type I the only conjugate to symmetric representation is the second one, for type II – the third one. Each type can be considered as a natural generalization of $U_q(sl_2)$ 6-j symbols because each tensor product decomposition for this case has no multiplicities and can be enumerated by an integer number rather than a whole Young diagram. We consider the expression for both types as an analytic function and study its special properties to obtain new symmetries. In section \[S3\] we simplify the expression. Firstly, we prove that the expression may be reduced and the series became much more similar to $U_q(sl_2)$ series. This was done for both types independently and as it appears they can be represented as one universal expression for both types. Then we express it in terms of q-hypergeometric function $_4\Phi_3$ with some factor. Also it is proven that this expression does not have any inequality restrictions on its arguments, as it was proposed in the original article. As a result, the expression becomes more convenient for studying symmetries. In section \[S4\] we analyze the hypergeometric expression of multiplicity-free 6-j symbol. We find the transformation between the multiplicity-free $U_q(sl_N)$ 6-j symbol and its $U_q(sl_2)$ counterpart. This result creates a lot of possibilities to generalize well-known $U_q(sl_2)$ 6-j symbol properties to the considered case. As an immediate output of such relation in section \[S5\] we derive the classical ($q=1$) 6-j symbol asymptotics, using known results for $U(sl_2)$. Originally it was written in terms of the associated tetrahedron [@Ponzano_Regge; @Roberts]. The $U(sl_N)$ generalization modifies the expression so that the tetrahedron now depends on $N$ and deforms differently for two types of 6-j symbols. In section \[S6\] the resulting 6-j symbol expression has been studied for symmetries. Obtained $_4\Phi_3$ series has two known symmetries: permutations of arguments in each row and the Sears’ transformation [@gaspar]. The total number of hypergeometric symmetries is 23040 for both types, it was obtained by manual computations on computer. However, only 24 form symmetry group of 6-j symbols for type I and 12 for type II. Some of them are tetrahedral, others can be described as the Regge symmetry generalization for $N\geq 2$. We also consider additional symmetries that equates $U_q(sl_N)$ and $U_q(sl_M)$ 6-j symbols in subsections \[SS4\],\[SS5\]. Being obtained as symmetries between hypergeometric series, they require a normalizing factor in terms of 6-j symbols. Non-trivial expressions are found for both types and examples are provided. The main results of these subsections are symmetries that generalize permutation in a different from tetrahedral way. They become usual well-known symmetries when $N=2$, but for $N>2$ they depend on $N$ explicitly. Racah coefficients, 6-j symbols and types I, II expression {#S2} ========================================================== To define 6-j symbols we need firstly to remind the Racah matrix definition. Here we work with q-deformed algebra $U_q(sl_N)$. Let us consider 3 irreducible $\mathbb{C}$-modules of representations $R_1,R_2,R_3$ acting in $V_{R_1},V_{R_2},V_{R_3}$. Due to a tensor product associativity, $(V_{R_1} \otimes V_{R_2}) \otimes V_{R_3} = V_{R_1} \otimes (V_{R_2} \otimes V_{R_3})$, hence there is a unitary transformation $$\begin{aligned} U:\ \ (R_1 \otimes R_2) \otimes R_3 &\rightarrow R_1 \otimes (R_2 \otimes R_3).\end{aligned}$$ On the other hand, we can rewrite it in irreducible components, where $M_{X}^{R_1,R_2}$ is a multiplicity space of all $X$’s in the decomposition $R_1\otimes R_2$: $$\begin{split} (R_1 \otimes R_2) \otimes R_3 &= \left(\bigoplus_i M_{X_i}^{R_1,R_2} \otimes X_i\right)\otimes R_3 = \bigoplus_{i,k} M_{X_i}^{R_1,R_2} \otimes M_{R_{4_k}}^{X_i,R_3} \otimes R_{4_k},\\ R_1 \otimes (R_2 \otimes R_3) &= R_1 \otimes \left(\bigoplus_j M_{Y_j}^{R_2,R_3} \otimes Y_j\right) = \bigoplus_{j,k} M_{R_{4_k}}^{R_1,Y_j} \otimes M_{Y_j}^{R_2,R_3} \otimes R_{4_k}. \end{split}$$ If we consider some particular $R_4$ in the decomposition, it corresponds to the vector space of representations. A basis constructed from the highest weights’ vectors differs for these two fusions. (1.5, 0) arc (0:-90:0.75); (0, 0) arc (-180:-90:0.75); (3.25, -0.75) arc (-90:0:0.75); (2.5, 0) arc (-180:-90:0.75); (0.75, -0.75) arc (-180:0:1.25); node at (0, 0.3) [$R_1$]{}; node at (1.5, 0.3) [$R_2$]{}; node at (2.5, 0.3) [$R_3$]{}; node at (4, 0.3) [$R_4$]{}; node at (2, -1.6) [$X_i$]{}; (4.5,-1) to (5.5,-1); node at (5, -0.7) [$U$]{}; (9, 0) arc (0:-90:1); (7, 0) arc (-180:-90:1); (8, -2) arc (-90:0:2); (6, 0) arc (-180:-90:2); (8,-1) to (8,-2); node at (6, 0.3) [$R_1$]{}; node at (7, 0.3) [$R_2$]{}; node at (9, 0.3) [$R_3$]{}; node at (10, 0.3) [$R_4$]{}; node at (8.4, -1.5) [$Y_i$]{}; Thus, there is a transformation between two vector spaces that is defined by the Racah matrix or Racah-Wigner 6-j symbols. Racah coefficients are elements of Racah matrix that is the map: $$\begin{aligned} \label{U_mat_def} U \left( \begin{matrix} R_1 & R_2 \\ R_3 & {R_4} \end{matrix} \right): \bigoplus_{i} M_{X_i}^{R_1,R_2} \otimes M_{R_{4}}^{X_i,R_3} \rightarrow \bigoplus_{j} M_{R_{4}}^{R_1,Y_j} \otimes M_{Y_j}^{R_2,R_3}. \end{aligned}$$ Wigner 6-j symbol is the element of a normalized Racah matrix: $$\begin{aligned} \left\lbrace \begin{matrix}\label{6j_def} R_1 & R_2 & X_i\\ R_3 & R_4 & Y_j \end{matrix} \right\rbrace = \frac{1}{\sqrt{\dim_q(X_i)\dim_q(Y_j)}} U_{i,j} \left( \begin{matrix} R_1 & R_2 \\ R_3 & R_4 \end{matrix} \right). \end{aligned}$$ Here $\dim_q$ means the quantum deformation of the usual expression for the dimension of the representation [@3SB]. It can be computed for every $U_q(sl_N)$ representation $R$ using the corresponding Young diagram $\lambda$ ($\lambda^T$ is a transposed Young diagram): $$\dim_q(\lambda)= \prod_{(i,j)\in \lambda} \frac{q^{\frac{1}{2}(N+i-j)} - q^{-\frac{1} {2} (N+i-j)}} {q^{\frac{1}{2}(\lambda_i-i+\lambda_j^T-j+1)} - q^{-\frac{1}{2}(\lambda_i-i+\lambda_j^T-j+1)}}.$$ In this paper we work with the special class of 6-j symbols, which can be seen as a natural generalization of $U_q(sl_2)$ case for $U_q(sl_N)$ 6-j symbols. The initial representations and the resulting one are either symmetric or conjugated to symmetric for this class. Further we will assume that $R_1,R_2,R_3,R_4$ representations are symmetric. Corresponding Young diagrams are $[r_1],[r_2],[r_3],[r_4]$, here $r_n$ are integers that denote numbers of boxes for $U_q(sl_N)$ symmetric representations. Conjugated Young diagram is written as $\overline{[r_n]}$ and correspond to $\overline{R}_n$. We shall call two 6-j symbols below type [I]{} and type [II]{}, $\ytableausetup{boxsize=0.6em, aligntableaux=top} \ytableaushort{\cdot}$ means $N-1$ vertical boxes. $$\begin{aligned} \text{I type: }\left\lbrace \begin{matrix} [r_1] & \overline{[r_2]} & X\\ [r_3] & [r_4] & Y \end{matrix} \right\rbrace &\equiv \left\lbrace \begin{matrix} \ydiagram{2}\dots \ydiagram{1} & \ytableaushort{\cdot}\dots \ytableaushort{\cdot} & \ytableaushort{\cdot\cdot}\dots \ytableaushort{\cdot\none}*{2}\dots \ydiagram{1}\\ \ydiagram{2}\dots \ydiagram{1} & \ydiagram{2}\dots \ydiagram{1} & \ytableaushort{\cdot\cdot}\dots \ytableaushort {\cdot\none}*{2} \dots \ydiagram{1} \end{matrix} \right\rbrace,\\ \text{II type: } \left\lbrace \begin{matrix} [r_1] & [r_2] & X\\ \overline{[r_3]} & [r_4] & Y \end{matrix} \right\rbrace &\equiv \left\lbrace \begin{matrix} \ydiagram{2}\dots \ydiagram{1} & \ydiagram{2}\dots \ydiagram{1} & \ydiagram{2,2}\dots \ydiagram{2,1}\dots \ydiagram{1}\\ \ytableaushort{\cdot}\dots \ytableaushort{\cdot}& \ydiagram{2}\dots \ydiagram{1} & \ytableaushort{\cdot\cdot}\dots \ytableaushort{\cdot\none}*{2}\dots \ydiagram{1} \end{matrix} \right\rbrace. \end{aligned}$$ Although arguments $R_1,R_2,R_3,R_4$ are very simple and can be parametrized by the width and $N$, the last pair of $X$ and $Y$ Young diagrams has more sophisticated expressions. There are two possible cases of tensor products: $[r_n]\otimes [r_m]$ and $[r_n]\otimes \overline{[r_m]}$. Each element in the decomposition depends on the initial pair of representations and the ordering number in the sum. From the Littlewood-Richardson rules [@harris] it is easy to see that the mentioned tensor products are multiplicity-free and all representations in a decomposition have different width. Similarly to $U_q(sl_2)$ case, where it is possible to enumerate diagrams by the only integer parameter $i$, for mentioned $U_q(sl_N)$ decompositions we have the enumerating parameter – the first row length. To shorten the notation we shall write 6-j symbol of type I and type II in a more compact form. Let us denote the type by variable $T \in \{1,2\}$. Type I 6-j symbol is: $$\left[ \begin{matrix}\label{MFS_nota_t1} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_{1}^N := \left\lbrace \begin{matrix} [r_1] & \overline{[r_2]} & \left[i, \dfrac{r_2-r_1+i}{2}^{N-2} \right]\\ [r_3] & [r_4] & \left[ j, \dfrac{r_2-r_3+j}{2}^{N-2} \right] \end{matrix} \right\rbrace,$$ and type II: $$\left[ \begin{matrix}\label{MFS_nota_t2} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_{2}^N := \left\lbrace \begin{matrix} [r_1] & [r_2] & \left[\dfrac{r_1+r_2+i}{2}, \dfrac{r_1+r_2-i}{2} \right]\\ \overline{[r_3]} & [r_4] & \left[ j, \dfrac{r_2-r_3+j}{2}^{N-2} \right] \end{matrix} \right\rbrace,$$ where $i,j$ is defined in such a way in order to have a nice $N=2$ limit. Let us note that the fusion rules restrictions require additional equalities: $$\begin{aligned} \label{fusion_rules} \begin{split} r_1+r_3&=r_2+r_4 \hspace{5mm}\text{ for type I,}\\ r_1+r_2&=r_3+r_4 \hspace{5mm}\text{ for type II.} \end{split}\end{aligned}$$ The equations (\[Regge\]) between $U_q(sl_2)$ 6-j symbols are called Regge symmetries or Regge transformations [@Regge] $(\rho = \frac{r_1 + r_2 + r_3 + r_4}{2}, \rho' = \frac{r_1 + r_3 + i + j}{2}, \rho'' = \frac{r_2 + r_4 + i + j }{2})$: $$\begin{gathered} \left\lbrace \begin{matrix}\label{Regge} r_1 & r_2 & i \\ r_3 & r_4 & j\end{matrix} \right\rbrace = \left\lbrace \begin{matrix} \rho - r_3 & \rho - r_4 & i \\ \rho - r_1 & \rho - r_2 & j \\ \end{matrix} \right\rbrace = \left\lbrace \begin{matrix} \rho' - r_3 & r_2 & \rho' - j \\ \rho' - r_1 & r_4 & \rho' - i \\ \end{matrix} \right\rbrace = \left\lbrace \begin{matrix} r_1 & \rho'' - r_4 & \rho'' - j\\ r_3 & \rho'' - r_2 & \rho'' - i \\ \end{matrix} \right\rbrace = \\ = \left\lbrace \begin{matrix} \rho-r_3 & \rho' - r_4 & \rho'' - j\\ \rho-r_1 & \rho' - r_2 & \rho'' - i \end{matrix} \right\rbrace = \left\lbrace \begin{matrix} \rho''-r_3 & \rho-r_4 & \rho' - j\\ \rho''-r_1 & \rho-r_2 & \rho' - i \end{matrix} \right\rbrace \nonumber. \end{gathered}$$ The tetrahedral symmetry is a known property of 6-j symbol to be invariant under row and column permutations [@WZW2] ($\lambda_i,\mu,\nu$ are arbitrary Young diagrams): $$\begin{aligned} \label{tetra} \left\{ \begin{matrix} \lambda_1 & \lambda_2 & \mu \\ \lambda_3 & \lambda_4 &\nu \end{matrix} \right\} &= \left\{ \begin{matrix} {\lambda_3} & {\lambda_2} &\overline{\nu} \\ {\lambda_1} & {\lambda_4} &\overline{\mu}\end{matrix} \right\} = \left\{ \begin{matrix} \lambda_3 & \overline{\lambda_4} & \overline{\mu} \\ \lambda_1 & \overline{\lambda_2} & \overline{\nu}\end{matrix} \right\} =\\ &= \left\{ \begin{matrix} \overline{\lambda_1} & \overline{\mu} & \overline{\lambda_2} \\ \lambda_3 & {\nu} & {\lambda_4}\end{matrix} \right\} =\left\{ \begin{matrix} \overline{\lambda_2} & \overline{\lambda_1} & \overline{\mu} \\ \lambda_4 & {\lambda_3} & {\nu}\end{matrix} \right\}\nonumber. \end{aligned}$$ 6-j symbol in $U_q(sl_N), N>2$ with symmetric and conjugate to symmetric representations is either trivial ($X$ and $Y$ has the only possible value) or may be equated by tetrahedral symmetry and conjugation to either type [I]{} or type [II]{}. There are only a few possible variants to write down a 6-j symbol with symmetric and conjugate to symmetric representations. By conjugation of 6-j symbol we can transform $R_4$ to a symmetric diagram. Thus, let us prove the proposition without loss of generality only for symmetric $R_4$. Let us now investigate how the first three arguments may be organized. There are four different cases that correspond to the number of conjugated representations in the product. - All three representations are conjugated. Let us conjugate all terms in the product $\overline{[r_1]} \otimes \overline{[r_2]} \otimes \overline{[r_3]} \supset [r_4]$, so we can consider ${[r_1]} \otimes {[r_2]} \otimes {[r_3]} \supset \overline{[r_4]}$ and $N>2$. It is obvious from the fusion rules [@harris] that for $N>4$ it is not possible to combine the representations into a conjugated one because there are no more than 3 rows in a resulting Young diagram, whereas $\overline{[r_4]}$ has $N-1>3$ rows. Now we need to prove that it is not possible even for $N={3,4}$. The $N=4$ case requires the rows of $R_4$ to be equal. The Littlewood-Richardson rules [@harris] say that the resulting diagram is constructed as the first multiplier with the second multiplier’s elements but with some restrictions. For symmetric diagrams they forbid to put the new elements in one column. Hence, if we need to combine diagrams into a rectangular one, the corresponding 6-j symbol is trivial. Indeed, the only way to combine the diagrams properly is to consider them equal and to put them under each other. Here and below we use some non-negative integer parameters $a,b,c$ that encode a Young diagram, the aim of these parameters is to specify the shape of a considered diagram. The $N=3$ case has a $\overline{[r_4]}$ diagram that may be written as $[a,a]$. The $[a,a]$ is trivial, because there is the only diagram $X = [r_1+r_2-b,b]$ that has width $a$. Indeed, if the width is smaller, the third multiplier can not make the second row width equal to $a$, if it is greater, we can not make $R_4$ anymore. Therefore, all $N>2$ 6-j symbols with 3 conjugated representations are trivial. - All three representations are symmetric. Obviously, if $R_1, R_2, R_3, R_4$ are symmetric in $U_q(sl_N), N>3$, then the corresponding 6-j symbol has the only $X = [r_1+r_2]$, the same for $Y$. If $N=3$, there is a possibility to make a Young diagram with columns of height $N$. However, the fusion rules restrict $X = [r_1+r_2-a, a] = [b+r_4,b]$, hence $X = [r_1+r_2+r_4, r_1+r_2-r_4]$ and this 6-j symbol is trivial. - Two representations are conjugated and one is symmetric. Note, that the multiplicity of $R_4$ in decomposition $R_1\otimes R_2\otimes R_3$ does not change under a permutation of multipliers. Hence we may always decompose the product of conjugated representations and then multiply it by the symmetric one. Without loss of generality we consider $\overline{[r_1]} \otimes \overline{[r_2]} \otimes {[r_3]}$. Let us firstly decompose the product of conjugated representations. In general, it has the diagram $[a^{N-2},b]$, where $b\le a$. It is obtained from $[(r_1+r_2)^{N-2},r_1+r_2-c,c]$ by reducing the column of height $N$. If $N>3$, the product $[a^{N-2},b] \otimes [r_3]$ may have a symmetric diagram in the decomposition only if $a=b$, but it will be trivial because $X = [(r_3-r_4)^{N-1}]$. If $N=3$, $[a,b]\otimes [r_3]$ easily makes symmetric diagram with condition $X=[a,a+r_3-r_4]$. But we can find $a$ from the $\overline{[r_1]} \otimes \overline{[r_2]}$ decomposition and it is unique for fixed $r_1$ and $r_2$. As a result, there are no non-trivial 6-j symbols with two conjugated symmetric representations and symmetric $R_4$. - One conjugated representation. There are three such 6-j symbols: $$\begin{aligned} \left\lbrace \begin{matrix} \overline{[r_1]} & [r_2] & X\\ [r_3] & [r_4] & Y \end{matrix} \right\rbrace, \left\lbrace \begin{matrix} [r_1] & \overline{[r_2]} & X\\ [r_3] & [r_4] & Y \end{matrix} \right\rbrace, \left\lbrace \begin{matrix} [r_1] & [r_2] & X\\ \overline{[r_3]} & [r_4] & Y \end{matrix} \right\rbrace. \end{aligned}$$ One can check that they may be nontrivial. We can apply a tetrahedral symmetry to these 6-j symbols, in particular, row permutation of arguments $(R_1, R_2) \leftrightarrow (R_3, R_4)$. After this transformation the first and the third 6-j symbols are swapped and the second one is invariant. Applying other symmetries, one can check that type I and type II are not equated by tetrahedral symmetries. It is worth mentioning that there are tetrahedral symmetries acting within each type. In particular, a type I 6-j symbol is still type I after row permutations and the swap of the first two columns. Type II is conserved only by the row permutation of the first two columns. These are the only tetrahedral symmetries that possible to derive if one consider symmetries of type I or type II. The others either were used earlier to transform 6-j symbol into one of the types, or transform any type into a completely different 6-j symbol, which has non-symmetric representations and much more complicated structure, so they are out of the scope of the present paper. The expression for 6-j symbol of type I and II was proposed in [@MFS]. It may be written as follows. [$$\begin{gathered} \label{MFS} \left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_{T}^N =\theta_N\left(r_1,r_2,i\right) \theta_N\left(r_3, r_4, i\right) \theta_N\left(r_1, r_4, j\right) \theta_N\left(r_2, r_3, j\right) [N-1]_q![N-2]_q!\sum_{z = z_{min}}^{z_{max}}\\\nonumber \dfrac{ (-1)^z [z+N-1]_q!\cdot A_{T,z} }{[ z-\frac{r_1+r_2+i}{2}]_q![z-\frac{r_3+r_4+i}{2}]_q! [z-\frac{r_1+r_4+j}{2}]_q! [z-\frac{r_2+r_3-j}{2}]_q! [\frac{r_1+r_2+r_3+r_4}{2}-z]_q! [\frac{i+j+r_1+r_3}{2}-z]_q! [\frac{i+j+r_2+r_4}{2}-z]_q!}, \\ \theta_N(a,b,c)=\sqrt{\dfrac{[\frac{a+b-c}{2}]_q! [\frac{c+a-b}{2}]_q! [\frac{b+c-a}{2}]_q!}{[\frac{a+b+c}{2}+N-1]_q!}}, \qquad A_{T,z} = \left[\begin{matrix} \dfrac{[k+z_{min}-z]_q!}{[k+z_{min}+N-2 - z]_q!} \hspace{5mm}\text{ for type I,}\\ \dfrac{[k-z_{max}+ z]_q!}{[k-z_{max} +N-2+ z]_q!} \hspace{5mm}\text{ for type II.} \end{matrix}\right. \end{gathered}$$]{} To write the 6-j symbol expression we use quantum numbers notations. It is by the definition $[n]_q= \frac{q^{\frac{n}{2}} -q^{-\frac{n}{2}}}{q^{\frac{1}{2}} -q^{-\frac{1}{2}}}$. Quantum generalization of factorials for non-negative integers can be written as $[n]_q! = \prod_{k=1}^{n}[k]_q$. Also $k = \frac{1}{2}\min(i-r_1+r_2, j-r_3+r_2)$ and $z_{min},z_{max}$ are defined as the smallest and the largest integers for which the summand is non-trivial , i.e. there are no factorials of negative integers. The expression differs for two types only in the $A_{T,z}$ expression. Also the following conditions were imposed in the original paper [@MFS] (as we show below, they are not necessary): $$\label{conds} \left\{\begin{matrix} 0 \le r_2\le r_1 \le r_3 \\ 0 \le r_1\le r_2 \end{matrix}\right. \hspace{5mm} \left.\begin{matrix} \text{for type I,}\\ \text{for type II.} \end{matrix}\right.$$ Hypergeometric expression for 6-j symbols {#S3} ========================================= In this section we express the 6-j symbol expression in terms of basic q-hypergeometric series $_4\Phi_3$. Firstly, we define the q-hypergeometric functions and remind their symmetric properties. After this we use the inequality properties (\[conds\]) to simplify the 6-j symbol expression. We prove with the help of tetrahedral symmetries that the 6-j symbol’s domain may be extended beyond the mentioned inequalities. Then we write the obtained series as a $_4\Phi_3$ function. As a result, both types can be written as q-hypergeometric $_4\Phi_3$ series multiplied by some factor. q-Hypergeometric symmetries --------------------------- A q-Pochhammer symbol is defined as $(a,q)_n = \prod_{k=0}^{n-1}(1-a q^k)$. The q-hypergeometric series are defined as: $$\begin{aligned} _{p+1}\phi_p\left( \begin{matrix} a_1, \ldots, a_{p+1}\\ b_1, \ldots, b_p \end{matrix};q,z \right) := \sum_{n=0}^{\infty} \dfrac{(a_1,q)_n\ldots (a_{p+1},q)_n}{(b_1,q)_n\ldots (b_p,q)_n (q,q)_n}z^n. \end{aligned}$$ It can be also rewritten in a form, which is more convenient for us: $$\begin{aligned} \label{hyp_def} {}_{p+1}\Phi_p\left( \begin{matrix} a_1, \ldots, a_p, a_{p+1} \\ b_1, \ldots, b_p \end{matrix};q,z \right) := {}_{p+1}\phi_p\left( \begin{matrix} q^{a_1}, \ldots, q^{a_p}, q^{a_{p+1}} \\ q^{b_1}, \ldots, q^{b_p} \end{matrix};q,z \right). \end{aligned}$$ It is far more convenient because it may be reformulated in terms of q-factorials: $$\begin{aligned} {}_{p+1}\Phi_p\left( \begin{matrix} a_1+1, \ldots, a_p+1, a_{p+1}+1 \\ b_1+1, \ldots, b_p+1 \end{matrix};q,z \right) = \sum_{n=0}^{\infty} \dfrac{[a_1+n]_q!}{[a_1]_q!}\ldots \dfrac{[a_{p+1}+n]_q!}{[a_{p+1}]_q!} \dfrac{[b_1]_q!}{[b_1+n]_q!}\ldots \dfrac{[b_p]_q!}{[b_p+n]_q!} \dfrac{z^n}{[n]_q!}.\end{aligned}$$ This expression evidently has the limit $\lim\limits_{q\rightarrow 1}[a]_q!= a!$, where the whole series becomes a usual hypergeometric function. There are a lot of known symmetries for $_4\Phi_3$ series. Here we consider only permutation symmetry and Sears’ transformation. Permutation symmetry is the evident property of ${}_r\Phi_p$ functions to be invariant under permutations $\omega \in \mathbb{S}_r$ and $u\in \mathbb{S}_p$: $$\label{trans_hyp_perm} {}_r\Phi_p\left( \begin{matrix} a_1,\ldots,a_r\\ b_1,\ldots,b_p \end{matrix} ;q,z\right)= {}_r\Phi_p\left( \begin{matrix} a_{\omega(1)},\ldots,a_{\omega(r)}\\ b_{u(1)},\ldots,b_{u(p)} \end{matrix} ; q,z \right).$$ Sears’ transformation [@gaspar] is the relation between two ${}_4\Phi_3$ functions: $$\label{trans_sears} \begin{split} {}_4\Phi_3\left( \begin{matrix} x,y,z,n\\ u,v,w \end{matrix} ;q,q\right)= \dfrac{[v{-}z{-}n{-}1]_q![u{-}z{-}n{-}1]_q![v{-}1]_q![u{-}1]_q!}{[v{-}z{-}1]_q![v{-}n{-}1]_q![u{-}z{-}1]_q![u{-}n{-}1]_q!}\ {}_4\Phi_3\left( \begin{matrix} w-x,w-y,z,n\\ 1{-}u{+}z{+}n,1{-}v{+}z{+}n,w \end{matrix} ; q,q \right), \end{split}$$ where $x+y+z+n+1=u+v+w$. 6-j symbol as   series ---------------------- Let us denote the sum (\[MFS\]) as $\left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_{T}^N = K'\cdot\sum_m I_m = K' \cdot I$, where $m = \frac{1}{2}(r_1+r_2+ r_3+r_4)-z$. Then it can be easily rewritten as: [$$\begin{gathered} I = \sum_{m = m_{min}}^{m_{max}}\dfrac{ (-1)^{\frac{r_1+r_2+r_3+r_4}{2}-m} [\frac{r_1+r_2+r_3+r_4}{2}-m+N-1]_q! \cdot A_{T,m}} {[m]_q![\frac{r_3+r_4-i}{2}-m]_q! [\frac{r_1+r_2-i}{2}-m]_q! [\frac{r_2+r_3-j}{2}-m]_q! [\frac{r_1+r_4-j}{2}-m]_q! [\frac{i+j-r_2-r_4}{2}+m]_q! [\frac{i+j-r_1-r_3}{2}+m]_q! },\\ K'=\theta_N\left(r_1,r_2,i\right) \theta_N\left(r_3, r_4, i\right) \theta_N\left(r_1, r_4, j\right) \theta_N\left(r_2, r_3, j\right) [N-1]_q![N-2]_q!\ ,\\ A_{T,m} = \left[\begin{matrix} \dfrac{[k-m_{max}+m]_q!}{[k-m_{max}+N-2 + m]_q!} \hspace{5mm}\text{ for type I,}\\ \dfrac{[k+m_{min}- m]_q!}{[k+m_{min} +N-2- m]_q!} \hspace{5mm}\text{ for type II.} \end{matrix}\right.\end{gathered}$$]{} The explicit relations for $m_{min}$ and $m_{max}$ can be easily found from the denominator factorials, because the summand is zero if and only if there is a negative factorial in the denominator: $$m_{max} = \frac{1}{2}\min\begin{pmatrix} r_1+r_2-i\\ r_3+r_4-i\\ r_1+r_4-j\\ r_2+r_3-j \end{pmatrix}, \hspace{10mm} m_{min} = \frac{1}{2}\max\begin{pmatrix} 0\\ r_1+r_3-i-j\\ r_2+r_4-i-j \end{pmatrix}.$$ As it can be derived from fusion rules, $k,m_{max},m_{min}$ are always integers when a 6-j symbol exists. Moreover, $k$ has a clear meaning in terms of Young diagrams – it is the minimum width among the conjugated parts of diagrams, corresponding to $X_i$ and $Y_j$. One can notice, that the considered expression fits the $_5\Phi_4$ definition (\[hyp\_def\]), if $z=q$. This allows us to claim the following. Both type [I]{} and type [II]{} may be written as $_5\Phi_4$ q-hypergeometric series multiplied by simple factors: [$$\begin{gathered} \left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_T^N = K'' \cdot {}_5\Phi_4 \left( \begin{matrix} a_1,a_2,a_3,a_4,a_5\\ b_1,b_2,b_3,b_4 \end{matrix}; q,q \right),\\ 2a_i = \left( \begin{matrix} 2\{k-m_{max}+1 , -k - m_{min}-N+2\}_T\\ -r_1-r_2+i \\ -r_3-r_4+i\\ -r_1-r_4+j\\ -r_2-r_3+j \end{matrix} \right), \qquad 2b_i = \left( \begin{matrix} -r_1-r_2-r_3-r_4-2(N-1)\\ i+j-r_2-r_4 + 2\\ i+j-r_1-r_3 + 2\\ 2\{k-m_{max}+N-1 , -k - m_{min}\}_T \\ \end{matrix} \right),\\ \begin{gathered} K'' = \dfrac{ K' \cdot A_{T,0} \cdot [\frac{r_1+r_2+r_3+r_4}{2}+N-1]_q! } {[\frac{r_3+r_4-i}{2}]_q! [\frac{r_1+r_2-i}{2}]_q! [\frac{r_2+r_3-j}{2}]_q! [\frac{r_1+r_4-j}{2}]_q! [\frac{i+j-r_2-r_4}{2}]_q! [\frac{i+j-r_1-r_3}{2}]_q! }, \end{gathered} \end{gathered}$$]{} where $\{e_1,e_2\}_T \equiv e_T$ is $e_1$ for type [I]{} and $e_2$ for type [II]{}. It can be proven straightforwardly by substitution of q-Pochhammer symbols. Expression of 6-j symbol as   series ------------------------------------ The obtained expression for 6-j symbol is not quite convenient to find its symmetries. Expressions for $k$, $m_{min}$ and $m_{max}$ may be simplified in the following way. \[L1\] For all type [I]{} 6-j symbols $k-m_{max} =\frac{i+j-r_1-r_3}{2}$ if the following conditions are satisfied: $$\label{type1_cond} \begin{cases} r_2\le r_1\le r_3,\\ r_1+r_3=r_2+r_4. \end{cases}$$ \[L2\] Let us consider $k-m_{max}=\frac{i+j-r_1-r_3}{2}$. One can check that there are 2 cases when it is so, hence they may be written as the union of two systems: $$\begin{cases} r_1+r_2-i\le r_3+r_4-i,\\ r_1+r_2-i\le r_2+r_3-j,\\ r_1+r_2-i\le r_1+r_4-j,\\ j-r_3\le i-r_1; \end{cases} \hspace{10mm} \begin{cases} r_2+r_3-j\le r_3+r_4-i,\\ r_2+r_3-j\le r_1+r_2-i,\\ r_2+r_3-j\le r_1+r_4-j,\\ i-r_1\le j-r_3. \end{cases}$$ If the conditions (\[type1\_cond\]) satisfied, the first three inequalities are true. The union of these two systems may be reduced to the next expression. $$\left[\begin{matrix} j-i\le r_4-r_2, \\ j-i\ge r_4-r_2. \end{matrix}\right.$$ Consequently, every 6-j symbol from type I is described by $k-m_{max} = \frac{i+j-r_1-r_3}{2}$. \[L3\] For all type [II]{} 6-j symbols $k+m_{min} = \frac{r_1+r_2-i}{2}$ if the conditions are satisfied: $$\label{17} \begin{cases} r_1\le r_2,\\ r_1+r_2=r_3+r_4. \end{cases}$$ The proof for type II is analogous to type I. Conditions on arguments of a 6-j symbol (\[conds\]) are redundant, i.e the expression (\[MFS\]) is valid even if the inequalities are not satisfied. We are able to obtain every possible 6-j symbol of types I and II by performing a tetrahedral symmetry (\[tetra\]) that leaves the type invariant: $$\begin{gathered} \left\{ \begin{matrix} [r_1] & \overline{[r_2]} & X \\ [r_3] & [r_4] & Y \end{matrix} \right\} = \left\{ \begin{matrix} {[r_3]} & \overline{[r_2]} &\overline{Y} \\ {[r_1]} & {[r_4]} &\overline{X}\end{matrix} \right\} = \left\{ \begin{matrix} {[r_2]} & \overline{[r_1]} & \overline{X} \\ [r_4] & {[r_3]} & {Y}\end{matrix} \right\}. \end{gathered}$$ One may immediately notice that these symmetries may transform a 6-j symbol from region $r_2\le r_1 \le r_3$ into all possible representations. The problem is that the expression for the transformed 6-j symbols may differ from the initial expression. We can check it by substituting arguments transformed by tetrahedral symmetries. Let us show that in our notations it acts on $r_1,r_2,r_3,r_4, i, j$ as a permutation. For $R_n$, the symmetry evidently acts as a permutation of $r_n$. There are also representations $X$ and $Y$ that is conjugated, we can consider only diagram $\left[ j, \frac{r_2-r_3+j}{2}^{N-2} \right]$ as an example. Under conjugation it transforms $\left[ j, \frac{r_2-r_3+j}{2}^{N-2} \right] \rightarrow \left[ j, \frac{r_3-r_2+j}{2}^{N-2} \right]$, but the expression depends only on $j$ that is invariant under conjugation. Therefore, tetrahedral symmetry acts on the expression as a permutation of arguments. One can check that it is invariant under written tetrahedral symmetry transformation. The same for type II, but we need only one relation (the inequality is $r_1\le r_2$): $$\begin{aligned} \left\{ \begin{matrix} [r_1] & {[r_2]} & X \\ \overline{[r_3]} & [r_4] & Y \end{matrix} \right\} = \left\{ \begin{matrix} {[r_3]} & {[r_4]} & {X} \\ \overline{[r_1]} & {[r_2]} & {Y}\end{matrix} \right\}. \end{aligned}$$ The symmetry acts non-trivially only on $r_1,r_2,r_3,r_4$, we already showed why it is a permutation. It is easy to see that the expression is invariant under such a transformation. Therefore, the expression does not change when we write a 6-j symbol without additional inequality conditions (\[conds\]). Then we can get rid of these conditions as even if they are not satisfied the expression is valid. We have proven in Lemma \[L1\] that for arguments satisfying the inequality condition (\[conds\]) there are only one combination of $k-m_{max}$ that is present for type I 6-j symbols. This results into the exact value of $A_{T,m}$ which allow us to reduce the whole series. Then we apply tetrahedral symmetries to prove that the statement is true for all type I 6-j symbols. The same procedure has been done for type II and this allows us to simplify both expressions and write down them as follows. [gather]{} \_T\^N = K’\_[m = m\_[min]{}]{}\^[m\_[max]{}]{}\ . We can express all factorials as q-Pochhammer symbols. The substitution differs for factorials with $+m$ and $-m$: $$\begin{aligned} \label{poch} [m_0+m]_q! = [m_0]_q!(q^{m_0+1},q)_{m}\cdot \dfrac{q^{-\frac{m}{4}(2m_0+m-1)}}{(1-q)^{m}}\ , \quad [m_0-m]_q! = \dfrac{(-1)^{m}[m_0]_q! } {(q^{-m_0},q)_{m}} \cdot \dfrac{(1-q)^{m}} { q^{\frac{m}{4}(2m_0+m-1)}}.\end{aligned}$$ By substituting this to the main expression one can check that among depending on $m$ terms only q-Pochhammer symbols remain. This allows us to write the series as a hypergeometric function: $$\begin{aligned} \left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_T \sim {}_4\Phi_3 \left( \begin{matrix} a_1,a_2,a_3,a_4\\ b_1,b_2,b_3 \end{matrix}; q,q \right).\end{aligned}$$ The ${}_4\Phi_3$ arguments may be easily obtained using (\[poch\]). Note, that there is the following relation on the arguments: $$\begin{aligned} \label{hyp_relation} a_1+a_2+a_3+a_4+1=b_1 + b_2 + b_3.\end{aligned}$$ And the factorizable part of the expression: [$$\label{coef_K} K_T = \dfrac{\theta_N\left(r_1,r_2,i\right) \theta_N\left(r_3, r_4, i\right) \theta_N\left(r_1, r_4, j\right) \theta_N\left(r_2, r_3, j\right) [N-1]_q![N-2]_q! [\frac{r_1+r_2+r_3+r_4}{2}+N-1]_q! }{[\frac{r_3+r_4-i}{2}]_q! [\frac{r_1+r_2-i}{2} + (N-2)\delta_{T,2}]_q! [\frac{r_2+r_3-j}{2}]_q! [\frac{r_1+r_4-j}{2}]_q! [\frac{i+j-r_2-r_4}{2}]_q! [\frac{i+j-r_1-r_3}{2}+(N-2)\delta_{T,1}]_q!}.$$]{} Combing all this into one, we come to the following statement. The considered 6-j symbol expression may me expressed as a $_4\Phi_3$ function for both types. The factor $K_T$ is as in (\[coef\_K\]). [gather]{} \_T\^N =K\_T\_4\_3 ( a\_1,a\_2,a\_3,a\_4\ b\_1,b\_2,b\_3 ; q,q )\[final\_expr\_hyp\],\ \[final\_expr\] 2 a\_i = ( -r\_1-r\_2+i - 2(N-2)\_[T,2]{}\ -r\_3-r\_4+i\ -r\_1-r\_4+j\ -r\_2-r\_3+j ), 2 b\_i = ( -r\_1-r\_2-r\_3-r\_4-2(N-1)\ i+j-r\_2-r\_4+2\ i+j-r\_1-r\_3 + 2 + 2(N-2)\_[T,1]{}\ ). This is the most suitable form of 6-j symbol for our aims. As it can be seen, we reduced the $_5\Phi_4$ series to the $_4\Phi_3$ one. This is a non-obvious result. In order to proceed with this reduction we used tetrahedral symmetry along with the special properties of the considered two types. Due to the fact that $U_q(sl_2)$ 6-j symbols are expressed via $_4\Phi_3$ too, we may use the same techniques to obtain new results, also the limit $N=2$ is very easy to apply. This result gives us an idea of a strong connection between 6-j symbols and q-hypergeometric series. For example, it is interesting whether all multiplicity-free 6-j symbols can be expressed as $_4\Phi_3$ series. It is interesting to analyze the number of independent parameters in the obtained expression. Neglecting $q$, on both sides we have 7 parameters: $\{r_1,r_2,r_3,r_4,i,j,N\}$ and $\{a_1,a_2,a_3,a_4, b_1,b_2,b_3\}$. They are not independent, it was mentioned that, on the one hand, each type has restrictions for $N>2$ that fix one parameter. On the other hand, obtained ${}_4\Phi_3$ series satisfies a balance condition $\sum_{i}a_i+1=\sum_{i}b_i$. Thus, for $N>2$ there are 6 parameters on both sides. For $N=2$, the fusion rules do not fix $r_n$, so there are 6 parameters on both sides. It is natural to ask whether there is a connection between the fusion rules and the balance condition. It seems like these equalities have different meaning, because the condition on $\{a_i,b_i\}$ is satisfied even if $r_1+r_3\neq r_2+r_4$. From this point of view another question arises: what class of 6-j symbols can be described in terms of ${}_4\Phi_3$ series with such equality? This question is out of our consideration in this paper, but it is still important and interesting to study. Relation with $U_q(sl_2)$ 6-j symbols {#S4} ===================================== In this section we investigate the relation between 6-j symbols in multiplicity-free $U_q(sl_N)$ and $U_q(sl_2)$ cases. As we have seen, the core of both expressions are $_4\Phi_3$ hypergeometric series. We have already mentioned the number of independent parameters in the series, but now we analyze it in details. Then we shall see the interesting connection between the usual $U_q(sl_2)$ 6-j symbol and considered one. Let us write down the $_4\Phi_3$ arguments as a vector space with the basis $(r_1,r_2,r_3,r_4,i,j,N)$. We put all the additional constants in $\vec{C}$ since they do not play any role in the next discussion: $$\begin{pmatrix} r_1 + r_2 + r_3 + r_4 + 2(N-1)\\ r_1 + r_2 - i + 2(N-2)\delta_{T,2}\\ r_3 + r_4 - i\\ r_1 + r_4 - j\\ r_2 + r_3 - j\\ -r_2 - r_4 + i + j + 2\\ i + j - r_1 - r_3 + 2(N-1)\delta_{T,1} \\ \end{pmatrix} = \begin{pmatrix} 1 & 1 & 1 & 1 & 0 & 0 & 1\\ 1 & 1 & 0 & 0 & -1 & 0 & 2\delta_{T,2}\\ 0 & 0 & 1 & 1 & -1 & 0 & 0\\ 1 & 0 & 0 & 1 & 0 & -1 & 0\\ 0 & 1 & 1 & 0 & 0 & -1 & 0\\ 0 & 1 & 0 & 1 & -1 & -1 & 0\\ -1 & 0 & -1 & 0 & 1 & 1 & 2\delta_{T,1}\\ \end{pmatrix} \begin{pmatrix} r_1\\ r_2\\ r_3\\ r_4\\ i\\ j\\ N \end{pmatrix} +\vec{C}.$$ The rank of this matrix is 6, so there is a kernel of dimension one. This kernel is described by a zero vector $\vec{v}$. Note that (\[hyp\_relation\]) is a completely different condition that does not depend on the values of parameters. The zero vector can be written as follows $$\begin{aligned} \label{zerotrans} \vec{v}=\begin{cases} \begin{pmatrix} 0,1,0,1,1,1,-1 \end{pmatrix}, \quad \text{Type I,}\\ \begin{pmatrix} 1,1,0,0,0,1,-1 \end{pmatrix}, \quad \text{Type II}, \end{cases}\end{aligned}$$ with the corresponding shift in the parameters being $$\begin{aligned} \alpha\vec{v}=\begin{cases} \alpha\vec{v}=\alpha (r_2 + r_4 + i + j - N), \quad \text{Type I,} \\ \alpha\vec{v}=\alpha (r_1 + r_2 + j - N), \quad \text{Type II}. \end{cases}\end{aligned}$$ This freedom allows to shift the arguments value without changing the actual value of the hypergeometric series, so it can be considered as a symmetry for 6-j symbol although for hypergeometric series it is tautological equality. If one examines the transformation for type I 6-j symbol, it can be seen that the fusion rules are in conflict with it. Indeed, the non-trivial transformation changes $r_2+r_4$, but leaves $r_1+r_3$ unchanged, thus (\[fusion\_rules\]) forbids such transformation for $N>2$, for either type I or type II. However for $N=2$ the fusion rules disappear and we can apply it without any problems. So we take $U_q(sl_N)$ 6-j symbol and make transformation in order to get the expression for $U_q(sl_2)$ 6-j symbol: $$\label{Nshift} \begin{split} _4\Phi_3(r_1,r_2,r_3,r_4,i,j,{\bf N})_{1} \ &= \ (-1)^N \cdot {}_4\Phi_3(r_1,r_2+N{-}2,r_3,r_4+N{-}2,i+N{-}2,j+N{-}2,{\bf 2}), \\ _4\Phi_3(r_1,r_2,r_3,r_4,i,j,{\bf N})_{2} \ &= \ (-1)^N \cdot {}_4\Phi_3(r_1+N{-}2,r_2+N{-}2,r_3,r_4,i,j+N{-}2,{\bf 2}). \end{split}$$ The only part of expression that differs is the factor $K_T$. It partly replicates the hypergeometric arguments, so only a few terms are left in the relation between of multiplicity free $U_q(sl_N)$ 6-j symbols and $U_q(sl_2)$ ones. For the sake of brevity, we will write the hypergeometric function from (\[final\_expr\_hyp\]) as $_4\Phi_3(r_1,r_2,r_3,r_4,i,j,N)_T$. The factor $K'$ changes after transformations, let us write it down explicitly. $$\begin{aligned} K'(N)=&\theta_N\left(r_1,r_2,i\right) \theta_N\left(r_3, r_4, i\right) \theta_N\left(r_1, r_4, j\right) \theta_N\left(r_2, r_3, j\right) [N-1]_q![N-2]_q!\ ,\\ \Theta_T(N):=&\dfrac{1}{[N{-}1]_q! [N{-}2]_q!}\dfrac{K'(N)}{K'(2)}.\end{aligned}$$ $$\begin{aligned} \label{coef_theta} \Theta_1(N)&= \left(\prod_{m=1}^{N-2} \left[\frac{i {-} r_1 {+} r_2}{2}+m\right]_q \left[\frac{j {+} r_2 {-} r_3}{2}+m\right]_q \left[\frac{j {-} r_1 {+} r_4}{2}+m\right]_q \left[\frac{i {-} r_3 {+} r_4}{2}+m\right]_q \right)^{-\frac{1}{2}},\\ \Theta_2(N)&= \left(\prod_{m=1}^{N-2} \left[\frac{r_1 {+} r_2{-}i}{2}+m\right]_q \left[\frac{j {+} r_2 {-} r_3}{2}+m\right]_q \left[\frac{j {+} r_1 {-} r_4}{2}+m\right]_q \left[\frac{i {+} r_3 {+} r_4}{2}{+}1+m\right]_q \right)^{-\frac{1}{2}}.\end{aligned}$$ The resulting relation between multiplicity-free $U_q(sl_N)$ and $U_q(sl_2)$ 6-j symbol is as follows. [align]{} \_1\^N &={ r\_1& r\_2+N-2 &i+N-2\ r\_3 & r\_4+N-2 & j+N-2 }(-1)\^N \[N-1\]\_q! \[N-2\]\_q! \_1(N),\ \_2\^N &={ r\_1+N-2 & r\_2+N-2 &i\ r\_3 & r\_4 & j+N-2 }(-1)\^N \[N-1\]\_q! \[N-2\]\_q! \_2(N). It can be easily checked that the remaining fusion rules for $N=2$ (triangle inequality, etc.) are always satisfied and the resulting 6-j symbol is non-trivial. On the other hand, if one tries to transform $U_q(sl_2)$ 6-j symbol into $N>2$ one, the number of problems arises and it is not possible in general. For example, if $r_1+r_3-r_2-r_4>0$, there is no corresponding $N>2$ 6-j symbol. This result is interesting not only because it reveals the hidden relation between two classes of 6-j symbols, but additionally it can be applied to extend a lot of known properties of $U_q(sl_2)$ to arbitrary $N$. In the next section we derive the asymptotics formula for the multiplicity-free case. Let us show an example of such a generalization. Asymptotics of 6-j symbol {#S5} ========================= The 6-j symbol asymptotics formula for $N=2, q=1$ was conjectured by G.Ponzano and T.Regge [@Ponzano_Regge] and later was proven by J. Roberts [@Roberts]. It is formulated in terms of tetrahedron that is combined from the edges of length $J_n:=r_n+1/2,J_5:=i+1/2,J_6:=j+1/2$ and approximates the limit $\lambda\rightarrow \infty $ for representations $\{\lambda r_n, \lambda i, \lambda j\}$: $$\label{tetr_sl2} \left\{\begin{matrix} r_1 & r_2 &i\\ r_3 & r_4 & j \end{matrix} \right\} \sim \dfrac{1}{\sqrt{12\pi |V(J_n)|}} \cos\left(\sum_{n=1}^{6} J_n \cdot \Omega(J_n) + \dfrac{\pi}{4} \right),$$ where $V$ is the tetrahedron volume, $\Omega_i$ is the external dihedral angle about the edge $J_i$. Let us consider 6-j symbols at $q=1$. Using we can find the asymptotics for $U(sl_N)$ 6-j symbol as an asymptotics for equal $U(sl_2)$ 6-j symbol. It looks very similar to , but with deformed expressions for edges, volume and angles. The tetrahedron is now made of $\widetilde{J}_n$ edges, which can be found from $U(sl_N)$ $J_n$: $$\begin{cases} \widetilde{J}_m = J_m,\\ \widetilde{J}_n = J_n+N-2, \end{cases}$$ where $m$ and $n$ are defined differently for two types: $$\begin{aligned} \label{J_def} m \in \text{\{1, 3\}}, \quad n \in \text{\{2, 4, 5, 6\}} \qquad \text{Type I},\\ m \in \text{\{3, 4, 5\} }, \quad n \in \text{\{1, 2, 6\}} \qquad \text{Type II}. \nonumber\end{aligned}$$ The corresponding volume and angles are denoted by $\widetilde{V}$ and $\widetilde{\Omega}_n$. The resulting asymptotics for 6-j symbol corresponding to arbitrary symmetric representations of $U_q(sl_N)$, thus, can be written in terms of the associated tetrahedron, but now the tetrahedron depends on $N$: [equation]{} \_T\^N \~ (\_[n=1]{}\^[6]{} \_n (\_n) + ). Although the factor is quite long for the general case, it becomes much simpler when all $r_n$ coincide, for example, for type I it looks like: $$\dfrac{\left(\frac{i}{2}+N{-}2\right)!}{\left(\frac{i}{2}\right)!}\dfrac{\left(\frac{j}{2}+N{-}2\right)!}{\left(\frac{j}{2}\right)!}\left[ \begin{matrix} r & r & i\\ r & r & j \end{matrix} \right]_{T=1}^N \sim \dfrac{(-1)^N(N-1)!(N-2)!}{\sqrt{12\pi |V(\widetilde{J}_n)|}} \cos\left(\sum_{i=1}^{6} \widetilde{J}_n \cdot \Omega(\widetilde{J}_n) + \dfrac{\pi}{4} \right).$$ Let us note, that the generalized formula when all parameters of 6-j symbol are the same does not correspond to the regular tetrahedron if $N>2$. Due to this fact we can not simplify the relation further. Interestingly, the resulting tetrahedron is deformed for every type differently. In particular, type II corresponds to the trigonal pyramid, whereas type I is a bent tetrahedron, which is combined of 4 equal isosceles triangles. Symmetries derivation {#S6} ===================== Hypergeometric symmetries group ------------------------------- In this subsection we do not write any symmetries explicitly. Here we are describing the structure of obtained symmetries. The statements in this subsection are given without analytical proof, but it has been checked manually. We use both permutation symmetry (\[trans\_hyp\_perm\]) and Sears’ transformation (\[trans\_sears\]) in order to get all possible 6-j symbol transformations. The arbitrary composition of Sears’ transformations and permutations can be written as: $$\begin{aligned} \label{om_sys} {}_4\Phi_3 \left( \begin{matrix} a_1,a_2,a_3,a_4\\ b_1,b_2,b_3 \end{matrix}; q,q \right)= \widetilde{C}\cdot {}_4\Phi_3 \left( \begin{matrix} \widetilde{a}_1,\widetilde{a}_2,\widetilde{a}_3,\widetilde{a}_4\\ \widetilde{b}_1,\widetilde{b}_2,\widetilde{b}_3 \end{matrix}; q,q \right),\end{aligned}$$ where variables with $ \ \widetilde{}\ $ denotes the resulting arguments. There is a factor $C$ that appears after Sears’ transformations, but we are not interested in it for now. The resulting symmetry has the following form: $$\left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_T^N = C\left[ \begin{matrix} \widetilde{r}_1 & \widetilde{r}_2 & \widetilde{i}\\ \widetilde{r}_3 & \widetilde{r}_4 & \widetilde{j} \end{matrix} \right]_T^M,$$ where $\widetilde{r}_n,\widetilde{i},\widetilde{j}$ are some linear combination of $r_n,i,j$ obtained by the mentioned transformations. Parameters $N,M$ denote the ranks of the corresponding algebras. To find the symmetries we have to solve the linear system of equations on arguments $\widetilde{r}_n,\widetilde{i},\widetilde{j},M$. Initially we consider $M=N$ to get a unique solution. The rank of the system is 6, because the hypergeometric function has 7 arguments with one additional constraint. Note, that we do not restrict them to the fusion rules when we solve the system. That is done because Sears’ transformation do not respect the fusion rules, but some of its combinations with permutations do. Hence we need to obtain all symmetries and then recover fusion rules using (\[Nshift\]). In this subsection we do not consider the relation (\[Nshift\]) as a symmetry because it is used to satisfy fusion rules by fixing parameter $M$. The overall set of symmetries $G$ that contains all compositions of permutations and Sears’ transformation is a group and it has 23040 elements in total [@23040]. For $N=M=2$ case this group was discovered in [@23040_Doyle], where it was called 22.5K group. They claimed that it is in fact Coxeter group $D6$, which arises in hyperbolic geometry as the group of hyperbolic tetrahedra symmetries. The volume of a hyperbolic tetrahedron is known to be connected with the quantum 6-j symbol of $U_q(sl_2)$ in an appropriate limit [@Murakami_Yano]. Our result was obtained via the computer algebra system. Permutations and Sears’ transformations were programmed explicitly and combined multiple times. By fixing all the constraints on permutations and Sears’ transformation, the program reached 23040 elements. It was checked that they are closed under composition. Each symmetry is non-degenerate due to the non-degeneracy of the initial equations, hence all elements are invertible. As a result, 23040 symmetries including identity form a group. Most of these symmetries cannot be applied in 6-j symbols because they often do not preserve the positiveness of $r_n,i,j$. Also the structure of its subgroups is not clear and it makes the analysis more complicated. Thus, we are interested only in the subgroup that generalizes the known set of symmetries from $N=M=2$ to arbitrary $N$ and $M$, let us call it $S\subset G$. There are 144 elements in $S$ and it is analogous to the $U_q(sl_2)$ group of permutations and Regge transformations, which we denote as $H=S\big|_{N=M=2}\big.$. Moreover, these groups are in one to one correspondence: each symmetry for $N\neq 2 \neq M$ may be transformed to a $N=2=M$ symmetry and vice versa. Note, that the found symmetries from $S$ are well-defined for hypergeometric series, but for 6-j symbols they require the positiveness of $r_n,i,j,M-2$. The other symmetries from $G$ are out of our scope in the next discussion. The reformulation of symmetries from $G$ in terms of 6-j symbol have some difficulties. On the one hand, the number of group elements is too large to analyze the symmetries manually, on the other hand the subgroups structure is still unclear. Also there are a lot of symmetries that do not preserve the positiveness of representation parameters, so a lot of symmetries can not be applied to 6-j symbols. Interestingly, the whole group may be obtained as a combination of symmetries $S$ and the following one: $$\left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_T^N = \left[ \begin{matrix} r_1 & r_2 & i\\ -r_3-1 & r_4 & j \end{matrix} \right]_T^N.$$ After the transformation (\[Nshift\]) is used to find $M$, it is natural to consider two classes of symmetries: one for $N=M$ and another for $N\neq M$. If the symmetry requires $N=M$, we call it the internal one, else we call it the external symmetry. The set of internal and external symmetries are denoted by $I$ and $E$ respectively. Let us provide this definition with examples of both internal and external symmetries. The internal symmetry: $$\left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_2^N = \left[ \begin{matrix} r_2 & r_1 & i\\ r_4 & r_3 & j \end{matrix} \right]_2^N.$$ The fusion rules (\[fusion\_rules\]) formally require two equalities for LHS and RHS. However, they are linearly dependent in this case, so the equality for one side yields the equality for the other side. Moreover, if $N\neq M\neq 2$, the conditions are in contradiction. The external symmetry: $$\left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_1^N = C\left[ \begin{matrix} r_1 & i+N-M & r_2+N-M\\ r_3 & j+N-M & r_4+N-M \end{matrix} \right]_1^M,$$ where $C$ is some factor. Here we have to restrict representations by two equalities: $r_1+r_3= r_2+r_4$ and $r_1+r_3 = i + j+2(N-M)$, so we should fix $2M=2N+i+j-r_1-r_3$: $$\left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_1^N = C\left[ \begin{matrix} r_1 & \dfrac{r_2+r_4+i-j}{2} & \dfrac{3r_2+r_4-i-j}{2}\\ r_3 & \dfrac{r_2+r_4-i+j}{2} & \dfrac{r_2+3r_4-i-j}{2} \end{matrix} \right]_1^{N+\frac{i+j-r_2-r_4}{2}}.$$ Parameters of the transformed 6-j symbol on the RHS have to be non-negative. Parameters $\widetilde{r}_n,\widetilde{i},\widetilde{j}$ are non-negative for each external symmetry, as it will be derived in Appendix. On the other hand, $M$ still have to be grater then or equal to 2, so not all 6-j symbols may be transformed by this symmetry. Each external symmetry induces a subset of 6-j symbols that has such a relation. \[st5\] For any non-trivial 6-j symbol $\left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_T^N$ the external symmetry of any type transforms it into the 6-j symbol with non-negative $\widetilde{r}_n,\widetilde{i},\widetilde{j}$. The proof of this statement uses explicit relations for 6-j symbol symmetries and it is proven in Appendix. The internal symmetries of 6-j symbols form group $I$ with the following structure. It is isomorphic to either $\mathbb{S}_4$ for type [I]{} or $\mathbb{S}_3 \times \mathbb{Z}_2$ for type [II]{}. If we consider only internal symmetries, we obtain subgroup $I\subset S$. One can check in a straightforward way that $|I|=24$ for type I, $|I|=12$ for type II and the symmetries are isomorphic to mentioned groups. $$\begin{aligned} &G \quad \supset \quad S \stackrel{N=M}{\supset} I, \qquad E:=S/I,\nonumber\\\nonumber \text{Type I:}\qquad&S \cong \mathbb{S}_4 \times \mathbb{S}_3, \qquad I \cong \mathbb{S}_4, \\\nonumber \text{Type II:}\qquad&S \cong \mathbb{S}_4 \times \mathbb{S}_3, \qquad I \cong \mathbb{S}_3 \times \mathbb{Z}_2,\\\nonumber &|G|=23040,\ |S|=144\end{aligned}$$ The explicit relations are written in the next subsections. The internal symmetries from $I$ may be applied to any 6-j symbol of the corresponding type. In other words, for every $r_n,i,j$ with the satisfied fusion rules it is possible to write down all symmetries from $I$. External symmetries relate 6-j symbols for different algebras. There are two important things to note here. Firstly, 6-j symbols and $_4\Phi_3$ differs by a factor that is not always invariant under external symmetries, so we need to add a normalizing factor to this symmetry. Secondly, since there are two group ranks $N$ and $M$, both of them should be greater than or equal to 2 for the symmetry to be valid. As a result, it may be applied only to the part of all type I and type II 6-j symbols. Let us note that for $U_q(sl_2)$ there are no restrictions from fusion rules, therefore $S$ coincides with $I$ and we have all $144$ symmetries [@klimyk]. Both internal and external symmetries can be derived using the relation (\[rollback\]) between $U_q(sl_2)$ 6-j symbols and MFS. This method may also be used to check the obtained equalities. If one expresses the list of MFS symmetries as $U_q(sl_2)$ 6-j symbols equalities, factors can be reduced and the equalities form the list of $U_q(sl_2)$ symmetries. Type I internal symmetries {#SS2} -------------------------- In this subsection we write down the internal symmetries of type I. These symmetries are very similar to the known ones and can be seen as a natural generalization of the symmetries from $U_q(sl_2)$, although in terms of Young diagrams it’s not obvious. In the shortened notation it is easy to see the correspondence between $U_q(sl_2)$ and $U_q(sl_N)$ symmetries. Although the internal symmetries of type I by definition need $r_1+r_3 = r_2 + r_4$ to be satisfied, we do not write it explicitly because in every equality either both 6-j symbols exist or both of them do not. The same idea is used for type II internal symmetries. To write down the symmetries in a more compact way, we use the following variables: $$\label{rho_def} \rho = \dfrac{r_1+r_2+r_3 +r_4}{2} \hspace{5mm} \rho' = \dfrac{r_2+i +r_4+j}{2} = \dfrac{r_1+i+r_3 +j}{2} = \rho''.$$ All 6-j symbols below are equal and form group $I$. Columns of the equality list correspond to row permutations, rows correspond to Regge symmetries analogue: [$$\begin{aligned} \left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_1^N \hspace{22pt} = \hspace{22pt} \left[ \begin{matrix} r_3 & r_4 & i\\ r_1 & r_2 & j \end{matrix} \right]_1^N \hspace{22pt} &= \hspace{22pt} \left[ \begin{matrix} r_1 & r_4 & j\\ r_3 & r_2 & i \end{matrix} \right]_1^N \hspace{22pt} = \hspace{22pt}\left[ \begin{matrix} r_3 & r_2 & j\\ r_1 & r_4 & i \end{matrix} \right]_1^N \\=\nonumber \left[ \begin{matrix} \rho-r_4 & \rho-r_3 & i\\ \rho-r_2 & \rho-r_1 & j \end{matrix} \right]_1^N \hspace{6pt} = \hspace{6pt} \left[ \begin{matrix} \rho-r_2 & \rho-r_1 & i\\ \rho-r_4 & \rho-r_3 & j \end{matrix} \right]_1^N \hspace{6pt} &= \hspace{6pt} \left[ \begin{matrix} \rho-r_4 & \rho-r_1 & j\\ \rho-r_2 & \rho-r_3 & i \end{matrix} \right]_1^N \hspace{6pt} = \hspace{6pt} \left[ \begin{matrix} \rho-r_2 & \rho-r_3 & j\\ \rho-r_4 & \rho-r_1 & i \end{matrix} \right]_1^N \\=\nonumber \left[ \begin{matrix} r_1 & \rho'-j & \rho'-r_4\\ r_3 & \rho'-i & \rho'-r_2 \end{matrix} \right]_1^N \hspace{3pt} = \hspace{3pt} \left[ \begin{matrix} r_3 & \rho'-i & \rho'-r_4\\ r_1 & \rho'-j & \rho'-r_2 \end{matrix} \right]_1^N \hspace{3pt} &= \hspace{3pt} \left[ \begin{matrix} r_1 & \rho'-i & \rho'-r_2\\ r_3 & \rho'-j & \rho'-r_4 \end{matrix} \right]_1^N \hspace{3pt} = \hspace{3pt} \left[ \begin{matrix} r_3 & \rho'-j & \rho'-r_2\\ r_1 & \rho'-i & \rho'-r_4 \end{matrix} \right]_1^N \\= \nonumber \left[ \begin{matrix} \rho''-j & r_2 & \rho''-r_3\\ \rho''-i & r_4 & \rho''-r_1 \end{matrix} \right]_1^N \hspace{1pt} = \hspace{1pt} \left[ \begin{matrix} \rho''-i & r_4 & \rho''-r_3\\ \rho''-j & r_2 & \rho''-r_1 \end{matrix} \right]_1^N \hspace{1pt} &= \hspace{1pt} \left[ \begin{matrix} \rho''-j & r_4 & \rho''-r_1\\ \rho''-i & r_2 & \rho''-r_3 \end{matrix} \right]_1^N \hspace{1pt} = \hspace{1pt} \left[ \begin{matrix} \rho''-i & r_2 & \rho''-r_1\\ \rho''-j & r_4 & \rho''-r_3 \end{matrix} \right]_1^N \\= \nonumber \left[ \begin{matrix} \rho''{-}j & \rho{-}r_3 & \rho'{-}r_4\\ \rho''{-}i & \rho{-}r_1 & \rho'{-}r_2 \end{matrix} \right]_1^N = \left[ \begin{matrix} \rho''{-}i & \rho{-}r_1 & \rho'{-}r_4\\ \rho''{-}j & \rho{-}r_3 & \rho'{-}r_2 \end{matrix} \right]_1^N &= \left[ \begin{matrix} \rho''{-}j & \rho{-}r_1 & \rho'{-}r_2\\ \rho''{-}i & \rho{-}r_3 & \rho'{-}r_4 \end{matrix} \right]_1^N = \left[ \begin{matrix} \rho''{-}i & \rho{-}r_3 & \rho'{-}r_2\\ \rho''{-}j & \rho{-}r_1 & \rho'{-}r_4 \end{matrix} \right]_1^N \\= \nonumber \left[ \begin{matrix} \rho{-}r_4 & \rho'{-}j & \rho''{-}r_3\\ \rho{-}r_2 & \rho'{-}i & \rho''{-}r_1 \end{matrix} \right]_1^N = \left[ \begin{matrix} \rho{-}r_2 & \rho'{-}i & \rho''{-}r_3\\ \rho{-}r_4 & \rho'{-}j & \rho''{-}r_1 \end{matrix} \right]_1^N &= \left[ \begin{matrix} \rho{-}r_4 & \rho'{-}i & \rho''{-}r_1\\ \rho{-}r_2 & \rho'{-}j & \rho''{-}r_3 \end{matrix} \right]_1^N = \left[ \begin{matrix} \rho{-}r_2 & \rho'{-}j & \rho''{-}r_1\\ \rho{-}r_4 & \rho'{-}i & \rho''{-}r_3 \end{matrix} \right]_1^N.\end{aligned}$$ ]{} These 24 symmetries form a representation of group $I$ mentioned above. It has two notable subgroups: row permutations and Regge transformations analogue. The isomorphism $I \cong \mathbb{S}_4$ is as follows. Permutations from the first row correspond to $\{(),(12)(34), (14)(23), (13)(24)\}$. The first column symmetries correspond to $\{(),(12), (23), (13), (123), (132)\}$. All others can be read from the table: $$\begin{tabular}{|c|c|c|c|} \hline () & (12)(34) & (14)(23) & (13)(24) \\ \hline (12) & (34) & (1324) & (1423) \\ \hline (23) & (1243) & (14) & (1342) \\ \hline (13) & (1432) & (1234) & (24) \\ \hline (123) & (243) & (134) & (142) \\ \hline (132) & (143) & (124) & (234) \\ \hline \end{tabular}$$ We can write down the generalization of Regge transformations (\[Regge\]): [align]{} \_1\^N =& \_1\^N= \_1\^N. Let us give a couple of examples of these symmetries: - Regge symmetry analogue, type I (1^st^ column is invariant, $N\ge2$): [$$\begin{aligned} \hspace{-5mm}\left\lbrace \begin{matrix} [8] & \overline{[4]} & [12,4^{N-2}]\\ [10] & [14] & [6] \end{matrix} \right\rbrace&=\left\lbrace \begin{matrix} [8] & \overline{[6]} & [14,6^{N-2}]\\ [10] & [12] & [4] \end{matrix} \right\rbrace,\ \left\lbrace \begin{matrix} [10] & \overline{[8]} & [18,8^{N-2}]\\ [12] & [14] & [6,5^{N-2}] \end{matrix} \right\rbrace=\left\lbrace \begin{matrix} [10] & \overline{[5]} & [15,5^{N-2}]\\ [12] & [17] & [9,8^{N-2}] \end{matrix} \right\rbrace,\\ \hspace{-5mm}\left\lbrace \begin{matrix} [12] & \overline{[6]} & [16,5^{N-2}]\\ [14] & [20] & [8] \end{matrix} \right\rbrace&=\left\lbrace \begin{matrix} [12] & \overline{[9]} & [19,8^{N-2}]\\ [14] & [17] & [5,5^{N-2}] \end{matrix} \right\rbrace,\ \left\lbrace \begin{matrix} [12] & \overline{[8]} & [10,3^{N-2}]\\ [14] & [18] & [6] \end{matrix} \right\rbrace=\left\lbrace \begin{matrix} [12] & \overline{[11]} & [13,6^{N-2}]\\ [14] & [15] & [3] \end{matrix} \right\rbrace. \end{aligned}$$]{} - Regge symmetry analogue, type I (2^nd^ column is invariant, $N\ge2$): [$$\begin{aligned} \left\lbrace \begin{matrix} [4] & \overline{[6]} & [2,2^{N-2}]\\ [3] & [1] & [5,4^{N-2}] \end{matrix} \right\rbrace&=\left\lbrace \begin{matrix} [2] & \overline{[6]} & [4,4^{N-2}]\\ [5] & [1] & [3,2^{N-2}] \end{matrix} \right\rbrace,\ \left\lbrace \begin{matrix} [6] & \overline{[5]} & [7,3^{N-2}]\\ [3] & [4] & [2,2^{N-2}] \end{matrix} \right\rbrace=\left\lbrace \begin{matrix} [7] & \overline{[5]} & [6,2^{N-2}]\\ [2] & [4] & [3,3^{N-2}] \end{matrix} \right\rbrace,\\ \left\lbrace \begin{matrix} [5] & \overline{[6]} & [7,4^{N-2}]\\ [4] & [3] & [8,5^{N-2}] \end{matrix} \right\rbrace&=\left\lbrace \begin{matrix} [4] & \overline{[6]} & [8,5^{N-2}]\\ [5] & [3] & [7,4^{N-2}] \end{matrix} \right\rbrace,\ \left\lbrace \begin{matrix} [4] & \overline{[6]} & [2,2^{N-2}]\\ [5] & [3] & [7,4^{N-2}] \end{matrix} \right\rbrace=\left\lbrace \begin{matrix} [2] & \overline{[6]} & [4,4^{N-2}]\\ [7] & [3] & [5,2^{N-2}] \end{matrix} \right\rbrace.\\ \end{aligned}$$]{} Type II internal symmetries {#SS3} --------------------------- One can similarly consider type II, there are only 12 symmetries. For brevity we use the following variables: $$\rho = \dfrac{r_1+r_2+r_3 +r_4}{2} \hspace{5mm} \rho' = \dfrac{r_2+i +r_4+j}{2} \hspace{5mm} \rho'' = \dfrac{r_1+i+r_3 +j}{2}.$$ All 6-j symbols below are equal. Columns of the table correspond to a column permutation, rows correspond to Regge symmetries. $$\begin{aligned} \left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_2^N &= \left[ \begin{matrix} r_2 & r_1 & i\\ r_4 & r_3 & j \end{matrix} \right]_2^N \\= \left[ \begin{matrix}\nonumber \rho-r_3 & \rho-r_4 & i\\ \rho-r_1 & \rho-r_2 & j \end{matrix} \right]_2^N &= \left[ \begin{matrix} \rho-r_4 & \rho-r_3 & i\\ \rho-r_2 & \rho-r_1 & j \end{matrix} \right]_2^N \\= \left[ \begin{matrix}\nonumber r_1 & \rho' - r_4 & \rho' - j\\ r_3 & \rho' - r_2 & \rho' - i \end{matrix} \right]_2^N &= \left[ \begin{matrix} \rho' - r_4 & r_1 & \rho' - j\\ \rho' - r_2 & r_3 & \rho' - i \end{matrix} \right]_2^N \\= \left[ \begin{matrix}\nonumber \rho'' - r_3 & r_2 & \rho'' - j\\ \rho'' - r_1 & r_4 & \rho'' - i \end{matrix} \right]_2^N &= \left[ \begin{matrix} r_2 & \rho'' - r_3 & \rho'' - j\\ r_4 & \rho'' - r_1 & \rho'' - i \end{matrix} \right]_2^N \\= \left[ \begin{matrix}\nonumber \rho-r_3 & \rho' - r_4 & \rho'' - j\\ \rho-r_1 & \rho' - r_2 & \rho'' - i \end{matrix} \right]_2^N &= \left[ \begin{matrix} \rho' - r_4 & \rho-r_3 & \rho'' - j\\ \rho' - r_2 & \rho-r_1 & \rho'' - i \end{matrix} \right]_2^N \\= \left[ \begin{matrix}\nonumber \rho''-r_3 & \rho-r_4 & \rho' - j\\ \rho''-r_1 & \rho-r_2 & \rho' - i \end{matrix} \right]_2^N &= \left[ \begin{matrix} \rho-r_4 & \rho' - r_3 & \rho' - j\\ \rho-r_2 & \rho' - r_1 & \rho' - i \end{matrix} \right]_2^N .\end{aligned}$$ The structure of isomorphism $I\cong \mathbb{S}_3 \times \mathbb{Z}_2$ is as follows: $$\begin{tabular}{|c|c|} \hline ()() & (12)() \\ \hline (12)(ab)& ()(ab) \\ \hline (13)(ab)& (132)(ab) \\ \hline (23)(ab)& (123)(ab) \\ \hline (123)() & (23)() \\ \hline (132)()& (13)() \\ \hline \end{tabular}$$ The Regge transformation is the only new relation here: [align]{} \_2\^N = \_2\^N = \_2\^N Let us give a couple of examples of these symmetries. - Regge symmetry analogue, type II (1^st^ column is invariant, $N\ge2$): $$\begin{aligned} \left\lbrace \begin{matrix} [5] & [6] & [10,1]\\ \overline{[3]} & [8] & [7,5^{N-2}] \end{matrix} \right\rbrace=\left\lbrace \begin{matrix} [5] & {[7]} & [10,2]\\ \overline{[3]} & [9] & [6,5^{N-2}] \end{matrix} \right\rbrace,\ \left\lbrace \begin{matrix} [5] & {[6]} & [11]\\ \overline{[1]} & [10] & [7,6^{N-2}] \end{matrix} \right\rbrace&=\left\lbrace \begin{matrix} [5] & {[7]} & [11,1]\\ \overline{[1]} & [11] & [6,6^{N-2}] \end{matrix} \right\rbrace,\\ \left\lbrace \begin{matrix} [4] & {[6]} & [10]\\ \overline{[1]} & [9] & [7,6^{N-2}] \end{matrix} \right\rbrace=\left\lbrace \begin{matrix} [4] & {[7]} & [10,1]\\ \overline{[1]} & [10] & [6,6^{N-2}] \end{matrix} \right\rbrace,\ \left\lbrace \begin{matrix} [3] & {[6]} & [8,1]\\ \overline{[4]} & [5] & [8,5^{N-2}] \end{matrix} \right\rbrace&=\left\lbrace \begin{matrix} [3] & {[8]} & [8,3]\\ \overline{[4]} & [7] & [6,5^{N-2}] \end{matrix} \right\rbrace. \end{aligned}$$ - Regge symmetry analogue, type II (2^nd^ column is invariant, $N\ge2$): $$\begin{aligned} \left\lbrace \begin{matrix} [4] & [2] & [6]\\ \overline{[1]} & [5] & [3,2^{N-2}] \end{matrix} \right\rbrace=\left\lbrace \begin{matrix} [6] & {[2]} & [6,2]\\ \overline{[3]} & [5] & [1] \end{matrix} \right\rbrace,&\ \left\lbrace \begin{matrix} [4] & {[3]} & [6,1]\\ \overline{[1]} & [6] & [2,2^{N-2}] \end{matrix} \right\rbrace=\left\lbrace \begin{matrix} [5] & {[3]} & [6,2]\\ \overline{[2]} & [6] & [1,1^{N-2}] \end{matrix} \right\rbrace,\\ \left\lbrace \begin{matrix} [5] & {[6]} & [10,1]\\ \overline{[4]} & [7] & [10,6^{N-2}] \end{matrix} \right\rbrace=\left\lbrace \begin{matrix} [10] & {[6]} & [10,6]\\ \overline{[9]} & [7] & [5,1^{N-2}] \end{matrix} \right\rbrace,&\ \left\lbrace \begin{matrix} [5] & {[6]} & [9,2]\\ \overline{[2]} & [9] & [4,4^{N-2}] \end{matrix} \right\rbrace=\left\lbrace \begin{matrix} [7] & {[6]} & [9,4]\\ \overline{[4]} & [9] & [2,2^{N-2}] \end{matrix} \right\rbrace. \end{aligned}$$ Type I external symmetries {#SS4} -------------------------- In this subsection we consider external symmetries from the group $S$. Let us denote by $\cong$ a external symmetry between two 6-j symbols with additional inequality restriction $M \ge 2$. For brevity we also drop out factors that occur in equalities and can be written as $C=(-1)^{N-M}\frac{K_T(r_1,r_2,r_3,r_4,i,j,N)}{K_T(\widetilde{r}_1,\widetilde{r}_2,\widetilde{r}_3,\widetilde{r}_4,\widetilde{i},\widetilde{j},M)}$. Let us consider external symmetries of type I. It is convenient to write down not the whole set $S\setminus I$, but the factor $E=S/I$. In $U_q(sl_2)$ we have the subgroups of Regge transformations, row and column permutations, one can notice that here we also have similar subgroups. The external symmetries for type I are analogous to column permutations and may be easily written with notations $\Delta_i=N-M_i$, $n_i=M_i-2$, $n_0=N-2$. $$\begin{aligned} \left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_1^N \cong &\left[ \begin{matrix}\label{w1s1} r_1 & i+\Delta_1 & r_2+\Delta_1\\ r_3 & j+\Delta_1 & r_4+\Delta_1 \end{matrix} \right]_1^{M_1} \cong \left[ \begin{matrix} i+n_0 & r_2+\Delta_2 & r_1-n_2\\ j+n_0 & r_4+\Delta_2 & r_3-n_2 \end{matrix} \right]_1 ^{M_2}\\ \cong &\left[ \begin{matrix}\label{w1s2} i+n_0 & r_1-n_3 & r_2+\Delta_3\\ j+n_0 & r_3-n_3 & r_4+\Delta_3 \end{matrix} \right]_1 ^{M_3} \cong \left[ \begin{matrix} r_2+n_0 & i+\Delta_4 & r_1-n_4 \\ r_4+n_0 & j+\Delta_4 & r_3-n_4 \end{matrix} \right]_1 ^{M_4}\\ \stackrel{N=M_5=2}{\cong} &\left[ \begin{matrix}\label{w1s3} r_2+n_0 & r_1-n_5 & i+\Delta_5\\ r_4+n_0 & r_3-n_5 & j+\Delta_5 \end{matrix} \right]_1^{M_5},\end{aligned}$$ where $n_i$, $\Delta_i$ and $M_i$ are fixed by fusion rules. We emphasize that $E$ is isomorphic to $\mathbb{S}_3$ only for $N=M_5=2$. In this case 6 elements from above are represented by $\{(),(23),(13),(132),(123),\textbf{(12)}\}$ correspondingly. In general, it is impossible to satisfy the fusion rules, so $E$ have only 4 transformations which are not closed under composition and $E \cong \mathbb{S}_3\setminus \{(12)\}$. These symmetries are interesting because they cannot be expressed as a combination of any known symmetries. From hypergeometric point of view these symbols have the same value of ${}_4\Phi_3$ but it’s still possible that $K_T$ is changed by this transformation. Let us write down a few examples of these symmetries: - The first symmetry, $N=M_1=4$: $$\begin{aligned} \left\lbrace \begin{matrix} [3] & \overline{[1]} & [4,1^{2}]\\ [6] & [8] & [5,1^{2}] \end{matrix} \right\rbrace& = \left\lbrace \begin{matrix} [3] & \overline{[4]} & [1,1^{2}]\\ [6] & [5] & [8,3^{2}] \end{matrix} \right\rbrace. \end{aligned}$$ - The second symmetry, $N=4, M_2=3$: $$\begin{aligned} \left\lbrace \begin{matrix} [5] & \overline{[4]} & [1]\\ [7] & [8] & [9,3^{2}] \end{matrix} \right\rbrace& = -\sqrt{\dfrac{[2]_q[3]_q}{[5]_q[8]_q}}\left\lbrace \begin{matrix} [3] & \overline{[5]} & [4,3]\\ [11] & [9] & [6] \end{matrix} \right\rbrace. \end{aligned}$$ - The third symmetry, $N=4,M_3=2$: $$\begin{aligned} \left\lbrace \begin{matrix} [7] & \overline{[3]} & [4]\\ [2] & [6] & [1,1^{2}] \end{matrix} \right\rbrace& = \sqrt{\dfrac{[2]_q[3]_q}{[5]_q[6]_q}}\left\lbrace \begin{matrix} [6] & \overline{[7]} & [5]\\ [3] & [2] & [8] \end{matrix} \right\rbrace. \end{aligned}$$ - The fourth symmetry, $N=4,M_4=5$: $$\begin{aligned} \left\lbrace \begin{matrix} [6] & \overline{[4]} & [8,3^{2}]\\ [5] & [7] & [9,4^{2}] \end{matrix} \right\rbrace& = -\dfrac{[2]_q[3]_q}{[7]_q}\sqrt{\dfrac{1}{[6]_q^3}}\left\lbrace \begin{matrix} [6] & \overline{[7]} & [3,2^{3}]\\ [9] & [8] & [2] \end{matrix} \right\rbrace. \end{aligned}$$ Type II external symmetries {#SS5} --------------------------- In a similar way we can consider type II symmetries $E=S/I$ and fix $M$ by transformation (\[Nshift\]). These symmetries are analogous to a column permutation and row permutations: $$\begin{aligned} \hspace{-5mm}&\left[ \begin{matrix}\label{w2s1} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_2^N \cong \left[ \begin{matrix} j + \Delta_1 & r_1 + \Delta_1 & r_4\\ i & r_3 & r_2 + \Delta_1 \end{matrix} \right]_2^{M_1} \cong \left[ \begin{matrix} r_2 + \Delta_2 & j + \Delta_2 & r_3\\ r_4 & i & r_1 + \Delta_2 \end{matrix} \right]_2^{M_2} \cong\\\hspace{-5mm}\cong &\left[ \begin{matrix}\nonumber i-n_3 & r_1+\Delta_3 & r_2+n_0\\ j+n_0 & r_3 & r_4-n_3 \end{matrix} \right]_2^{M_3} \cong \left[ \begin{matrix}\nonumber r_2+\Delta_4 & i-n_4 & r_1+n_0\\ r_4 & j+n_0 & r_3-n_4 \end{matrix} \right]_2^{M_4} \cong \left[ \begin{matrix}\nonumber r_4-n_5 & j+\Delta_5 & r_1+n_0\\ r_2+n_0 & i & r_3-n_5 \end{matrix} \right]_2^{M_5} \cong\\\hspace{-5mm}\cong &\left[ \begin{matrix}\nonumber j+\Delta_6 & r_3-n_6 & r_2+n_0\\ i & r_1+n_0 & r_4-n_6 \end{matrix} \right]_2^{M_6} \cong \left[ \begin{matrix}\nonumber r_1+\Delta_7 & r_4-n_7 & j+n_0\\ r_3 & r_2+n_0 & i-n_7 \end{matrix} \right]_2^{M_7} \cong \left[ \begin{matrix}\nonumber r_3-n_8 & r_2+\Delta_8 & j+n_0\\ r_1+n_0 & r_4 & i-n_8 \end{matrix} \right]_2^{M_8} \cong\\\hspace{-5mm}\cong &\left[ \begin{matrix}\nonumber r_4-n_9 & i-n_9 & r_3\\ r_2+n_0 & j+n_0 & r_1+\Delta_9 \end{matrix} \right]_2^{M_9} \cong \left[ \begin{matrix}\nonumber i-n_{10} & r_3-n_{10} & r_4\\ j+n_0 & r_1+n_0 & r_2+\Delta_{10} \end{matrix} \right]_2^{M_{10}} \stackrel{N=M_{11}=2}{\cong} \left[ \begin{matrix} r_3-n_{11} & r_4-n_{11} & i\\ r_1+n_0 & r_2+n_0 & j+\Delta_{11} \end{matrix} \right]_2^{M_{11}} \nonumber.\end{aligned}$$ We emphasize that the last 6-j symbol exists only for $N=M_{11}=2$ as it is impossible to satisfy the inequalities otherwise. The isomorphism $E\big{|}_{N=M_{11}=2}\cong \mathbb{A}_4$ is as follows. The first row correspond to elements $\{(),(143), (134)\}$. The first column is presented by $\{(), (132), (234), (243)\}$. Other elements can be read from the table: $$\begin{tabular}{|c|c|c|} \hline () & (143) & (134) \\ \hline (132) & (123) & (142) \\ \hline (234) & (14)(23) & (13)(24) \\ \hline (243) & (124) & \textbf{(12)(34)} \\ \hline \end{tabular}$$ If we consider arbitrary $N$, $E$ is not closed under compositions and $E \cong \mathbb{A}_4 \setminus \{(12)(34)\}$. Let us write down a few examples of these symmetries: - The first symmetry, $N=M_1=4$: $$\begin{aligned} \left\lbrace \begin{matrix} [5] & [2] & [7]\\ \overline{[4]} & [3] & [6,2^{2}] \end{matrix} \right\rbrace& = \left\lbrace \begin{matrix} [6] & [5] & [7,4]\\ \overline{[7]} & [4] & [2,2^{2}] \end{matrix} \right\rbrace. \end{aligned}$$ - The third symmetry, $N=4, M_3=5$: $$\begin{aligned} \left\lbrace \begin{matrix} [8] & [1] & [9]\\ \overline{[5]} & [4] & [6,1^{2}] \end{matrix} \right\rbrace& = -\sqrt{\dfrac{[10]_q[8]_q}{[4]_q[3]_q}}\left\lbrace \begin{matrix} [6] & [7] & [8,5]\\ \overline{[8]} & [5] & [1,1^{3}] \end{matrix} \right\rbrace. \end{aligned}$$ - The seventh symmetry, $N=4,M_7=5$: $$\begin{aligned} \left\lbrace \begin{matrix} [8] & [1] & [9]\\ \overline{[5]} & [4] & [6,1^{2}] \end{matrix} \right\rbrace& = -\sqrt{\dfrac{[10]_q[8]_q}{[4]_q[3]_q}}\left\lbrace \begin{matrix} [7] & [1] & [8]\\ \overline{[5]} & [3] & [6,1^{3}] \end{matrix} \right\rbrace. \end{aligned}$$ - The ninth symmetry, $N=4,M_9=3$: $$\begin{aligned} \left\lbrace \begin{matrix} [8] & [4] & [6,3]\\ \overline{[3]} & [9] & [5,3^{2}] \end{matrix} \right\rbrace& = -\sqrt{\dfrac{[2]_q^2[3]_q[6]_q^2[10]_q}{[4]_q^3[5]_q[11]_q[12]_q}}\left\lbrace \begin{matrix} [8] & [5] & [8,5]\\ \overline{[6]} & [7] & [9,5] \end{matrix} \right\rbrace. \end{aligned}$$ Main results ============ In this section we collect the most important results obtained in the paper. We are using the special notation (\[MFS\_nota\_t1\],\[MFS\_nota\_t2\]) for MFS. - Expression (\[final\_expr\_hyp\]) for MFS via q-hypergeometric series: $$\begin{gathered} \left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_T^N =K_T\cdot {}_4\Phi_3 \left( \begin{matrix} a_1,a_2,a_3,a_4\\ b_1,b_2,b_3 \end{matrix}; q,q \right),\\ 2 a_i = \left( \begin{matrix} -r_1-r_2+i - 2(N-2)\delta_{T,2}\\ -r_3-r_4+i\\ -r_1-r_4+j\\ -r_2-r_3+j \end{matrix} \right), \qquad 2 b_i = \left( \begin{matrix} -r_1-r_2-r_3-r_4-2(N-1)\\ i+j-r_2-r_4+2\\ i+j-r_1-r_3 + 2 + 2(N-2)\delta_{T,1}\\\end{matrix} \right).\end{gathered}$$ Factor $K_T$ depends on type $T$ and defined as in (\[coef\_K\]): [$$\hspace{-10mm} K_T = \dfrac{\theta_N\left(r_1,r_2,i\right) \theta_N\left(r_3, r_4, i\right) \theta_N\left(r_1, r_4, j\right) \theta_N\left(r_2, r_3, j\right) [N-1]_q![N-2]_q! [\frac{r_1+r_2+r_3+r_4}{2}+N-1]_q! }{[\frac{r_3+r_4-i}{2}]_q! [\frac{r_1+r_2-i}{2} + (N-2)\delta_{T,2}]_q! [\frac{r_2+r_3-j}{2}]_q! [\frac{r_1+r_4-j}{2}]_q! [\frac{i+j-r_2-r_4}{2}]_q! [\frac{i+j-r_1-r_3}{2}+(N-2)\delta_{T,1}]_q!}.$$]{} - Relation (\[rollback\]) between MFS and $U_q(sl_2)$ 6-j symbols: $$\begin{aligned} \left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_1^N &=\left\{\begin{matrix} r_1& r_2+N-2 &i+N-2 \\ r_3 & r_4+N-2 & j+N-2 \end{matrix} \right\}(-1)^N [N-1]_q! [N-2]_q! \cdot \Theta_1(N),\\ \left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_2^N &=\left\{\begin{matrix} r_1+N-2 & r_2+N-2 &i\\ r_3 & r_4 & j+N-2 \end{matrix} \right\}(-1)^N [N-1]_q! [N-2]_q! \cdot \Theta_2(N),\end{aligned}$$ with factors $\Theta_1,\Theta_2$ defined in (\[coef\_theta\]): [$$\begin{aligned} \Theta_1(N)&= \left(\prod_{m=1}^{N-2} \left[\frac{i {-} r_1 {+} r_2}{2}+m\right]_q \left[\frac{j {+} r_2 {-} r_3}{2}+m\right]_q \left[\frac{j {-} r_1 {+} r_4}{2}+m\right]_q \left[\frac{i {-} r_3 {+} r_4}{2}+m\right]_q \right)^{-\frac{1}{2}},\\ \Theta_2(N)&= \left(\prod_{m=1}^{N-2} \left[\frac{r_1 {+} r_2{-}i}{2}+m\right]_q \left[\frac{j {+} r_2 {-} r_3}{2}+m\right]_q \left[\frac{j {+} r_1 {-} r_4}{2}+m\right]_q \left[\frac{i {+} r_3 {+} r_4}{2}+1+m\right]_q \right)^{-\frac{1}{2}}.\end{aligned}$$]{} - The asymptotics (\[asympt\]) of MFS for $U(sl_N)$: $$\dfrac{1}{\Theta_T(N)}\left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_T^N \sim \dfrac{(-1)^N\cdot(N-1)!\cdot(N-2)!}{\sqrt{12\pi\cdot |V(\widetilde{J}_k)|}} \cos\left(\sum_{n=1}^{6} \widetilde{J}_k \cdot \Omega(\widetilde{J}_k) + \dfrac{\pi}{4} \right),$$ where $\widetilde{J}_k$ are defined in (\[J\_def\]). Symmetries of 6-j symbols ------------------------- There is a group of MFS symmetries that has 144 elements in total. It is convenient to split them into the internal and external symmetries. The internal ones always act in $U_q(sl_N)$, the external ones connect $U_q(sl_N)$ and $U_q(sl_M)$ 6-j symbols. - Counterpart of the Regge transformations (\[regge\_I\]), type I ($\rho' = \frac{r_1+r_3+i +j}{2} = \frac{r_2+r_4+i +j}{2}$): $$\begin{aligned} \left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_1^N =& \left[ \begin {matrix} { r_1}&\rho' - j&\rho' - r_4\\ { r_3}&\rho' - i&\rho' - r_2\end {matrix} \right]_1^N= \left[ \begin {matrix} \rho' - j &r_2&\rho' - r_3\\ \rho' - i&r_4&\rho' - r_1\end {matrix} \right]_1^N. \end{aligned}$$ - Counterpart of the Regge transformations (\[regge\_II\]), type II ($\rho' = \frac{r_1+r_3+i +j}{2}, \rho''= \frac{r_2+r_4+i +j}{2}$): $$\begin{aligned} \left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_2^N = \left[ \begin {matrix} { r_1}&\rho' - r_4&\rho' - j\\ { r_3}&\rho' - r_2&\rho' - i\end {matrix} \right]_2^N = \left[ \begin {matrix} \rho'' - r_3 &r_2&\rho'' - j\\ \rho'' - r_1&r_4&\rho'' - i\end {matrix} \right]_2^N.\end{aligned}$$ The next symmetries are between $U_q(sl_N)$ and $U_q(sl_M)$ 6-j symbols. Values of $M_i$ are fixed by fusion rules. For brevity we use the notations $\Delta_i=N-M_i$, $n_i=M_i-2$, $n_0=N-2$. - Type I external symmetries (\[w1s1\]): [$$\begin{aligned} &\left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_1^N \cong \left[ \begin{matrix} r_1 & i+\Delta_1 & r_2+\Delta_1\\ r_3 & j+\Delta_1 & r_4+\Delta_1 \end{matrix} \right]_1^{M_1} \cong \left[ \begin{matrix} i+n_0 & r_2+\Delta_2 & r_1-n_2\\ j+n_0 & r_4+\Delta_2 & r_3-n_2 \end{matrix} \right]_1 ^{M_2}\cong\\ \cong &\left[ \begin{matrix}\nonumber i+n_0 & r_1-n_3 & r_2+\Delta_3\\ j+n_0 & r_3-n_3 & r_4+\Delta_3 \end{matrix} \right]_1 ^{M_3} \cong \left[ \begin{matrix} r_2+n_0 & i+\Delta_4 & r_1-n_4 \\ r_4+n_0 & j+\Delta_4 & r_3-n_4 \end{matrix} \right]_1 ^{M_4} \stackrel{N=M_5=2}{\cong} \left[ \begin{matrix} r_2+n_0 & r_1-n_5 & i+\Delta_5\\ r_4+n_0 & r_3-n_5 & j+\Delta_5 \end{matrix} \right]_1^{M_5}.\end{aligned}$$]{} - Type II external symmetries (\[w2s1\]): [$$\begin{aligned} \hspace{-5mm}&\left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_2^N \cong \left[ \begin{matrix} j + \Delta_1 & r_1 + \Delta_1 & r_4\\ i & r_3 & r_2 + \Delta_1 \end{matrix} \right]_2^{M_1} \cong \left[ \begin{matrix} r_2 + \Delta_2 & j + \Delta_2 & r_3\\ r_4 & i & r_1 + \Delta_2 \end{matrix} \right]_2^{M_2} \cong\\\hspace{-5mm}\cong &\left[ \begin{matrix}\nonumber i-n_3 & r_1+\Delta_3 & r_2+n_0\\ j+n_0 & r_3 & r_4-n_3 \end{matrix} \right]_2^{M_3} \cong \left[ \begin{matrix}\nonumber r_2+\Delta_4 & i-n_4 & r_1+n_0\\ r_4 & j+n_0 & r_3-n_4 \end{matrix} \right]_2^{M_4} \cong \left[ \begin{matrix}\nonumber r_4-n_5 & j+\Delta_5 & r_1+n_0\\ r_2+n_0 & i & r_3-n_5 \end{matrix} \right]_2^{M_5} \cong\\\hspace{-5mm}\cong &\left[ \begin{matrix}\nonumber j+\Delta_6 & r_3-n_6 & r_2+n_0\\ i & r_1+n_0 & r_4-n_6 \end{matrix} \right]_2^{M_6} \cong \left[ \begin{matrix}\nonumber r_1+\Delta_7 & r_4-n_7 & j+n_0\\ r_3 & r_2+n_0 & i-n_7 \end{matrix} \right]_2^{M_7} \cong \left[ \begin{matrix}\nonumber r_3-n_8 & r_2+\Delta_8 & j+n_0\\ r_1+n_0 & r_4 & i-n_8 \end{matrix} \right]_2^{M_8} \cong\\\hspace{-5mm}\cong &\left[ \begin{matrix}\nonumber r_4-n_9 & i-n_9 & r_3\\ r_2+n_0 & j+n_0 & r_1+\Delta_9 \end{matrix} \right]_2^{M_9} \cong \left[ \begin{matrix}\nonumber i-n_{10} & r_3-n_{10} & r_4\\ j+n_0 & r_1+n_0 & r_2+\Delta_{10} \end{matrix} \right]_2^{M_{10}} \stackrel{N=M_{11}=2}{\cong} \left[ \begin{matrix} r_3-n_{11} & r_4-n_{11} & i\\ r_1+n_0 & r_2+n_0 & j+\Delta_{11} \end{matrix} \right]_2^{M_{11}} \nonumber. \end{aligned}$$]{} Conclusion ========== The 6-j symbols beyond $U_q(sl_2)$ are rapidly becoming very complicated to analyze. Even in the case of symmetric and conjugate to symmetric representations where we know the analytic expression, there are many features that hide from our sight. Firstly, 6-j expression in its original form [@MFS] is the q-factorial series that can be written as a function $_5\Phi_4$, but after some manipulations it became clear that the expression is very similar to $U_q(sl_2)$ one and may be written as (\[final\_expr\]) via $_4\Phi_3$. Secondly, the hypergeometric function has a relation (\[hyp\_relation\]) that is necessary to use the Sears’ transformation. This allow us to think that there is an important class of 6-j symbols with 6 free parameters that is connected with $_4\Phi_3$ series. Considered expression (\[final\_expr\]) is already applicable to $N=2$ case and types I, II. It is an interesting question what else may be expressed via $_4\Phi_3$. The relation (\[rollback\]) between multiplicity-free $U_q(sl_N)$ and $U_q(sl_2)$ symbols reveals the nature of multiplicity-free case. In fact, multiplicity-free 6-j symbols tends to be very similar to $U_q(sl_2)$ one. As was found in [@3SB; @Alekseev:2019], the other class of 6-j symbol with symmetric incoming representations may be expressed via $U_q(sl_2)$ one. The further study of more difficult classes can tell us more about the structure of 6-j symbols, but now we can vividly see that q-hypergeometric series play the main role in this problem. Obtained symmetries show that there are much more relations for 6-j symbols in $U_q(sl_N)$ than tetrahedral symmetries. As the most bright example of this statement, we show that the Regge symmetry is generalizable to both types as (\[regge\_I\], \[regge\_II\]). External symmetries, on the other hand, are less convenient to use, but they provide a lot of new relations that depend on $N$ explicitly and connects 6-j symbols from different $N$. Appendix {#appendix .unnumbered} ======== In this Appendix we write down explicitly the new symmetries mentioned in select results. Also we prove that external symmetries always preserve the non-negativeness of $\widetilde{r}_n,\widetilde{i},\widetilde{j}$. - Counterpart of the Regge transformations (\[regge\_I\]) in terms of Young diagrams, type I: [$$\begin{aligned} \left\lbrace \begin{matrix} [r_1] & \overline{[r_2]} & \left[ i, \frac{r_2-r_1+i}{2}^{N-2} \right]\\ [r_3] & [r_4] & \left[ j, \frac{r_2-r_3+j}{2}^{N-2} \right] \end{matrix} \right\rbrace =& \left\lbrace \begin {matrix} [r_1]& \overline{\left[\dfrac{r_2+r_4-i+j}{2}\right]} &\left[\dfrac{-r_2+r_4+i+j}{2},\dfrac{r_3-r_2+j}{2}^{N-2}\right]\\ [r_3]&\left[\dfrac{r_2+r_4+i-j}{2}\right]&\left[\dfrac{r_2-r_4+i+j}{2}, \dfrac{r_2-r_3+j}{2}^{N-2}\right] \end {matrix} \right\rbrace=\\ =&\left\lbrace \begin {matrix} \left[\dfrac{r_1+r_3-i+j}{2}\right] &\overline{[r_2]}&\left[\dfrac{-r_1+r_3+i+j}{2} , \dfrac{r_2-r_1+i}{2}^{N-2} \right]\\ \left[\dfrac{r_1+r_3+i-j}{2}\right]& [r_4] &\left[\dfrac{r_1-r_3+i+j}{2}, \dfrac{r_2-r_3+j}{2}^{N-2}\right] \end {matrix} \right\rbrace\nonumber. \end{aligned}$$]{} - Counterpart of the Regge transformations (\[regge\_II\]) in terms of Young diagrams, type II: [$$\begin{aligned} \left\lbrace \begin{matrix} [r_1] & [r_2] & \left[\frac{r_1+r_2+i}{2}, \frac{r_1+r_2-i}{2} \right]\\ \overline{[r_3]} & [r_4] & \left[ j, \frac{r_2-r_3+j}{2}^{N-2} \right] \end{matrix} \right\rbrace = & \left\lbrace \begin {matrix} [r_1]& \left[\dfrac{r_2-r_4+i+j}{2}\right] &\left[\dfrac{r_1+r_2+i}{2},\dfrac{r_1-r_4+j}{2}\right]\\ \overline{[r_3]}&\left[\dfrac{r_4-r_2+i+j}{2}\right]&\left[\dfrac{r_2+r_4-i+j}{2}, \dfrac{r_2-r_3+j}{2} ^{N-2}\right] \end {matrix} \right\rbrace = \\ = &\left\lbrace \begin {matrix} \left[\dfrac{r_1-r_3+i+j}{2}\right] &[r_2]&\left[\dfrac{r_1+r_2+i}{2} , \dfrac{r_2-r_3+j}{2} \right]\\ \overline{\left[\dfrac{r_3-r_1+i+j}{2}\right]}& [r_4] &\left[\dfrac{r_1+r_3-i+j}{2}, \dfrac{r_2+r_1-i}{2} ^{N-2}\right] \end {matrix} \right\rbrace\nonumber. \end{aligned}$$]{} - Type I external symmetries (\[w1s1\]): $$\begin{aligned} \left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_1^N \cong &\left[ \begin{matrix} r_1 & \dfrac{r_2+r_4+i-j}{2} & \dfrac{3r_2+r_4-i-j}{2}\\ r_3 & \dfrac{r_2+r_4-i+j}{2} & \dfrac{r_2+3r_4-i-j}{2} \end{matrix} \right]_1^{N+\frac{i+j-r_2-r_4}{2}} \\\cong &\left[ \begin{matrix} i+N-2 & \dfrac{i+j+r_2-r_4}{2} +N-2& \dfrac{i+j+r_1-r_3}{2}\\ j+N-2 & \dfrac{i+j-r_2+r_4}{2}+N-2 & \dfrac{i+j-r_1+r_3}{2} \end{matrix} \right]_1^{2+\frac{r_2+r_4-i-j}{2}}\\ \cong &\left[ \begin{matrix} i+N-2 & \dfrac{i+j+r_1-r_3}{2} & \dfrac{i+j+r_2-r_4}{2}+N-2\\ j+N-2 & \dfrac{i+j-r_1+r_3}{2} & \dfrac{i+j-r_2+r_4}{2}+N-2 \end{matrix} \right]_1 ^{2+\frac{r_1+r_3-i-j}{2}} \\ \cong&\left[ \begin{matrix} r_2+N-2 & \dfrac{r_2+r_4+i-j}{2} +N-2& \dfrac{3r_1+r_3-i-j}{2}\\ r_4+N-2 & \dfrac{r_2+r_4-i+j}{2}+N-2 & \dfrac{r_1+3r_3-i-j}{2} \end{matrix} \right]_1 ^{2+\frac{i+j-r_2-r_4}{2}}\\ \cong &\left[ \begin{matrix} r_2+N-2 & r_1+N-2 & i+2N-4\\ r_4+N-2 & r_3+N-2 & j+2N-4 \end{matrix} \right]_1^{4-N}. \end{aligned}$$ - Type II external symmetries (\[w2s1\]): [$$\begin{aligned} \left[ \begin{matrix} r_1 & r_2 & i\\ r_3 & r_4 & j \end{matrix} \right]_2^N \cong &\left[ \begin{matrix} \frac{r_3+i-r_1+j}{2} & \frac{r_3+i+r_1-j}{2} & r_4\\ i & r_3 & \frac{3r_2+i-r_4-j}{2} \end{matrix} \right]_2^{N+\frac{r_1+j-r_3-i}{2}} \\\cong & \left[ \begin{matrix} \frac{r_4+i+r_2-j}{2} & \frac{r_4+i-r_2+j}{2} & r_3\\ r_4 & i & \frac{3r_1+i-r_3-j}{2} \end{matrix} \right]_2^{N+\frac{r_2+j-r_4-i}{2}} \\\cong &\left[ \begin{matrix} \frac{r_3+j-r_1+i}{2} & \frac{r_3+j+r_1-i}{2}+N-2 & r_2+N-2\\ j+N-2 & r_3 & \frac{r_4+j+r_2-i}{2} \end{matrix} \right]_2^{2+\frac{r_1+i-r_3-j}{2}} \\ \cong &\left[ \begin{matrix} \frac{r_4+j+r_2-i}{2}+N-2 & \frac{r_4+j-r_2+i}{2} & r_1+N-2\\ r_4 & j+N-2 & \frac{r_3+j+r_1-i}{2} \end{matrix} \right]_2^{2+\frac{r_2+i-r_4-j}{2}} \\\cong &\left[ \begin{matrix} \frac{r_2+i+r_4-j}{2} & \frac{r_2+i-r_4+j}{2}+N-2 & r_1+N-2\\ r_2+N-2 & i & \frac{3r_3+i-r_1-j}{2} \end{matrix} \right]_2^{2+\frac{r_4+j-r_2-i}{2}} \\\cong & \left[ \begin{matrix} \frac{r_1+i-r_3+j}{2}+N-2 & \frac{r_1+i+r_3-j}{2} & r_2+N-2\\ i & r_1+N-2 & \frac{3r_4+i-r_2-j}{2} \end{matrix} \right]_2^{2+\frac{r_3+j-r_1-i}{2}} \\\cong & \left[ \begin{matrix} r_3+N-2 & r_2 & j+N-2\\ r_3 & r_2+N-2 & i+r_3-r_1 \end{matrix} \right]_2^{2+r_1-r_3} \\\cong & \left[ \begin{matrix} r_1 & r_4+N-2 & j+N-2\\ r_1+N-2 & r_4 & i+r_1-r_3 \end{matrix} \right]_2^{2+r_1-r_3} \\\cong &\left[ \begin{matrix} \frac{r_2+j+r_4-i}{2}+N-2 & \frac{r_2+j-r_4+i}{2}+4-N & r_3\\ r_2+N-2 & j+N-2 & \frac{r_1+j+r_3-i}{2}+2N-4 \end{matrix} \right]_2^{4-N+\frac{r_4+i-r_2-j}{2}} \\\cong & \left[ \begin{matrix} \frac{r_1+j-r_3+i}{2}+2-N & \frac{r_1+j+r_3-i}{2}+2-N & r_4\\ j+N-2 & r_1+N-2 & \frac{r_2+j+r_4-i}{2}+2N-4 \end{matrix} \right]_2^{4-N+\frac{r_3+i-r_1-j}{2}} \\\cong &\left[ \begin{matrix} r_3+N-2 & r_4+N-2 & i\\ r_1+N-2 & r_2+N-2 & j+2N-4 \end{matrix} \right]_2^{4-N}. \end{aligned}$$]{} Proof of statement \[st5\] {#proof-of-statement-st5 .unnumbered} -------------------------- Let us firstly prove that the following expressions are non-negative: $$\begin{aligned} \begin{cases} r_1+r_3+i-j \ge 0, \\ 3r_2+r_4-i-j \ge 0, \end{cases} \qquad &T\in\{1,2\},\\ i+r_1-r_3 \ge 0, \quad\qquad &T=2. \end{aligned}$$ The non-negativity can be proven using the inequalities on $i,j$. These inequalities are tautological generalization of the $U_q(sl_2)$ case [@Alekseev:2019]: $$\begin{aligned} \max\begin{pmatrix} |r_1-r_2|\\ |r_3-r_4| \end{pmatrix} \le i \le \min\begin{pmatrix} r_1+r_2\\ r_3+r_4 \end{pmatrix}, \qquad \max\begin{pmatrix} |r_2-r_3|\\ |r_1-r_4| \end{pmatrix} \le j \le \min\begin{pmatrix} r_2+r_3\\ r_1+r_4 \end{pmatrix}. \end{aligned}$$ With the suitable substitution the proof is obvious: $$r_1+r_3+i-j \ge r_1+r_3+(-r_3+r_4)-(r_1+r_4) \ge 0,$$ $$3r_2+r_4-i-j \ge 3r_2+r_4-(r_1+r_2)-(r_2+r_3) \ge r_2+r_4-r_1-r_3 = 0,$$ $$i+r_1-r_3 \ge \max\begin{pmatrix} r_2-r_1+r_1-r_3\\ r_3-r_4+r_1-r_3 \end{pmatrix} =\max\begin{pmatrix} r_2-r_3\\ r_1-r_4 \end{pmatrix} =\max\begin{pmatrix} r_2-r_3\\ r_3-r_2 \end{pmatrix} \ge 0.$$ Similarly one can derive non-negativeness of all expressions from external symmetries. Since the derivation is analogous in these cases, they are omitted. Acknowledgments {#acknowledgments .unnumbered} =============== We are deeply indebted to Andrei Mironov and Alexei Morozov for numerous stimulating discussions. V.A. is also grateful to Satoshi Nawata for clarifications on fusion rules, to Andrei Zotov and Victor Mishnyakov for useful discussions and comments. Our work was partly supported by the grant of the Foundation for the Advancement of Theoretical Physics “BASIS" (A.M., A.S. and A.V.), by RFBR grants 19-01-00680 (V.A.), 17-01-00585 (A.M.), 18-31-20046 (A.S.), by joint RFBR grants 19-51-18006 (A.M.), 19-51-50008-Yaf-a (A.M.), 18-51-05015-Arm-a (A.M, A.S.), 18-51-45010-Ind-a (A.M, A.S.), 19-51-53014-GFEN-a (A.M, A.S.), 19-51-18006-Bolg-a (A.M.), by President of Russian Federation grant MK-2038.2019.1 (A.M.). [^1]: alekseev.va@phystech.edu [^2]: Andrey.Morozov@itep.ru [^3]: sleptsov@itep.ru

--- abstract: | Recognizing how objects interact with each other is a crucial task in visual recognition. If we define the context of the interaction to be the objects involved, then most current methods can be categorized as either: (i) training a single classifier on the combination of the interaction and its context; or (ii) aiming to recognize the interaction independently of its explicit context. Both methods suffer limitations: the former scales poorly with the number of combinations and fails to generalize to unseen combinations, while the latter often leads to poor interaction recognition performance due to the difficulty of designing a context-independent interaction classifier. To mitigate those drawbacks, this paper proposes an alternative, context-aware interaction recognition framework. The key to our method is to explicitly construct an interaction classifier which combines the context, and the interaction. The context is encoded via word2vec into a semantic space, and is used to derive a classification result for the interaction. The proposed method still builds one classifier for one interaction (as per type (ii) above), but the classifier built is adaptive to context via weights which are context dependent. The benefit of using the semantic space is that it naturally leads to zero-shot generalizations in which semantically similar contexts (subject-object pairs) can be recognized as suitable contexts for an interaction, even if they were not observed in the training set. Our method also scales with the number of interaction-context pairs since our model parameters do not increase with the number of interactions. Thus our method avoids the limitation of both approaches. We demonstrate experimentally that the proposed framework leads to improved performance for all investigated interaction representations and datasets. author: - | [ Bohan Zhuang, Lingqiao Liu, Chunhua Shen, Ian Reid]{}\ School of Computer Science, University of Adelaide, Australia bibliography: - 'CSRef.bib' title: 'Towards Context-aware Interaction Recognition[^1] ' --- Introduction ============ Object interaction recognition is a fundamental problem in computer vision and it can serve as a critical component for solving many visual recognition problems such as action recognition [@mallya2016learning; @ramanathan2015learning; @wang2015action; @bilen2016dynamic; @Zhang_2016_CVPR], visual phrase recognition [@hu2016modeling; @rohrbach2016grounding; @li2017vip], sentence to image retrieval [@ma2015multimodal; @karpathy2015deep] and visual question answering [@wu2016ask; @lu2016knowing; @wu2016value]. Unlike object recognition in which the object appearance and its class label have a clear association, the interaction patterns, e.g. “eating”, “playing”, “stand on”, usually have a vague connection to visual appearance. This phenomenon is largely caused by the same interaction being involved with different objects as its context, i.e. the subject and object of an interaction type. For example, “cow eating grass” and “people eating bread” can be visually dissimilar although both of them have the same interaction type “eating”. Thus the subject and object associated with the interaction – also known as the *context* of the interaction – could play an important role in interaction recognition. In existing literature, there are two ways to model the interaction and its context. The first one treats the combination of interaction and its context as a single class. For example, in this approach, two classifiers will be built to classify “cow eating grass" and “people eating bread." To recognize the interaction “eating”, images that are classified as either “cow eating grass” or “people eating bread” will be considered as having interaction “eating". This treatment has been widely used in defining action (interaction) classes in many action (interaction) recognition benchmarks [@mallya2016learning; @ramanathan2015learning; @wang2015action; @bilen2016dynamic; @Zhang_2016_CVPR]. This approach, however, suffers from poor scalability and generalization ability. The number of possible combinations of the interaction and its context can be huge, and thus it is very inefficient to collect training images for each combination. Also, this method fails to generalize to an unseen combination even if both its interaction type and context are seen in the training set. To handle these drawbacks, another way is to model the interaction and the context separately [@lu2016visual; @desai2011discriminative; @gupta2008beyond; @sadeghi2015viske]. In this case, the interaction is classified independently of its context, which can lead to poor recognition performance due to the difficulty of associating the interaction with certain visual appearance in the absence of context information. To overcome the imperfection of interaction classification, some recent works employ techniques such as language priors [@lu2016visual] or structural learning [@li2017vip; @Liang2017VRD] to avoid generating an unreasonable combination of interaction and context. However, the context-independent interaction classifier is still used as a building block, and this prevents the system from gaining more accurate recognition from visual cues. The solution proposed in this paper aims to overcome the drawbacks of both methods. To avoid the explosion of the number of classes, we still separate the classification of the interaction and the context into two stages. However, different to the second method, the interaction classifier in our method is designed to be adaptive to its context. In other words, for the same interaction, different contexts will result in different classifiers and our method will encourage interactions with similar contexts to have similar classifiers. By doing so, we can achieve context-aware interaction classification while avoiding treating each combination of context and interaction as a single class. Based on this framework, we investigate various feature representations to characterize the interaction pattern. We show that our framework can lead to performance improvements for all the investigated feature representations. Moreover, we augment the proposed framework with an attention mechanism, which leads to further improvements and yields our best performing recognition model. Through extensive experiments, we demonstrate that the proposed methods achieve superior performance over competing methods. Related work ============ *Action recognition:* Action is one of the most important interaction patterns and action recognition in images/videos has been widely studied  [@mallya2016learning; @ramanathan2015learning; @wang2015action; @bilen2016dynamic; @Zhang_2016_CVPR]. Various action recognition datasets such as Stanford 40 actions [@yao2011human], UCF-101 [@soomro2012ucf101] and HICO [@chao2015hico] have been proposed, but most of them focus on actions (interactions) with limited number of context. For example, in the relatively large HICO [@chao2015hico] dataset, there are only 600 categories of human-object interactions. Thus the interplay of the interaction and its context has not been explored in the works of this direction. *Visual relationships:* Some recent works focus on the detection of visual relationships. A visual relationship is composed of an interaction and its context, i.e. subject and object. Thus this direction is most relevant to this paper. In fact, the interaction recognition can be viewed as the most challenging part of the visual relationship detection. Some recent works in visual relationship detection have made progress in improving the detection performance and the detection scalability. The work in [@lu2016visual] leveraged language priors to produce relationship detections that make sense to human beings. The latest approaches [@Liang2017VRD; @li2017vip; @zhang2017visual] attempt to learn the visual relationship detector in an end-to-end manner and explicitly reason the interdependency among relationship components at the visual feature level. *Language-guided visual recognition:* Our method uses language information to guide the visual recognition. This corresponds to the recent trend in utilizing language information for benefiting visual recognition. For example, language information has also been incorporated in phrase grounding [@plummer2016phrase; @hu2016modeling; @rohrbach2016grounding] tasks. In [@hu2016modeling; @rohrbach2016grounding], attention model is employed to extract linguistic cues from phrases. Language guided attention has also been widely used in visual question answering [@donahue2015long; @karpathy2015deep; @malinowski2015ask; @ren2015image] and has recently been applied to one-shot learning [@vinyals2016matching]. Methods ======= Context-aware interaction classification framework -------------------------------------------------- In general, an interaction and its context can be expressed as a triplet $\left\langle \emph{O1-P-O2} \right\rangle$, where $P$ denotes the interaction, and $O1$ and $O2$ denote its subject and object respectively. In our study, we assume the interaction context (*O1,O2*) has been detected by a detector (i.e. we are given bounding boxes and lables for both subject $O1$ and object $O2$) and the task we are addressing is to classify their interaction type $P$. To recognize the interaction, existing works take two extremes in designing the classifier. One is to directly build a classifier for each $P$ and assume that the same classifier applies to $P$ with different context. Another takes the combination of $\left\langle \emph{O1-P-O2} \right\rangle$ as a single class and build a classifier for each combination. As discussed in the introduction section, the former does not fully leverage the contextual information for interaction recognition while the latter suffers from the scalability and generalization issues. Our proposed method lies between those two extremes. Specifically, we still allocate one classifier for each interaction type, however we make the classifier parameters adaptive to the context of the interaction. In other words, the classifier is a function of the context. The schematic illustration of this idea is shown in Figure \[fig:overview\]. Formally, we assume that the interaction classifier takes a linear classifier form $y_p = \mathbf{w}_p^{\top}\phi(I),~~\mathbf{w}_p \in \mathbb{R}^d$, where $y_p$ is the classification score for the $p$-th interaction and $\phi(I)$ is the feature representation extracted from the input image. The classifier parameters for the $p$-th interaction $\mathbf{w}_p$ are a function of $(O1,O2)$, that is, the context of the $p$-th interaction. It is designed as the summation of the following two terms: $$\label{Eq:combine} \mathbf{w}_p(O1,O2) = \mathbf{\bar{w}}_p + r_p(O1,O2),$$ where the first term $\mathbf{\bar{w}}_p$ is independent of the context; it plays a role which is similar to the traditional context-independent interaction classifier. The second term $r_p(O1,O2)$ can be viewed as an auxiliary classifier generated from the information of context $(O1,O2)$. Note that the summation of two classifiers has been widely used in transfer learning [@patricia2014learning; @arnold2007comparative; @do2005transfer] and multi-task learning [@evgeniou2004regularized; @parameswaran2010large], e.g. one term corresponds to the classifier learned in the target domain and another corresponds to the classifier learned in the source domain. Intuitively, for two interaction-context combinations, if both of them share the same interaction and their contexts are similar, the interaction in those combinations tends to be associated with similar visual appearance. For example, $\left\langle \emph{boy, playing, football} \right\rangle$ and $\left\langle \emph{man, playing, soccer} \right\rangle$ share similar context, so the interaction “playing” should suggest similar visual appearance for these two combinations. This inspires us to design $\mathbf{w}_p(O1,O2)$ to allow semantically similar contexts to generate similar interaction classifiers, as demonstrated in Figure \[fig:context\]. To realize this idea, we first represent the object and subject through their word2vec embedding which maps semantically similar words into similar vectors and then generate the auxiliary classifier $r_p$ by concatenating their embeddings. Formally, $r_p$ is designed as: $$\begin{aligned} \label{eq:relation_vec} r_p(O1,O2) = {{\mathbf{V}_p}}f(\mathbf{Q}E(O1,\,O2)),\end{aligned}$$ where $E(O1,\,O2) \in \mathbb{R}^{2e}$ is the concatenation of the $e$-dimensional word2vec embeddings of $(O1,O2)$, and $\mathbf{Q} \in \mathbb{R}^{m \times 2e}$ is a projection matrix to project $E(O1,\,O2)$ to a low-dimensional (e.g. 20) semantic embedding space. $f(\cdot)$ is the RELU function and $\bf{V}_p$ transforms the context embedding to the auxiliary classifier. Note that $\mathbf{V}_p$ and $\mathbf{\bar{w}}_p$ in Eq. (\[Eq:combine\]) are distinct per interaction type $p$ while the projection matrix $\mathbf{Q}$ is shared across all interactions. All of these parameters are learnt at training time. **Remark:** Many recent works [@Liang2017VRD; @li2017vip; @zhang2017visual; @plummer2016phrase] on visual relationship detection takes a structural learning alike formulation to simultaneously predict $O1,O2$ and $P$. The unary term used in their framework is still a context-independent classifier and such choice may lead to poor recognition accuracy in identifying interaction from the visual cues. To improve these techniques, one could replace their unary terms with our context-aware interaction recognition module. On the other hand, their simultaneous prediction framework could also benefit our method in achieving better visual relationship performance. Since our focus is to study the interaction part, we do not pursue this direction in this paper and leave it for future work. Feature representations for interactions recognition ---------------------------------------------------- One remaining issue in implementing the framework in Eq. (\[Eq:combine\]) is the design of $\phi(I)$, that is, the feature representation of the interaction. It is clear that the choice of the feature representation can have significant impact on the interaction prediction performance. In this section, we investigate two types of feature representations to characterize the interaction. We evaluate these feature representations in Sec. \[sec:VRD\]. ### Spatial feature representation {#sec:spatial} Our method assumes that the context has been detected and therefore the interaction between the subject and the object could be characterized by the spatial features of the detection bounding boxes. These kind of features have been previously employed  [@hu2016modeling; @plummer2016phrase; @zhang2017visual] to recognize the visual relationship of objects. In our study, we use both the spatial features from each bounding box and the spatial features from their mutual relationship. Formally, let $(x,y,w,h)$ and $(x',y',w',h')$ be the bounding box coordinates of the *subject* and *object*, respectively. Given the bounding boxes, the spatial feature for a single box is a 5-dimentional vector represented as $[\frac{x}{{{W_I}}},\frac{y}{{{H_I}}},\frac{{x + w}}{{{W_I}}},\frac{{y + h}}{{{H_I}}},\frac{{{S_b}}}{{{S_I}}}]$, where $S_b$ and $S_I$ are the areas of region $b$ and image $I$, $W_I$ and $H_I$ are the width and height of the image $I$. And the pairwise spatial vector is denoted as $[\frac{{x - x'}}{{w'}},\frac{{y - y'}}{{h'}},\log \frac{w}{{w'}},\log \frac{h}{{h'}}]$. We concatenate them together to get a 14-dimentional feature representation (using both subject and object bounding boxes). Then the spatial feature directly passes through the context-aware classifier defined in Eq. (\[Eq:combine\]) for the interaction classification. ### Appearance feature representation {#sec:appearance} Besides spatial features, we can also use appearance features, e.g. the activations of a deep neural network to depict the interaction. In our study, we first crop the union region of the subject and object bounding boxes, and rescale the region to $224 \times 224 \times 3$ as the input of a VGG-16 [@simonyan2014very] CNN. We then apply the mean-pooling to the activations of the $conv5\_3$ layer as our feature representation $\phi(I)$. This feature is then fed into our context-aware interaction classifier in Eq. (\[Eq:combine\]). To improve the performance, we treat the context-aware interaction classifier as a newly added layer and fine-tune this layer with the VGG-16 net in an end-to-end fashion. Improving appearance representation with attention and context-aware attention {#sec:attention} ------------------------------------------------------------------------------ The discriminative visual cues for interaction recognition may only appear in a small region of the input image or the image region. For example, to see if “man riding bike” occurs, one may need to focus on the region near human feet and bike pedal. This consideration motivates us to use attention module to encourage the network “focus on” discriminative regions. Specially, we can replace the mean-pooling layer in Sec. \[sec:appearance\] with an attention-pooling layer. Formally, let ${{\bf{h}}_{ij}} \in {R^c}$ denote the last convolutional layer activations at the spatial location $(i,j)$, where $i=1,2,...,M$ and $j=1,2,...,N$ are the coordinates of the feature map and $M$, $N$ are the height and width of the feature map respectively, c is the number of channels. The attention pooling layer pools the convolutional layer activations into a $c$-dimensional vector through: $$\begin{aligned} \begin{array}{l} {\bar{a}(\mathbf{h}_{ij})} = \frac{{a(\mathbf{h}_{ij}) + \varepsilon }}{{\sum\limits_i {\sum\limits_j {(a(\mathbf{h}_{ij}) + \varepsilon )} } }},\\ {\widetilde {\mathbf{h}}} = \frac{1}{MN} \sum\limits_{ij}{\bar{a}(\mathbf{h}_{ij})} {{\mathbf{h}}_{ij}}, \end{array}\end{aligned}$$ where $a(\mathbf{h}_{ij})$ is the attention generation function which produces an attention value for each location $(i,j)$. The attention value is then normalized ($\varepsilon$ is a small constant) and used as a weighting factor to pool the convolutional activations ${\mathbf{h}}_{ij}$. We consider two designs of $a(\mathbf{h}_{ij})$. **Direct attention**: The first attention generation function is simply designed as $a(\mathbf{h}_{ij}) = f(\mathbf{w}_{att}^{\top} \mathbf{h}_{ij} + b)$, where ${{\bf{w}}_{att}}$ and $b$ are the weight and bias of the attention model. **Context-aware attention**: In the above attention generation function, the attention value is solely determined by $\mathbf{h}_{ij}$. Intuitively, however, it makes sense that different attention is required for different classification tasks. For example, to examine “man riding bike” and examine “man playing football", different regions-of-interest should be focused on. We therefore propose to use a context-aware attention generator; i.e. we design $\mathbf{w}_{att}$ as a function of $(P, O1, O2)$. We can follow the framework in Eq. (\[Eq:combine\]) to calculate: $$\begin{aligned} \mathbf{w}_{att}(P, O1,O2) = \mathbf{\bar{w}^a}_{p} + {{\mathbf{V}^a_p}}f(\mathbf{Q}E(O1,\,O2)),\end{aligned}$$ where $\mathbf{\bar{w}^a}_{p}$ is the attention weight for the $p$-th interaction independent of its context and ${\mathbf{V}^a_p}$ transforms the semantic embedding of the context to the auxiliary attention weight for the $p$-th interaction. Note that in this case $\mathbf{w}_{att}$ depends on the interaction class $P$ and therefore different attention-pooling vectors $\widetilde {\mathbf{h}}_p$ will be generated for different $P$. $\widetilde {\mathbf{h}}_p$ will be then sent to the context-aware classifier for interaction $P$ to obtain the decision value for $P$ and the class that produces the maximal decision value will be considered as the recognized interaction. This structure is illustrated in Figure \[fig:attention\]. Implementation details {#sec:implementation} ---------------------- For all the above methods, we use the standard multi-class cross-entropy loss to train the models. The Adam algorithm [@kingma2014adam] is applied as the optimization method. The methods that use appearance features involve convolutional layers from the standard VGG-16 network together with some newly added layers. For the former we initialize those layers with the parameters pretrained on ImageNet [@russakovsky2015imagenet] and for the latter we randomly initialize the parameters. We set the learning rate to 0.001 and 0.0001 for the new layers and VGG-16 layers respectively. Experiments {#sec:experiment} =========== To investigate the performance of the proposed methods, we analyse the effects of the context-aware interaction classifier, the attention models and various feature representations. Eight methods are implemented and compared: 1. “**Baseline1-app**”: We directly fine-tune the VGG-16 model to classify the interaction categories. Inputs are the union of subject and object boxes. This baseline models the interaction and its context separately, which corresponds to the approach described in Figure \[fig:overview\] (c). 2. “**Baseline1-spatial**”: We directly train a linear classifier to classify the spatial features described in Sec. \[sec:spatial\] into multiple interaction categories. 3. “**Baseline2-app**”: We treat the combination of the interaction and its context as a single class and fine-tune the VGG-16 model for classification. This corresponds to using appearance feature to implement the method in Figure \[fig:overview\] (a). 4. “**Baseline2-spatial**”: Similar to “Baseline2-app”. We train a linear classifier to classify the spatial features into the classes derived from the combination of the interaction and its context. 5. “**AP+C**”: We apply the context-aware classifier to the appearance representation described in Sec. \[sec:appearance\]. 6. “**AP+C+AT**”: The basic attention-pooling representation described in Sec. \[sec:attention\] with the classifier in **AP+C**. 7. “**AP+C+CAT**”: The context-aware attention-pooling representation described in Sec. \[sec:attention\] with the classifier in **AP+C**. 8. “**Spatial+C**”: We apply the context-aware classifier to the spatial features described in Sec. \[sec:spatial\]. Besides those methods, we also compare the performance of our methods against those reported in the related literature. However, it should noted that these methods may use different feature representation, detectors or pre-training strategies. Evaluation on the Visual Relationship dataset --------------------------------------------- We first conduct experiments on the Visual Relationship Detection (VRD) dataset [@lu2016visual]. This dataset is designed for evaluating the visual relationship ($\left\langle \emph{subject, predicate, object} \right\rangle$) detection, where the “predicate” in those datasets is equivalent to the “interaction” in our paper and we will use them interchangeably thereafter. It contains 4000 training and 1000 test images including 100 object classes and 70 predicates. In total, there are 37993 relationship instances with 6672 relationship types, out of which 1877 relationships occur only in the test set but not in the training set. Following [@lu2016visual], we evaluate on three tasks: (1) For **predicate detection**, the input is an image and a set of ground-truth object bounding boxes. The task is to predict the possible interactions between pairs of objects. Since the interaction recognition is the main focus of this paper, the performance of this task provides the most relevant indication of the quality of the proposed method. (2) In **phrase detection**, we aim to predict $\left\langle \emph{subject-predicate-object} \right\rangle$ and localize the entire relationship in one bounding boxes. (3) For **relationship detection**, the task is to recognize $\left\langle \emph{subject-predicate-object} \right\rangle$ and localize both subject and object bounding boxes. Both boxes should have at least 0.5 overlap with the ground truth bounding boxes in order to be regarded as a correct prediction. For the second and third tasks, we use the object detection results (both bounding boxes and corresponding detection scores) provided in [@lu2016visual]. This allows us to fairly compare the performance of the proposed interaction recognition framework without the influence of detection. We use the Recall@100 and Recall@50 as our evaluation metric following [@lu2016visual]. Recall@x computes the fraction of times the correct relationship is calculated in the top $x$ predictions, which are ranked by the product of the objectness confidence scores and the classification probabilities of the interactions. As discussed in [@lu2016visual], we do not use the mean average precision (mAP), which is a pessimistic evaluation metric because it cannot exhaustively annotate all possible relationships in an image. ### Detection results comparison {#sec:VRD} In this section, we evaluate the performance of three detection tasks on the Visual Relationship Detection (VRD) benchmark dataset and provide the comprehensive analysis. We compare all the eight methods and the results in [@sadeghi2011recognition; @lu2016visual]. The results are shown in Table \[tab:relationship\]. From it we can make the following observations: *The effect of context-aware modeling:* To validate the main point in this paper, we compare the proposed method against two context-interaction modeling baselines, i.e. baseline1-app, baseline2-app, baseline1-spatial and baseline2-spatial). By analysing the results, we can see that the proposed context-aware modeling methods (methods with “AP”) achieves much better performance than the four baselines. The improvement achieved by use context-aware modeling is consistently observed for both spatial features and appearance features. This justifies that the context information is crucial for interaction prediction. *Various feature representations:* We also quantitatively investigate the performance of the proposed context-aware framework under various feature types. As can be seen in Table \[tab:relationship\], the appearance feature representation performs consistently better than the spatial feature representation, especially for the baseline2 setting. This may be because the visual feature representation has richer discriminative power than the 14-dimensional spatial feature. Also, with our context-aware recognition framework, we can significantly boost the performance of both features and interestly in this case the gap between two types of features is largely diminished, e.g. AP+C+CAT vs. Spatial+C. *The effect of attention models:* We also investigate the impacts of the attention scheme employed in our model by comparing AP+C, AP+C+AT and AP+C+CAT. The best results are obtained by utilizing the context-aware attention model. This justifies our postulate that it is better to make the network attend on the discriminative regions of feature maps. *Comparison with  [@sadeghi2011recognition] and [@lu2016visual]:* Finally, we compare our methods with the methods in  [@sadeghi2011recognition] and  [@lu2016visual]. As seen, our methods achieve better performance than these two competing methods. Since our methods use the same object detection in [@lu2016visual], our result is most comparable to it. Note that our model does not employ explicit language priors modeling as in [@lu2016visual] and our improvement purely comes from the visual cue. This again demonstrates the power of context-aware interaction recognition. To better evaluate our approach, we further visualize some test examples of AP+C+CAT in Figure \[fig:qualitative1\]. We can see that our predictions are reasonable in most cases. ### Zero-shot learning performance evaluation {#sec:zero-shot} An important motivation of our method is to make the interaction classifier generalizable to unseen combinations of the interaction and context. In this section, we report the performance of our method on a zero-shot learning setting. Specifically, we train our models on the training set and evaluate their interaction classification performance on the 1877 unseen visual relationships in the test set. The results are reported in Table \[tab:zeroshot\]. From the table, we can see that the proposed methods work especially well in the zero-shot learning. For example, our best performed method (AP+C+CAT) almost doubled the performance on predicate detection in comparison with the Language Priors [@lu2016visual] method. This big improvement can be largely attributed to the advantage of using the context-aware scheme to model the interaction. In the Language Priors [@lu2016visual] method, the visual term for recognizing interaction is context-independent. Without context information to constrain the appearance variations, the learned interaction classifier tends to overfit the training set and fails to generalize to images with unseen interaction-context combinations. In comparison, with context-aware modeling, we explicitly consider the visual appearance variations introduced by changing context, thus more accurate and generalizable interaction classifier can be learned. One interesting observation made in Table \[tab:zeroshot\] is that the spatial feature representation produces better performance than the appearance based representation, as is evident from the superior performance of Spatial+C over AP methods. We speculate this is because spatial relationship features are more object independent and are less prone to overfiting the training set. To intuitively evaluate zero-shot performance, we add some test examples of AP+C+CAT in Figure \[fig:qualitative2\]. We can make reasonable predictions on unseen interaction-context combinations in most cases. ### Extensions and comparison with the state-of-the-art methods Since the main focus of above experiments is to validate the advantage of the proposed methods over four competing baselines, we did not explore some techniques which could potentially further improve the visual relationship detection performance on the VRD dataset. To make our method achieve more comparable performance on the visual relationship and visual phrase detection tasks, we may consider two straightfoward extensions for our method: (1) use a better detector and (2) incorporate the language term trained in [@lu2016visual]. In the following part, we will examine the performance attained by applying these extensions and compare the resultant performance against the very latest state-of-the-art approaches [@Liang2017VRD; @li2017vip; @zhang2017visual; @plummer2016phrase] on the VRD dataset. *Improved detector:* We first examine the effect of using a better detector by replacing the detection results obtained in [@lu2016visual] with that obtained by a Faster-RCNN detector [@girshick2015fast]. Note that the Faster-RCNN detector has also been used in [@Liang2017VRD; @li2017vip; @zhang2017visual; @plummer2016phrase] and using it will make our method comparable with the current state-of-the-arts. In our implementation, only the top 50 candidate object proposals, ranked by objectness confidence scores are extracted for mining relationships in per test image. The result of this modification is reported in Table \[tab:VRD\] with our method annotated as Faster-RCNN + (AP+C+CAT). As seen, our method achieves best performance on phrase detection R@50, relationship detection, zero-shot phrase and relationship detection. Note that our method can be further incorporated into the end-to-end relationship detection framework such as [@li2017vip] to achieve even better performance. *Language priors:* Language priors make significant contribution to [@lu2016visual] and in this section we apply the language priors released by [@lu2016visual] to investigate its impact. Following [@lu2016visual], we multiply our best performed model Faster-RCNN + (AP+C+CAT) with the language priors for interactions to obtain the final detection scores and the result is shown in Table \[tab:VRD\] with the annotation Faster-RCNN + (AP+C+CAT) + Language Priors. Interestingly, the introduction of the language priors only introduces a marginal performance improvement. We suspect that is due to that our method builds a classifier with the information of both the interaction and context, and the correlation of interaction and context has been implicitly encoded. Therefore adding the language priors does not bring further benefit. Evaluation on the Visual Phrase dataset --------------------------------------- Following [@lu2016visual], we also run additional experiments on the Visual Phrase [@sadeghi2011recognition] dataset. It has 17 phrases, out of which 12 of these phrases can be represented as triplet relationships as in the VRD dataset. We use the setting of [@lu2016visual] to conduct the experiment and report the R@50 and R@100 results in Table \[tab:VP\]. Since the Visual Phrase dataset does not provide detection results, we apply the RCNN [@girshick2014rich] model to produce a set of candidate object regions and corresponding detection scores. As seen from Table \[tab:VP\], AP+C+CAT again achieves the best performance. In comparison with the performance of [@lu2016visual], our method improves most in the zero-shot learning setting. This is consistent with the observation made in Sec. \[sec:zero-shot\]. Conclusion ========== In this paper, we study the role of context in recognizing the object interaction pattern. After identifying the importance of using context information, we propose a context-aware interaction classification framework which is accurate, scalable and enjoys good generalization ability to recognize unseen context-interaction combinations. Further, we investigate various ways to derive the visual representation for interaction patterns and extend the context-aware framework to design a new attention-pooling layer. With extensive experiments, we validate the advantage of the proposed methods and produce the state-of-the-art performance on two visual relationship detection datasets. [^1]: The first two authors contributed equally to this work. This work was in part supported by an ARC Future Fellowship to C. Shen. Corresponding author: C. Shen (e-mail: chunhua.shen@adelaide.edu.au).

--- abstract: | We review the current status of the study of rotation curve (RC) of the Milky Way, and present a unified RC from the Galactic Center to the galacto-centric distance of about 100 kpc. The RC is used to directly calculate the distribution of the surface mass density (SMD). We then propose a method to derive the distribution of dark matter (DM) density in the in the Milky Way using the SMD distribution. The best-fit dark halo profile yielded a local DM density of We also review the estimations of the local DM density in the last decade, and show that the value is converging to a value at $\rho_\odot=0.39\pm 0.09$ .\ [**Key words**]{} galaxies: DM—galaxies: individual (Milky Way)—galaxies: rotation curve\ ([*Invited review accepted for Galaxies to appear in special issue on “Debate on the Physics of Galactic Rotation and the Existence of Dark Matter”*]{}) author: - | Yoshiaki Sofue\ Institute of Astronomy, The University of Tokyo, Mitaka, Tokyo 181-0015, Japan\ E-mail: sofue@ioa.s.u-tokyo.ac.jp title: Rotation Curve of the Milky Way and the Dark Matter Density --- 0[ V\_0 ]{} Introduction ============ The rotation curve (RC) of the Milky Way was obtained by observations of galactic objects in the non-MOND (MOdified Newtonian Dynamics)frame work. The existence of the dark halo (DH) has been confirmed by the analysis of the observed RCs, assuming that  Newtonian dynamics applies evenly to the result of the observations. In this article, current works of RC observations are briefly reviewed, and a new estimation of the local dark matter (DM) density is presented in the framework of Newtonian dynamics. An RC is defined as the mean circular velocity $\Vrot$ around the nucleus plotted as a function of the galacto-centric radius $R$. Non-circular streaming motion due to the triaxial mass distribution in a bar is crucial for kinematics in the innermost region, though it does not affect the mass determination much in the disk and halo. Spiral arms are another cause for local streaming, which affect the mass determination by several percent, while they do not influence the mass determination of the dark halo much. There are several reviews on RCs and mass determination of galaxies \[[@SofueRubin2001; @Sofue2017; @Salucci2019]\]. In this review, we revisit recent RC studies and determination of the local DM density in our Milky Way. In Section \[Section2\], we briefly review the current status of the RC determinations along with the methods. We adopt the galactic constants: $(\rzero,\vzero)$=(8.0 kpc, 238 ) \[[@Honma+2012; @Honma+2015]\], where $\rzero$ is the distance of the Sun from the galactic center (GC) and $\vzero$ is the circular velocity of the local standard of rest (LSR) at the Sun \[[@Fich+1991]\]. Rotation Curve of the Milky Way {#Section2} =============================== Progress in the Last Decades ---------------------------- The galactic RC is dependent on the galactic constants. Accordingly, the uncertainty and error in the RC include uncertainties of the constants. Currently recommended, determined, or measured values are summarized in Table \[tabGal\], where they appear to be converging to around $\sim 8.0-8.3$ and . In this paper, we adopt $R_0= 8.0$ kpc and $V_0 =238$ from the recent measurements with VERA (VLBI Experiments for Radio Astrometry) \[[@Honma+2012; @Honma+2015]\]. **Authors (Year)** **(kpc)** **()** ---------------------------------------------------------------- ----------------- ------------- IAU recommended (1982) 8.2 220 Review before 1993 (Reid 1993) \[[@Reid1993]\] $8.0 \pm 0.5$ Olling and Dehnen 2003 \[[@Olling+2003]\] $7.1\pm 0.4$ $184\pm 8$ VLBI Sgr A$^*$ (Ghez et al. 2008) \[[@Ghez+2008]\] $8.4 \pm 0.4$ ibid (Gillessen et al. 2009) \[[@Gillessen+2009]\] $8.33 \pm 0.35$ Maser astrometry (Reid et al. 2009) \[[@Reid+2009]\] $8.4\pm 0.6$ $254\pm 16$ Cepheids (Matsunaga et al. 2009) \[[@Matsunaga+2009]\] $8.24 \pm 0.42$ VERA (Honma et al. 2012, 2015) \[[@Honma+2012; @Honma+2015]\]. $8.05\pm 0.45$ $238\pm 14$ Adopted in this paper 8.0 238 : Galactic constants ($R_0,V_0$). []{data-label="tabGal"} The RC of the galaxy has been obtained by various methods as described in the next subsection, and many authors presented their results based on different galactic constants (Table \[tabrcmw\]). **Authors (Year)** **Radii (kpc)** **Method** ----------------------------------------------------------------------------------------------------------- ----------------- ------------------------- Burton and Gordon (1978)\[[@Burton+1978]\] 0–8 HI tangent Blitz et al. (1979) \[[@Blitz+1979]\] 8–18 OB-CO assoc. Clemens (1985)\[[@Clemens1985]\] 0 -18 CO/compil. Dehnen and  Binney (1998)\[[@Dehnen+1998]\] 8–20 compil. + model Genzel et al. (1994–), Ghez et al. (1998–)\[[@Genzel+2010; @Ghez+2008]\] 0–0.0001 GC IR spectr. Battinelli, et al. (2013)\[[@Battinelli+2013]\] 9–24 C stars Bhattacharjee et al.(2014)\[[@Bhattacharjee+2014]\] 0–200 Non-disk objects Lopez-Corredoira (2014)\[[@Lopez2014]\] 5–16 Red-clump giants $\mu$ Boby et al. (2012)\[[@Bovy+2012b]\] 4-14 NIR spectroscopy Bobylev (2013); — & Bajkova (2015)\[[@Bobylev2013; @Bobylev+2015]\] 5–12 Masers/OB stars Reid et al. (2014)\[[@Reid+2014]\] 4-16 Masers SF regions, VLBI Honma et al. (2012, 2015)\[[@Honma+2012; @Honma+2015]\] 3–20 Masers,VLBI Iocco et al. (2015, 2016); Pato & Iocco (2017a,b)\[[@Iocco+2015; @Iocco+2016; @Pato+2017a; @Pato+2017b]\] 1–25 kpc CO/HI/opt/maser/compil. Huang et al. (2016)\[[@Huang+2016]\] 4.5–100 HI/opt/red giants Kre[ł]{}owski et al (2018)\[[@Krelowski+2018]\] 8–12 GAIA Lin and Li (2019)\[[@Lin+2019]\] 4–100 compil. Eilers et al (2019)\[[@Eilers+2019]\] 5–25 Wise, 2Mass, GAIA Mróz et al. (2019)\[[@Mroz+2019]\] 4–20 Classical cepheids Sofue et al. (2009); Sofue (2013, 2015, this work)\[[@Sofue+2009; @Sofue2013; @Sofue2015]\] 0.01–1000 CO/HI/maser/opt/compil. : Rotation curves (RCs) of the Milky Way galaxy.[]{data-label="tabrcmw"} In the 1970–1980s, the inner RC was extensively measured using the terminal-velocities of HI (neutral hydrogen) and CO (carbon monoxide) gases \[[@Burton+1978; @Clemens1985; @Fich+1989]\]. In the late 1980s to the 2000s, outer rotation velocities were measured by combining optical distances of OB \[[@Blitz+1979; @Demers+2007]\]. The HI thickness method was also useful to measure rotation of the entire disk \[[@Merrifield1992; @Honma+1997]\]. The innermost mass distributions inside the GC have been obtained extensively since the 1990s using the motion of infrared stellar objects \[[@Genzel+2010; @Ghez+2008; @Lindqvist+1992; @Gillessen+2009]\]. Trigonometric determinations of both the 3D positions and velocities have provided the strongest tool to date for measurement of the galactic rotation \[[@Honma+2007; @Honma+2012; @Honma+2015; @Sakai+2015; @Nakanishi+2015]\]. A number of optical parallax measurements of stars such with GAIA have been obtained for RC determination \[[@Lopez2014; @Krelowski+2018]\]. The total mass of the galaxy, including the extended dark halo, has been measured by analyzing the outermost RC and motions of satellite galaxies orbiting the galaxy, and the mass up to kpc has been estimated to be $\sim 3 \times 10^{11}\Msun$ \[[@Sofue2015; @Callingham+2019]\]. Methods to Determine the Galactic RC ------------------------------------ The particular location of the Sun inside the Milky Way makes it difficult to measure the rotation velocity of the galactic objects. Sophisticated methods have been developed to solve this problem, as briefly described below. ### Tangent-Velocity Method Inside the solar circle ($-90^\circ \le l \le 90^\circ$), the galactic gas disk has tangential points, at which the rotation velocity is parallel to the line of sight and attains the maximum radial velocity $ {\vr}_{\rm ~max}$ (terminal or tangent-point velocity). The rotation velocity $V(R)$ at galacto-centric distance $R=\Rsun \sin~ l$ is calculated simply correcting for the solar motion. ### Radial-Velocity + Distance Method If the distance $r$ of the object is measured by spectroscopic and/or trigonometric observations, the rotation velocity is obtained by geometric conversion of the radial velocity, distance, and the longitude. The distance has to be measured independently, often using spectroscopic distances of OB stars, and the distances are assumed to be the same as those of associated molecular clouds and HII (ionized hydrogen) regions, whose radial velocities are observed by radio lines. Since the photometric distances have often large errors, obtained RC plots show large scatter. ### Trigonometric Method If the proper motion and radial velocity along with the distance are measured at the same time, or from different observations, the 3D velocity vector, and therefore the rotation velocity, of any source is uniquely determined without being biased by assumption of circular motion as well as the galactic constants. VLBI (very long baseline interferometer) measurements of maser sources \[[@Honma+2007; @Honma+2012; @Honma+2015; @Nakanishi+2015]\] and optical/IR trigonometry of stars \[[@Roeser+2010; @Lopez2014]\] have given the most accurate RC. ### Disk-Thickness Method The errors in the above methods are mainly caused by the uncertainty of the distance measurements. This disadvantage is eased by the HI-disk thickness method \[[@Merrifield1992; @Honma+1997]\]. The angular thickness of the HI disk along an annulus ring is related to can be used to determine the rotation velocity by combining with radial velocity distribution along the longitude. ### Pseudo-RC from Non-Disk Objects Beyond or outside the galactic disk, globular clusters and satellite galaxies are used to estimate the pseudo-circular velocity from their radial velocities based on the Virial theorem, assuming that their motions are at random, or the  rotation velocity is calculated by $\Vrot \sim \sqrt{2} v_g$, where $v_g$ is the galacto-centric radial velocity. [On the other hand, Huang et al. (2016) \[[@Huang+2016]\] have recently employed more sophisticated, probably more reliable, method to solve the Jeans equations for the non-disk stars and clusters.]{} Unified RC ---------- A RC covering a wide region of the galaxy has been obtained by compiling the existing data by re-scaling the distances and velocities to the common galactic constants \[[@Sofue+2009]\], and later to (8.0 kpc, 238 ) \[[@Sofue2013; @Sofue2017]\]. In these works, the central RC inside the GC has been obtained from analyses of the kinematics of the molecular gas and infrared stellar motions as well as the supposed Keplerian motion representing the central massive black hole. Outer RC beyond $R\sim 30$ kpc has been determined from the radial motions of satellite galaxies and globular clusters. The RC determination has been improved recently by compiling a large amount of data from a variety of spectroscopic as well as trigonometric measurements from radio to optical wavelengths. An extensive compilation of the data of rotation velocities of the galactic disk has been published recently, and is available as an internet data base \[[@Iocco+2015; @Iocco+2016; @Pato+2015a; @Pato+2015b; @Pato+2017a; @Pato+2017b]\]. Figure \[mwrc\]a shows the presently obtained unified RC using the curves from\[[@Sofue2015; @Sofue2017]\]and RC by Huang et al. (2016)\[[@Huang+2016]\] between $R=4.6$ and $\sim 100$ kpc. Although Huang et al. employed the galactic constants of (8.34 pc, 240 ), we did not apply rescaling to (8.0, 238), because the galacto-centric distances of off-plane objects are less dependent on the solar position compared to the disk objects as used for our RC at $<\sim 20$ kpc where the rotation velocity is rather flat, and also because their $V_0=240$ is close to our 238 . ![ (**a**) Unified RC of the Milky Way used in this paper for the mass distribution obtained by averaging the RCs from references \[[@Sofue2015; @Sofue2017; @Huang+2016]\]. The bars are standard deviations within each Gaussian-averaging bin. The plotted values are listed in the tables in Appendix \[AppendixA\]. (**b**) Logarithmic RC of the Milky Way from \[[@Sofue2015; @Sofue2017]\] (circles), compared with those from the recent literature: Green circles with error bars are from the compilation by \[[@Pato+2017a; @Pato+2017b]\]and blue triangles are their running averages. Red triangles stand for data from \[[@Krelowski+2018]\]based on GAIA data. [These two data are re-scaled to ($R_0, V_0$)=(8.0 kpc, 238 ).]{} Pink rectangles are the RC by \[[@Huang+2016]\].[without re-scaling]{}. (**c**) Same, but in linear scale. (**d**) Same, but close up in the solar vicinity.[]{data-label="mwrc"}](fig1.eps){width="8.5cm"} The unified RC was obtained by taking Gaussian running averages of rotation velocities from the used RCs in each of newly settled radius bins, where the  statistical weight of each input point was given by the inverse of the squared error. In Figure \[mwrc\]b,c we compare the unified RC with the recent measurements by \[[@Pato+2017a; @Pato+2017b; @Krelowski+2018]\]re-scaled to the galactic constants of (8.0 kpc, 238 ) following the method described in \[[@Sofue+2009]\].Although individual data points are largely scattered, their averages well coincide with the unified RC. In the figures we also compare the data with the RC by \[[@Huang+2016]\]up to $\sim 100$ kpc without rescaling, which also coincides with the other data within the scatter. We here comment on the property of the unified RC built by averaging the published data. It must be remembered that the averaging procedure does not satisfy the condition of statistics in the strict meaning, because the data are compiled from different authors using a variety of instruments and analysis methods, which makes it difficult to evaluate common statistical weights for the used data points. So, remembering such a property, in view that the unified RC well approximates the original curves as well as for its convenience for the determination of the mass distribution by the least-squares and/or $\chi^2$ fitting, we shall employ it in our present analysis. Mass Components --------------- The rotation velocity is related to the gravitational potential, hence to the mass distribution, as V(R)==, where $\Phi_i$ is the gravitational potential of the $i$-th component and $V_i$ is the corresponding circular velocity. The rotation velocity is often represented by superposition of the central black hole (BH), bulge, disk, and the dark halo as V(R)= . Here, the subscript BH represents black hole, b stands for bulge, d for disk, and h for the dark halo. The contribution from the black hole can be neglected in sufficiently high accuracy, when the dark halo is concerned. The mass components are usually assumed to have the following functional forms. ### Massive Black Hole The GC of the Milky Way is known to nest a massive black hole of mass of $M_{\rm BH}\sim 4 \times 10^6\Msun$ \[[@Genzel+2010; @Ghez+2008; @Gillessen+2009]\].The RC is assumed to be expressed by a curve following the Newtonian potential of a point mass at the nucleus. ### De Vaucouleurs Bulge The commonly used SMD profile to represent the central bulge, which is assumed to be proportional to the empirical optical profile of the surface brightness, is the  law \[[@deV1958]\], \_[b]{}(R) = \_[be]{} [exp]{} , \[eq-smdb\] where $ \Sigma_{\rm be} $ is the value at radius $R_{\rm b}$ enclosing a half of the integrated surface mass \[[@Sofue2017]\].Note that the surface profile, also the exponential disk, has a finite value at the center. The volume mass density $\rho(r)$ at radius $r$ for a spherical bulge is calculated using the SMD by (r) = \_r\^ dx, \[eq-rhob\] and the mass inside $R$ is M(R) =4\_0\^R r\^2(r)dr. The circular velocity is thus obtained by V\_[b]{}(R) = . More general form $e^{-(R/r_e)^n}$ called the law is discussed in relation to its dynamical relation to the galactic structure based on the more general profile \[[@Ciotti1991; @Trujillo2002]\]. ### Exponential Disk The galactic disk is generally represented by an exponential disk \[[@Freeman1970]\],where the SMD is expressed as \_[d]{} (R)=\_[d]{} [exp]{}(-R/R\_[d]{}). \[eq-smdd\] Here, $\Sigma_{\rm d}$ is the central value, $R_d$ is the scale radius. The total mass of the exponential disk is given by $M_{\rm disk}= 2 \pi \Sigma_{dc} R_{\rm d}^2$. The RC for a thin exponential disk is expressed by \[[@Binney+1987]\]V\_[d]{}(R)=, where $y=R/ (2R_{\rm d}) $, and $I_i$ and $K_i$ are the modified Bessel functions. The dark halo is described in the next section Dark Halo {#Section3} ========= The existence of dark halos in spiral galaxies has been firmly evidenced from the well established difference between the galaxy mass predicted by the luminosity and the mass predicted by the rotation velocities \[[@SofueRubin2001; @Sofue2017; @Salucci2019]\]. In the Milky Way, extensive analyses of RC and motions of non-disk objects such as globular clusters and dwarf galaxies in the Local Group have shown flat rotation up to $\sim 30$ kpc, beyond which the RC declines smoothly up to $\sim 300$ kpc \[[@Sofue2013; @Sofue2015]\]. Further analyses of non-disk tracer objects have also shown that the outer RC declines in a similar manner \[[@Bhattacharjee+2014; @Huang+2016; @Li+2017]\]. The fact that the rotation velocity beyond $R\sim 30$ kpc declines monotonically indicates that the isothermal model can be ruled out in representing the Milky Way’s halo. Dark Halo Models ---------------- There have been various proposed DH models, which may be categorized into two types: The cored halo models \[[@Burkert1995; @Salucci+2000; @Brownstein+2006]\] are a modification of the isothermal model with a steeper decrease of density at large radii. The central cusp models \[[@Navarro+1995; @Navarro+1997; @Moore+1999; @Fukushige+2004]\] are based on extensive $N$-body numerical simulations of the structural evolution in the cold dark matter scenario in the expanding universe, which predict an infinitely increasing central peak. In either type, all the DH models predict decreasing DM density beyond $h$ as $\rho \propto R^{-3}$, or declining rotation velocity as $\Vrot \propto \sqrt{{\rm ln} \ R/R}$. The cored halo models exhibit a central plateau of finite density with scale radius, or the core radius, $h$, and are often represented by the following functions, where $x=R/h$. [**Isothermal halo**]{}: \_[Iso]{} (x)=, \[eq-iso\] [**Beta model with $\beta=1$** ]{} \[[@Navarro+1995]\] :\_(x)= . [**Burkert model**]{} \[[@Burkert1995; @Salucci+2000]\] : \_[Bur]{} (x)=, \[eq-bur\] [**Brownstein model**]{} \[[@Brownstein+2006]\] : \_[Bro]{} (x)=. \[eq-bro\] On the other hand, the central cusp models are often represented by the following functions. [**NFW model**]{} \[[@Navarro+1996; @Navarro+1997]\] : \_[NFW]{} (x)=, \[eq-nfw\] [**Moore model**]{} \[[@Moore+1999; @Fukushige+2004]\] with $\alpha=1.5$: \_[Moo]{} (x) = =. \[eq-nfw\] Figure \[rhoModels\] shows schematic density profiles for various DH models with $h=10$ kpc combined with the bulge and exponential disk, where the halo density is normalized at $R=20$ kpc.. ![ (**Top**) Schematic density profiles of [NFW]{} (Navarro, Frenk, White) (thick solid), Moore (upper long dash), Burkert (long dash), Brownstein (dot), $\beta$ (dash), and isothermal (thin solid) models with $h=10$ kpc normalized at 20 kpc, compared with the disk (straight line) and bulge (inner thick dash). Uppermost thin lines are the sum of bulge, disk and halo. (**Bottom**) Same, but in log–log plot. The NFW cusp and cored halos do not much contribute to the mass density in the GC, whereas the Moore cusp somehow resembles the bulge profile. []{data-label="rhoModels"}](fig2.eps){width="10cm"} Cusp vs Cored Halo ------------------ The density profiles for the NFW (Navarro, Frenk and White), Moore, Burkert, $\beta$, and Brownstein models are almost identical beyond the core radius $h$, where they tend to $\propto R^{-3}$. Differences among the models appear within the Solar circle. The cusp models (NFW and Moore models) predict steep increase of density toward the center with a singularity. The cored halo models predict a mild and low density plateau in the center with the peak densities not much differing from each other within a factor of two. However, the Burkert model has a singularity with the density gradient being not continuous across the nucleus. Most of the DH models predict lower density in the innermost galaxy by two to several orders of magnitudes than the bulge’s density. This implies that the DH does not much influence the kinematics in the inner galaxy. Namely, it is practically impossible to detect the DM cusp by analyzing the RC. Only the Moore model predicts cusp density exceeding the bulge’s density in the very center at $R<\sim 0.1$ pc, whereas the applicability of the model to such small sized region is not obvious \[[@Fukushige+2004]\]. Central DM Density ------------------ If we assume that the functional form of the NFW model is valid in the very central region, the SMD at $R\sim 100$ pc could be estimated to be about $\Sigma \sim 2.2\times 10^3\Msun {\rm pc}^{-2}$. This yields an approximate volume density on the order of $\rho \sim \Sigma/R\sim 11\Msun {\rm pc}^{-3} \sim 840$ GeV cm$^{-3}$ for a detector of $\sim 1.4\deg$ resolution. Such estimations could be a key to the indirect detection experiments of DM in the GC . However, it is stressed that the DM density in the GC is two to several orders of magnitudes smaller than the bulge’s density on the order of $10^4-10^5$ GeV cm$^{-3}$, making the kinematical detection of DM difficult. Interestingly, the column density of DM, hence brightness (flux/steradian) of self-annihilation emission ($\gamma$-ray) stays almost constant against the radius and is therefore constant regardless the resolution of the detector. On the other hand, the emission measure $\sim \rho^2 R$ varies as $\propto R^{-1}$, hence, the brightness of collision-origin emission ($\gamma$ or microwave haze) increases toward the center \[e.g.,[@Finkbeiner2004]\], so that the detection rate will increase with the detector’s resolution. Another concern about the DM cusp is the kinetic energy of individual particles. In order for the cusp to be stationary, the particles must be bound to the gravitational potential, so that the particle’s speed must be lower than the escaping velocity $v\sim \sqrt{2}\Vrot \sim 300$ . This will give a constraint on the cross section $\sigma_A$ of the DM annihilation, if the collision rate $\sigma_A v$ is fixed by the detection of DM-origin emissions. The cored halo models (isothermal, Burkert, Brownstein, and the $\beta$ models) predict a mild and finite-density plateau with scale radius of $h$ ($\sim 10$ kpc). Their central densities are also several orders of magnitude less than the bulge’s density, hence do not contribute to the kinematics of the gas and stars in the GC. DM Density from Direct SMD {#Section4} ========================== SMD from RC ----------- In the decomposition method of the RC, the resulting mass distribution depend on the assumed functional forms of the model profiles. In order to avoid this inconvenience, the RC can be used to directly calculate the surface mass distribution without employing any functional form. Only an assumption has to be made, either if the galaxy’s shape is a sphere or a flat disk. On the assumption of spherical distribution, the mass inside radius $R$ is given by $$M(R)=\frac{R {V(R)}^{2}}{G}. \label{masssphere}$$ Then the surface-mass density (SMD) ${\Sigma}_{S}(R)$ at $R$ is calculated by \_[S]{}(R) = 2 \_0\^ (r) dz, \[smdsphere\] where $$\rho(r) =\frac{1}{4 \pi r^2} \frac{dM(r)}{dr}. \label{rhosphere}$$ If the galaxy is assumed to be a flat thin disk, the SMD ${\Sigma}_{\rm d}(R)$ is calculated by solving Poisson’s equation (Freeman 1970; Binney and  Tremaine 1987) by $${\Sigma}_{\rm d}(R) =\frac{1}{{\pi}^2 G} \left[ \frac{1}{R} \int\limits_0^R {\left(\frac{dV^2}{dr} \right)}_x K \left(\frac{x}{R}\right)dx + \int\limits_R^{\infty} {\left(\frac{dV^2}{dr} \right)}_x K \left (\frac{R}{x}\right) \frac{dx}{x} \right]. \label{smdflat}$$ Here, $K$ is the complete elliptic integral, which becomes very large when $x\simeq R$. The SMD distributions in the galaxy for the sphere and flat-disk cases have been calculated for the recent RCs \[[@Sofue2017]\].In this paper we apply the same method to the here obtained unified RC (Figure \[mwrc\]). Since we aim at studying the dark halo, which is postulated to be rather spherical than a flat disk, we assume spherical mass distribution. The calculated SMD distribution is shown in Figure \[smd\_fit\]. ![(**Top**) Direct surface-mass density (SMD) calculated for the unified RC in figure \[mwrc\] in spherical symmetry assumption (dots with error bars) in semi-logarithmic representation. The solid line is the $\chi^2$ fit, and red, blue, and dashed lines represent the NFW halo, disk, and bulge, respectively. (**Bottom**) Same, but in log–log plots. The semi-logarithmic plot makes it easier to discriminate the dark halo from exponential disk, which appears as a straight line. The plotted values are listed in the tables in Appendix \[AppendixA\]. []{data-label="smd_fit"}](fig3.eps){width="10cm"} The SMD is strongly concentrated toward the center, reaching a value as high as $\sim 10^5 \Msun~{\rm pc}^{-2}$ within $R\sim 10$ pc, representing the core of the central bulge with the extent of several hundred pc. It is followed by a straightly declining profile from $R\sim 2$ to 8 kpc in the semi-logarithmic plot, representing the exponential nature of the galactic disk. In the outer galaxy beyond $\sim 8$ kpc, the SMD profile tends to be displaced from the straight disk profile, and is followed by an extended outskirt with a slowly declining profile, representing a massive halo extending to the end of the RC measurement at . Fitting by Bulge, Disk, and Dark Halo ------------------------------------- In order to separate the dark halo from the disk and bulge components, the well established RC decomposition method has been extensively applied to the RCs \[[@Sofue2017; @Salucci2019]\].Besides this traditional method, we here propose to use the SMD distribution. For this, we assume three mass components of bulge, exponential disk, and dark halo. In order to represent the, we employ the NFW profile as a ’tool’ for its popularity and for the dynamics background based on the extensive numerical simulations. We employ the least $\chi^2$ fitting method, where $\chi^2$ is defined by \^2=\_i\[(SMD\_i\^[direct]{} - SMD\_i\^[calc]{})/\_i\]\^2, with $i$ denoting the value at the $i$-th data point, and $\sigma_i$ is the standard deviation around each data point in the running averaging procedure of the SMD distribution. Fitting parameters are the scale radius $a_d$ and central SMD $\Sigma_d^0$ for the disk, and the scale (core) radius $h$ and representative DM density $\rho_{\rm model}^0$ for the halo. The bulge SMD is fixed to an assumed profile, which is negligible in the present fitting range at $R\ge 1$ kpc. The fitting was obtained between $R=1$ and 100 kpc. The fitting result for the NFW halo model is shown in Figure \[smd\_fit\]. The solid line is the $\chi^2$ fit to SMD, and red, blue, and dashed lines represent the halo, disk, and bulge components, respectively. Note that the semi-logarithmic plot makes it visually easier to recognize the dark halo significantly displaced from the exponential disk, which appears as a straight line. Local DM Density ---------------- We thus obtained the NFW DM halo parameters to be $h=10.94\pm 1.05$ kpc, $\rho_{\rm NFW}^0=0.787\pm 0.037$ , which yields the local DM density $\rho_\odot=0.359\pm 0.017$ . The best-fit parameters for the disk are determined to be $a_d=4.38\pm 0.35 $ kpc and $\Sigma_0=(1.28\pm 0.09)\times 10^3 \Msun {\rm pc}^{-2}$. Table \[tab\_fit\] lists the fitted result along with the minimized $\chi^2$ value. **Component** **Parameter** **Fitted Value** ------------------------- -------------------- ------------------------------------------------- -------- Expo. disk $a_d$ $4.38\pm 0.35 $ kpc $\Sigma_0$ $(1.28\pm 0.09)\times 10^3 \Msun {\rm pc}^{-2}$ NFW dark halo $h$ $10.94\pm 1.05$ kpc $\rho_{\rm NFW}^0$ $0.787\pm 0.037$ $\rho_\odot$ $0.359\pm 0.017$ 11.9 Burkert$^\dagger$ $\rho_\odot$ $\sim 0.30\pm 0.02$ $17.3$ Brownstein$^\dagger$ $\rho_\odot$ $\sim 0.40\pm 0.02$ $17.9$ $\beta$ model$^\dagger$ $\rho_\odot$ $\sim 0.31\pm 0.02$ $17.3$ : Best-fit parameters of the direct SMD by NFW halo and exponential disk.[]{data-label="tab_fit"} $^\dagger$ Rough fitting, not conclusive. We also obtained $\chi^2$ fitting using the Burkert, Brownstein, and $\beta$ profiles, and listed the local DM density and minimized $\chi^2$ in Table \[tab\_fit\]. In these three models, the $\chi^2 \sim 17-18$ were found to be systematically greater than that for the NFW model ($\chi^2=11.9$). The reason for the difference is due to the systematic difference in the functional behavior between NFW and the other three models: NSF has a cusp steeply increasing toward the center with sharpening scale radius, which results in the possibility of finer fitting to the slightly curved SMD profile at $R<\sim 10$ kpc in the semi-log plot. On the contrary, the other three models predict almost negligible SMD there, so that halo parameters contribute less intensively to the fitting in the innermost region, or the fitting must be done only by the disk’s two parameters there, resulting in worse fitting. ----------------------------------------------------------------------- ------------------- ----------- -------- -- [**Reference**]{} **(kpc)** **()** Weber and de Boer (2010)\[[@Weber+2010]\] 0.2 - 0.4 Catena and Ulio (2010)\[[@Catena+2010]\] $0.389 \pm 0.025$ Bovy and Tremaine (2012) \[[@Bovy+2012]\] $0.3\pm 0.1$ Piffl et al. (2014) \[[@Piffl+2014]\] 0.58 Pato et al (2015), Pato & Iocco (2015) \[[@Pato+2015a; @Pato+2015b]\] $0.42\pm 0.25$ 230 Huang et al. (2016)\[[@Huang+2016]\] $0.32 \pm 0.02$ 8.34 240 McMillan (2017)\[[@McMillan2017]\] $0.38\pm 0.04$ 8.21 233.1 Lin and Li (2019)\[[@Lin+2019]\] $0.51 \pm 0.09$ 8.1 240 Salucci et al. (2010, 2019) \[[@Salucci+2010; @Salucci2019]\] $0.43\pm 0.06$ 8.29 239 Eilers et al (2019) \[[@Eilers+2019]\] $0.3\pm 0.03$ 8.1 229 de Salas et al. (2019) \[[@deSalas+2019]\] $0.3 - 0.4$ Cautun et al (2019) \[[@Cautun+2019]\] $0.34\pm 0.02$ 8 229 Karukes et al (2019) \[[@Karukes+2019]\] $0.43 \pm 0.02$ 8.34 240 Sofue (2013) \[[@Sofue2013]\] $0.40 \pm 0.04$ 8.0 238 —– (2020 this paper) $0.36\pm 0.02$ 8.0 238 Average$^\ddagger$ $0.387 \pm 0.080$ ----------------------------------------------------------------------- ------------------- ----------- -------- -- : Current determinations of the local DM density and the literature.[]{data-label="tab_localdm"} $^\dagger$ =38.2 $\Msun\ {\rm pc}^{-3}$. $^\ddagger$ Simple average of the listed values with equal weighting.\ The local DM density is a key quantity in laboratory experiments by the direct detection of DM, and has been estimated by a number of authors with a variety of methods. In Table \[tab\_localdm\] we list the local DM densities from the literature along with the present value for NFW profile. They are also plotted in Figure \[localDMauthors\] against publication years. The $\rho_\odot$ values seem to be nearly constant in the decade. Averaging all the listed values with an equal weighting yields $\rho_\odot=0.39 \pm 0.09$ , which may be taken as a ’canonical’ value. ![Local dark matter (DM) density from the literature (Table \[tab\_localdm\]) plotted against publication year. The dashed line indicates a simple mean of the plots at $\rho_\odot=0.39 \pm 0.09$ .[]{data-label="localDMauthors"}](fig4.eps){width="9cm"} Dependence on the Galactic Constants ------------------------------------ We have re-scaled the adopted RC to $(R_0,V_0)=(8.0, 238)$ (kpc, ), which may vary within several %. The resulting local DM density will vary accordingly, depending on the constants. The local mass density of the spherical component is dependent on the constants as $\rho_0 \propto R_0 V_0^2/R_0^3 \sim V_0^2 R_0^{-2}$. For small corrections $\delta V_0$ and $\delta R_0$, the DM density will change as $\delta \rho_0/\rho_0 \sim 2(\delta V_0/ V_0-\delta R_0/R_0)$. For example, for $\delta V_0 \sim \pm $ 10 , the estimated local density varies by $\delta \rho_0/\rho_0 \sim \pm 0.08$, or for $\delta R_0 \sim \pm 0.1$ kpc, $\delta \rho_0/\rho_0 \sim \mp 0.025$. Summary ======= We reviewed the current status of determination of the RC of the Milky Way, and presented a unified RC from the GC to outer halo at $R\sim 100$ kpc. The RC was used to directly calculate the SMD without assuming any functional form. The disk appears as a straight line on the semi-logarithmic plot of SMD against $R$, and is visually well discriminated from the DH having an extended outskirt. The SMD distribution was fitted by a bulge, disk, and NFW dark halo using the $\chi^2$ method. The best-fit DH profile yielded the local DM density of $0.359 \pm 0.017$ . We also reviewed the current estimations from the literature in the last decade, which appear to be converging to a mean value of $\rho_\odot=0.39 \pm 0.09$ . [[**Acknowledgments**]{}]{} [The data analysis was performed at the Center of Astronomical Data Analysis of the National Astronomical Observatory of Japan. The author is grateful to Professor A. Hofmeister for inviting him to this special issue. ]{} Tables concerning the RC and SMD of the Milky Way {#AppendixA} ================================================= Tables \[tabrcA\] and \[tabrcB\] list the running-averaged RC of the Milky Way using the data from \[[@Sofue2015; @Sofue2017; @Huang+2016]\],which is used to calculate the SMD in Figure \[smd\_fit\]. Tables \[tab\_smdsA\] and \[tab\_smdsB\] lists the directly calculated SMD from the RC \[0.7\] ------------ ------- ------------------- **Radius** **Standard Dev**. (kpc) () () 0.100 144.9 3.7 0.110 147.4 4.2 0.121 150.4 4.8 0.133 153.8 6.1 0.146 158.9 10.3 0.161 167.4 16.1 0.177 180.1 22.4 0.195 196.6 27.1 0.214 213.6 26.9 0.236 227.8 22.7 0.259 237.9 17.0 0.285 244.4 11.8 0.314 248.2 7.6 0.345 250.2 4.7 0.380 251.0 2.9 0.418 250.7 2.1 0.459 249.7 2.3 0.505 248.0 2.9 0.556 245.9 3.7 0.612 243.2 4.6 0.673 239.8 5.7 0.740 235.8 6.4 0.814 231.7 6.5 0.895 227.8 6.0 0.985 224.5 5.2 1.083 221.7 4.5 1.192 219.1 4.0 1.311 216.8 3.7 1.442 214.7 3.4 1.586 212.7 3.1 1.745 210.9 2.8 1.919 209.5 2.3 2.111 208.5 1.8 2.323 208.2 1.6 ------------ ------- ------------------- : Rotation curve of the Milky Way used in Figure \[mwrc\].[]{data-label="tabrcA"} \[0.7\] ------------ ------- ------------------- **Radius** **Standard Dev**. (kpc) () () 2.555 208.9 2.2 2.810 210.7 3.6 3.091 213.4 4.8 3.400 217.2 5.9 3.740 222.0 6.6 4.114 226.6 5.7 4.526 229.5 4.4 4.979 231.6 4.3 5.476 234.1 5.3 6.024 237.2 5.7 6.626 239.5 5.0 7.289 240.1 4.1 8.018 239.0 4.4 8.820 236.7 5.4 9.702 234.5 6.0 10.672 234.2 7.1 11.739 237.1 9.8 12.913 242.8 12.4 14.204 248.5 13.3 15.625 249.7 14.8 17.187 246.2 17.4 18.906 243.3 18.3 20.797 243.9 17.5 22.876 245.6 15.6 25.164 243.7 15.2 27.680 237.3 16.1 30.448 229.6 15.5 33.493 222.5 14.1 36.842 215.0 14.0 40.527 207.1 13.8 44.579 200.3 12.7 49.037 194.7 11.9 53.941 189.8 11.3 59.335 186.2 10.4 65.268 184.7 9.6 71.795 183.9 9.3 78.975 181.4 11.0 86.872 175.5 14.6 95.560 167.7 16.3 ------------ ------- ------------------- : Continued from Table \[tabrcA\].[]{data-label="tabrcB"} \[0.7\] ------------ --------- ------------------- **Radius** **Standard Dev.** **(kpc)** 0.100 29933.0 861.3 0.110 29054.0 654.8 0.121 28384.0 666.0 0.133 28160.0 570.8 0.146 28319.0 637.2 0.161 28203.0 1406.6 0.177 27368.0 2481.1 0.195 25014.0 3514.2 0.214 21548.0 4357.7 0.236 17908.0 4806.8 0.259 14804.0 4733.1 0.285 12369.0 4231.5 0.314 10489.0 3549.3 0.345 8978.9 2929.7 0.380 7736.5 2384.1 0.418 6700.8 1959.1 0.459 5830.8 1636.9 0.505 5090.6 1374.2 0.556 4452.0 1158.1 0.612 3899.9 973.5 0.673 3464.9 803.4 0.740 3145.2 644.7 0.814 2904.3 510.8 0.895 2701.7 415.1 0.985 2510.1 354.4 1.083 2320.9 319.0 1.192 2144.8 291.9 1.311 1985.0 266.7 1.442 1843.6 240.7 1.586 1718.4 214.5 1.745 1611.4 188.2 1.919 1519.3 164.5 2.111 1440.6 144.4 2.323 1368.3 130.9 ------------ --------- ------------------- : Directly calculated SMD by spherical assumption of the mass distribution.[]{data-label="tab_smdsA"} \[0.7\] ------------ ------------------------- ------------------------- **Radius** Standard Dev. (kpc) $\Msun \ {\rm pc}^{-2}$ $\Msun \ {\rm pc}^{-2}$ 2.555 1296.4 125.1 2.810 1220.5 126.3 3.091 1139.8 134.1 3.400 1055.2 146.1 3.740 944.3 157.4 4.114 824.6 161.5 4.526 734.9 150.5 4.979 668.3 133.7 5.476 600.8 123.8 6.024 523.5 119.3 6.626 446.8 113.6 7.289 383.7 101.6 8.018 339.1 83.4 8.820 314.1 62.2 9.702 303.4 43.9 10.672 293.2 39.0 11.739 272.0 48.2 12.913 229.5 58.5 14.204 170.5 65.2 15.625 127.5 62.4 17.187 114.7 46.4 18.906 110.0 35.2 20.797 91.8 33.5 22.876 61.2 34.5 25.164 37.2 33.2 27.680 28.0 25.4 30.448 24.7 17.1 33.493 20.3 11.6 36.842 17.3 7.8 40.527 17.1 4.7 44.579 16.7 3.0 49.037 15.2 2.3 53.941 14.4 2.3 59.335 13.4 3.3 65.268 10.6 4.3 71.795 6.2 4.9 ------------ ------------------------- ------------------------- : Continued from Table \[tab\_smdsA\].[]{data-label="tab_smdsB"} [**References**]{} [999]{} 1\. Sofue, Y.; Rubin, V.  Rotation Curves of Spiral Galaxies. *Annu. Rev. Astron. Astrophys.* **2001**, *39*, 137. 2\. 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--- abstract: 'Polarization-resolved magneto-luminescence, together with simultaneous magneto-transport measurements, have been performed on a two-dimensional electron gas (2DEG) confined in CdTe quantum well in order to determine the spin-splitting of fully occupied electronic Landau levels, as a function of the magnetic field (arbitrary Landau level filling factors) and temperature. The spin splitting, extracted from the energy separation of the $ \sigma^+$ and $\sigma^-$ transitions, is composed of the ordinary Zeeman term and a many-body contribution which is shown to be driven by the spin-polarization of the 2DEG. It is argued that both these contributions result in a simple, rigid shift of Landau level ladders with opposite spins.' author: - 'J.' - 'K.' - 'F. J.' - 'P.' - 'B. A.' - 'D. K.' - 'M.' - 'V.' - 'G.' - 'T.' title: 'Enhancement of the spin-gap in fully occupied two-dimensional Landau levels' --- A number of experiments on two-dimensional electron gases (2DEGs) [@Nicholas88; @Usher90; @Leadley98; @Maude98] clearly show that the thermal activation of carriers across the Fermi energy, located between the spin split Landau levels at odd integer filling factors ($\nu$), is governed by a gap which can significantly surpass the single particle Zeeman energy included in band structure models. This phenomenon, referred to in the literature as $g$ factor or spin gap enhancement, [@Fang68; @Janak69; @Ando74] is thought to be driven by the spin polarization of a 2DEG and is a primary manifestation of the interactions between two-dimensional electrons in the integer quantum Hall effect (QHE) regime. It is a result of the specific character of the spin-excitation spectra of a 2DEG at odd integer $\nu$-QHE states [@Bychkov81; @Kallin84]. It can be seen as arising from the contribution of Coulomb interactions (including exchange terms) to the energy which is required to remove, or inject, an electron from, or to, a given spin resolved Landau level (LL). To date, the effect of the spin-gap enhancement has been generally limited to experiments [@Nicholas88; @Usher90; @Maude98; @Dolgopolov97; @Wang92; @Wiegers97] which probe the spin splitting at the Fermi level, for QHE states at exactly odd filling factors. This limitation has been thought to be overcome with spectroscopic methods such as, for example, interband optics [@Kukushkin93; @Kukushkin96; @Potemski98] or tunneling experiments [@Dial07], which, within their trivial description, permit to investigate the processes of removing/adding an electron from/to a 2DEG, at arbitrary energy, filling factor and temperature. Among the different spectroscopic methods, magneto-luminescence measurements has been widely invoked to investigate electron-electron correlation in the QHE regime, however, measurements to probe the spin-gap enhancement are rather scarce [@Kukushkin93; @Dial07]. Here, we report on magneto-photoluminescence studies of a 2DEG confined in a high quality CdTe quantum well, and, show that the enhancement of the spin splitting is not only a property of spin excitations at the Fermi level, but that it is also relevant for fully occupied spin Landau levels, located well below the Fermi energy. We have measured the many body contribution to the spin gap for fully populated spin Landau levels over a wide range of filling factors and temperatures, and show that it is driven by Coulomb interaction, apparent via the spin polarization of the investigated 2DEG with its relatively large bare Zeeman splitting. The increasingly high quality of GaAs/GaAlAs structures has been driving advances in the physics of interacting 2D electrons. Notably, 2D electrons in a GaAs matrix are characterized by a relatively small bare $g$ factor (-0.44) and therefore by a small value of the interaction parameter $\eta=E_z/\mathcal{D}$, where $E_z=g\mu_BB$, $\mathcal{D}=e^2/ \epsilon l_B$, and, $l_B=\sqrt{\hbar/eB}$ is the magnetic length. The small value of $\eta$ is responsible for the rich physics exhibited by interacting 2D electrons in the QHE regime, for example the occurrence of competing spin polarized/unpolarized many body ground states [@Clark89] or Skyrmion-type spin texture excitations [@Sondhi93; @Schmeller95; @Maude96]. However, this complex physics often masks the appearance of simpler and basic many body effects, which should emerge more clearly when $\eta$ is sufficiently large. Disorder is an additional source of complications in ascertaining the spin polarization in systems with small $g$ factors. While high electron mobilities are obviously advantageous, GaAs-based structures are also rather fragile, displaying, for example, metastable effects upon illumination, with an associated decrease in mobility and homogeneity, which frequently prevents the simultaneous basic characterization of such structures using magneto-optics and magneto-transport. A 2DEG in a CdTe matrix [@Karczewski98], used in our experiments, is characterized by relatively large (bare) $g$ factor (-1.6) and the $\eta$-parameter in this system exceeds by a factor of $\approx 3$ its value in GaAs structures (the dielectric screening $\epsilon=10$ is slightly less efficient in CdTe). CdTe, which has a conduction band as simple as the one in GaAs, appears to be an almost ideal model system to study the QHE physics of the primary spin-polarized states. The significant progress in the crystal growth of CdTe quantum wells permits nowadays to attain a 2DEG with reasonably high mobilities. As shown in Fig. \[Fig1\], the sample studied here shows a well pronounced fractional QHE and permits a trouble-free, simultaneous measurement of high quality magneto-photoluminescence and magneto-transport. The active part of the investigated structure consist of a $20$ nm-wide CdTe quantum well (QW), modulation doped on one side with iodine, and embedded between Cd$_{0.74}$Mg$_{0.26}$Te barriers. The sample, in form of 1.5$\times$6mm rectangle, was equipped with electrical contacts in a Hall bar configuration to permit simultaneous optical and electric measurements. Experiments have been carried out using either a $^3$He/$^4$He dilution refrigerator or a variable temperature $^4$He cryostat, in magnetic fields supplied by a resistive (28 T) or superconducting (11 T) magnets. A standard, low frequency ($\approx 10$ Hz) lock-in technique has been applied for the resistance measurements. Polarization resolved, $\sigma^{+}$ and $\sigma^{-}$ photoluminescence (PL) spectra have been measured using a single 600 $\mu$m-diameter optical fiber to transmit the excitation beam (514 nm-line of Ar$^+$ laser) and to collect the photoluminescence signal for the spectrometer (spectral resolution $\approx100\mu eV$) equipped with a CCD camera. An appropriate linear polarizer and $\lambda$/4-plate were placed directly between the end of the fiber and the sample. The $\sigma^{+}$ and $\sigma^{-}$ PL components were measured by reversing the polarity of the magnetic field. Special attention has been paid to assure a low level of laser excitation ($\approx50$ $\mu$W/cm$^2$), to precisely calibrate the magnetic field, and to measure the spectra at small intervals (down to 5 mT) of the magnetic field. Under our experimental conditions (continuous laser illumination), the 2DEG density of $\approx4.5\times 10^{11}$ cm$^{-2}$ and mobility of $\mu=2.6 \times 10^{5}$ cm$^2$/Vs were well reproduced in different experimental runs. The representative results of simultaneous magneto-PL and magneto-resistivity measurements of our sample are shown in Fig. \[Fig1\]. As can be seen in Fig. \[Fig1\](b), the investigated 2DEG shows all typical attributes of the QHE in a system with fairly high mobility and relatively high electron concentration; well developed integer QHE states and the appearance of $5/3$, $4/3$ and $2/3$ fractional states (which will be discussed elsewhere). From the field at which the Shubnikov de Haas (SdH) oscillations ($B_{1}\approx94$ mT), and spin-splitting appears ($B_{2}\approx0.51$ T), we obtain a first estimate of the enhanced $g$ factor, $g^*\approx3.7$ using the condition ($\hbar eB_1/m^{*} \approx g^{*}\mu _{B}B_2$) where the electron effective mass $m^{*}=0.1m_{e}$ was derived from cyclotron resonance absorption measured on a parent sample. A Dingle analysis of the SdH oscillations gives a quantum lifetime $\tau_q=\hbar/2\Gamma=(3.0\pm0.3)$ ps (broadening of Lorentzian Landau levels $\Gamma\approx110~\mu$eV) as compared to the transport lifetime $\tau_{\tau}\approx15$ ps (derived from the measured mobility). The evolution of the PL with the magnetic field (Fig. \[Fig1\]) resembles spectra reported in numerous PL investigations, widely applied in the past to GaAs-based structures [@Asano02]. Peaks in the magneto-PL spectra are due to the recombination of electrons from occupied conduction band LLs ($L_{N}, E_{N}=(N+1/2)\hbar\omega_{c}, N=0,1,..$) with photo-excited holes from valence band LLs ($L^{h}_{N}, E^{h}_{N}=(N+1/2)\hbar\omega^{h}_{c}, N=0,1,...)$, where $\omega_{c}$ and $\omega^{h}_{c}$ is the cyclotron frequency of the electrons and holes respectively. The energy of the main peaks, due to $L_{N} \rightarrow L^{h}_{N}$ ($N_{e}-N_{h}=0$) transitions which scale as $E_{0}+(N+1/2)(\hbar\omega_{c}+\hbar\omega^{h}_{c})$, are shown as black dots in Fig. \[Fig1\]. Since $\hbar\omega^{h}_{c}/\hbar\omega_{c}\approx5$ [@Romestain80] the magneto-PL spectra reflect largely the characteristic fan chart of electronic LLs (with respect to band-edge energy, $E_{0}$) including their occupation factor. Opposite LL spin components are resolved in the $\sigma^{+}$ and $\sigma^{-}$ spectra. The exchange of the intensity between the $\sigma^{+}$ and $\sigma^{-}$ PL when sweeping through filling factor $\nu=1$ is typical of the 2DEG studied here and results from the selection rules which are specific to CdTe (see Fig. \[Fig1\](d)). The non-monotonic variation, with magnetic field, of the transition energies and intensities (oscillations which correlate with filling factor) and possible appearance of line splitting (see Fig. \[Fig1\]) are other common features of magneto-PL investigations of a 2DEG. Electron-electron interactions, combined with different perturbations induced by the presence of the valence band hole, are almost certainly at the origin of these features [@Asano02]. The understanding these features is far from universal and a detailed analysis of the energy and intensity of each individual magneto-PL transitions is beyond the scope of our paper. We have found, however, that information on the effects of electron-electron interactions can be extracted from the relative positions of polarization-resolved PL peaks arising from different LL spin components. We focus our attention on the two lowest energy $\sigma^{+}$ and $\sigma^{-}$ magneto-PL transitions (Fig. \[Fig2\]) which are due to electrons, with different spins, recombining from the fully populated ($L_{0}$) LL. While the energy of each of these peaks displays a non-trivial dependence on the magnetic field, here we focus on the evolution of the energy separation $\Delta E$ between the $\sigma ^{+}$ and $\sigma ^{-}$ transitions plotted in Fig. \[Fig2\](c). The splitting $\Delta E$ does not follow a linear field dependence which is expected for the case of an ordinary Zeeman effect. This can be even seen in the raw data in Fig. \[Fig2\](a-b); The splitting $\Delta E$ observed at higher field $B=4.63$ T is clearly smaller than the splitting at lower fields $B=3.7$ T. The $\Delta E$ versus $B$ dependence in Fig. \[Fig2\](c) naturally suggests that this dependence is composed of two terms; a Zeeman term ($\Delta E _{Z}$) linear with $B$ and a many body term ($\Delta E _{\uparrow\downarrow}$), which is non-monotonic with $B$, having maxima at odd integer $\nu$ and zeros at even integer $\nu$. The linear term can be extracted from the splitting at even integer $\nu$ and it is in agreement with the ordinary Zeeman effect expected in our structure. Taking into account the selection rules depicted in Fig. \[Fig1\](d), the splitting $\Delta E _{Z}=(|g|-|g_{h}|)\mu_{B}B=g_{eff}\mu_{B}B$, which requires $g_{eff}=1.1$ to fit the data in agreement with the reported values of $g=-1.6$ and $g_{h}\approx0.5$, for electronic and valence hole $g$ factors in CdTe QWs [@Zhao96]. To further clarify the origin of the $\Delta E_{\uparrow\downarrow}$ term, we plot this term as a function of filling factor and show its characteristic evolution with temperature (Fig. \[Fig3\]). We have extracted $\Delta E_{\uparrow\downarrow}$ from different experimental runs, by subtracting the ordinary Zeeman term which is assumed to be temperature independent. The electron concentration (filling factor scale) was determined using simultaneous magneto-resistance measurements. An inspection of the results presented in Fig. \[Fig3\] strongly suggest that $\Delta E_{\uparrow\downarrow}$ is ruled by the spin polarization $\mathcal{P}=\frac{n_\downarrow-n_\uparrow}{n_\downarrow+n_\uparrow}$ of the 2DEG. A quantitative verification of this hypothesis is provided by the following simple model. We consider the ideal case of a 2DEG with discrete Landau levels separated by $\hbar\omega_{c}$ and spin split by $$\label{Spingap} \Delta_{s}=|g|\mu_{B}B + \Delta E_{\uparrow\downarrow}=|g|\mu_{B}B+ \Delta_{0}'\varphi(B)\cdot\frac{n_\downarrow-n_\uparrow}{n_\downarrow+n_\uparrow}.$$ In particular, we assume that the enhanced part ($\Delta E _{\uparrow\downarrow}$) of the spin splitting is common for all Landau levels, including the lowest LL ($L_{0}$) which we probe with PL and the LL in the vicinity of the Fermi energy, the occupation of which determines the spin polarization. Furthermore, we suppose that $\varphi(B)=\sqrt[4]{B^{2}+B^{2} _{0}}$ in order to phenomenologically account for the expected behavior of $\Delta E_{\uparrow\downarrow}$ in the limit of high magnetic fields ($\varphi(B) \sim \sqrt{B}$) and when $B$ tends to zero (($\varphi(B)$=constant) [@NOTE]. Finally, we self-consistently calculate $\Delta E _{\uparrow\downarrow}$ (and $\mathcal{P}$) and obtain agreement with the data by adjusting the two fitting parameters, $\Delta_0=\Delta_0'\sqrt{B_{0}}=2.1$ meV and $B_0=3.7$ T. Despite the rather crude approximations, the calculations well reproduce the experimental data (Fig. \[Fig3\]) over a wide range of filling factors ($4\leq \nu \leq 10$) and for different temperatures up to the temperature for which $E_{\uparrow\downarrow}$ (and $\mathcal{P}$) vanishes. The agreement is less satisfactory in the vicinity of $\nu=3$, and completely fails around $\nu=1$ where difference between $\sigma^+$ and $\sigma^-$ peaks shows almost no enhancement effect. These discrepancies are due to the fact that the physics of PL processes for a 2DEG at low filling factors is far more complex [@Asano02] compared to our temptingly simple picture of electrons which recombine (are extracted) from the homogenous Fermi sea of a 2DEG. The use of discrete LLs in our calculations is justified by the large bare Zeeman energy which exceeds the LL width (110 $\mu$eV, extracted from low field transport data) already at fields of $\sim2$ T ($\nu \sim 9$). The assumption that $E_{\uparrow\downarrow}$ does not depend on LL index is probably also realistic. When modelling the data, we have investigated various scenarios for a LL index dependence of the spin-gap enhancement but found that a constant value reproduces the data fairly well. Although it is more difficult to accurately determine the energy of the weak magneto-PL peaks for the higher $N>0$ LLs, it is possible to follow the separation between $\sigma+$ and $\sigma-$ transitions associated with the $L_{1}$-level in the vicinity of $\nu=5$. As shown in Fig. \[Fig3\] (right panel), the extracted enhancement of the spin gap in the $L_{1}$-level is practically the same as in the $L_{1}$-level. Moreover, we find a fair agreement between the spin gaps $|g|\mu_{B}B + \Delta E_{\uparrow\downarrow}$, extracted from PL, for fully populated LLs and the activation gaps, of $0.95$, $0.63$ and $0.36$ meV, for spin excitations across the Fermi energy, which we have estimated from resistance measurements at filling factors $\nu=5,7$, and $9$, respectively. Finally, let us speculate about a possible extension of the assumed model to the limit of low magnetic fields and to the particular case of $\nu=1$. When $B\rightarrow0$, the extrapolation of Eq. \[Spingap\] (at $T=0$ K) yields a linear $\Delta_{S}$ versus $B$ dependence; $\Delta_{S}=|g|\mu_{B}B+\Delta_{0}/\nu=(|g|+\Delta_{0}/\mu_{B}B_{\nu=1}) \mu_{B}B =g^{*}\mu_{B}B$, where $B_{\nu=1}$ corresponds to the magnetic field for $\nu=1$. With $B_{\nu=1}=18.5$ T ($n=4.5\times 10^{11}$ cm$^{-2}$) and $\Delta_{0}=2.1$ meV we extract $g^{*}=3.6$ for the enhanced $g$ factor in good agreement with the estimation of $g^{*}\sim3.7$ from the low field onset of spin splitting in the SdH oscillations. Setting $\nu=1$ (and $\mathcal{P}=1$) in Eq. ( \[Spingap\]) we extrapolate $\Delta_{S}=|g|\mu_{B}B+\Delta_{0}\sqrt[4]{1+B^{2} _{\nu=1}/ B^{2} _{0}}$ and calculate $\Delta_{S}=6.4$ meV. This value is a factor of $\sim4$ smaller than its ultimate limit of $\sqrt{\pi/2}e^2/\epsilon l_B$ [@Bychkov81; @Kallin84] but in good agreement with the reported values in GaAs structures from optical and capacitance measurements  [@Kukushkin96; @Dolgopolov97]. In conclusion, spectroscopic polarization-resolved magneto-PL studies of a 2DEG confined in CdTe quantum well reveal the many-body enhancement of the spin-splitting of fully occupied 2D Landau levels well below the Fermi energy. The enhancement is mainly determined by the spin polarization of the 2DEG, since the spin gap is maximized at odd filling factors, but vanishes at even filling factors or high temperatures. We argue that the spin polarization simply induces, in addition to the ordinary Zeeman splitting, a rigid shift of the spin up Landau levels with respect to spin-down Landau levels. This simple picture for the many body spin-gap enhancement emerges from magneto-PL studies of a 2DEG with relatively large (single particle) $g$ factor. [25]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (), . , ****, (). , ****, ().

--- abstract: 'Let $\left(a_{n}\right)_{n}$ be a strictly increasing sequence of positive integers, denote by $A_{N}=\left\{ a_{n}:\,n\leq N\right\} $ its truncations, and let $\alpha\in\left[0,1\right]$. We prove that if the additive energy $E\left(A_{N}\right)$ of $A_{N}$ is in $\Omega\left(N^{3}\right)$, then the sequence $\left(\left\langle \alpha a_{n}\right\rangle \right)_{n}$ of fractional parts of $\alpha a_{n}$ does not have Poissonian pair correlations (PPC) for almost every $\alpha$ in the sense of Lebesgue measure. Conversely, it is known that $E\left(A_{N}\right)=\mathcal{O}\left(N^{3-\varepsilon}\right)$, for some fixed $\varepsilon>0$, implies that $\left(\left\langle \alpha a_{n}\right\rangle \right)_{n}$ has PPC for almost every $\alpha$. This note makes a contribution to investigating the energy threshold for $E\left(A_{N}\right)$ to imply this metric distribution property. We establish, in particular, that there exist sequences $\left(a_{n}\right)_{n}$ with $$E\left(A_{N}\right)=\Theta\left(\frac{N^{3}}{\log\left(N\right)\log\left(\log N\right)}\right)$$ such that the set of $\alpha$ for which $\left(\alpha a_{n}\right)_{n}$ does not have PPC is of full Lebesgue measure. Moreover, we show that for any fixed $\varepsilon>0$ there are sequences $\left(a_{n}\right)_{n}$ with $E\left(A_{N}\right)=\Theta\left(\frac{N^{3}}{\log\left(N\right)\left(\log\log N\right)^{1+\varepsilon}}\right)$ satisfying that the set of $\alpha$ for which the sequence $\left(\bigl\langle\alpha a_{n}\bigr\rangle\right)_{n}$ does not have PPC is of full Hausdorff dimension.' author: - 'Thomas Lachmann[^1], and Niclas Technau[^2]' title: On Exceptional Sets in the Metric Poissonian Pair Correlations problem --- Introduction ============ The theory of uniform distribution modulo $1$ dates back, at least, to the seminal paper of Weyl [@Weyl:; @=0000DCber; @die; @Gleichverteilung; @von; @Zahlen; @mod.; @Eins]. Weyl showed, inter alia, that for any fixed irrational $\alpha\in\mathbb{R}$ and integer $d\geq1$ the sequences $\left(\bigl\langle\alpha n^{d}\bigr\rangle\right)_{n}$ are uniformly distributed modulo $1$. However, in recent years various authors have been investigating a more subtle distribution property of such sequences - namely, whether the asymptotic distribution of the pair correlations has a property which is called Poissonian, and defined as follows: Let $\left\Vert \cdot\right\Vert $ denote the distance to the nearest integer. A sequence $\left(\theta_{n}\right)_{n}$ in $\left[0,1\right]$ is said to have (asymptotically) Poissonian pair correlations, if for each $s\geq0$ the pair correlation function[^3] $$R_{2}\left(\left[-s,s\right],\left(\theta_{n}\right)_{n},N\right)\coloneqq\frac{1}{N}\#\left\{ 1\leq i\neq j\leq N:\,\left\Vert \theta_{i}-\theta_{j}\right\Vert \leq\frac{s}{N}\right\} \label{eq: definition of the Pair Correlation Counting function}$$ tends to $2s$ as $N\rightarrow\infty$. Moreover, let $\left(a_{n}\right)_{n}$ denote a strictly increasing sequence of positive integers. If no confusion can arise, we write $$R\left(\left[-s,s\right],\alpha,N\right)\coloneqq R_{2}\left(\left[-s,s\right],\left(\alpha a_{n}\right)_{n},N\right)$$ and say that a sequence $\left(a_{n}\right)_{n}$ has metric Poissonian pair correlations if $\left(\alpha a_{n}\right)_{n}$ has Poissonian pair correlations for almost all $\alpha\in\left[0,1\right]$ in the sense of Lebesgue measure. It is known that if a sequence $\left(\theta_{n}\right)_{n}$ has Poissonian pair correlations, then it is uniformly distributed modulo $1$, cf. [@Aistleitner; @Lachmann; @Pausinger:; @Pair; @correlations; @and; @equidistribution; @Larcher; @Grepstad:; @On; @pair; @correlation; @and; @discrepancy]. Yet, the sequences $\left(\left\langle \alpha n^{d}\right\rangle \right)_{n}$ do *not* have Poissonian pair correlations for *any* $\alpha\in\mathbb{R}$ if $d=1$. For $d\geq2$, Rudnick and Sarnak [@Rudnick; @Sarnak:; @The; @pair; @correlation; @function; @of; @fractional; @parts; @of; @polynomials] proved that $\left(n^{d}\right)_{n}$ has metric Poissonian pair correlations (metric PPC). For alternative proofs, we refer the reader to Heath-Brown and the work of Marklof and Strömbergsson [@Marklof; @Str=0000F6mbergsson:; @Equidistribution; @of; @Kronecker; @sequences; @along; @closed; @horocycles].[^4] Given these results, it is natural to investigate which properties of a sequence of integers $\left(a_{n}\right)_{n}$ implies the metric PPC of $\left(a_{n}\right)_{n}$. Partial answers are known, e.g. it follows from work of Boca and Zaharescu [@Boca; @Zaharescu:; @Pair; @correlation; @of; @values; @of; @rational; @functions; @(mod; @p)] that $\left(P\left(n\right)\right)_{n}$ has metric PPC if $P$ is any polynomial with integer coefficients of degree at least two. An interesting general result in this direction is due to Aistleitner, Larcher, and Lewko [@Aistleitner; @Larcher; @Lewko:; @Additive; @Energy; @and; @the; @Hausdorff; @Dimension; @of; @the; @Exceptional; @Set; @in; @Metric; @Pair; @Correlation; @Problems] who used a Fourier analytic approach combined with a bound on GCD sums of Bondarenko and Seip [@Bondarenko; @Seip:; @GCD; @sums; @and; @complete; @sets; @of; @square-free; @numbers] to relate the metric PPC of $\left(a_{n}\right)_{n}$ with its combinatoric properties. For stating it, let $\left(a_{n}\right)_{n}$ denote henceforth a strictly increasing sequence of positive integers and denote the set of the first $N$ elements of $\left(a_{n}\right)_{n}$ by $A_{N}$. Moreover, define the additive energy $E\left(I\right)$ of a finite set integers $I$ via $$E\left(I\right)\coloneqq\sum_{\underset{a+b=c+d}{a,b,c,d\in I}}1.$$ In the following, let $\mathcal{O}$ and $o$ denote the standard Landau symbols/O-notation.\ \ A main finding of [@Aistleitner; @Larcher; @Lewko:; @Additive; @Energy; @and; @the; @Hausdorff; @Dimension; @of; @the; @Exceptional; @Set; @in; @Metric; @Pair; @Correlation; @Problems] is the implication that if the truncations $A_{N}$ satisfy $$E\left(A_{N}\right)=\mathcal{O}\left(N^{3-\varepsilon}\right)\label{eq: Aistleitner bound}$$ for some fixed $\varepsilon>0$, then $\left(a_{n}\right)_{n}$ has metric PPC. Note that $\left(\#I\right)^{2}\leq E\left(I\right)\leq\left(\#I\right)^{3}$ where $\#I$ denotes the cardinality of $I\subset\mathbb{Z}$. Roughly speaking, a set $I$ has large additive energy if and only if it contains a “large” arithmetic progression like structure. Indeed, if $\left(a_{n}\right)_{n}$ is a geometric progression or of the form $\left(n^{d}\right)_{n}$ for $d\geq2,$ then (\[eq: Aistleitner bound\]) is satisfied. Furthermore, note that the metric PPC property may be seen as a sort of pseudorandomness; in fact, for a given sequence of $\left[0,1\right]$-uniformly distributed, and independent random variables $\left(\theta_{n}\right)_{n}$, one has $$\lim_{N\rightarrow\infty}R\left(\left[-s,s\right],\left(\theta_{n}\right)_{n},N\right)=2s\label{eq: counting function asymtotically Poissonian}$$ for every $s\geq0$ almost surely.\ \ Wondering about the optimal bound for the additive energy of the truncations $A_{N}$ to imply the metric PPC property of $\left(a_{n}\right)_{n}$, the two following questions were raised in [@Aistleitner; @Larcher; @Lewko:; @Additive; @Energy; @and; @the; @Hausdorff; @Dimension; @of; @the; @Exceptional; @Set; @in; @Metric; @Pair; @Correlation; @Problems] where we use the convention that $f=\Omega\left(g\right)$ means for $f,g:\mathbb{N}\rightarrow\mathbb{R}$ there is a constant $c>0$ such that $g\left(n\right)>cf\left(n\right)$ holds for infinitely many $n$. Is it possible for a strictly increasing sequence $\left(a_{n}\right)_{n}$ of positive integers with $E\left(A_{N}\right)=\Omega\left(N^{3}\right)$ to have metric PPC? Do all increasing strictly sequences $\left(a_{n}\right)_{n}$ of positive integers with $E\left(A_{N}\right)=o\left(N^{3}\right)$ have metric PPC? Both questions were answered in the negative by Bourgain whose proofs can be found in [@Aistleitner; @Larcher; @Lewko:; @Additive; @Energy; @and; @the; @Hausdorff; @Dimension; @of; @the; @Exceptional; @Set; @in; @Metric; @Pair; @Correlation; @Problems] as an appendix, without giving an estimate on the measure of the set that was used to answer Question 1, and without a quantitative bound on $E\left(A_{N}\right)$ appearing in the negation of Question 2. However, a quantitative analysis, as noted in [@Walker:; @The; @Primes; @are; @not; @Metric; @Poissonian], shows that the sequence Bourgain constructed for Question 2 satisfies $$E\left(A_{N}\right)=\mathcal{O}_{\varepsilon}\left(\frac{N^{3}}{\left(\log\log N\right)^{\frac{1}{4}+\varepsilon}}\right)\label{eq: Bourgains bound for the sequence of non PPC}$$ for any fixed $\varepsilon>0$. Moreover, Nair posed the problem[^5] whether the sequence of prime numbers $\left(p_{n}\right)_{n}$, ordered by increasing value, has metric PPC. Recently, Walker [@Walker:; @The; @Primes; @are; @not; @Metric; @Poissonian] answered this question in the negative. Thereby he gave a significantly better bound than (\[eq: Bourgains bound for the sequence of non PPC\]) for the additive energy $E\left(A_{n}\right)$ for a sequence $\left(a_{n}\right)_{n}$ not having metric PPC - since the additive energy of the truncations of $\left(p_{n}\right)_{n}$ is in $\Theta\bigl(\left(\log N\right)^{-1}N^{3}\bigr)$ where $f=\Theta\left(g\right)$, for functions $f,g$, means that $f=\mathcal{O}\left(g\right)$ and $g=\mathcal{O}\left(f\right)$ holds. The main objective of our work is to improve upon these answers to Questions 1, and Question 2. For a given sequence $\left(a_{n}\right)_{n}$, we denote by $\NPPC\left(\left(a_{n}\right)_{n}\right)$ the (“exceptional”) set of all $\alpha\in\left(0,1\right)$ such that the pair correlation function (\[eq: definition of the Pair Correlation Counting function\]) does not tend to $2s$, as $N$ tends to infinity, for some $s\geq0$. \[thm: Bourgain’s Result concerning the measure of the set of counterexamples\]Suppose $\left(a_{n}\right)_{n}$ is a strictly increasing sequence of positive integers. If $E(A_{N})=\Omega\left(N^{3}\right)$, then $\NPPC\left(\left(a_{n}\right)_{n}\right)$ has positive Lebesgue measure. We prove the following sharpening. \[thm: full measure of set of counterexamples\]Suppose $\left(a_{n}\right)_{n}$ is a strictly increasing sequence of positive integers. If $E(A_{N})=\Omega\left(N^{3}\right)$, then $\NPPC\left(\left(a_{n}\right)_{n}\right)$ has full Lebesgue measure. Moreover, we lower the known energy threshold, and estimate the Hausdorff dimension of the exceptional set from below. For stating our second main theorem, we denote by $\mathbb{R}_{>x}$ the set of real numbers exceeding a given $x\in\mathbb{R}$, and recall that for a function $g:\mathbb{R}_{>1}\rightarrow\mathbb{R}_{>0}$ the lower order of infinity $\lambda\left(g\right)$ is defined by $$\lambda\left(g\right)\coloneqq\liminf_{x\rightarrow\infty}\frac{\log g\left(x\right)}{\log x}.$$ This notion arises naturally in the context of Hausdorff dimensions. Roughly speaking, it quantifies the (lower) asymptotic growth rate of a function. \[thm: lowering the known Energy threshold\]Let $f:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>2}$ be a function increasing monotonically to $\infty$, and satisfying $f\left(x\right)=\mathcal{O}\bigl(\left(\log x\right)^{-\nicefrac{7}{3}}x^{\nicefrac{1}{3}}\bigr)$. Then, there exists a strictly increasing sequence $\left(a_{n}\right)_{n}$ of positive integers with $E(A_{N})=\Theta\bigl((f\left(N\right))^{-1}N^{3}\bigr)$ such that if $$\sum_{n\geq1}\frac{1}{nf(n)}\label{eq: divergence of the reciprocal of (f(n) times n)}$$ diverges, then for Lebesgue almost all $\alpha\in\left[0,1\right]$ $$\limsup_{N\rightarrow\infty}R\left(\left[-s,s\right],\alpha,N\right)=\infty\label{eq: divergence of the Pair Correlation Function}$$ holds for any $s>0$; additionally, if (\[eq: divergence of the reciprocal of (f(n) times n)\]) converges and $\sup\left\{ f\left(2x\right)/f\left(x\right):\,x\geq x_{0}\right\} $ is strictly less than $2$ for some $x_{0}>0$, then $\NPPC\left(\left(a_{n}\right)_{n}\right)$ has Hausdorff dimension at least $\left(1+\lambda\right)^{-1}$ where $\lambda$ is the lower order of infinity of $f$. We record an immediate consequence of Theorem \[thm: lowering the known Energy threshold\] by using the convention that the $r$-folded iterated logarithm is denoted by $\log_{r}\left(x\right)$, i.e. $\log_{r}\left(x\right)\coloneqq\log_{r-1}\left(\log\left(x\right)\right)$ and $\log_{1}\left(x\right)\coloneqq\log\left(x\right)$. \[cor: order of magnitude for the additive energy of the sequence of counter examples\]Let $r$ be a positive integer. Then, there is a strictly increasing sequence $\left(a_{n}\right)_{n}$ of positive integers with $$E\left(A_{N}\right)=\Theta\left(\frac{N^{3}}{\log\left(N\right)\log_{2}\left(N\right)\ldots\log_{r}\left(N\right)}\right)$$ such that $\NPPC\left(\left(a_{n}\right)_{n}\right)$ has full Lebesgue measure. Moreover, for any $\varepsilon>0$ there is a strictly increasing sequence $\left(a_{n}\right)_{n}$ of positive integers with $$E\left(A_{N}\right)=\Theta\left(\frac{\left(\log_{r}\left(N\right)\right)^{-\varepsilon}N^{3}}{\log\left(N\right)\log_{2}\left(N\right)\ldots\log_{r}\left(N\right)}\right)$$ such that $\NPPC\left(\left(a_{n}\right)_{n}\right)$ has full Hausdorff dimension. The proof of Theorem \[thm: lowering the known Energy threshold\] connects the metric PPC property to the notion of “optimal regular systems” from Diophantine approximation. It uses, among other things, a Khintchine-type theorem due to Beresnevich. Furthermore, despite leading to better bounds, the nature of the sequences underpinning Theorem \[thm: lowering the known Energy threshold\] is much simpler than the nature of those sequences previously constructed by Bourgain [@Aistleitner; @Larcher; @Lewko:; @Additive; @Energy; @and; @the; @Hausdorff; @Dimension; @of; @the; @Exceptional; @Set; @in; @Metric; @Pair; @Correlation; @Problems] (who used, inter alia, large deviations inequalities form a probability theory), or the sequence of prime numbers studied by Walker [@Walker:; @The; @Primes; @are; @not; @Metric; @Poissonian] (who relied on estimates, derived by the circle-method, on the exceptional set in Goldbach-like problems).\ \ In the converse direction, there has been remarkable progress, due to a work of Bloom, Chow, Gafni, Walker - who improved under the assumption that the sequence is not “too sparse” the power saving bound (\[eq: Aistleitner bound\]) to a saving of a little more than the square of a logarithm. More precisely, their result is as follows. \[thm: Bloom, Chow, Gafni, Walker theorem\]Let $\left(a_{n}\right)_{n}$ be a strictly increasing sequence of positive integers. Suppose there is an $\varepsilon>0$ and a $C=C\left(\varepsilon\right)>0$ such that $$E\left(A_{N}\right)=\mathcal{O}_{\varepsilon}\left(\frac{N^{3}}{\left(\log N\right)^{2+\varepsilon}}\right),\qquad\delta\left(N\right)\geq\frac{C}{\left(\log N\right)^{2+2\varepsilon}}$$ where $\delta\left(N\right)\coloneqq N^{-1}\#\left(A_{N}\cap\left\{ 1,\ldots,N\right\} \right)$. Then, $\left(a_{n}\right)_{n}$ has metric PPC. First main theorem ================== Let us give an outline of the proof of Theorem \[thm: full measure of set of counterexamples\]. For doing so, we begin by sketching the reasoning of Theorem A: As it turns out, except for a set of neglectable measure, the counting function in (\[eq: definition of the Pair Correlation Counting function\]) can be written as a function that admits a non-trivial estimate for its $L^{1}$-mean value. The $L^{1}$-mean value is infinitely often too small on sets whose measure is uniformly bounded from below. Thus, there exists a sequence of set $\left(\Omega_{r}\right)_{r}$ of $\alpha\in\left[0,1\right]$ such that $R\left(\left[-s,s\right],\alpha,N\right)$ is too small for every $\alpha\in\Omega_{r}$ for having PPC and Theorem A follows. Our reasoning for proving Theorem \[thm: full measure of set of counterexamples\] is building upon this argument of Bourgain while we introduce new ideas to construct a sequence of sets $\left(\Omega_{r}\right)_{r}$ that are “quasi (asymptotically) independent” - meaning that for every fixed $t$ the relation $\lambda(\Omega_{r}\cap\Omega_{t})\leq\lambda(\Omega_{r})\lambda(\Omega_{t})+o\left(1\right)$ holds as $r\rightarrow\infty$. Roughly speaking, applying a suitable version of the Borel-Cantelli lemma, combined with a sufficiently careful treatment of the $o\left(1\right)$ term, will then yield Theorem \[thm: full measure of set of counterexamples\]. However, before proceeding with the details of the proof we collect in the next paragraph some tools from additive combinatorics that are needed. Preliminaries ------------- We start with a well-know result relating, in a quantitative manner, the additive energy of a set of integers with the existence of a (relatively) dense subset with small difference set where the difference set $B-B\coloneqq\left\{ b-b':\,b,b'\in B\right\} $ for a set $B\subseteq\mathbb{R}$. \[Balog-Szem=0000E9redi-Gowers\]Let $A\subseteq\mathbb{Z}$ be a finite set of integers. For any $c>0$ there exist $c_{1},c_{2}>0$ depending only on $c$ such that the following holds. If $E(A)\geq c\left(\#A\right)^{3}$, then there is a subset $B\subseteq A$ such that 1. $\#B\geq c_{1}\#A,$ 2. $\#\left(B-B\right)\leq c_{2}\#A.$ Moreover, we recall that for $\delta>0$ and $d\in\mathbb{Z}$ the set $$B\left(d,\delta\right)\coloneqq\left\{ \alpha\in\left[0,1\right]:\,\left\Vert d\alpha\right\Vert \leq\delta\right\}$$ is called Bohr set. These appear frequently in additive combinatorics. The following two simple observation will be useful. \[lem: upper estimate for measure of Omega\_varepsilon,n\]Let $B\subseteq\mathbb{Z}$ be a finite set of integers. Then, $$\lambda\Biggl(\Biggl\{\alpha\in\left[0,1\right]:\underset{d\in\left(B-B\right)\setminus\left\{ 0\right\} }{\min}\left\Vert d\alpha\right\Vert <\frac{\varepsilon}{\#\left(B-B\right)}\Biggr\}\Biggr)\leq2\varepsilon$$ for every $\varepsilon\in(0,1)$ where $\lambda$ is the Lebesgue measure. By observing that the set under consideration is contained in $$\bigcup_{\underset{m\not=n}{m,n\in B}}B\left(m-n,\frac{\varepsilon}{\#\left(B-B\right)}\right),$$ and $\lambda\left(B\left(m-n,\frac{\varepsilon}{\#\left(B-B\right)}\right)\right)=\frac{2\varepsilon}{\#\left(B-B\right)}$, the claim follows at once. \[lem: Omega\_n has only finitely many connected components\]Suppose $A$ is a finite intersection of Bohr sets, and $B$ is a finite union of Bohr sets. Then, $A\setminus B$ is the union of finitely many intervals. Furthermore, we shall use the Borel-Cantelli lemma in a version due to Erdős-Rényi. \[lem: Erdos Renyi version of Borel Cantelli\]Let $\left(A_{n}\right)_{n}$ be a sequence of Lebesgue measurable sets in $\left[0,1\right]$ satisfying $$\sum_{n\geq1}\lambda\left(A_{n}\right)=\infty.$$ Then, $$\lambda\left(\limsup_{n\rightarrow\infty}A_{n}\right)\geq\limsup_{N\rightarrow\infty}\frac{\left(\sum_{n\leq N}\lambda\left(A_{n}\right)\right)^{2}}{\sum_{m,n\leq N}\lambda\left(A_{n}\cap A_{m}\right)}.$$ Moreover, let us explain the main steps in the proof of Theorem \[thm: full measure of set of counterexamples\]. Let $\varepsilon\coloneqq\varepsilon\left(j\right)\coloneqq\frac{1}{10^{j}}c_{1}^{2}$ be for $j\in\mathbb{N}$ where the constant $c_{1}$ is specified later-on, and fix $j$ for now. In the first part of the argument, we show how a sequence - that is constructed in the second part of the argument - with the following crucial (but technical) properties implies the claim. For every fixed $j$, we find a corresponding $s=s(j)$ and construct a sequence $\left(\Omega_{r}\right)_{r}$ of exceptional values $\alpha$ satisfying the following properties: 1. \[enu:Pair correlations functions too small on exceptional set\]\[enu:First property of exceptional sets\]For all $\alpha\in\Omega_{r}$, the pair correlation function admits the upper bound $$R\left(\left[-s,s\right],\alpha,N\right)\leq2\tilde{c}s\label{eq: Pair correlations function too small on exceptional set}$$ for some absolute constant $\tilde{c}\in\left(0,1\right)$, depending on $\left(a_{n}\right)$ only. 2. \[enu:exceptional sets get upper asymptotically independent\]For all integers $r>t\ge1$, the relation $$\lambda\left(\Omega_{r}\cap\Omega_{t}\right)\leq\lambda\left(\Omega_{r}\right)\lambda\left(\Omega_{t}\right)+2\varepsilon\lambda\left(\Omega_{t}\right)+\mathcal{O}\left(r^{-2}\right)\label{eq: exceptional sets get upper asymptotically independent}$$ holds. 3. \[enu:Each exceptional set has only finitely many connected components\]Each $\Omega_{r}$ is the union of finitely many intervals (hence measurable). 4. \[enu: absolute lower bound for the measure of Omega\]\[enu: last property of exceptional sets\]For all $r\geq1$, the measure $\lambda\left(\Omega_{r}\right)$ is uniformly bounded from below by $$\lambda\left(\Omega_{r}\right)\geq\frac{c_{1}^{2}}{8}.\label{eq: absolute lower bound for the measure of Omega}$$ Proof of Theorem \[thm: full measure of set of counterexamples\] ---------------------------------------------------------------- 1\. Suppose there is $\left(\Omega_{r}\right)_{r}$ satisfying \[enu:Pair correlations functions too small on exceptional set\]-\[enu: absolute lower bound for the measure of Omega\]. Then, by using (\[eq: exceptional sets get upper asymptotically independent\]), we get $$\begin{aligned} \sum_{r,t\leq N}\lambda\left(\Omega_{r}\cap\Omega_{t}\right) & \le2\sum_{2\leq t\leq N}\,\sum_{1\leq r<t}\left(\lambda\left(\Omega_{r}\right)\lambda\left(\Omega_{t}\right)\right)+2\varepsilon N^{2}+\mathcal{O}\left(N\right)\\ & \leq\left(\sum_{t\leq N}\lambda\left(\Omega_{t}\right)\right)^{2}+2\varepsilon N^{2}+\mathcal{O}\left(N\right).\end{aligned}$$ By recalling that $\Omega_{r}=\Omega_{r}\left(\varepsilon\right)=\Omega_{r}\left(j\right)$, we let $\Omega(j)\coloneqq\limsup_{r\rightarrow\infty}\Omega_{r}$. By using the inequality above in combination with Lemma \[lem: Erdos Renyi version of Borel Cantelli\] and the bound (\[eq: absolute lower bound for the measure of Omega\]), we obtain that the set $\Omega(j)$ has measure at least $$\begin{aligned} \limsup_{N\rightarrow\infty}\frac{\left(\sum_{r\leq N}\lambda\left(\Omega_{r}\right)\right)^{2}}{\sum_{r,t\leq N}\lambda\left(\Omega_{r}\cap\Omega_{t}\right)} & \geq\limsup_{N\rightarrow\infty}\frac{1}{1+\frac{4\varepsilon N^{2}}{\left(\sum_{r\leq N}\lambda\left(\Omega_{r}\right)\right)^{2}}}\\ & \geq\limsup_{N\rightarrow\infty}\frac{1}{1+\frac{256}{c_{1}^{2}}\varepsilon}=\frac{1}{1+\frac{256}{c_{1}^{2}}\varepsilon}.\end{aligned}$$ Note that due to (\[eq: Pair correlations function too small on exceptional set\]) every $\alpha\in\Omega\left(j\right)$ does not have PCC. Now, letting $j\rightarrow\infty$ proves the assertion.\ 2. For constructing $\left(\Omega_{r}\right)_{r}$ with the required properties, let $c>0$ such that $E\left(A_{N}\right)>cN^{3}$ for infinitely many integers $N$. By choosing an appropriate subsequence $\left(N_{i}\right)_{i}$ and omitting the subscript $i$ for ease of notation, $E\left(A_{N}\right)>cN^{3}$ holds for every $N$ occurring in this proof. Moreover, let $c_{1},c_{2}$ and $B_{N}$ be as in Lemma \[Balog-Szem=0000E9redi-Gowers\], corresponding to the $c$ just mentioned. Arguing inductively, while postponing the base step,[^6] we assume that for $1\leq r<R$, and $s=\frac{\varepsilon}{2c_{2}}$ there are sets $\left(\Omega_{r}\right)_{1\leq r<R}$ that satisfy the properties \[enu:First property of exceptional sets\]-\[enu: last property of exceptional sets\] for all distinct integers $1\leq r,t<R$. Let $N\geq R$. Lemma \[lem: upper estimate for measure of Omega\_varepsilon,n\] implies that the set $\Omega_{\varepsilon,N}\subseteq[0,1]$ of all $\alpha\in\left[0,1\right]$ satisfying $\left\Vert \left(r-t\right)\alpha\right\Vert <N^{-1}s$ for some distinct $r,t\in B_{N}$ has measure at most $2\varepsilon$. Setting $$\mathcal{D}_{N}:=\left\{ \left(r,t\right)\in\left(A_{N}\times A_{N}\right)\setminus\left(B_{N}\times B_{N}\right):\,r\not=t\right\} ,$$ we get for $\alpha\notin\Omega_{\varepsilon,N}$ that $$R\left(\left[-s,s\right],\alpha,N\right)=\frac{1}{N}\#\left\{ \left(r,t\right)\in\mathfrak{\mathcal{D}}_{N}:\,\left\Vert \left(r-t\right)\alpha\right\Vert <N^{-1}s\right\} .$$ Let $\ell_{R}$ denote the length of the smallest subinterval of $\Omega_{r}$ for $1\leq r<R$, and define $C\left(\Omega_{r}\right)$ to be the set of subintervals of $\Omega_{r}$. Note that $\ell_{R}>0$, and $\max_{1\leq r<R}\#C\left(\Omega_{r}\right)<\infty$. We divide $\left[0,1\right)$ into $$P\coloneqq\left\lfloor 1+2\ell_{R}^{-1}R^{2}\max_{1\leq r<R}\#C\left(\Omega_{r}\right)\right\rfloor$$ parts $\mathcal{P}_{i}$ of equal lengths, i.e. $\mathcal{P}_{i}\coloneqq\left[\frac{i}{P},\frac{i+1}{P}\right)$ where $i=0,\ldots,P-1$. After writing $$\begin{aligned} & \frac{1}{N}\underset{\mathcal{P}_{i}}{\int}\#\left\{ \left(r,t\right)\in\mathfrak{\mathcal{D}}_{N}:\,\left\Vert \left(r-t\right)\alpha\right\Vert \leq N^{-1}s\right\} \text{d}\alpha\label{eq: counting function integrated on an atom}\\ & =\frac{1}{N}\underset{\left(r,t\right)\in\mathfrak{\mathcal{D}}_{N}}{\sum}\underset{\mathcal{P}_{i}}{\int}\mathbf{1}_{\left[-\frac{s}{N},\frac{s}{N}\right]}\left(\left\Vert \left(r-t\right)\alpha\right\Vert \right)\text{d}\alpha,\nonumber \end{aligned}$$ we split the sum into two parts: one part containing differences $\left|r-t\right|>R^{k}P$, and a second part containing differences $\left|r-t\right|\leq R^{k}P$ where $$k\coloneqq\left\lfloor \frac{1}{\log2}\log\frac{20}{c_{1}^{2}\left(1-2^{-1}c_{1}^{2}\right)s}\right\rfloor +1.$$ Letting $\mathbf{1}_{B}$ denote the characteristic function of $X\subseteq\left[0,1\right]$, the Cauchy-Schwarz inequality implies $$\underset{\mathcal{P}_{i}}{\int}\mathbf{1}_{\left[-\frac{s}{N},\frac{s}{N}\right]}\left(\left\Vert \left(r-t\right)\alpha\right\Vert \right)\text{d}\alpha\leq\sqrt{\frac{1}{P}\frac{2s}{N}}.$$ Since for any $x>0$ there are at most $2xN$ choices of $\left(r,t\right)\in\mathcal{D}_{N}$ such that $\left|r-t\right|\leq x$, we obtain $$\frac{1}{N}\underset{\underset{\left|r-t\right|\leq PR^{k}}{\left(r,t\right)\in\mathcal{D}_{N}}}{\sum}\underset{\mathcal{P}_{i}}{\int}\mathbf{1}_{\left[-\frac{s}{N},\frac{s}{N}\right]}\left(\left\Vert \left(r-t\right)\alpha\right\Vert \right)\text{d}\alpha\leq2PR^{k}\sqrt{\frac{1}{P}\frac{2s}{N}}$$ which is $\leq P^{-1}R^{-k}$ if $N$ is sufficiently large. Moreover, for any $\left|r-t\right|>PR^{k}$ we observe that $$\underset{\mathcal{P}_{i}}{\int}\mathbf{1}_{\left[-\frac{s}{N},\frac{s}{N}\right]}\left(\left\Vert \left(r-t\right)\alpha\right\Vert \right)\text{d}\alpha\leq\frac{2s}{PN}+\frac{4}{PR^{k}N}$$ and $\#\mathcal{D}_{N}\leq N^{2}-\bigl(\#B_{N}\bigr)^{2}\leq\tilde{c}N^{2}$ where $\tilde{c}\coloneqq1-c_{1}^{2}$.Therefore, the mean value (\[eq: counting function integrated on an atom\]) on $\mathcal{P}_{i}$ of the counting function $R$ is bounded from above by $$\begin{aligned} \frac{1}{N}\left(\#\mathcal{D}_{N}\right)^{2}\left(\frac{2s}{PN}+\frac{4}{PR^{k}N}\right)+\frac{1}{PR^{k}}\leq\frac{2\tilde{c}s}{P}+\frac{5}{PR^{k}}.\end{aligned}$$ Hence, it follows that the measure of the set $\Delta_{N}\left(i\right)$ of $\alpha\in\mathcal{P}_{i}$ with $$\frac{1}{N}\#\left\{ \left(r,t\right)\in\mathfrak{\mathcal{D}}_{N}:\,\left\Vert \left(r-t\right)\alpha\right\Vert \leq N^{-1}s\right\} \leq2\left(1-\frac{c_{1}^{2}}{2}\right)s\label{eq: modified counting too small for being Poissonian}$$ admits, by the choice of $k$, the lower bound $$\lambda\left(\Delta_{N}\left(i\right)\right)\geq\frac{1}{P}-\frac{1}{P}\frac{2\tilde{c}s+5R^{-k}}{2\left(1-\frac{c_{1}^{2}}{2}\right)s}\geq\frac{1}{P}\left(\frac{c_{1}^{2}}{2}-\frac{c_{1}^{2}}{8}\right).\label{eq: absolute lower bound for Omega_N on partition}$$ Note that $\Delta_{N}\left(i\right)$ is the union of finitely many intervals, due to Lemma \[lem: Omega\_n has only finitely many connected components\]. So, we may take $\Delta_{N}'\left(i\right)\subset\Delta_{N}\left(i\right)$ being a finite union of intervals such that $\lambda\left(\Delta_{N}'\left(i\right)\right)$ equals the lower bound in (\[eq: absolute lower bound for Omega\_N on partition\]). Let $$\Omega_{R}\coloneqq\Omega_{R}\left(N\right)\coloneqq\Delta_{N}\setminus\Omega_{\varepsilon,N}\qquad\mathrm{where}\qquad\Delta_{N}\coloneqq\bigcup_{i=0}^{P-1}\Delta_{N}'\left(i\right).$$ We are going to show now that $\Omega_{R}$ satisfies the properties \[enu:First property of exceptional sets\] - \[enu: last property of exceptional sets\]. Now, $\Omega_{R}$ satisfies property \[enu: absolute lower bound for the measure of Omega\] with $r=R$ since $$\lambda\left(\Omega_{R}\right)\geq\lambda\left(\Delta_{N}\right)-\lambda\left(\Omega_{\varepsilon,N}\right)=\frac{c_{1}^{2}}{2}-\frac{c_{1}^{2}}{8}-2\varepsilon\geq\frac{c_{1}^{2}}{8}.$$ Furthermore, $\Omega_{R}$ satisfies property \[enu:Pair correlations functions too small on exceptional set\] by construction and also property \[enu:Each exceptional set has only finitely many connected components\] since all sets involved in the construction of $\Omega_{R}$ were a finite union of intervals. Let $1\leq r<R$, and $I$ be a subinterval of $\Omega_{r}$. Then, $$\begin{aligned} \lambda\left(I\cap\Delta_{N}\right) & =\sum_{i:\mathcal{P}_{i}\cap I\neq\emptyset}\lambda\left(\mathcal{P}_{i}\cap I\cap\Delta_{N}\right)\\ & \leq\frac{2}{P}+\sum_{i:\mathcal{P}_{i}\subsetneq I}\lambda\left(\mathcal{P}_{i}\cap\Delta_{N}\right)\\ & \leq\frac{2}{P}+\sum_{i:\mathcal{P}_{i}\subsetneq I}\lambda\left(\Delta_{N}'\left(i\right)\right).\end{aligned}$$ By summing over all subintervals $I\in C\left(\Omega_{r}\right)$, we obtain that $$\begin{aligned} \lambda\left(\Omega_{r}\cap\Delta_{N}\right) & \leq\sum_{I\in C\left(\Omega_{r}\right)}\left(\frac{2}{P}+\sum_{i:\mathcal{P}_{i}\subsetneq I}\lambda\left(\Delta_{N}'\left(i\right)\right)\right)\\ & \leq\frac{1}{R^{2}}+\sum_{I\in C\left(\Omega_{r}\right)}P\lambda\left(I\right)\frac{\lambda\left(\Omega_{N}\right)}{P}\\ & =\lambda\left(\Omega_{r}\right)\lambda\left(\Omega_{N}\right)+\frac{1}{R^{2}}.\end{aligned}$$ We deduce property \[enu:exceptional sets get upper asymptotically independent\] from this estimate and Lemma \[lem: upper estimate for measure of Omega\_varepsilon,n\] via $$\begin{aligned} \lambda\left(\Omega_{r}\cap\Omega_{R}\right) & \leq\lambda\left(\Omega_{r}\cap\Delta_{N}\right)\\ & \leq\lambda\left(\Omega_{r}\right)\left(\lambda\left(\Omega_{N}\right)-\lambda\left(\Omega_{\varepsilon,N}\right)\right)+\frac{1}{R^{2}}+\lambda\left(\Omega_{r}\right)\lambda\left(\Omega_{\varepsilon,N}\right)\\ & \leq\lambda\left(\Omega_{r}\right)\lambda\left(\Omega_{R}\right)+2\varepsilon\lambda\left(\Omega_{r}\right)+\mathcal{O}\left(R^{-2}\right)\end{aligned}$$ This concludes the induction step. The only part missing now is the base step of the induction. For realizing it, let $N$ denote the smallest integer $m$ with $E\left(A_{m}\right)>cm^{3}$. We replace $\mathcal{P}_{i}$ in (\[eq: counting function integrated on an atom\]) by $\left[0,1\right]$ to directly derive $$\int_{0}^{1}\frac{1}{N}\#\left\{ \left(r,t\right)\in\mathfrak{\mathcal{D}}_{N}:\,\left\Vert \left(r-t\right)\alpha\right\Vert \leq N^{-1}s\right\} \mathrm{d}\alpha\leq2\tilde{c}s,$$ and conclude that the set $\Omega_{1}'$ of $\alpha\in\left[0,1\right]$ satisfying (\[eq: modified counting too small for being Poissonian\]) has a measure at least $\frac{c_{1}^{2}}{2}$. Thus, $\Omega_{1}\coloneqq\Omega_{1}'\setminus\Omega_{N,\varepsilon}$ has measure at least as large as the right hand side of (\[eq: absolute lower bound for the measure of Omega\]). For property (\[eq: exceptional sets get upper asymptotically independent\]) is nothing to check and that $\Omega_{1}$ is a finite union of intervals follows from Lemma \[lem: Omega\_n has only finitely many connected components\] by observing that $$\Omega_{1}'=\bigcap_{d_{1},\ldots,d_{L\left(N\right)}}\left(B\left(d_{1},N^{-1}s\right)^{C}\cup\ldots\cup B\left(d_{L\left(N\right)},N^{-1}s\right)^{C}\right)$$ where the intersection runs through any set of $L\left(N\right)=\left\lfloor N2\tilde{c}s\right\rfloor $ tuples of differences $d_{i}=r_{i}-t_{i}\neq0$ of components of $\left(r_{i},t_{i}\right)\in\mathcal{D}_{N}$ for $i=1,\ldots,L\left(N\right)$. Thus, the proof is complete. Second main theorem =================== The sequences $\left(a_{n}\right)_{n}$ enunciated in Theorem \[thm: lowering the known Energy threshold\] are constructed in two steps. In the first step, we concatenate (finite) blocks, with suitable lengths, of arithmetic progressions to form a set $P_{A}$. In the second step, we concatenate (finite) blocks, with suitable lengths, of geometric progressions to form a set $P_{G}$ and then define $a_{n}$ to be the $n$-th element of $P_{A}\cup P_{G}$. On the one hand, the arithmetic progression like part $P_{A}$ serves to ensure, due to considerations from metric Diophantine approximation, the divergence property (\[eq: divergence of the Pair Correlation Function\]) on a set with full measure or controllable Hausdorff dimension; on the other hand, the geometric progression like part $P_{G}$ lowers the additive energy, as much as it can. For doing so, a geometric block will appear exactly before and after an arithmetic block, and have much more elements.\ \ For writing the construction precisely down, we introduce some notation. Let henceforth $\left\lfloor x\right\rfloor $ denote the greatest integer $m$ that is at most $x\in\mathbb{R}$. Suppose trough-out this section that $f$ is as in Theorem \[thm: lowering the known Energy threshold\]. We set $P_{A}^{\left(1\right)}$ to be the empty set while $P_{G}^{\left(1\right)}\coloneqq\left\{ 1,2\right\} $. Moreover, for $j\geq2$ we let $P_{A}^{\left(j\right)}$ denote the set of $\bigl\lfloor2^{j}\bigl(f(2^{j})\bigr)^{-\beta}\bigr\rfloor$ consecutive integers that start with $C_{j}=2\max\bigl\{ P_{G}^{(j-1)}\bigr\}$, and $P_{G}^{\left(j\right)}$ is such that the difference set $P_{G}^{\left(j\right)}-2C_{j}$ is the geometric progression $2^{i}$ for $1\leq i\leq\bigl\lfloor\bigl(f(2^{j})\bigr)^{-\gamma}2^{j}\bigl(1-\bigl(f(2^{j})\bigr)^{\gamma-\beta}\bigr)\bigr\rfloor$ where $0<\gamma<\beta<\nicefrac{3}{4}$ are parameters[^7] to be chosen later-on. In this notation, we take $$P_{A}\coloneqq\bigcup_{j\geq1}P_{A}^{\left(j\right)},\qquad P_{G}\coloneqq\bigcup_{j\geq1}P_{G}^{\left(j\right)},$$ and denote by $a_{n}$ the $n$-th smallest element in $P_{A}\cup P_{G}$. For $d\in\mathbb{Z}$ and finite sets of integers $X,Y$, we abbreviate the number of representation of $d$ as a difference of an $x\in X$ and a $y\in Y$ by $\text{rep}_{X,Y}(d)\coloneqq\#\{(x,y)\in X\times Y:\,x-y=d\}$; observe that $$E\left(X\right)=\sum_{d\in\mathbb{Z}}\left(\mathrm{rep}_{X,X}\left(d\right)\right)^{2},\label{eq: additive Energy in terms of number of representation}$$ and $$R\left(\left[-s,s\right],\alpha,N\right)=\frac{1}{N}\underset{d\neq0}{\sum}\text{rep}_{A_{N},A_{N}}(d)\mathbf{1}_{\left[0,\frac{s}{N}\right]}\left(\left\Vert \alpha d\right\Vert \right).\label{eq: lower bound for counting function fo the pair correlations}$$ Preliminaries ------------- We begin to determine the order of magnitude of $E\left(A_{N}\right)$ for the truncations $A_{N}$ of the sequence constructed above. Since the cardinality of elements in the union of the blocks $P_{G}^{\left(j\right)},P_{A}^{\left(j\right)}$ has about exponential growth, it is reasonable to expect $E\left(A_{N}\right)$ to be of the same order of magnitude as the additive energy of the last block $P_{G}^{\left(J\right)}\cup P_{A}^{\left(J\right)}$ that is fully contained in $A_{N}$ - note that $J=J\left(N\right)$; i.e. to expect the magnitude of $E\bigl(P_{G}^{\left(J\right)}\cup P_{A}^{\left(J\right)}\bigr)$ which is roughly equal to $E\bigl(P_{A}^{\left(J\right)}\bigr)$. The following proposition verifies this heuristic considerations. \[prop: additive energy of good-guy-bad-guy sequence\]Let $\left(a_{n}\right)_{n}$ be as in the beginning of Section 3, and $f$ be as in one of the two assertions in Theorem \[thm: lowering the known Energy threshold\]. Then, $E\left(A_{N}\right)=\Theta\bigl(N^{3}\bigl(f\bigl(N\bigr)\bigr){}^{-3\left(\beta-\gamma\right)}\bigr)$. For the proof of Proposition \[prop: additive energy of good-guy-bad-guy sequence\], we need the next technical lemma. \[lem: auxiliary lemma for calculating additive energy\]Let $\FJ\coloneqq2^{j}\bigl(f\bigl(2^{j}\bigr)\bigr)^{-\delta}$, for $j\geq1$ and fixed $\delta\in\left(0,1\right)$, where $f$ is as in Proposition \[prop: additive energy of good-guy-bad-guy sequence\]. Then, $\sum_{i\leq j}F_{i}=\mathcal{O}\bigl(F_{j}\bigr)$ and $$\sum_{d\in\mathbb{Z}}\biggl(\sum_{j,i\leq J}\mathrm{rep}_{P_{G}^{\left(j\right)},P_{A}^{\left(i\right)}}\left(d\right)\biggr)^{2}=\mathcal{O}\left(J^{6}2^{2J}\right).$$ Suppose that $f\left(x\right)=\mathcal{O}\bigl(x^{\nicefrac{1}{3}}\left(\log x\right)^{-\nicefrac{7}{3}}\bigr)$ is such that (\[eq: divergence of the reciprocal of (f(n) times n)\]) diverges. Because $$\sum_{j\leq J+1}\frac{1}{f\bigl(2^{j}\bigr)}\geq\sum_{k\leq2^{J}}\frac{1}{kf\left(k\right)}$$ diverges as $J\rightarrow\infty$ and $\left(f\bigl(2^{j}\bigr)/f\bigl(2^{j+1}\bigr)\right)_{j}$ is non-decreasing, we conclude that $\lim_{j\rightarrow\infty}\bigl(f\bigl(2^{j}\bigr)/f\bigl(2^{j+1}\bigr)\bigr)=1$. Therefore, there is an $i_{0}$ such that the estimate $\bigl(f\bigl(2^{i}\bigr)\bigr)^{-1}f\bigl(2^{i+h}\bigr)<\bigl(\nicefrac{3}{2}\bigr)^{\frac{h}{\delta}}$ holds for any $i\geq i_{0}$ and $h\in\mathbb{N}$. Hence, $$\frac{1}{F_{j}}\sum_{i\leq j}F_{i}\leq o\left(1\right)+\sum_{i_{0}\leq i\leq j}2^{i-j}\left(\frac{3}{2}\right)^{j-i}=\mathcal{O}\bigl(1\bigr).$$ If $f$ is such that (\[eq: divergence of the reciprocal of (f(n) times n)\]) converges and $f\left(2x\right)\leq\left(2-\varepsilon\right)f\left(x\right)$ for $x$ large enough, then we obtain by a similar argument that $\sum_{i\leq j}F_{i}$ is in $\mathcal{O}\bigl(F_{j}\bigr)$. Furthermore, $\mathrm{rep}_{P_{G}^{\left(j\right)},P_{A}^{\left(i\right)}}\left(d\right)=\mathcal{O}\left(i\right)$, for every $j\geq1$, and non-vanishing for $\mathcal{O}\bigl(2^{2j}\bigr)$ values of $d$ which implies the last claim. We can now prove the proposition. Let $\FJ=2^{j}\bigl(f\bigl(2^{j}\bigr)\bigr)^{-\beta}$, $N\geq1$ be large and denote by $J=J\left(N\right)\geq0$ the greatest integer $j$ such that $P_{G}^{\left(j-1\right)}\subseteq A_{N}$. Since $$E\bigl(A_{N}\bigr)\geq E\bigl(P_{A}^{\left(J-1\right)}\bigr)=\Omega\bigl(N^{3}\bigl(f\bigl(N\bigr)\bigr){}^{-3\left(\beta-\gamma\right)}\bigr),$$ it remains to show that $E\bigl(A_{N}\bigr)=\mathcal{O}\bigl(N^{3}\bigl(f\bigl(N\bigr)\bigr){}^{-3\left(\beta-\gamma\right)}\bigr)$. By exploiting (\[eq: additive Energy in terms of number of representation\]), $$E\bigl(A_{N}\bigr)\leq\sum_{d\in\mathbb{Z}}\bigl(\mathrm{rep}_{A_{T_{J}},A_{T_{J}}}\left(d\right)\bigr)^{2}\quad\text{where}\quad T_{J}\coloneqq\#\bigcup_{j\leq J}\left(P_{A}^{\left(j\right)}\cup P_{G}^{\left(j\right)}\right).$$ Moreover, $\mathrm{rep}_{A_{T_{J}},A_{T_{J}}}\left(d\right)=S_{1}\left(d\right)+S_{2}\left(d\right)$ where $S_{1}\left(d\right)$ abbreviates the mixed sum $\sum_{i,j\leq J}\bigl(\mathrm{rep}_{P_{A}^{\left(j\right)},P_{G}^{\left(i\right)}}\left(d\right)+\mathrm{rep}_{P_{G}^{\left(i\right)},P_{A}^{\left(j\right)}}\left(d\right)\bigr)$ and $S_{2}\left(d\right)$ abbreviates the sum $\sum_{i,j\leq J}\bigl(\mathrm{rep}_{P_{G}^{\left(i\right)},P_{G}^{\left(j\right)}}\left(d\right)+\mathrm{rep}_{P_{A}^{\left(i\right)},P_{A}^{\left(j\right)}}\left(d\right)\bigr)$. Using that for any real numbers $a,b$ the inequality $\left(a+b\right)^{2}\leq2\bigl(a^{2}+b^{2}\bigr)$ holds, we obtain $$E\bigl(A_{N}\bigr)=\mathcal{O}\biggl(\sum_{d\in\mathbb{Z}}\bigl(S_{1}\left(d\right)\bigr)^{2}+\sum_{d\in\mathbb{Z}}\bigl(S_{2}\left(d\right)\bigr)^{2}\biggr).$$ Lemma \[lem: auxiliary lemma for calculating additive energy\] implies that $\sum_{d\in\mathbb{Z}}\bigl(S_{2}\left(d\right)\bigr)^{2}=\mathcal{O}\bigl(\left(\log N\right)^{6}N^{2}\bigr)$ due to $J=\mathcal{O}\left(\log N\right)$. Moreover, we note that $\mathrm{rep}_{P_{A}^{\left(i\right)},P_{A}^{\left(j\right)}}\left(d\right)$ is non-vanishing for at most $4F_{J}$ values of $d$ as $i,j\leq J$. Since $\mathrm{rep}_{P_{A}^{\left(i\right)},P_{A}^{\left(j\right)}}\left(d\right)\leq F_{\min\left(i,j\right)}$ holds, we deduce that $$\sum_{i,j\leq J}\mathrm{rep}_{P_{A}^{\left(i\right)},P_{A}^{\left(j\right)}}\left(d\right)=\mathcal{O}\biggl(\sum_{j\leq J}\sum_{i\leq j}F_{i}\biggr).$$ By Lemma \[lem: auxiliary lemma for calculating additive energy\], the right hand side is in $\mathcal{O}\bigl(F_{J}\bigr)$. Since $\mathrm{rep}_{P_{G}^{\left(i\right)},P_{G}^{\left(j\right)}}\left(d\right)\leq1$, where $i,j\leq J$, is non-vanishing for at most $\mathcal{O}\bigl(T_{J}^{2}\bigr)=\mathcal{O}\left(N^{2}\right)$ values of $d$, we obtain that $$\sum_{d\in\mathbb{Z}}\bigl(S_{1}\left(d\right)\bigr)^{2}=\mathcal{O}\bigl(F_{J}^{3}+\left(\log N\right)^{6}N^{2}\bigr)$$ which is in $\mathcal{O}\bigl(N^{3}\bigl(f\bigl(N\bigr)\bigr){}^{-3\left(\beta-\gamma\right)}\bigr)$. Hence, $E\bigl(A_{N}\bigr)=\mathcal{O}\bigl(N^{3}\bigl(f\bigl(N\bigr)\bigr){}^{-3\left(\beta-\gamma\right)}\bigr)$. For estimating the measure or the Hausdorff dimension of $\NPPC\left(\left(a_{n}\right)_{n}\right)$ from below, we recall the notion of an optimal regular system. This notion, roughly speaking, describes sequences of real numbers that are exceptionally well distributed in any subinterval, in a uniform sense, of a fixed interval. Let $J$ be a bounded real interval, and $S=\left(\alpha_{i}\right)_{i}$ a sequence of distinct real numbers. $S$ is called an optimal regular system in $J$ if there exist constants $c_{1},\,c_{2},\,c_{3}>0$ - depending on $S$ and $J$ only - such that for any $I\subseteq J$ there is an index $Q_{0}=Q_{0}\left(S,I\right)$ such that for any $Q\geq Q_{0}$ there are indices $$c_{1}Q\leq i_{1}<i_{2}<\ldots<i_{t}\leq Q\label{eq: property of c1 in optimal regular system definition}$$ satisfying $\alpha_{i_{h}}\in I$ for $h=1,\ldots,t$, and $$\left|\alpha_{i_{h}}-\alpha_{i_{\ell}}\right|\geq\frac{c_{2}}{Q}\label{eq: property of c2 in optimal regular system definition}$$ for $1\leq h\neq\ell\leq t$, and $$c_{3}\lambda\left(I\right)Q\leq t\leq\lambda\left(I\right)Q.\label{eq: property of c3 in optimal regular system definition}$$ Moreover, we need the following result(s) due to Beresnevich which may be thought of as a far reaching generalization of Khintchine’s theorem, and Jarník-Besicovitch theorem in Diophantine approximation. \[thm: Khintchine a la Victor\]Suppose $\psi:\mathbb{\mathbb{R}}_{>0}\rightarrow\mathbb{R}_{>0}$ is a continuous, non-increasing function, and $S=\bigl(\alpha_{i}\bigr)_{i}$ an optimal regular system in $\left(0,1\right)$. Let $\mathcal{K}_{S}\left(\psi\right)$ denote the set of $\xi$ in $\left(0,1\right)$ such that $\left|\xi-\alpha_{i}\right|<\psi\left(i\right)$ holds for infinitely many $i$. If $$\sum_{n\geq1}\psi\left(n\right)\label{eq: sum over psi values}$$ diverges, then $\mathcal{K}_{S}\left(\psi\right)$ has full measure.\ Conversely, if (\[eq: sum over psi values\]) converges, then $\mathcal{K}_{S}\left(\psi\right)$ has measure zero and the Hausdorff dimension equals the reciprocal of the lower order of $\frac{1}{\psi}$ at infinity. For a rational $\alpha=\frac{p}{q}$, where $p,q\in\mathbb{Z}$, $q\neq0$, we denote by $H\left(\alpha\right)$ its (naive) height, i.e. $H\left(\alpha\right)\coloneqq\max\left\{ \left|p\right|,\left|q\right|\right\} $. It is well-known that the set of rational numbers in $\left(0,1\right)$, ordered in classes by increasing height and in each class ordered by numerically values, gives rise to an optimal regular system in $\left(0,1\right)$. The following lemma says, roughly speaking, that this assertion remains true for the set of rationals in $\left(0,1\right)$ whose denominators are members of a special sequence that is not too sparse in the natural numbers. The proof can be given by modifying the proof of the classical case, compare [@Bugeaud:; @Approximation; @by; @algebraic; @numbers Prop. 5.3]; however, we shall give the details for making this article more self-contained. \[lem: optimal regular system\]Let $\vartheta:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>1}$ be monotonically increasing to infinity with $\vartheta\left(x\right)=\mathcal{O}\bigl(x^{\nicefrac{1}{4}}\bigr)$ and $\vartheta\left(2^{j+1}\right)/\vartheta\left(2^{j}\right)\rightarrow1$ as $j\rightarrow\infty$. For each $j\in\mathbb{N}$, we let $$B_{j}\coloneqq\frac{2^{j}}{f\left(2^{j}\right)\sqrt{\vartheta\left(2^{j}\right)}},\qquad b_{j}\coloneqq\frac{2}{3}B_{j}.$$ Let $S=\bigl(\alpha_{i}\bigr)_{i}$ denote a sequence running through all rationals in $\left(0,1\right)$ whose denominators are in $M\coloneqq\bigcup_{j\geq1}\bigl\{ n\in\mathbb{N}:\,b_{j}\leq n\leq B_{j}\bigr\}$ such that $i\mapsto H\bigl(\alpha_{i}\bigr)$ is non-decreasing. Then, $S$ is an optimal regular system in $\left(0,1\right)$. Let $X\geq2$. There are strictly less than $2X^{2}$ rational numbers in $\left(0,1\right)$ with height bounded by $X$. We take $J=J\left(X\right)$ to be the largest integer $j\geq1$ such that $B_{j}\leq X$. Then, for $X$ large enough, there are at least $$\begin{aligned} \sum_{j\leq J}\sum_{b_{j}\leq q\leq B_{j}}\varphi\left(q\right) & \geq\sum_{j\leq J}\left(\frac{1}{3\pi^{2}}B_{j}^{2}+\mathcal{O}\left(B_{j}\log B_{j}\right)\right)\\ & \geq\frac{1}{6\pi^{2}}\frac{2^{2J}}{f^{2}\left(2^{J}\right)\vartheta\left(2^{J}\right)}+\mathcal{O}\left(J2^{J}\right)\\ & >\left(\frac{X}{5\pi}\right)^{2}\end{aligned}$$ distinct such rationals in $\left(0,1\right)$ with height not exceeding $X$. Hence, we obtain $\frac{\sqrt{i}}{2}\leq H\left(\alpha_{i}\right)\leq\sqrt{25\pi^{2}\left(i+1\right)}+1$ for $i$ sufficiently large. Let $Q\in\mathbb{N}$, $I\subseteq\left[0,1\right]$ be a non-empty interval, and let $F$ denote the set of $\xi\in I$ satisfying the inequality $\left\Vert q\xi\right\Vert <Q^{-1}$ with some $1\leq q\leq\frac{1}{1000}Q$. Note that $F$ has measure at most $$\sum_{q\leq\frac{1}{1000}Q}\left(\frac{2}{qQ}q\lambda\left(I\right)+\frac{2}{qQ}\right)=\frac{1}{500}\lambda\left(I\right)+\mathcal{O}\left(\frac{\log Q}{Q}\right)<\frac{1}{400}\lambda\left(I\right)$$ for $Q\geq Q_{0}$ where $Q_{0}=Q_{0}\left(S,I\right)$ is sufficiently large. Let $\bigl\{\nicefrac{p_{j}}{q_{j}}\bigr\}_{1\leq j\leq t}$ be the set of all rationals $\nicefrac{p_{j}}{q_{j}}\in\left(0,1\right)$ with $q_{j}\in M$, $\frac{1}{1000}Q<q_{j}<Q$ that satisfy $$\left|\frac{p_{j}}{q_{j}}-\frac{p_{j'}}{q_{j'}}\right|>\frac{2000}{Q^{2}}$$ whenever $1\leq j\neq j'\leq t$. Observe that for $J$ as above with $X=Q$ sufficiently large, it follows that $$\left\{ q\in M:\,b_{J}\leq q\leq B_{J}\right\} \subseteq\left\{ \frac{Q}{1000},\frac{Q}{1000}+1,\ldots,Q\right\}$$ holds and there are hence at least $\frac{1}{3\pi^{2}}B_{J}^{2}+\mathcal{O}\left(B_{J}\log B_{J}\right)>\frac{1}{400}Q^{2}$ choices of $\nicefrac{p_{j}}{q_{j}}\in\left(0,1\right)$ with $q_{j}\in M$ and $\frac{1}{1000}Q<q_{j}<Q$. Due to $\lambda\left(I\setminus F\right)>\frac{399}{400}\lambda\left(I\right)$, we conclude $t\geq400\frac{Q^{2}}{4000}\frac{399}{400}\lambda\left(I\right)$. Thus, taking $c_{1}\coloneqq\nicefrac{1}{1000}$, $c_{2}\coloneqq2000$, and $c_{3}\coloneqq\frac{399}{4000}$ in (\[eq: property of c1 in optimal regular system definition\]), (\[eq: property of c2 in optimal regular system definition\]) and (\[eq: property of c3 in optimal regular system definition\]), respectively, $S$ is shown to be an optimal regular system. Now we can proceed to the proof of Theorem \[thm: lowering the known Energy threshold\]. Proof of Theorem \[thm: lowering the known Energy threshold\] ------------------------------------------------------------- We argue in two steps depending on whether or not the series (\[eq: divergence of the reciprocal of (f(n) times n)\]) converges. Proposition (\[prop: additive energy of good-guy-bad-guy sequence\]) implies the announced $\Theta$-bounds on the additive energy of $A_{N}$, in both cases.\ \ (i) Suppose (\[eq: divergence of the reciprocal of (f(n) times n)\]) diverges, and fix $s>0$. Let $\vartheta:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>1}$ be monotonically increasing to infinity with $\vartheta\left(x\right)=\mathcal{O}\left(x^{\nicefrac{1}{4}}\right)$ such that $$\psi\left(n\right)\coloneqq\frac{1}{nf\left(n\right)\vartheta\left(n\right)}\label{eq: specification of the appxomation function-1}$$ satisfies the divergence condition (\[eq: sum over psi values\]). Thus, $\vartheta\left(2^{j}\right)/\vartheta\left(2^{j-1}\right)\rightarrow1$ as $j\rightarrow\infty$. Hence, $S=\left(\alpha_{i}\right)_{i}$ from Lemma \[lem: optimal regular system\] is an optimal regular system. Furthermore, if $\alpha_{i}=\frac{m}{n}$, then $i\geq cn^{2}$ holds true with a constant $c=c\left(f,\vartheta\right)>0$ due to $b_{J}\leq n\leq B_{J}$, for some integer $J$, and $$\sum_{j\leq J-1}\sum_{b_{j}\leq m\leq B_{j}}\varphi\left(m\right)=\Theta\bigl(B_{J}^{2}\bigr).$$ Therefore, $\psi\left(i\right)\leq c^{-1}n^{-2}\bigl(f\bigl(cn^{2}\bigr)\vartheta\bigl(cn^{2}\bigr)\bigr)^{-1}$. The growth assumption on $f$ and the growth bound $\vartheta\left(x\right)=\mathcal{O}\bigl(x^{\nicefrac{1}{4}}\bigr)$ yields that if $j$ is large enough, then $b_{j}\leq n\leq B_{j}$ implies $cn^{2}>2^{j}$ and hence we obtain $\psi\left(i\right)\leq c^{-1}n^{-2}\bigl(f\bigl(2^{j}\bigr)\vartheta\bigl(2^{j}\bigr)\bigr)^{-1}$. Combining these considerations, we have established that $$n\psi\left(i\right)=\mathcal{O}\left(2^{-j}\left(\vartheta\bigl(2^{j}\bigr)\right)^{-\nicefrac{1}{2}}\right).$$ Moreover, for a function $g:\mathbb{N}\rightarrow\mathbb{R}_{>0}$, we let $E_{g}$ denote the set of $\alpha\in\left(0,1\right)$ such that for infinitely many $j$ there is some $n$ with $b_{j}\leq n\leq B_{j}$ satisfying $\left\Vert n\alpha\right\Vert =\mathcal{O}\left(2^{-j}g\left(j\right)\right)$. Set $h\left(j\right)\coloneqq\bigl(\vartheta\bigl(2^{j}\bigr)\bigr)^{-\nicefrac{1}{2}}$. Applying Theorem \[thm: Khintchine a la Victor\] with $\psi$ as in (\[eq: specification of the appxomation function-1\]), implies that $E_{h}$ has full measure. Therefore, for any $\alpha\in E_{h}$ we get $$\left\Vert n\alpha\right\Vert \leq n\left|\alpha-\alpha_{i}\right|=\mathcal{O}\Bigl(2^{-j}\left(\vartheta\bigl(2^{j}\bigr)\right)^{-\nicefrac{1}{2}}\Bigr)\label{eq: good approximation to alpha in terms of psi-1}$$ for infinitely many $j$. Now if $b_{j}\leq n\leq B_{j}$ for $j$ sufficiently large and $n,\alpha$ as in (\[eq: good approximation to alpha in terms of psi-1\]), then it follows that by taking any integer $m\leq\left(f\left(2^{j}\right)\right)^{\gamma}\bigl(\vartheta\bigl(2^{j}\bigr)\bigr)^{\frac{1}{3}}$ also the multiples $$nm\leq2^{j}\left(f\left(2^{j}\right)\right)^{\gamma-1}\left(\vartheta\left(2^{j}\right)\right)^{-\nicefrac{1}{6}}$$ satisfy that $\mathbf{1}_{\left[0,sT_{j}\right]}\left(\left\Vert \alpha(mn)\right\Vert \right)=1$ where $T_{j}=\mathcal{O}\bigl(2^{j}\left(f\left(2^{j}\right)\right)^{-\gamma}\bigr)$ is as in the Proof of Proposition \[prop: additive energy of good-guy-bad-guy sequence\]. If additionally $\gamma-1\geq-\beta$ holds, then we obtain that $\text{rep}_{A_{T_{j}},A_{T_{j}}}(mn)\geq\nicefrac{1}{2}2^{j}\left(f\left(2^{j}\right)\right)^{-\beta}$ holds for $j$ sufficiently large. By (\[eq: lower bound for counting function fo the pair correlations\]), we obtain $$R\bigl(\left[-s,s\right],\alpha,T_{j}\bigr)\geq C\left(f\left(2^{j}\right)\right)^{2\gamma-\beta}\bigl(\vartheta\bigl(2^{j}\bigr)\bigr)^{\nicefrac{1}{3}}$$ for infinitely many $j$ where $C>0$ is some constant. For the optimal choice of the parameters $\beta,\gamma>0$, we are therefore led to find the maximal $\beta$ such that $2\gamma-\beta\geq0$ and $\gamma-1\geq-\beta$ is satisfied. The (unique) solution is $\beta=\nicefrac{2}{3}$ and $\gamma=\nicefrac{1}{3}$. Hence, (\[eq: divergence of the Pair Correlation Function\]) follows for $\alpha\in E_{h}$.\ \ (ii) Suppose the series (\[eq: divergence of the reciprocal of (f(n) times n)\]) converges. We keep the same sequence as in step (i) while taking $\vartheta\left(x\right)=1+\log\left(x\right)$, as we may. The arguments of step (i) show that any $\alpha\in E_{h}$, where $h\left(j\right)=j^{-\nicefrac{1}{2}}$, satisfies (\[eq: divergence of the Pair Correlation Function\]); now the conclusion is that $E_{h}$ has Hausdorff dimension equal to the reciprocal of $$\liminf_{x\rightarrow\infty}\frac{-\log\left(\psi\left(x\right)\right)}{\log x}=1+\liminf_{x\rightarrow\infty}\frac{\log f\left(x\right)}{\log x}.$$ Thus, the proof is complete. #### Concluding remarks {#concluding-remarks .unnumbered} We would like to mention two open problems related to this article. The first problem concerns extensions of Theorem \[thm: full measure of set of counterexamples\]. Let $\left(a_{n}\right)_{n}$ be an increasing sequence of positive integers with $E\left(A_{N}\right)=\Omega\left(N^{3}\right)$. Has the complement of $\NPPC\left(\left(a_{n}\right)_{n}\right)$ Hausdorff dimension zero; or is it, in fact, empty? The second problem is related to Corollary \[cor: order of magnitude for the additive energy of the sequence of counter examples\]. How large has $E\left(A_{N}\right)$ to be for ensuring that $\NPPC\left(\left(a_{n}\right)_{n}\right)$ has full Lebesgue measure? #### Acknowledgements {#acknowledgements .unnumbered} Both authors would like to express their gratitude towards C. Aistleitner for introducing us to the topic of this article, and valuable discussions. #### Addresses\ {#addresses .unnumbered} Thomas Lachmann, 5010 Institut für Analysis und Zahlentheorie 8010 Graz, Steyrergasse 30/II email: lachmann@math.tugraz.at\ \ Niclas Technau, 5010 Institut für Analysis und Zahlentheorie 8010 Graz, Steyrergasse 30/II, email: technau@math.tugraz.at [10]{} C. Aistleitner, T. Lachmann, F. Pausinger: *Pair correlations and equidistribution,* [arXiv:1702.07365](https://arxiv.org/abs/1612.05495), J. of Num. Th., to appear. C. Aistleitner, G. Larcher and M. Lewko: *Additive Energy and the Hausdorff Dimension of the Exceptional Set in Metric Pair Correlation Problems,* [arXiv:1606.03591 ](https://arxiv.org/abs/1606.03591), Israel J. Math., to appear. F. P. Boca and A. Zaharescu: *Pair correlation of values of rational functions (mod p)*, Duke Math. J., 105(2):267307, 2000. A. Bondarenko and K. Seip: *GCD sums and complete sets of square-free numbers*, Bull. Lond. Math. Soc., 47(1):2941, 2015. T. Bloom, S. Chow, A. Gafni and A. Walker: Additive Energy and the Metric Poissonian Property, Work in Progress Y. Bugeaud: *Approximation by algebraic numbers,* Vol. 160, Cambridge University Press, 2004. D. R. Heath-Brown: *Pair correlation for fractional parts of $n^{2}$*, Math. Proc. Cambridge Philos. Soc., 148(3):385407, 2010. G. Larcher and S. Grepstad: *On pair correlation and discrepancy, [arXiv:1612.08008](https://arxiv.org/abs/1612.08008)*, Arch. Math., to appear. J. Marklof and A. Strömbergsson: *Equidistribution of Kronecker sequences along closed horocycles*, Geom. Funct. Anal., 13(6):12391280, 2003. Z. Rudnick and P. Sarnak: *The pair correlation function of fractional parts of polynomials*, Comm. Math. Phys., 194(1):6170, 1998. Z. Rudnick, P. Sarnak, and A. Zaharescu: *The distribution of spacings between the fractional parts of $n^{2}$*, Invent. Math., 145(1):3757, 2001. Z. Rudnick and A. Zaharescu: *A metric result on the pair correlation of fractional parts of sequences*, Acta Arith., 89(3):283293, 1999. T. Tao and V. Vu: *Additive combinatorics*, Vol. 105, Cambridge University Press, 2006. J. L. Truelsen: *Divisor problems and the pair correlation for the fractional parts of $n^{2}$*, Int. Math. Res. Not., (16):31443183, 2010. A. Walker: *The Primes are not Metric Poissonian,* [arXiv:1702.07365](https://arxiv.org/abs/1702.07365) H. Weyl*: Über die Gleichverteilung von Zahlen mod. Eins*, Math. Ann., 77(3):313352, 1916. [^1]: The first author is supported by the Austrian Science Fund (FWF): Y-901. [^2]: The second author is supported by the Austrian Science Fund (FWF) projects: W1230, and (for part of the time) by Y-901. [^3]: The subscript $2$ in $R_{2}$ indicates that relations of second order, i.e. pair correlations, are counted. [^4]: It is worthwhile to mention that the case $d=2$ is of particular interest for its connection to mathematical physics, see [@Rudnick; @Sarnak:; @The; @pair; @correlation; @function; @of; @fractional; @parts; @of; @polynomials] for further references. [^5]: This problem was posed at the problem session of the ELAZ conference in 2016. [^6]: The bases step uses simplified versions of the arguments exploited in the induction step, and will therefore be postponed. [^7]: No particular importance should be attached to requiring $\beta<\nicefrac{3}{4}$, or using “dyadic steps lengths $2^{j}$”. Doing so is for simplifying the technical details only - eventually, it will turn out that $\beta=\nicefrac{2}{3}=2\gamma$ is the optimal choice of parameters in this approach. For proving this to the reader, we leave $\gamma,\beta$ undetermined till the end of this section.

--- author: - Maria Silvia Garelli - Feodor V Kusmartsev date: 'Received: date / Revised version: date' title: 'Buckyball Quantum Computer: Realization of a Quantum Gate' --- Introduction ============ During recent years there is a strong progress in modeling physical realizations of a quantum computer. Many quantum physical systems have been investigated for the realization of quantum gates. The most remarkable studies were related to systems associated to Quantum Optics Ion Traps, to Quantum Electrodynamics in Optical Cavities and to Nuclear Magnetic Resonance. All these experiments are aimed to realize a quantum gate. The first type of experiments is based on trapping ions in electromagnetic traps, where the ions, which encode the qubit in the charge degrees of freedom, are subjected to the mutual electrostatic interaction and to a state selective displacement generated by an external state dependent force [@Cirac; @Steane; @Sasura; @Calarco]. Cavity quantum electrodynamics (QED) techniques are based on the coherent interaction of a qubit, generally represented by an atom or semiconductor dot system, with a single mode or a few modes of the electromagnetic field inside a cavity. Depending on the particular system, the qubit can be represented by the polarization states of a single photon or by two excited states of an atom. Although cavity QED experiments are very promising, they have been accomplished for few qubits [@Pellizzari; @van; @Rauschenbeutel; @Duan]. In the third experiment, nuclear spins represent qubits. These spins can be manipulated using nuclear magnetic resonance techniques, and through the study of the quantum behavior of spins, quantum operations are realized. However, the number of spins which can be collected in a system is very limited, and this forbids the building up of a scalable quantum computer [@Gershenfeld; @Schmidt; @Leibfried; @Nielsen]. From the study of such systems, we learn that the decoherence phenomenon is the main issue which prevents the realization of quantum gates. Here we will focus on a physical systems, which will be able to produce a realistic quantum gate. The basic elements of our system are fullerene molecules with encapsulated atoms or ions, which are called *buckyballs* or *endohedral fullerenes*. Each of the trapped atoms carries a spin. This spin, associated with electronic degrees of freedom, encodes the qubit. It has been shown [@Greer], that these endohedral systems provide a long lifetime for the trapped spins and that the fullerene molecules represent a good sheltering environment for the very sensible spins trapped inside. These endohedral systems are typically characterized by two relaxation times. The first is $T_1$, which is due to the interactions between a spin and the surrounding environment. The second one is $T_2$ and it is due to the dipolar interaction between the qubit encoding spin and the surrounding endohedral spins randomly distributed in the sample. While $T_1$ is dependent on temperature, $T_2$ is practically independent of it. The experimental measure of the two relaxation times shows that $T_1$ increases with decreasing temperature from about $100\mu s$ at $T=300 K$ to several seconds below $T=5K$, and that the value of the other relaxation time, $T_2$, remains constant, that is $T_2\simeq 20\mu s$ [@Knorr1; @Knorr2]. In comparison with $T_2$ the value of $T_1$ is very large, therefore the system decoherence is determined by the spin-spin relaxation processes. It is supposed that the value of $T_2$ can be increased, if it will be possible to design a careful experimental architecture, which could screen the interaction of the spins with the surrounding magnetic moments. It should be possible to reduce the relaxation time of the system due to the random spin-spin interactions, if we consider a system composed by arrays of endohedrals encapsulated in a nanotube [@Khlobystov], this system is also called as *peapod*, or considering buckyballs embedded on a substrate. These should be reliable systems for the realization of quantum gates. In such architectures the decoherence time for each encapsulated spin should be longer. Quantum computing through the study of doped fullerene systems has been investigated in many works [@Harneit; @Harneit1; @Feng; @Suter; @Twam]. Although we have followed many ideas suggested in these previous papers, we consider a different approach for the realization of quantum gates. Our study is focused on a system composed by two buckyballs. Our aim is the realization of a quantum *$\pi$-gate*, which is a generalization of the *phase gate*, this will be treated in Sec. \[phasegate\]. To perform the $\pi$-gate, we need to know the time evolution of the coefficients of the standard computational basis states over which we expand the wave function of our system. The two particle phases are evaluated through the numerical solution of the Schr[ö]{}dinger equation, see Secs. \[ind\]-\[dip\]. We have used two approaches: a time independent Hamiltonian, see Sec. \[ind\], and a time dependent one, see Sec. \[dip\]. The main result of our study is the gate time, that is the time required by the system in order to perform the $\pi$-gate. The values obtained are around $\tau\simeq1\times10^{-8}s$, which is a few orders smaller than the shortest relaxation time, $T_2$. From the comparison of the gate time, $\tau$, to the relaxation time, $T_2$, we get that it is theoretically possible to realize some thousands of basic gate operations before the system decoheres. We have also checked the reliability of our gate through the analysis of the *concurrence* of the two-qubit state, see Sec. \[concurr\]. The best value for the concurrence is obtained in the case of a time dependent Hamiltonian, while the gate time is nearly the same in both cases. Physical Features of the System =============================== The system under consideration is composed by two interacting buckyballs. Several experimental and theoretical studies on buckyballs [@Greer; @Harneit; @Heath; @Shinohara; @Saunders; @Weid], show that many different types of atoms can be encased in fullerenes molecules. However, in most of the studied endohedral\ fullerenes, there is a charge transfer from the encapsulated atom to the fullerene cage, with a resulting considerable alteration of the electronic properties of the cage. This is not the case for group V encased atoms. These atoms reside just at the center of the fullerene molecule, therefore there is no hybrididazion of the electron cloud of the encased atom and there is no Coulomb interaction with the fullerene cage. In particular, the most promising endohedral molecule should be the $N@C_{60}$, which is characterized by many interesting chemical-physical properties. Following Refs. [@Greer; @Harneit; @Weid] , experiments and theoretical calculations suggest that there is a repulsive exchange interaction between the fullerene and the electronic cloud of the encapsulated atom. The electrons in the cloud of the encased nitrogen are tighter bound than in a free nitrogen atom, which allow the encased nitrogen to be less reactive even at room temperature. These results, together with the location of the nitrogen atom in the central site, suggest that in $N@C_{60}$ the nitrogen can be considered as an independent particle, with all the properties of the free atom. Since any charge interaction is screened, the fullerene cage does not take any part in the interaction process and it can be considered just as a trap for the nitrogen atom. Therefore, the only physical quantity of interest is the spin of the trapped particle. A nitrogen atom can be effectively described as a $\frac{3}{2}$-spin particle. This spin is associated with the electronic degrees of freedom. Taking into account also the nuclear spin, which is $\frac{1}{2}$ for the $N@C_{60}$, the number of relevant degrees of freedom will be not increased [@Meher]. We will consider a more simple model assuming that the encased atoms are described as $\frac{1}{2}$-spin particles. In absence of any mutual interaction and without any applied magnetic field, the energy levels associated with these spin particles are degenerate. If we apply a static magnetic field, this degeneracy is lifted. As a result, due to the Zeeman effect, a two level system arises for each $\frac{1}{2}$-spin particle. Each of these two levels encodes the qubit. The spin-up component, $m_s=+\frac{1}{2}$, encodes the computational basis state $\mid1\rangle$, and the spin-down component, $m_s=-\frac{1}{2}$, represents the state $\mid0\rangle$. Gate Operation: The Phase Gate {#phasegate} ============================== Quantum computers operate with the use of *Quantum Gates*. Quantum gates are defined as fundamental quantum computational operations. They are presented as unitary transformations, which act on the quantum states, which describe the qubits. Therefore a quantum computer must operate with the use of many quantum gates. The simplest gates are the single-qubit gates. Since our system is composed by two qubits, we will consider a two-qubit quantum gate. One of the most important quantum gates is the *Universal Two-Qubit Quantum Gate* [@Nielsen], which is called the CNOT-gate. The CNOT operation is defined by the following four by four unitary matrix $$\label{cnot} U_{CNOT}= \left(% \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\\ \end{array}% \right),$$ and its action over the computational basis states reads: $$\begin{aligned} \mid 00\rangle &\rightarrow &\mid 00\rangle ;\\ \mid 01\rangle &\rightarrow &\mid 01\rangle ;\\ \mid 10\rangle &\rightarrow &\mid 11\rangle ;\\ \mid 11\rangle &\rightarrow &\mid 10\rangle .\end{aligned}$$ The CNOT gate is given by the composition of a single-qubit Hadamard gate followed by a two-qubit $\pi$-gate, finally followed by another single-qubit Hadamard gate. The representation of the Hadamard gate in the Bloch sphere is a $\frac{\pi}{2}$ rotation about the $y$ axis, followed by a reflection of the $x-y$ plane. In this paper we will focus on the realization of the two-qubit $\pi$-gate. It is a particular choice of the general *phase gate*, represented by the following matrix $$G= \left(% \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & e^{\imath\vartheta} \\ \end{array}% \right),$$ and its action on the computational basis states is the following: $$\begin{aligned} \label{phasegate1} |00\rangle &\rightarrow& |00\rangle \\ |01\rangle&\rightarrow& |01\rangle \\ |10\rangle&\rightarrow& |10\rangle \\ \label{phasegate4} |11\rangle&\rightarrow&e^{\imath \vartheta}|11\rangle.\end{aligned}$$ When $\vartheta=\pm\pi$, the resulting quantum gate is called a $\pi -$*gate*. In general, the time evolution of the four states of the standard computational basis can be described as follows: $$\begin{aligned} \label{phase1} |00\rangle &\rightarrow&e^{i \phi_{00}} |00\rangle \\ |01\rangle&\rightarrow&e^{i \phi_{01}} |01\rangle \\ |10\rangle&\rightarrow&e^{i \phi_{10}} |10\rangle \\ \label{phase4} |11\rangle&\rightarrow&e^{i \phi_{11}}\mid 11\rangle.\end{aligned}$$ In order to obtain the action of the ideal quantum phase gate, equations (\[phasegate1\]-\[phasegate4\]), see Ref. [@Calarco] , we have to apply the following local operator: $$\hat S=\hat S_1\otimes \hat S_2,$$ where $$\begin{aligned} \hat S_1=\mid 0\rangle_1\langle 0\mid e^{\imath s^0_1}+\mid 1\rangle_1\langle 1 \mid e^{\imath s^1_1}\\ \hat S_2=\mid 0\rangle_2\langle 0\mid e^{\imath s^0_2}+\mid 1\rangle_2\langle 1 \mid e^{\imath s^1_2}\end{aligned}$$ and the phases $s^j_i$ are defined as follows:: $$\begin{aligned} s^0_1&=&-\phi_{00}/2\\ s^1_1&=&-\phi_{10}+\phi_{00}/2\\ s^0_2&=&-\phi_{00}/2 \\ s^1_2&=&-\phi_{01}+\phi_{00}/2,\end{aligned}$$ After a straightforward calculation we obtain the desirable phase: $$\label{vartheta} \vartheta=\phi_{11}-\phi_{10}-\phi_{01}+\phi_{00}.$$ In our system, in order to realize a $\pi$-gate, we need to know the time evolution of the wave function. The time evolved wave function, expanded on the standard computational basis, is given by the following equation: $$\begin{aligned} \label{wave} \mid \psi(t)\rangle&=c_1(t)\mid 00\rangle+c_2(t)\mid 01\rangle\\\nonumber &+c_3(t)\mid 10\rangle+c_4(t)\mid 11\rangle.\end{aligned}$$ Each coefficient $c_i(t)$, $i=1,..4$, is a complex number, whose phase, arranged as in equation (\[vartheta\]), is used for the realization of the $\pi$-gate. Concurrence {#concurr} =========== When we consider a $\frac{1}{2}$-spin particle as the encoding system for the qubit, it may incur to a *spin-flip* process. This phenomenon consists in the swapping between the spin-up and spin-down components $$\begin{aligned} \mid0\rangle & \rightarrow & \mid1\rangle,\\ \mid1\rangle & \rightarrow & \mid0\rangle.\end{aligned}$$ If we consider the two-qubit state, known as *EPR pair*, $$\frac{\mid00\rangle+\mid11\rangle}{\sqrt{2}},$$ we can see that it is unaffected by the spin-flip of both qubits. This state, for this feature, is called maximally *entangled*. Therefore, we can define the *entanglement* as the property of quantum states, which shows if the state is good for carrying quantum information. The most entangled a quantum state is, the most reliable it is for transferring quantum information. In our study we have considered the *concurrence*, see Ref. [@Wootters], as a measure of the entanglement of the state describing the two-qubit system. A *pure* state of two particles quantum system is called entangled if it cannot be factorisable, that is it cannot be written as the direct product of the states describing each particle. A *mixed* state if it cannot be represented as a mixture of factorisable pure states. In this Section we will refer to the *entanglement of formation*, which quantifies the resources needed for the creation of an entangled state. For a complete treatment about the entanglement of formation of pure and mixed states see Refs. [@Bennet; @Hill]. The entanglement of formation of a quantum state can be evaluated through the concurrence [@Wootters]. Since the state describing our system is a pure state, the degree of entanglement of our system can be quantified through the definition of the concurrence for a pure state [@Wootters], which is defined by $$\label{concurrence} C(\psi)=\mid\langle\psi\mid\tilde\psi\rangle\mid,$$ where $\mid \tilde\psi\rangle$ is the spin-flipped state of system. The spin-flip transformation, which for a $\frac{1}{2}$-spin particle is the standard time reversal transformation [@sakurai], is defined as follows $$\label{spflip} \mid \tilde\psi\rangle=\hat\sigma_y\mid\psi^*\rangle,$$ where $\hat\sigma_y$ is the Pauli y-matrix and $\mid\psi^*\rangle$ is the complex conjugate of $\mid \psi\rangle$. The entanglement, see [@Wootters], is defined as a function of concurrence, through the following equation $$E(\psi)=f(C(\psi)),$$ where function $f(C(\psi))$ is given by $$\begin{aligned} f(C(\psi))&=&h(\frac{1+\sqrt{1-C(\psi)^2}}{2}),\\ h(x)&=&-x\log_2 x-(1-x)\log_2(1-x).\end{aligned}$$ Function $f(C(\psi))$ increases monotonically from $0$ to $1$ as $C(\psi)$ ranges from $0$ to $1$. Therefore, the concurrence can be considered as a measure of the entanglement.\ The state describing our two-qubit system, written as a superposition of the standard two-qubit computational basis states, is given by $$\label{normal} \mid\psi\rangle=c_1\mid00\rangle+c_2\mid01\rangle+c_3\mid10\rangle+c_4\mid11\rangle.$$ Following eq. (\[spflip\]), the spin-flip transformation over the state (\[normal\]) gives $$\label{sflip} \mid\tilde{\psi}\rangle=-c_1^*\mid00\rangle+c_2^*\mid01\rangle+c_3^*\mid10\rangle-c_4^*\mid11\rangle.$$ Finally, we obtain the concurrence of our system, see eq. (\[concurrence\]), by performing the state product between states (\[normal\]) and (\[sflip\]). The normalized concurrence of the system is given by the following equation $$\label{concnorm} C(\psi)=\frac{2\mid c_2^*c_3^*-c_1^*c_4^*\mid}{\mid c_1\mid^2+\mid c_3\mid^2+\mid c_3\mid^2+\mid c_4\mid^2}.$$ The result obtained in eq. (\[concnorm\]) will be used to evaluate the degree of entanglement of our system during the gate operation. When the concurrence related to a wave function reaches its maximum value, the state is maximally entangled. Therefore, we require that the concurrence of the wave function of the system, at the end of the gate operation, reaches a value next to its maximum. Phase Gate: Time Independent Case {#ind} ================================= Preliminary Setup ----------------- Our system is composed by two spins, which interact with a static magnetic field. Applying a static magnetic field oriented in the $z$ direction, for the Zeeman effect, we get the splitting of the spin z component into the spin-up and spin-down components. The energy difference between the two levels give the resonance frequency of the particle. However, when we apply a static magnetic field on the whole sample, all the particles will have the same resonance frequency. To perform manipulations on each buckyball, we need to be able to distinguish each of them. This setup leads to the most relevant experimental disadvantage for systems composed by arrays of buckyballs, which is the difficulty in the individual addressing of each qubit particle. This problem can be overcome with the use of external field gradients, which can shift the electronic resonance frequency of the qubit-encoding spins [@Harneit1; @Suter]. Magnetic field gradients can be generated by considering wires through which flows current. If we place two parallel wires outside our two buckyball system, it is generated an additional magnetic field in the space between the wires. Following a paper by Groth [*et al.*]{} [@Groth], with the help of atom chip technology, wires with a high current density can be built. The magnetic field amplitude generated by the two wires is given by $$B_g=\frac{\mu_0}{2\pi}I(\frac{1}{x+\rho+d/2}+\frac{1}{x-\rho-d/2}),$$ where $I$ is the current intensity, $d$ is the distance between the two wires, $\rho$ is the radius of each wire and $x$ is the distance of a buckyball with respect to the origin of the axes. With the choice $I=0.6 A$, $d=1\mu m$ and $\rho=1 \mu m$, through a numerical computation, we obtain the magnetic field distribution shown in Fig. (\[gradfield\]). We could not consider a current greater than $I=0.6A$ because the wires would face a too high heating process, and eventually they could be destroyed. On the other hand, we could not consider currents smaller than $10^{-1}A$, because the arising magnetic field gradient would be too small for each buckyball. In this case, the resonance frequencies related to the buckyballs would differ for only few $MHz$, which could be a too small gap to be realized by a frequency resonator. Realization of the Phase Gate ----------------------------- Choosing a static magnetic field in the z direction, the Hamiltonian of the system is given by the following equation ($\hbar=1$) $$\label{hamtind} \begin{array}{lll} H&=J_0\vec{\hat\sigma}_1\cdot\vec{\hat\sigma}_2+g(r)[\vec{\hat\sigma}_1\cdot \vec{\hat\sigma}_2-3(\vec{\hat\sigma}_1\cdot\vec n)(\vec{\hat\sigma}_2\cdot\vec n)]\\\label{hamtind1} &-\mu_B[((B_{z_1}+B_{g_1})\hat\sigma_{z_1})\otimes I_2\\ &+I_1\otimes((B_{z_2}+B_{g_2})\hat\sigma_{z_2})], \end{array}$$ In the previous equation, $J_0$ is the exchange spin-spin interaction coupling constant, ${\hat\sigma}_1$ and ${\hat\sigma}_2$ are the Pauli spin matrices, $g(r)=\gamma_1\gamma_2\frac{\mu_0\mu_B^2}{8\pi r^3}$, where $\mu_0$ is the diamagnetic constant, $\mu_B$ is the Bohr magneton and $r$ is the distance between the two trapped atoms, $\vec n$ is the unit vector in the direction of the line which joins the centers of the two encased atoms, $B_{z_1}=B_{z_2}$ is the static magnetic field in the $z$ direction, $B_{g_1}$ and $B_{g_2}$ are the additional magnetic fields due to the field gradient. We make an assumption, considering the trapped particles as electrons. Therefore the gyromagnetic ratio $\gamma\simeq2$, and $g(r)=\frac{\mu_0\mu_B^2}{2\pi r^3}$. Through the study of fullerenes’ spectra in ESR (Electron Spin Resonance) experiments, and also through theoretical studies, it has been shown [@Greer; @Waiblinger; @Harneit], that the exchange interaction is very small. Therefore, in eq. (\[hamtind\]), we can neglect the exchange term proportional to $J_0$, leaving the spin dipole-dipole interaction as the leading term of the mutual interaction between the two endohedrals. Choosing the direction of vector $\vec n$ parallel to the $x$ axis, the dipole-dipole interaction term is simplified as follows $$\hat D=g(r)(\hat\sigma_{z_1}\hat\sigma_{z_2}+\hat\sigma_{y_1}\hat\sigma_{y_2}-2\hat\sigma_{x_1}\hat\sigma_{x_2}).$$ The Hamiltonian matrix form is given by the following matrix $$\label{matind} \left(% \begin{array}{cccc} g(r)+m_1 & 0 & 0 & -3g(r) \\ 0 & -g(r)+m_2 & -g(r) & 0 \\ 0 & -g(r) & -g(r)-m_2 & 0 \\ -3g(r) & 0 & 0 & g(r)-m_1 \\ \end{array}% \right),$$ where $$m_1=-\mu_B(B_{z_1}+B_{g_1}+B_{z_2}+B_{g_2})$$ and $$m_2=-\mu_B(B_{z_1}+B_{g_1}-B_{z_2}-B_{g_2}),$$ are the static magnetic field terms. Solving the\ Schr[ö]{}dinger equation $$\label{schrod} \imath\frac{\partial}{\partial t}\mid \psi(t)\rangle=H\mid \psi(t)\rangle,$$ where the wave function is a superposition of the standard two-qubit computational basis, given by equation (\[wave\]), we get the four differential equation system $$\begin{aligned} \label{systemTindip1} \dot c_1(t)&=&-\imath[(g(r)+m_1)c_1(t)-3g(r)c_4(t)];\\ \dot c_2(t)&=&-\imath[(-g(r)+m_2)c_2(t)-g(r)c_3(t)];\\ \dot c_3(t)&=&-\imath[ -g(r)c_2(t)+(-g(r)-m_2)c_3(t)];\\\label{systemTindip4} \dot c_4(t)&=&-\imath[-3g(r) c_1(t)+(g(r)-m_1)c_4(t)],\end{aligned}$$ which allows us to evaluate the phases acquired by each computational basis state during the time evolution. Applying eq. (\[vartheta\]) to the present time evolved phases, we get the desirable $\pi$-gate $$\begin{aligned} \label{vartheta1} \vartheta &=&Arg(c_1(t))-Arg(c_2(t))\\\nonumber &-&Arg(c_3(t))+Arg(c_4(t))=\pm\pi,\end{aligned}$$ where $Arg(c_i(t))$, $i=1,..,4$, which correspond to phases $\phi_{jl}$, $j,l=0,1$, in eq. (\[vartheta\]), are the phases of coefficients $c_i(t)$ of equation (\[wave\]). We have numerically solved the differential equation system (\[systemTindip1\]-\[systemTindip4\]), with the use of a Mathematica programme. The numerical quantities used for the numerical calculations are $r=1.14 nm$, $B_{z1}=B_{z2}=10\times 10^{-2}T$, $B_{g_1}=6.08 \times 10^{-5}T$ and $B_{g_2}=-6.08 \times 10^{-5}T$, which give the resonance frequencies $\omega_1=1.7599\times10^{10}Hz$ and $\omega_2=1.7577\times10^{10}Hz$. The time evolution of the phase $\vartheta$ is shown in Fig. \[tind\]. The gate time, which corresponds to the case $\vartheta=-\pi$ is $\tau\simeq 9.1\times 10^{-9}s$. This result has been found for a chosen set of initial conditions $c_i(0)$, $i=1,..,4$. However, we did many trials for different numerical values of the set $c_i(0)$, $i=1,..,4$. In all these cases, phase $\theta$ shows a linear behavior and the resulting gate times are all in the same range, which is of the order of $10^{-8} s$. If the set of initial conditions is real, the starting value of phase $\theta$ is always equal to zero. If the set of initial conditions is complex, the starting value of $\theta$ is in the range $[-\pi,+\pi]$, but it can always be rescaled to zero. The numerical value of the distance between the two buckyballs, $r$, is a fixed value, which depends on the substrate where the buckyballs reside. The amplitude of the static magnetic field has been found by considering the allowed experimental limits for its realization. The chosen value for this amplitude has been found by checking the response of the system, i.e. the gate time, after some trials. Therefore, we can say that the phase gate time depends on the distance between the two buckyballs and on the amplitude of the static magnetic field, but it is independent of the choice of the initial values $c_i(0)$, $i=1,..4$.\ If we compare the gate-time, $\tau$, to the shortest decoherence time, $T_2\simeq20\mu m$, we can deduce that it will be theoretically possible to realize about thousands gate operations before the system relaxes. To know the fidelity of the gate and the reliability of the results, we need to evaluate the concurrence during the time evolution. With the use of a Mathematica programme we have plotted the time evolution of the concurrence, equation (\[concnorm\]), from $t=0s$ to the gate time $t=\tau$, see Fig. \[conctind\]. Analyzing picture (\[conctind\]), we can see that the concurrence shows a smooth behavior. It monotonically ranges from zero and its maximum is reached at time $t=\tau$, with the respective value $C(\psi(\tau))=0.88$. Even if the maximum concurrence does not coincides with the ideal value $1$, it is near to this value and the system shows an acceptable degree of entanglement. It is convenient to investigate other system configurations, in order to check if it is possible to improve the concurrence. In the next Section we will analyze the case of an additional magnetic field, oscillating in time in the x-y plane. Phase gate: Time Dependent Case. {#dip} ================================ In this Section, we apply to our system an additional time dependent magnetic field. To induce the transitions between the two Zeeman energy levels, we need to apply an oscillating magnetic field in the $x-y$ plane with angular frequency, $\omega$, equal to the spin resonance frequency. In the case of a transverse linear oscillating magnetic field, the total applied magnetic field is given by $$\vec B(t)=(B_l\cos \omega t,B_l\cos \omega t,(B_z+B_g)).$$ The Hamiltonian of the system reads $$\begin{array}{lll} H&=g(r)(\sigma_{z_1}\sigma_{z_2}+\sigma_{y_1}\sigma_{y_2}-2\sigma_{x_1}\sigma_{x_2})\\ &-\mu_B (B_{z_1}+B_{g_1})\sigma_{z_1}\otimes I_2\\ &- \mu_B( B_{z_2}+B_{g_2})I_1\otimes \sigma_{z_2}\\ &-\mu_B B_{l_1}(\sigma_{x_1}\cos \omega_1 t+\sigma_{y_1}\cos \omega_1t)\otimes I_2\nonumber\\ &+I_1\otimes(-\mu_B B_{l_2}(\sigma_{x_2}\cos \omega_2 t+\sigma_{y_2}\cos \omega_2t)). \end{array}$$ Like in the time independent case, solving the Schr[ö]{}dinger equation, we get a four differential equation system, whose solution give the time evolution of the phase for each computational basis state. Arranging the phases as prescribed in equation (\[vartheta\]), we have obtained the $\pi$-gate. In the numerical computation we have used the additional quantity $B_{l_1}=B_{l_2}=5\times 10^{-4}T$. shown in Fig. \[tdip\], and the numerical value of the gate time is $\tau\simeq 9.8\times 10^{-9}s$. Also in this case, comparing the gate time, $\tau$, to the decoherence time $T_2$, we observe that it will be possible to perform about thousands gate operations before the system relaxes. The relevant result in the treatment of the time dependent case is the concurrence. In Fig. \[conctdip\], it is represented the time evolution of the concurrence, $C(\psi(t))$, which has been numerically evaluated with a Mathematica programme. It shows a monotonic behavior and its maximum, evaluated at time $t=\tau$, corresponds to $C(\psi(\tau))=0.96$. Therefore, an additional linearly polarized oscillating field in the $x-y$ plane allows the system to be characterized by a better concurrence degree. Conclusions =========== To model quantum gates we considered a system composed by two endohedral fullerene molecules, subjected to external magnetic fields. We assume that each molecule may be treated as a $\frac{1}{2}$-spin particle, where the spin is associated to the encapsulated atoms. In the magnetic field the spin degeneracy of the spin up and down components is lifted and it arises the Zeeman splitting. As the result, there two two-level system are arising. Each of these two-level systems corresponds to a single qubit. If the applied static magnetic field to the whole sample is homogeneous, each of these qubits will be characterized by the same resonance frequency. This leads to the difficulty in the individual addressing of each single qubit. To overcome this problem, we have to apply inhomogeneous magnetic fields. in this paper we have used a magnetic field generated by two metallic wires. Each wire is carrying a current, therefore the magnetic field is decreasing with the distance from a wire. In the proposed configuration of two parallel wires, there arises a gradient of the magnetic field when we are moving from a wire to the other one. If we place two buckyballs in the space between these two wires, they will be subjected to the gradient of this field, and therefore the associated resonance frequencies of the related two-level system are different. In this paper we have performed a quantum $\pi$-phase gate. To realize this particular quantum gate we have estimated the phase of each computational basis state, see equation (\[vartheta\]). The leading mutual interaction between the two qubits is the spin dipole-dipole interaction. First we studied the time evolution of our system taking into account this mutual interaction between the qubits and considering the qubits subjected to static magnetic fields only. Then we applied to the system also time dependent magnetic fields. The wave function of the system is given by the superposition of the four computational basis states, see equation (\[wave\]). The time evolution of the coefficients of each computational basis state is determined via the solution of the Schr[ö]{}dinger equation. With the use of these coefficients and of equation (\[vartheta1\]), we can evaluate the operational gate time for the $\pi$-phase gate. Its numerical value is $\tau\simeq9.1\times 10^{-9}s$ for the time independent case, and $\tau\simeq9.8\times 10^{-9}s$ for the time dependent one. Comparing the gate time, $\tau$, to the shortest relaxation time, $T_2$, we have observed that in both cases it will be possible to perform about thousands quantum gate operations before the system decoheres. This is our main result. As far as we are aware, this result indicates that our system could be the most favorable for the realization of a quantum gate. Obviously, for realistic models of quantum computers, the ratio of the decoherence time and the operational time must be very large, otherwise the system relaxes before the completing of the quantum computation. The goal of any quantum computational proposal is the entanglement of the state of the system under consideration. At this purpose, we have studied the concurrence, see Sec. \[concurr\]. The concurrence gives information about the entanglement of the state, therefore it is related to the reliability of the gate operation. A maximally entangled state is left unchanged under a spin-flip operation and its concurrence is maximum. In our system, at the end of the gate operation, the value of the concurrence is $C\simeq0.88$ in the time independent case, and $C\simeq0.96$ in the time dependent one. Both values are acceptable because they are both related to a very good degree of entanglement for the state describing our system. 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--- abstract: 'We are interested in finite groups acting orientation-preservingly on $3$–manifolds (arbitrary actions, ie not necessarily free actions). In particular we consider finite groups which contain an involution with nonempty connected fixed point set. This condition is satisfied by the isometry group of any hyperbolic cyclic branched covering of a strongly invertible knot as well as by the isometry group of any hyperbolic $2$–fold branched covering of a knot in $S^3$. In the paper we give a characterization of nonsolvable groups of this type. Then we consider some possible applications to the study of cyclic branched coverings of knots and of hyperelliptic diffeomorphisms of $3$–manifolds. In particular we analyze the basic case of two distinct knots with the same cyclic branched covering.' address: | Università degli Studi di Trieste\ Dipartimento di Matematica e Informatica\ 34100 Trieste\ Italy author: - Mattia Mecchia bibliography: - 'link.bib' title: 'Finite groups acting on $3$–manifolds and cyclic branched coverings of knots' --- We are interested in finite groups acting orientation-preservingly on 3-manifolds (arbitrary actions, ie not necessarily free actions). In particular we consider finite groups which contain an involution with nonempty connected fixed point set. This condition is satisfied by the isometry group of any hyperbolic cyclic branched covering of a strongly invertible knot as well as by the isometry group of any hyperbolic 2-fold branched covering of a knot in the 3-sphere. In the paper we give a characterization of nonsolvable groups of this type. Then we consider some possible applications to the study of cyclic branched coverings of knots and of hyperelliptic diffeomorphisms of 3-manifolds. In particular we analyze the basic case of two distinct knots with the same cyclic branched covering. We are interested in finite groups acting orientation-preservingly on 3&ndash;manifolds (arbitrary actions, ie not necessarily free actions). In particular we consider finite groups which contain an involution with nonempty connected fixed point set. This condition is satisfied by the isometry group of any hyperbolic cyclic branched covering of a strongly invertible knot as well as by the isometry group of any hyperbolic 2&ndash;fold branched covering of a knot in S&lt;sup&gt;3&lt;/sup&gt;. In the paper we give a characterization of nonsolvable groups of this type. Then we consider some possible applications to the study of cyclic branched coverings of knots and of hyperelliptic diffeomorphisms of 3&ndash;manifolds. In particular we analyze the basic case of two distinct knots with the same cyclic branched covering. Introduction ============ The following problem has been diffusely studied in the literature: which finite groups admit an action on a homology $3$–sphere. The choice of the coefficients of the homology changes completely the situation. If a finite group $G$ acts freely on an integer homology $3$–sphere (and in particular on the standard $3$–sphere $S^3$), the group $G$ has periodic cohomology of period four. Milnor [@Mn] gave a list of groups which are candidates for free actions on integer homology $3$–spheres. This list consists of the finite subgroups of ${\rm SO}(4)$ and the Milnor groups $Q(8n,k,l)$. The recent results of Perelman imply that no group of type $Q(8n,k,l)$ acts on $S^3$ [@P1; @P2]. On the contrary some Milnor groups admit an action on an integer homology $3$–sphere [@Mg]. If we admit arbitrary actions, the list of candidates is again comparable with the list of finite subgroups of ${\rm SO}(4)$. For example Reni and Zimmermann (see Zimmermann [@Z] and Mecchia and Zimmermann [@MZ1]) characterized the nonsolvable groups acting on integer homology $3$–spheres; the unique simple group that admits an action on an integer homology $3$–sphere is $\A_5$ (and it cannot act freely). For the standard $3$–sphere, Thurston’s orbifold geometrization theorem [@BLPo] implies that the finite groups with nonfree actions are exactly the subgroups of ${\rm SO}(4)$. On the other hand, Cooper and Long [@CL] proved that every finite group admits an action on a rational homology $3$–sphere (and even a free action). The class of $\Bbb{Z}_2$–homology $3$–spheres is intermediate between these two cases. This class is interesting also because $\Bbb{Z}_2$–homology $3$–spheres appear more frequently than integer homology $3$–spheres; for example $2$–fold branched coverings of knots in $S^3$ are $\Bbb{Z}_2$–homology $3$–spheres. Dotzel and Hamrick [@DH] proved that every finite $2$–group acting on a $\Bbb{Z}_2$–homology $3$–sphere acts orthogonally on $S^3$. This property is not true in general for solvable groups (already for integer homology $3$–spheres). In [@MZ1] a list of nonsolvable groups which are candidates for actions on $\Bbb{Z}_2$–homology $3$–spheres was given; in this case the only simple groups, that occur, are the projective special linear groups ${\rm PSL}(2,q)$. In the present paper we consider finite groups acting orientation-preservingly on $3$–manifolds which contain an involution with nonempty connected fixed point set. We recall that any involution acting on a $\Bbb{Z}_2$–homology $3$–sphere has connected fixed point set (maybe empty), so there are some relations with our situation. For example the $2$–fold branched coverings of knots satisfy both assumptions but in general the two conditions give different classes of $3$–manifolds. In fact not all $\Bbb{Z}_2$–homology $3$–spheres admit the action of an involution with nonempty fixed point set. For example if $K$ is a hyperbolic knot in $S^3$ without symmetries, for coefficients sufficiently large, Dehn surgery along the knot gives a hyperbolic manifold with trivial isometry group (by Thurston’s hyperbolic surgery theorem [@T]); moreover for $p$ odd a $p/q$–surgery gives a $\Bbb{Z}_2$–homology $3$–sphere. On the other hand all the $3$–manifolds that are the $n$–fold cyclic branched covering of a strongly invertible knot admit the action of an involution with nonempty and connected fixed point set; it is easy to find examples of $n$–fold cyclic branched coverings of strongly invertible knots that have nontrivial first $\Bbb{Z}_2$–homology group (some computation of first homology group can be found in [@Go]). The possibility to study the $n$–fold cyclic branched coverings of strongly invertible knots is one of the motivations of this paper. Another example of a $3$–manifold admitting an involution with nonempty connected fixed point set can be obtained by a $3$–component link $L$ admitting a symmetry $t$ with nonempty fixed point set which acts as a reflection on one component while exchanging the remaining two (eg the Borromean rings); the $2$–fold branched covering $M$ of $L$ has nontrivial first $\Bbb{Z}_2$–homology group (see Sakuma [@Sa Sublemma 15.4]) and the lift of $t$ is an involution with the desired property. When we consider finite groups acting on $3$–manifolds, the two different assumptions imply different analyses. In fact for $\Bbb{Z}_2$–homology $3$–spheres we have some global information about $2$–groups which admit an action. In our case we can control directly only the centralizer of the involution with nonempty connected fixed point set, thus it is more difficult to pass to a global description of the group, even in the case of $2$–groups. A first step in this direction was obtained by Reni and Zimmermann. \[thm0\][[@RZ1]]{}Let $G$ be a finite group of orientation-preserving diffeomorphisms of a closed orientable $3$–manifold; if $G$ contains an involution with nonempty connected fixed point set, then $G$ has sectional $2$–rank at most four (ie every $2$–subgroup is generated by at most four elements). In this paper we try to analyze the whole group. We describe the structure of the group “up to solvable sections”. The interest for nonsolvable groups is also motivated by geometry. For example, if two knots have the same hyperbolic cyclic branched covering $M$ and the isometry group of $M$ is solvable, then it is possible to describe the relation between the two knots [@RZ1]. The problem is not completely solved if the isometry group is not solvable. We summarize part of the description in the following theorem; we recall that a group $E$ is *semisimple* if it is perfect and the factor group of $E$ by its center is a direct product of nonabelian simple groups (see Suzuki [@S2 Chapter 6.6] or Gorenstein, Lyons and Solomon [@GLS1 p16]). \[thm1\] Let $G$ be a finite group of orientation-preserving diffeomorphisms of a closed orientable $3$–manifold; we denote by $\O(G)$ the maximal normal subgroup of odd order and by $E$ the maximal semisimple normal subgroup of $G/\O(G)$. Suppose that $G$ contains an involution with nonempty connected fixed point set. 1. If the semisimple group $E$ is not trivial, it has at most two components and the factor group of $G/\O(G)$ by $E$ is solvable. Moreover the factor group of $E$ by its center is either a simple group of sectional $2$–rank at most four or the direct product of two simple groups with sectional $2$–rank at most two. 2. If $E$ is trivial, there exists a normal subgroup $N$ of $G$ such that $N$ is solvable and $G/N$ is isomorphic to a subgroup of ${\rm GL}(4,2)$, the general linear group of $4\times 4$ matrices over the finite field with 2 elements. The simple groups of sectional $2$–rank at most four are classified by the Gorenstein–Harada Theorem [@G p6], an important part of the classification of finite simple groups. A well-known part of the classification, which was proved before then the Gorenstein–Harada Theorem, is the classification of finite simple groups of $2$–rank at most two (ie every elementary $2$–subgroup is generated by at most two elements) [@G p6]; obviously sectional $2$–rank at most two implies $2$–rank at most two. More details are given in where is proved. If $E$ is not trivial, the structure of the solvable group $(G/\O(G))/E$ is well understood. Also in the second case, if we suppose that the group $G$ is not solvable, a short list of candidates for the group $G/N$ can be produced (the nonsolvable subgroups of ${\rm GL}(4,2)\cong \A_8$ can be easily deduced from [@A]). In the study of cyclic branched coverings of knots, we are mainly interested in the case when the projection of the involution with nonempty connected fixed point set is contained in $E$, the maximal semisimple normal subgroup. Under this condition the list of candidates is much shorter. \[thm2\] Let $G$ be a finite group of orientation-preserving diffeomorphisms of a closed orientable $3$–manifold; we denote by $\O(G)$ the maximal normal subgroup of odd order and by $E$ the maximal semisimple normal subgroup of $G/\O(G)$. Suppose that $G$ contains an involution $h$ with nonempty and connected fixed point set such that the coset $h\O(G)$ is contained in $E$; then $G/\O(G)$ has a normal subgroup $D$ isomorphic to one of the following groups: $${\rm PSL}(2,q), \quad{\rm PSL}(2,q)\times \Z_2 \quad \hbox{or} \quad {\rm SL}(2,q) \times_{\Z_2} {\rm SL}(2,q')$$ where $q$ and $q'$ are odd prime powers greater than four. The factor group $(G/\O(G))/D$ contains, with index at most two, an abelian subgroup of rank at most four. The group ${\rm SL}(2,q)$ is the special linear group of $2\times 2$ matrices of determinant one over the finite Galois field with $q$ elements. The group ${\rm SL}(2,q)$ is a perfect group which has a unique involution; this involution generates its center $Z$, and the factor group ${\rm SL}(2,q)/Z$ is the projective special linear group ${\rm PSL}(2,q)$ (which is a simple group for $q\ge 4$). The group ${\rm SL}(2,q) \times_{\Z_2} {\rm SL}(2,q')$ is a central product where the involutions in the centers of ${\rm SL}(2,q)$ and $ {\rm SL}(2,q')$ are identified. is not simply a specialization of . We have to do some new work to prove properties of $E$, using directly the fact that $E$ contains the projection of $h$; we need also more precise information about finite simple groups in the Gorenstein–Harada list. Probably it is possible to exclude some groups with sectional $2$–rank at most four also in the general case considered in . A possible approach is to suppose that the special involution is not in $E$ and consider $\Z_2$–extensions of the simple groups in the Gorenstein–Harada list; some $\Z_2$–extensions may have again sectional $2$–rank at most four. At the moment we are not sure if this approach case by case, that might be rather technical and long, can produce a relevant reduction of the list of the possible groups. As a corollary of we can consider the case of semisimple groups (see Reni and Zimmermann [@RZ1] for the case of simple groups). We focus now on some applications. We describes first some results concerning actions of finite groups on homology $3$–spheres. Let $f$ be a nontrivial orientation-preserving periodic diffeomorphism of a $3$–manifold $M$. We say that $f$ is *hyperelliptic* if the quotient orbifold $M/f$ has underlying topological space homeomorphic to $S^3$. Using the structure of the finite $2$–subgroups acting on $\Bbb{Z}_2$–homology $3$–spheres, Reni [@R] proved that, up to conjugacy, there are at most nine hyperelliptic involutions acting on a hyperbolic $\Bbb{Z}_2$–homology $3$–sphere; we recall that a hyperelliptic involution on a $\Bbb{Z}_2$–homology $3$–sphere has nonempty connected fixed point set. This is equivalent to say that there exist at most nine inequivalent $\pi$–hyperbolic knots with the same $2$–fold branched covering. Boileau, Paoluzzi and Zimmermann [@BPaZ] proved that, up to conjugacy, at most four cyclic groups generated by a hyperelliptic diffeomorphism of odd prime order can act on an irreducible integer homology $3$–sphere. Thus an irreducible integer homology $3$–sphere can be the cyclic branched covering with odd prime order of at most four inequivalent knots. Also in this case a hyperelliptic diffeomorphism of prime order has nonempty connected fixed point set. The characterization of the finite nonsolvable groups which act on integer homology $3$–spheres plays an important role in the proof in the hyperbolic case. We remark that one of the basic steps in the proof of the upper bound is the fact that hyperelliptic diffeomorphisms often commute and nonabelian situations are, in some sense, exceptions that can be described. The commutativity of hyperelliptic diffeomorphisms corresponds in the language of knots to the standard abelian construction. Suppose $M$ is the $n$–fold and $m$–fold cyclic branched covering of two knots $K$ and $K^{\prime}$, respectively. We denote by $H$ and $H'$ the cyclic transformation groups of $K$ and $K^{\prime }$, respectively; the preimage $\tilde K$ (resp. $\tilde K^{\prime} $) of $K$ (resp. $K^{\prime}$) in $M$ is the fixed point set of $H$ (resp. $H'$). The groups $H$ and $\smash{H'}$ commute and they generate a group $A$ of diffeomorphisms of $M$ isomorphic to $\Bbb{Z}_n\times \Bbb{Z}_m$; when $n=m$ the group $A$ has rank two and it is isomorphic to $\Bbb{Z}_n\times \Bbb{Z}_n$. Each element of the transformation group $H$ (resp. $H'$) induces a rotation on $\smash{\tilde K ^{\prime}}$ (resp. $\smash{\tilde K}$), and the quotient orbifold $M/A$ is the $3$–sphere whose singular set is a link $L=\bar K \cup \bar K'$, where $\bar K$ (resp. $\bar K'$) is the projection of $K$ (resp. $K'$). We remark that by the positive solution to the Smith Conjecture both components of $L$ are trivial knots. On the other hand, starting from $L$, we can obtain $K$ (resp. $\smash{K'}$) taking the preimage of $\bar K$ (resp. $\bar K'$) in the $m$–fold (resp. $n$–fold) cyclic branched covering of $\bar K'$ (resp. $\bar K$). This construction serves to study the relation between two links with the same hyperbolic cyclic branched covering (see Reni and Zimmermann [@RZ2] and Mecchia [@M]). The standard abelian construction is the unique possibility in many different situations. [[@RZ2]]{}Let $M$ be a hyperbolic $3$–manifold. Suppose that $M$ is the $n$–fold and $m$–fold cyclic branched covering of inequivalent knots $K$ and $K'$, respectively, such that $m$ and $n$ are not powers of two. Suppose that one of the following conditions holds: 1. $n$ and $m$ have a common prime divisor different from two; 2. $K$ is not strongly invertible and $K$ is not self-symmetric with order $n$; 3. The orientation-preserving isometry group of $M$ is solvable. Then $K$ and $K'$ arise from the standard abelian construction. A $2$–component link is called *symmetric* if there exists an orientation-preserving diffeomorphism of $S^3$ which exchanges the $2$–components of the link. A *cyclic symmetry* of a knot $K$ is a diffeomorphism of $(S^3,K)$ of finite order and with nonempty fixed point set $F$ disjoint from $K$. The set $F$ is an unknotted circle by the positive solution to the Smith Conjecture. The quotient of $S^3$ by a cyclic symmetry is again the $3$–sphere and $F$ and $K$ project to a $2$–component link. We call a knot $K$ *self-symmetric* with order $n$ if $K$ admits a cyclic symmetry $f$ of order $n$ such that the associated quotient link is symmetric. We use to generalize point 2 of the previous Theorem. We want to include also the class of strongly invertible knots that is largely studied in knot theory. Unfortunately the standard abelian construction does not remain the unique possibility. \[thm3\] Let $M$ be a hyperbolic $3$–manifold. Suppose that $M$ is the $n$–fold and $m$–fold cyclic branched covering of two hyperbolic knots $K$ and $K'$, respectively, such that $m$ and $n$ are not powers of two. Let $G$ be the orientation-preserving isometry group of $M$ and $\O(G)$ the maximal normal subgroup of odd order. If the knot $K$ is not self-symmetric with order $n$, then one of the following cases occurs: 1. $K$ and $K'$ arise from the standard abelian construction; 2. $G$ contains $h$, an involution with nonempty connected fixed point set, such that $h\O(G)$ is contained in the maximal normal semisimple subgroup of $G/\O(G)$ (in particular applies to $G$); 3. All prime divisors of $n$ and $m$ are contained in $\{2,3,5,7\}$ and there exists a normal subgroup $N$ of $G$ such that $N$ is solvable and $G/N$ is isomorphic to a subgroup of ${\rm GL}(4,2)$. The knots $K$ and $K'$ in are inequivalent. It follows from volume considerations if $n\neq m$, and from the fact that $K$ is not self-symmetric if $n=m$. As in the case of integer homology $3$–spheres, the noncommuting situations are, in some sense, exceptional. For integer homology $3$–spheres there exists an universal bound to the number of cyclic groups generated by a hyperelliptic diffeomorphism (with connected nonempty fixed point set) of odd prime order; we propose the following: There exists a universal bound $C$ such that any hyperbolic orientable closed $3$–manifold admits at most $C$ nonconjugate cyclic groups generated by a hyperelliptic diffeomorphism with connected nonempty fixed point set. We remark that the condition about the fixed point set is necessary; in general, there is no universal bound for hyperelliptic diffeomorphisms in hyperbolic $3$–manifolds [@RZ3]. We recall that Cooper and Long [@CL] proved that every finite group admits an action on a hyperbolic rational homology $3$–sphere; to prove the conjecture the use of homology may be insufficient. Probably we have to consider directly conditions about the fixed point sets of the diffeomorphisms, for example the existence of involutions with connected nonempty fixed point set (the hypothesis considered in this paper). Preliminary results {#Section 2} =================== In this section we present some preliminary results about finite groups acting on $3$–manifolds. \[prop1\] Let $G$ be a finite group of orientation-preserving diffeomorphisms of a closed orientable $3$–manifold and $f$ an element in $G$ with nonempty connected fixed point set $K$. Then the normalizer $N_G (f)$ of the subgroup generated by $f$ in $G$ is isomorphic to a subgroup of a semidirect product $${\Bbb Z}_2 \ltimes ({\Bbb Z}_a\times {\Bbb Z}_b),$$ for some nonnegative integers $a$ and $b$, where a generator of ${\Bbb Z}_2$ (an $f$–reflection, ie acting as a reflection on $K$ ) acts on the normal subgroup ${\Bbb Z}_a \times {\Bbb Z}_b$ of $f$–rotations (ie the elements acting as rotations on $K$) by sending each element to its inverse. In particular, $N_G(f)$ is solvable. See Mecchia and Zimmermann [@MZ2 Lemma 1]. \[prop2\] Let $G$ be a finite group of orientation-preserving diffeomorphisms of a closed orientable $3$–manifold. If $G$ is isomorphic to ${\Bbb Z}_2 \times {\Bbb Z}_2 \times {\Bbb Z}_2$, there exists in $G$ an involution that either acts freely or has nonconnected fixed point set. By contradiction we suppose that the seven involutions in $G$ have connected and nonempty fixed point set. Let $f$ be one of the involutions, by the group $G$ contains four $f$–rotations and four $f$–reflections. We denote by $r$ one $f$–rotation of order two different from $f$ and we denote by $t$ one $f$–reflection, so the four $f$ rotations are Id, $f,\,r$ and $rf$ and the four $f$ reflections are $t,\,tf,\,tr$ and $tfr$. Since the fixed point sets of $f$ and $t$ have nonempty intersection then the fixed point sets of $t$ and $tf$ have nonempty intersection, then $t$ is a $tf$–reflection. Now we consider $r$ and $rf$, both of them have nonempty connected fixed point set and both the subgroups of $r$ and $rf$–rotations coincide with the subgroup of $f$–rotations. We deduce that $t$ is an $r$–reflection and an $rf$–reflection and consequently $t$ is an $rt$–reflection and an $rft$–reflection. It turns out that $t$ acts as a reflection on the fixed point set of each involution in $G$ different from $t$ and viceversa each involution in $G$ different from $t$ acts as a reflection on the fixed point set of $t$; so in $G$ we have six $t$–reflections and this is impossible by . We conclude the section with a purely algebraic proposition that describes the centralizer of an involution in the factor groups by odd order normal subgroups. This proposition shows that, for an involution $t$, the quotient of the centralizer is the centralizer of the projection of $t$ in the quotient; its proof is elementary but we often use this fact. \[prop3\] Let $t$ be an involution in a finite group $G$ and let $N$ be a normal subgroup of $G$ of odd order. Then the centralizer $C_{G/N}(tN)$ of the coset $tN$ in $G/N$ is isomorphic to $C_{G}(t)/(C_{G}(t)\cap N),$ that is the factor group of the centralizer of $t$ in $G$ by the intersection $C_{G}(t)\cap N$. Indeed we prove the equality $\{cN|c\in C_{G}(t)\}=C_{G/N}(tN)$ and then the thesis follows from the Second Isomorphism Theorem. The inclusion $\{cN|c\in C_{G}(t)\}\subseteq C_{G/N}(tN)$ is trivial. We suppose that $fN$ is contained in $C_{G/N}(tN)$ that is $ftf^{-1}N=tN$, so there exists $k\in N$ such that $ftf^{-1}=tk$. The subgroup $\langle t,N \rangle $ of $G$ generated by $t$ and $N$ has a Sylow $2$–subgroup of order two, so all the involutions in $\langle t,N\rangle$ are conjugate, in particular there exists an element $g\in N$ such that $gtg^{-1}=tk$. It follows that $g^{-1}f$ is contained in $C_{G}(t)$ and, since $g\in N$, we have that $f$ is contained in $fN=Nf=N(g^{-1}f)=(g^{-1}f)N$. The coset $fN$ is contained in $\{cN|c\in C_{G}(t)\}$ and the inclusion $\{cN|c\in C_{G}(t)\}\supseteq C_{G/N}(tN)$ is proved. Proof of {#Section 3} ========= We denote by $\bar G$ the factor group $G/\O(G)$ and by $\tilde E$ the factor group of $E$ by its center $Z(E)$. The maximal semisimple normal subgroup $E$ has sectional $2$–rank at most four and it has at most two components. If $E$ has two components, $\tilde E$ is the direct product of two simple groups with sectional $2$–rank two. By , $E$ has sectional $2$–rank at most four and consequently $\tilde E$ has sectional $2$–rank at most four. We recall that a minimal set of generators of a group means a set of generators such that any proper subset does not generate the group. In general we can have minimal sets of generators with different numbers of elements for the same finite group but, by Burnside’s basis theorem [@S1 Theorem 1.16, p92], any two minimal sets of generators of a $p$–group contain the same number of elements. Moreover, in the direct product of two groups, the union of a minimal set of generators of the first group with a minimal set of generators of the second group is a minimal set of generators of the direct product. It follows that the sectional $2$–rank of the direct product of two groups is equal or greater then the sum of the sectional $2$–ranks of the two direct factors. Since simple groups have sectional $2$–rank at least two [@S2 p.144] we get the thesis. We denote by $C$ the centralizer $C_{\bar G}(E)$ of $E$ in $\bar G$. If $E$ is not trivial, then $C$ is solvable. Since $C$ is the centralizer of a normal subgroup, $C$ is normal in $G$. The intersection of $C$ and $E$ is $Z(E)$, the center of $E$. The center of $E$ has order a power of two, otherwise $\O(G)$ is not maximal. We denote by $D$ the group generated by $C$ and $E$; the group $D$ is a central product of $E$ and $C$. By , the sectional $2$–rank of $D$ is equal or smaller then four; it follows that $D/Z(E)$ has sectional $2$–rank equal or smaller then four. The factor group $D/Z(E)$ is isomorphic to $E/Z(E)\times C/Z(E)$; the sectional $2$–rank of $E/Z(E)$ is at least two, so the sectional $2$–rank of $C/Z(E)$ is at most two (see Step 1). The maximal semisimple normal subgroup of $C$ is trivial, otherwise $E$ is not maximal. We consider $F(C)$ the generalized Fitting subgroup of $C$. We recall that the generalized Fitting subgroup is the subgroup generated by the maximal semisimple normal subgroup and by the Fitting subgroup; the Fitting subgroup is the maximal nilpotent normal subgroup [@S2 p452]. In this case, since the maximal semisimple normal subgroup of $C$ is trivial, $F(C)$ coincides with the Fitting subgroup. Note that, since $F(C)$ is nilpotent, its Hall subgroup of maximal odd order is unique. Since $F(C)$ is characteristic in $C$, the generalized Fitting subgroup $F(C)$ is a $2$–group, otherwise $\O(G)$ is not maximal. The group $C$ acts on $F(C)$ by conjugation. The centralizer $C_C(F(C))$ of $F(C)$ in $C$ is contained in $F(C)$ [@S2 Theorem 6.11, p452] and in particular it is a $2$–group; the factor group $C/C_C(F(C))$ is a subgroup of the automorphism group of $F(C)$. Let $\Phi$ be the Frattini subgroup of $F(C)$; the factor group $F(C)/\Phi$ is an elementary abelian group. The totality of automorphisms that leave every element of $F(C)/\Phi$ invariant is a normal $2$–subgroup of ${\rm Aut}(F(C))$ . Let $T$ be the subgroup of $C$ of elements that act trivially on $F(C)/\Phi$; then $T$ is a normal $2$–subgroup and $C/T$ is a subgroup of ${\rm GL}(d,2)$, where $d$ is the rank of $F(C)/\Phi$. Since $F$ has sectional $2$–rank at most four we have $d\leq 4$; if $d\leq 2$ the group ${\rm GL}(d,2)$ is solvable and the proof is finished. Suppose that $d=3$. The group ${\rm GL}(3,2)$ has order $2^3\cdot 3 \cdot 7$; any automorphism of order seven permutes cyclically all the involutions in ${\Bbb Z}_2\times {\Bbb Z}_2\times{\Bbb Z}_2 $. In this case $\Phi$ cannot contain $Z(E)$ because $F(C)/Z(E)$ must have sectional $2$–rank at most two. At least one involution in $F(C)/\Phi$ is the projection of an element in $Z(E)$ and it is contained in the center of $C$; this involution is fixed by conjugation by each element of $C$ and $C/T$ cannot contain any element of order seven; $C/T$ has order at most 24 and it is solvable. Suppose finally that $d=4$. The group ${\rm GL}(4,2)$ has order $2^6\cdot 3^2 \cdot 5\cdot 7$; an automorphism of order five does not centralize any involution of ${\Bbb Z}_2\times {\Bbb Z}_2\times{\Bbb Z}_2 \times{\Bbb Z}_2 $ (we have three orbits with five elements) and an automorphism of order seven centralizes exactly one involution (two orbits with seven elements and one orbit with only one element). We consider the group $Z(E)\cdot \Phi$ generated by $Z(E)$ and $\Phi$. The group $F(C)/(Z(E)\cdot \Phi)$ must have rank at most two. It follows that at least three involutions in $F(C)/\Phi$ are projections of elements in the center of $E$. The group $C/T$ cannot contain elements of order five or seven and hence the order of $C/T$ is product of powers of 2 and 3. By Burnside’s Theorem [@S2 Theorem 4.25, p216], any such group is solvable. This finishes the proof of Step 2. If $E$ is not trivial, $\bar G/E$ is solvable. The normal subgroup $D$ is the subgroup generated by $E$ and $C=C_{\bar G}(E)$; we consider the factor group $\bar G/D$ that is isomorphic to a subgroup of Out$(E)$, the outer automorphism group of $E$. If an automorphism of $E$ acts trivially on $\tilde E=E/Z(E)$, it acts trivially on $E$; this is a consequence of the three subgroups lemma [@S2 (6.3), p447], [@GLS2 Lemma 3.8, p7] and of the fact that $E$ is perfect. It follows that the group Out($E$) is a subgroup of Out($\tilde E$). We recall that the outer automorphism group of a simple group is solvable (for a discussion about this property, called the Schreier property, see [@GLS2 p4]). The group $\tilde E$ is either a simple group with sectional $2$–rank at most four or the direct product of two simple groups with sectional $2$–rank at most two; in this last case Out($\tilde E$) contains, with index at most two, the direct product of the outer automorphism groups of the two components [@GLS2 Lemma 3.23, p13]. In any case Out($\tilde E$) is solvable; it follows that $\bar G/D$ and hence $\bar G/E$ are solvable. If $E$ is trivial, there exists a normal subgroup $N$ of $G$ such that $N$ is solvable and $G/N$ is isomorphic to a subgroup of ${\rm GL}(4,2)$. We consider $F(\bar G)$ the generalized Fitting subgroup of $\bar G$; since $E$ is trivial, $F(\bar G)$ coincides with the Fitting subgroup. The subgroup $F(\bar G)$ does not contain any element with odd order, otherwise $\O(G)$ is not maximal. The generalized Fitting subgroup contains $C_{\bar G}(F(\bar G))$ its centralizer in $\bar G$ [@S2 Theorem 6.11, p452] and in particular $C_{\bar G}(F(\bar G))$ is a $2$–group; the factor group of $\bar G$ by $C_{\bar G}(F(\bar G))$ is isomorphic to a subgroup of ${\rm Aut}(F(\bar G))$, the automorphism group of $F(\bar G)$. We consider $\Phi$, the Frattini subgroup of $F(\bar G)$. As a consequence, the factor group $F(\bar G)/\Phi$ is an elementary group. The totality of automorphisms that leave every element of $F(\bar G)/\Phi$ invariant is a normal $2$–subgroup of ${\rm Aut}(F(\bar G))$ [@S1, p93]. The factor group $\bar G$ contains $\bar N$, a normal $2$–subgroup, such that $\bar G/\bar N$ is isomorphic to a subgroup of ${\rm GL}(d,2)$ where $d$ is the rank of $F(\bar G)/\Phi$. We denote by $N$ the preimage of $\bar N$ with respect to the projection of $G$ onto $\bar G$; we remark that $G/N$ is isomorphic to a subgroup of ${\rm GL}(d,2)$ and $N/\O(G)$ is a $2$–group. If $G$ contains an involution with nonempty connected fixed point set, implies that $G$ has sectional $2$–rank at most four and hence we can set $d=4$. Proof of {#Section 4} ========= To simplify the notation we denote by $\bar G$ the factor group $G/\O(G)$ and we denote by $\bar g$ the coset $gO(G)$ where $g$ is an element of $G$; by hypothesis we have an involution $h$ in $G$ with connected and nonempty fixed point set such that $\bar h$ is contained in $E$, the maximal semisimple normal subgroup of $\bar G$. In the proof we often use the following fact: by and , if we have an involution $t$ in $G$ with nonempty fixed point set, the centralizer $C_{\bar G}(\bar t)$ of $\bar t$ is isomorphic to subgroup of a semidirect product ${\Bbb Z}_2 \ltimes ({\Bbb Z}_a\times {\Bbb Z}_b)$ where a generator of ${\Bbb Z}_2$ acts on the normal subgroup ${\Bbb Z}_a \times {\Bbb Z}_b$ by sending each element to its inverse. In particular $C_{\bar G}(\bar h)$ is isomorphic to $C_{G}(h)/(C_{G}(h)\cap \O(G))$ and we call $\bar h$–rotations (resp. $\bar h$–reflections) the elements of $C_{\bar G}(\bar h)$ that are projections of $h$–rotations (resp. $h$–reflections); since $\O(G)$ cannot contain $h$–reflections, this notation is not ambiguous. In the proof we call a group *admissible* if it has a subgroup of index at most two that is isomorphic to a subgroup of a semidirect product ${\Bbb Z}_2 \ltimes ({\Bbb Z}_a\times {\Bbb Z}_b)$, where a generator of ${\Bbb Z}_2$ acts on the normal subgroup ${\Bbb Z}_a \times {\Bbb Z}_b$ by sending each element to its inverse. We note that an admissible group is solvable and a subgroup or a factor group of an admissible group is again admissible. We remark also that subgroups of $\bar G$, that contain the centralizer $C_{\bar G}(\bar h)$ with index at most two, are admissible. The order of $Z(E)$, the center of $E$, is a power of two, the involution $\bar h$ is not contained in $Z(E)$ and either $Z(E)$ is cyclic or $Z(E)$ is elementary abelian of order four and $C_E (\bar h)$ is elementary abelian of order eight. The order of $Z(E)$ is a power of two, otherwise $\O(G)$ is not maximal. Since $C_E (\bar h)$ is solvable, the center $Z(E)$ does not contain $\bar h$. Since $\bar h \in E$, the center $Z(E)$ is a subgroup of $C_E (\bar h)$, that is isomorphic to a subgroup of the semidirect product ${\Bbb Z}_2 \ltimes ({\Bbb Z}_{2^n}\times {\Bbb Z}_{2^m})$. If $Z(E)$ contains an element with order strictly greater than two, then $Z(E)$ can contain only $\bar h$–rotations and, since $\bar h \notin Z(E)$, the center $Z(E)$ can contain only one involution; this fact implies that $Z(E)$ has to be cyclic. Suppose now that all the nontrivial elements in $Z(E)$ have order two. If $Z(E)$ is not cyclic, it must contain an $\bar h$–reflection which thus must commute with the whole group and we have only one possibility: $Z(E)\cong {\Bbb Z}_{2}\times {\Bbb Z}_{2}$ and $C_E (\bar h)\cong {\Bbb Z}_{2}\times {\Bbb Z}_{2}\times {\Bbb Z}_{2}$. We denote by $\tilde E$ the factor group $E/Z(E)$ and we denote by $\tilde h$ the coset $\bar h Z(E)$. We consider $C_{\smash{\tilde E}} (\tilde h)$ the centralizer of $\tilde h$ in $\tilde E$. 1. If $Z(E)$ is not cyclic, then $C_{\smash{\tilde E}} (\tilde h)$ has order at most eight. 2. If $Z(E)$ is cyclic, then $C_{\smash{\tilde E}} (\tilde h)$ contains with index at most two the factor group $C_E (\bar h)/Z(E)$. In both cases $C_{\smash{\tilde E}} (\tilde h)$ is admissible. We denote by $P$ the subgroup $\{\bar f\in E|\,\exists \, \bar g\in Z(E) \,\,\, {\rm such} \,\,\, {\rm that } \,\,\, \bar f\bar h \bar f^{-1}=\bar h \bar g \}$ that is the preimage of $C_{\smash{\tilde E}} (\tilde h)$ with respect to the standard projection of $E$ onto $\tilde E=E/Z(E)$; we recall that $\smash{C_{\smash{\tilde E}} (\tilde h)=P/Z(E)}$. If $Z(E)$ is the trivial group the thesis trivially holds. We suppose that $Z(E)$ is cyclic and nontrivial; we denote by $\bar z$ the unique involution in $Z(E)$ and we get the following equality: $$P=\{\bar f\in E|\,\,\, {\rm either} \,\,\, \bar f\bar h \bar f^{-1}=\bar h \,\,\, {\rm or} \,\,\, \bar f\bar h \bar f^{-1}=\bar h \bar z \}.$$ In this case we obtain that $C_{\smash{\tilde E}} (\tilde h)$ contains with index at most two the factor group $C_E (\bar h)/Z(E)$. Finally we suppose that $Z(E)$ is not cyclic. By Step 1, we have that $Z(E)\cong {\Bbb Z}_{2}\times {\Bbb Z}_{2}$ and $C_E (\bar h)\cong {\Bbb Z}_{2}\times {\Bbb Z}_{2}\times {\Bbb Z}_{2}$ for the center contains an $\bar h$–reflection. The centralizer $C_E(\bar h)$ is a normal subgroup of $P$; since $C_E(\bar h)$ contains its centralizer in $P$, the factor group $P/C_E(\bar h)$ acts effectively on $C_E(\bar h)$ by conjugation and $P/C_E(\bar h)$ is isomorphic to a subgroup of Aut $({\Bbb Z}_{2}\times {\Bbb Z}_{2}\times {\Bbb Z}_{2})$, the automorphism group of the elementary abelian group of order eight. Moreover $P/C_E(\bar h)$ leaves invariant elementwise $Z(E)\cong {\Bbb Z}_{2}\times {\Bbb Z}_{2}$ that is a subgroup of index two in $C_E(\bar h)$; this fact implies that $P/C_E(\bar h)$ is a subgroup of ${\Bbb Z}_{2}\times {\Bbb Z}_{2}$. So $C_{\smash{\tilde E}} (\tilde h)=P/Z(E).$ has order at most eight. This finishes the proof of Step 2. If $E$ has one component, $\tilde E$ has only one conjugacy class of involutions. In this case $\tilde E$ is a simple group with sectional $2$–rank at most four and we apply the Gorenstein–Harada classification of finite simple groups of sectional $2$–rank at most four (see Gorenstein [@G p6] and Suzuki [@S2 Theorem 8.12, p513]). The group $\tilde E$ contains $\tilde h$ and the centralizer of $\tilde h$ is admissible. We will show that no group in the Gorenstein–Harada list which has more than one conjugacy class of involutions contains an involution with an admissible centralizer. The following list of groups contains all the simple groups with sectional $2$–rank at most four and more than one conjugacy class of involutions (the algebraic properties of the simple groups can be found in the *Atlas of finite groups* [@A], Sakuma [@S2 Chapter 6.5] or Gorenstein [@G]): $$\begin{array}{l}M_{12};\ {\rm PSp}(4,q), \hbox{for } q \hbox{ odd};\ J_2;\ \Bbb A_n, \hbox{for } 8\leq n\leq 11;\\ {\rm PSL} (4,q),\ {\rm PSU} (4,q),\ {\rm PSL} (5,q) \hbox{ and }{\rm PSU} (5,q), \hbox{for } q \hbox{ odd}.\end{array}$$ We can rule out directly the following groups because the centralizer of any involution in these groups is not solvable (for the groups of Lie type see Suzuki [@S2 6.5.2, 6.5.7, 6.5.15]; for $J_2$ see Gorenstein [@G p99] or the *Atlas of finite groups* [@A]): $$\begin{array}{l}J_2;\ {\rm PSp}(4,q);\ {\rm PSL} (4,q) \hbox{ and }{\rm PSU} (4,q), \hbox{ for }q\hbox{ odd, } q\geq 5;\\ {\rm PSL} (5,q)\hbox{ and }{\rm PSU} (5,q),\hbox{ for }q\hbox{ odd.}\end{array}$$ The Mathieu group $M_{12}$ and the alternating groups $\Bbb A_n$, for $8\leq n\leq 11$, contain some involutions with solvable centralizer but the centralizers of such involutions contain $\Bbb S_4$, that is not admissible ([@A] for $M_{12}$). Finally, in the groups ${\rm PSp}(4,3)$, ${\rm PSL} (4,3)$ and ${\rm PSU} (4,3)$, the centralizer of each involution contains a subgroup with a factor group isomorphic to the non admissible group $\A_4 \cong {\rm PSL} (2,3) \cong {\rm PSU} (2,3)$ [@S2 6.5.2, 6.5.7, 6.5.15]. This concludes the proof. We denote by $\tilde S_2$ a Sylow $2$–subgroup of $\tilde E$; if $E$ has one component either $\tilde S_2$ has sectional $2$–rank two or $\tilde S_2$ is an elementary abelian group with eight elements. Since by Step 3 the involutions in $\tilde E$ are all conjugate, we can suppose that $\tilde h$ is central in $\tilde S_2$, this implies that $\smash{\tilde S_2=C_{\smash{\tilde S_2}}(\tilde h)}$. We denote by ${\cal E}$ the preimage of $E$ in $G$ with respect to the projection of $G$ onto $\bar G$. We recall that we described $Z(E)$ in Step 1; we consider three cases according to the structure of $Z(E)$. We remark also that a $2$–group with order at most eight which is not elementary abelian of rank three, has sectional $2$–rank at most two, so when we will obtain that $\tilde S_2$ has order at most eight, we will get the thesis. Suppose first that $Z(E)$ is trivial. In this case $\tilde S_2$ is isomorphic to the Sylow $2$–subgroup of ${\cal E}$. The involutions in ${\cal E}$ are all conjugate. In fact if we consider $t$ and $t'$ two involutions in ${\cal E}$, we know that $\bar t$ and $\bar t'$ are conjugate in $E$, so there exists $g$ in $\O(G)$ such that $t$ is conjugate to $t'g$. Since $\O(G)$ has odd order, the group generated by $t'$ and $\O(G)$ has Sylow $2$–subgroup of order two and all the involutions in the group are conjugate; in particular $t'$ and $t'g$ are conjugate. We can conclude that $t$ and $t'$ are conjugate. All the involutions in ${\cal E}$ are conjugate to $h$, so all the involutions have nonempty connected fixed point set; by , the group ${\cal E}$ cannot contain a subgroup isomorphic to ${\Bbb Z}_2 \times {\Bbb Z}_2 \times {\Bbb Z}_2$. Since, by , the Sylow $2$–subgroup of ${\cal E}$ is a subgroup of the semidirect product ${\Bbb Z}_2 \ltimes ({\Bbb Z}_{2^a}\times {\Bbb Z}_{2^b})$, we obtain that $\tilde S_2$ is dihedral or abelian of rank two. If $Z(E)$ is elementary abelian of order four, by Step 2 we have that $\tilde S_2=C_{\tilde S_2} (\tilde h)$ has order at most eight and we get thesis. Finally we suppose that $Z(E)$ is cyclic and nontrivial. We consider $S_2$ the Sylow $2$–subgroup of $E$, the center $Z(E)$ is contained in $S_2$ and $\tilde S_2$ is the projection of $S_2$. By Step 2 we can assume that $(C_{S_2}(\bar h)/Z(E))$ has index at most two in $\tilde S_2$. If $Z(E)$ contains an $\bar h$–reflection, by the centralizer $C_{E}(\bar h)$ has order at most eight, and we conclude that $(C_{S_2}(\bar h)/Z(E))$ has order at most four and $\tilde S_2$ has order at most eight. So we can suppose that $Z(E)$ contains only $\bar h$–rotations. We denote by $R$ the subgroup of $\bar h$–rotations contained in the Sylow $2$–subgroup of $E$; the subgroup $R$ contains $Z(E)$. We obtain that the factor group $R/Z(E)$ is cyclic. In fact, if $R/Z(E)$ has rank two, we have an $\bar h$–rotation $\bar f$ of order different than two such that $\bar f\notin Z(E)$ and $\bar f^2\in Z(E)$. The coset $\bar f Z(E)$ contains no involution and the coset $\bar hZ(E)$ contains two involutions for $\bar h$ is not in the center; on the other hand since $\bar f Z(E)$ and $\bar hZ(E)$ represent two involutions in $\tilde E$, they are conjugate and this gives a contradiction. This concludes the proof in the case that $(C_{S_2}(\bar h)/Z(E))=\tilde S_2$. On the other hand, if $C_{S_2}(\bar h)$ does not contain any $\bar h$–reflection, $\tilde S_2$ contains a cyclic subgroup of index at most two, so it has sectional $2$–rank at most two and the proof is finished. So we can suppose the following two facts: 1. $C_{S_2}(\bar h)$ contains $\bar t$ an $\bar h$–reflection; 2. $(C_{S_2}(\bar h)/Z(E))$ has index two in $\tilde S_2=C_{\tilde S_2} (\tilde h)$; in this case there exist two nontrivial elements $\bar s$ in $S_2$ and $\bar c$ in $Z(E)$ such that $\smash{\bar s \bar h\bar s^{-1}=\bar h \bar c}$. Since $\bar t Z(E)$ is conjugate to $\bar hZ(E)$ and $\bar t Z(E)$ contains a number of involutions equal to the order of $Z(E)$ (these elements are all reflections), we obtain that $Z(E)$ has order two. Since $R/Z(E)$ is cyclic, we have $R\cong {\Bbb Z}_2 \times {\Bbb Z}_{2^m}$. Moreover we obtain that $m=1$; in fact, if $R$ contains an element of order strictly greater than two, one involution between $\bar h$ and $\bar h \bar c$ is characteristic in $C_{S_2}(\bar h)$ (all the involutions which are obtained as powers of elements of order strictly greater than two coincide) and this is in contradiction with the existence of $\bar s$. We can conclude that $C_{S_2}(\bar h)$ is an elementary group of order eight and $\tilde S_2$ has order eight. If $E$ has one component, $E$ is isomorphic to ${\rm PSL}(2,q)$, with $q\geq 5$. By Step 3 the simple group $\tilde E=E/Z(E)$ has only one conjugacy class of involutions and by Step 4 $\tilde E$ has sectional $2$–rank smaller than two or $\tilde S_2$ is an elementary abelian group of order eight. In the Gorenstein–Harada list we find the following groups that satisfy these properties and that were not already excluded in Step 3: $$\begin{array}{l}{\rm PSL} (2,q),\hbox{ for }q\hbox{ odd and }q\geq 5;\ {\rm PSL} (3,q) \hbox{ and }{\rm PSU} (3,q) \hbox{ for }q\hbox{ odd;}\\ M_{11}; \Bbb A_7;\ J_1;\ {\rm PSL} (2,8);\ ^{2}G_2(3^n), \hbox{ for }n>1. \end{array}$$ We recall that $\tilde E$ has to contain $\tilde h$ an involution with admissible centralizer. In the groups $J_1$ and $^{2}G_2(3^n)$ for $n>1$ the centralizer of an involution is isomorphic to the non admissible group $ {\Bbb Z}_2 \times \Bbb{\rm PSL} (2,q)$ with $q>5$ [@S2 p514]. We can rule out ${\rm PSL} (3,q)$ and ${\rm PSU} (3,q)$ for $q$ odd, $q\geq 5$, because the centralizer of any involution in these groups is not solvable [@S2 6.5.2, 6.5.15]. We consider the groups ${\rm PSL} (3,3)$, ${\rm PSU} (3,3)$ and $M_{11}$. The centralizer of any involution in these groups has a subgroup which has the alternating group $\A_4 \cong {\rm PSL} (2,3) \cong {\rm PSU} (2,3)$ as a factor group and so it is not admissible [@S2 6.5.2, 6.5.15], [@A]. The group $PSL(2,8)$ does not admit central perfect extension [@A]; in this case $Z(E)$ should be trivial. The Sylow $2$–subgroup of $PSL(2,8)$ is elementary abelian of order eight, by the same argument used in Step 4 for the case of $Z(E)$ trivial we can exclude this group. We consider $\Bbb A_7$. If $Z(E)$ is not trivial, the unique central extension of $\Bbb A_7$ with center of order a power of two is $\A^*_7$. The Sylow $2$–subgroup of $\A^*_7$ is a quaternion group of order eight and it contains a unique involution that is central in the group and this is impossible. We can suppose that $Z(E)$ is trivial and $E\cong \Bbb A_7$. We consider the centralizer of the involution $\bar h$ in $ \Bbb A_7$; we can suppose up to conjugation that $\bar h$ is the permutation $(1,2)(3,4)$. The centralizer contains $(5,6,7)$, $(1,3)(2,4)$ and $(1,2)(5,6)$; the involution $(1,3)(2,4)$ commutes with the element of order three $(5,6,7)$, so $(1,3)(2,4)$ is an $\bar h$–rotation. On the other hand $(1,3)(2,4)$ and $(1,2)(5,6)$ do not commute and by , this cannot occur. Finally we consider the groups ${\rm PSL}(2,q)$, with $q\geq 5$. The only central perfect extension of ${\rm PSL}(2,q)$ with nontrivial center of order a power of two is ${\rm SL}(2,q)$, that contains a unique involution that is central in the group and this is not possible. The only remaining possibility is that $Z(E)$ is trivial and $E$ is isomorphic to ${\rm PSL}(2,q)$, with $q\geq 5$. If $E$ has two components, $E$ is isomorphic to $ {\rm SL}(2,q) \times_{\Z_2} {\rm SL}(2,q')$, with $q$ and $q'$ odd prime powers greater than four. We consider $\tilde E=\tilde A \times \tilde B$ where $\tilde A$ and $\tilde B$ are two simple groups. By Step 3 $\tilde A$ and $\tilde B$ have sectional $2$–rank two. The simple groups with this property are: $$\hbox{${\rm PSL} (2,q)$, for $q$ odd and $q\geq 5$; $M_{11}$; $\Bbb A_7$; ${\rm PSL} (3,q)$ and ${\rm PSU} (3,q)$, for $q$ odd.}$$ By Step 2 we recall that we have an involution $\tilde h$ in $\tilde E$ such that its centralizer is admissible. We have that $\tilde h= (\tilde h_A, \tilde h_B)$ where $\tilde h_A\in \tilde A$ and $\tilde h_B\in \tilde B$. The centralizer of $C_{\smash{\tilde E}}(\tilde h)$ is the direct product of $\smash{C_{\tilde A}(\tilde h_A)}$ and $\smash{C_{\tilde B}(\tilde h_B)}$. We remark that $\tilde h_A$ and $\tilde h_B$ cannot be the identity of the group otherwise the centralizer of $\tilde h$ is not solvable, so they are involutions. The two centralizers $C_{\tilde A}(\tilde h_A)$ and $ C_{\tilde B}(\tilde h_B)$ must be admissible groups. This condition excludes as components $M_{11}$, ${\rm PSL} (3,q)$ and ${\rm PSU} (3,q)$, with $q$ odd, because they do not contain any involution with admissible centralizer (see Step 5). So we obtain that $\tilde A$ an $\tilde B$ are isomorphic to ${\rm PSL} (2,q)$ or $\Bbb A_7$. If $Z(E)$ is trivial, the centralizer of each involution in $E$ contains an elementary abelian group of order sixteen; moreover the group $\tilde E=E$ contains the involution $\bar h$ and the centralizer of $\bar h$ cannot contain any elementary abelian group of order sixteen. We can suppose that $Z(E)$ is not trivial, that is at least one between the components of $E$ is not simple. By Step 1, the center $Z(E)$ is a $2$–group. The central perfect extensions of ${\rm PSL} (2,q)$ and $\A_7$ with center with order a power of two are ${\rm SL} (2,q)$ and $\A_7^*$ that contain a unique involution that is central in the groups. So $E$ cannot be a direct product of its components otherwise the centralizer of each involution in $E$ contains a nonsolvable group. We obtain that $E=A\times_{{\Bbb Z}_2}B$ where $A,B\cong {\rm SL} (2,q)$ or $\A^*_7$. Finally we exclude $\smash{\A^*_7}$ as a possible component. We consider $\smash{\bar h=(\bar h_A, \bar h_B)}$, where $\smash{\bar h_A} \in A$ and $\smash{\bar h_B} \in B$. The centralizer of $\bar h$ contains the centralizer of $\bar h_A$ in $ A$ and the centralizer of $\bar h_B$ in $ B$. If one between $\bar h_A$ and $\bar h_B$ is the identity or is an element of order two, the centralizer of $\bar h$ is not solvable. To have an admissible centralizer for $\bar h$, we have to suppose that both $\bar h_A$ and $\bar h_B$ have order four (note that $\bar h$ has order two). Any element of order four in $\A^*_7$ contains in its centralizer noncommuting elements of order eight and three which contradicts and (see the *Atlas of finite groups* [@A] for the structure of $ \A^*_7$). We denote by $C=\smash{C_{\bar G}(E)}$ the centralizer of $E$ in $\bar G$. Since $E$ is normal in $G$, the group $C$ is normal in $G$. Since $C$ is contained in the normalizer of $\bar h$ it is isomorphic to a subgroup of the semidirect product ${\Bbb Z}_2 \ltimes ({\Bbb Z}_{a}\times {\Bbb Z}_{b})$. The maximal subgroup of odd order in $C$ is unique and so is characteristic, thus it is normal in $G$. It follows that $C$ has to be a $2$–group otherwise $\O(G)$ is not maximal. We denote by $D$ the subgroup generated by $E$ and $C$; the subgroup $D$ is a central product $E\times_{Z(E)}C$. The factor group of $\bar G$ by $D$ is a subgroup of ${\rm Out} (E)$, the outer automorphism group of $E$. Consider first the case $E\cong {\rm PSL}(2,q)$; in this case $D=E\times C$. In $E$ all the involutions are conjugate, so the centralizer in $\smash{\bar G}$ of each involution of $E$ is isomorphic to a subgroup of the semidirect product ${\Bbb Z}_2 \ltimes ({\Bbb Z}_{a}\times {\Bbb Z}_{b})$. The subgroup $E$ contains an elementary subgroup isomorphic to $ {\Bbb Z}_2 \times {\Bbb Z}_2$; the subgroup $C$ centralizes each involution in $E$. Since the only possible abelian $2$–group with rank at least three, contained in ${\Bbb Z}_2 \ltimes ({\Bbb Z}_{a}\times {\Bbb Z}_{b})$, is the elementary abelian subgroup of order eight, then either $C$ is trivial or $C\cong {\Bbb Z}_2$. The outer automorphism group of ${\rm PSL}(2,q=p^n)$ is isomorphic to ${\Bbb Z}_2\times{\Bbb Z}_n$ [@S2 p509]. Suppose now that $E$ has two components; the factor group $D/Z(E)$ is isomorphic to $ E/Z(E)\times C/Z(E)$. Since $E/Z(E)$ has sectional $2$–rank four and $D/Z(E)$ has sectional $2$–rank at most four, it follows that $C/Z(E)$ has to be trivial and $E=D$. The set of the components of $E$ is uniquely determined by the group and any automorphism of $E$ induces a permutation on the set of its components [@GLS2 Theorem 3.5, p7]; if $E=A\times_{{\Bbb Z}_2}B$, then the outer automorphism group of $E$ contains with index at most two a subgroup isomorphic to ${\rm Out}(A)\times {\rm Out} (B)$ [@GLS2 Lemma 3.23, p13]. The outer automorphism group of ${\rm SL}(2,q)$ is the same as that of ${\rm PSL}(2,q)$ that is isomorphic to ${\Bbb Z}_2\times{\Bbb Z}_n$. This concludes the proof. Proof of {#Section 5} ========= We denote by $H$ (resp.  $H'$) the transformation group of $K$ (resp. $K'$); each nontrivial element of $H$ (resp.  $H'$) fixes pointwise the same simple connected curve $\tilde K$ (resp. $\tilde K'$) in $M$ that is the preimage of $K$ (resp. $K'$) in $M$. Since $M$ is hyperbolic, by Thurston’s orbifold geometrization theorem [@BLPo], we can suppose, up to conjugation, that the transformation groups are contained in $G$. We note that $\tilde K$ and $\tilde K'$ do not coincide, even after conjugation. If $n=m$ it follows from the fact that $K$ and $K'$ are inequivalent. If $n\neq m$ and $H'$ fixes pointwise $\tilde K$ we obtain some nontrivial symmetries of the knot $K$ which fix pointwise the knot and this is impossible by the positive solution to the Smith Conjecture. For each prime divisor $p$ of $n$ (resp. $m$) we denote by $H_p$ (resp. $H'_p$) the Sylow p-subgroup of $H$ (resp. $H'$). Suppose that a subgroup of $H$ with order strictly greater than two normalizes a subgroup of $H'$ with order strictly greater than two, then $H$ commutes elementwise with $H'$; in particular $K$ and $K'$ arise from the standard abelian construction. The same statement holds inverting the roles of $K$ and $K'$. We denote by $B$ the subgroup of $H$ and by $B'$ the subgroup of $H'$. The subgroup $B$ normalizes $B'$; since $B$ has order strictly greater than two, implies that $B$ commutes elementwise with $B'$. The group $B'$ fixes setwise $\tilde K$; if $f\in B'$ we obtain that $fH f^{-1}$ fixes pointwise $\tilde K$. Since there exists at most one cyclic group of given order that fixes pointwise a connected curve, we obtain that $B'$ normalizes $H$. By , the groups $B'$ and $H$ commute elementwise. Using the same argument as before we obtain that $H$ and $H'$ commute elementwise, this concludes the proof. Let $B$ be a subgroup of $G$ and let $p$ be an odd prime number such that $p$ divides the order of $B\cap H$ or the order of $B\cap H'$. Then $S_p$, the Sylow $p$–subgroup of $B$, is abelian of rank one or two; there are exactly one or two simple closed curves in $M$ that are fixed by some nontrivial element of $S_p$ with connected fixed point set; the normalizer $N_G(S_p)$ of $S_p$ in $G$ is solvable. The statement of Step 2 may appear rather technical but it has the advantage that, in this form, it applies directly throughout the remaining steps. Without loss of generality, we suppose that $p$ divides the order of $B\cap H$. Up to conjugation we can suppose that $H_p\cap B$ is contained in $S_p$. We consider $N=N_{S_p}(H_p\cap B)$ the normalizer of $H_p\cap B$ in $S_p$. By the group $N$ is abelian of rank at most two. By Step 1 the group $N$ projects to a group of symmetries of $K$. Since $M$ is hyperbolic, $K$ is a hyperbolic knot and in particular is not the unknotted circle. By the positive solution of the Smith conjecture, $N/(H_p\cap B)$ is cyclic and there exists at most one connected simple closed curve fixed pointwise by elements of $N/(H_p\cap B)$. An element of $N$, that is not contained in $H_p$ and has nonempty fixed point set, projects to a nontrivial symmetry of $K$ with nonempty fixed point set; moreover $H_p\cap B$ fixes setwise the fixed point set of any element of $N$. Thus in $N$ there exists at most one maximal cyclic subgroup different from $H_p\cap B$ with nonempty connected fixed point set. If $f$ is an element of $S_p$ that normalizes $N$, it acts by conjugation on the set of maximal cyclic subgroups with nonempty connected fixed point set. Since these groups are at most two and $p$ is odd, the action must be trivial and $f$ normalizes $ H_p\cap B$. We have that $N_{S_p}(N)=N$ and by [@S1 Theorem 1.6, p88] $S_p=N$. Finally we consider the normalizer $N_{G}(S_p)$ of $S_p$ in $G$. The group $N_{G}(S_p)$ acts by conjugation on the set of maximal cyclic subgroups of $S_p$ with nonempty connected fixed point set. Since these groups are at most two, the normalizer $N_{G}(S_p)$ contains with index at most two $N_{G}(H_p)$ that is solvable. This concludes the proof. Let $B$ (resp. $B'$) be a subgroup of $H$ (resp. $H'$) such that the order of $B$ (resp. $B'$) is not a power of two. If $B$ and $B'$ generate a subgroup $B\cdot B'$ of $G$ that does not contain any involution with connected and nonempty fixed point set, then $K$ and $K'$ arise from the standard abelian construction. In particular if $G$ does not contain any involution with connected and nonempty fixed point set, then $K$ and $K'$ arise from the standard abelian construction. Let $p$ be an odd prime number that divides the order of $B$, the subgroup $H\cap B$ contains a nontrivial $p$–group. We denote by $S_p$ a $p$–Sylow of $B\cdot B'$, $S_p$ is abelian of rank at most two. If $p$ divides also the order of $B'$, we can suppose that a nontrivial subgroup of $B$ and a nontrivial subgroup of $B'$ are contained in the same Sylow $p$–subgroup of $B\cdot B'$. By Step 1 this implies that $K$ and $K'$ arise from the standard abelian construction. We can suppose that an odd prime number $q$, different from $p$, divides the order of $B'$. By Step 2, we deduce that $S_p$ contains exactly one or two maximal cyclic subgroups with nonempty connected fixed point set; up to conjugation we can suppose that one of these groups is $S_p\cap H$. We consider $N$ the normalizer of $S_p$ in $B\cdot B'$. The group $N$ acts by conjugation on the set of the maximal cyclic subgroups with nonempty connected fixed point set; $N$ contains with index at most two $N_0$, the normalizer of $S_p\cap H$ in $B\cdot B'$. We recall that $H$ fixes pointwise $\tilde K$ that is a simple closed curve. We prove that $N_0$ is abelian. Suppose that $N_0$ contains $t$, an involution with nonempty fixed point set that acts as a reflection on $\tilde K$. Since $t$ fixes setwise $\tilde K$, it normalizes $H$ and projects to a strong inversion of the knot $K$. Any strong inversion of $K$ has connected fixed point set; since $K$ is connected, also $t$ has connected fixed point set. We suppose that $B\cdot B'$ does not contain any involution with nonempty connected fixed point set, so each element of the group $N_0$ acts as a rotation on $\tilde K$ and it is abelian. Now we prove that $N=N_0$. If $N\neq N_0$, there exists an element $f\in N$ such that $f (S_p\cap H) f^{-1}\neq (S_p \cap H)$. The group $f (S_p\cap H)f^{-1}$ and $S_p \cap H$ commute elementwise. The fixed point set of $f (S_p\cap H) f^{-1}$ is $f(\smash{\tilde K})$, a simple closed curve that is distinct from $\smash{\tilde K}$. We consider the group $\smash{fH f^{-1}}$ that fixes pointwise the simple closed curve $f(\tilde K)$; since $f(\tilde K)$ is distinct from $\tilde K$ the groups $fH f^{-1}$ and $H$ intersect trivially. Moreover by Step 1, the groups $fH f^{-1}$ and $H$ commute elementwise. A generator of $fH f^{-1}$ projects to a cyclic symmetry of $K$ with order $n$ and with nonempty connected fixed point set. Finally we obtain also that the associated quotient link is symmetric; in fact $f$ normalizes the group generated by $fH f^{-1}$ and $H$ and it projects to the quotient link exchanging the two components. If $N\neq N_0$, the knot $K$ should be self-symmetric and this is excluded by hypothesis. We have obtained that the normalizer of $S_p$ in $B\cdot B'$ is abelian, in particular $S_p$ is contained in the center of its normalizer. By [@S2 Theorem 2.10, p143] $B\cdot B'$ splits as a semidirect product $U\rtimes S_p$. We have supposed that there exists $q$ different from $p$ such that $q$ divides the order of $B'$. Any Sylow $q$–subgroup is contained in $U$ and $S_p$ acts by conjugation on the set of Sylow $q$–subgroups. Since $p$ does not divide the order of $U$, it follows that some orbit has only one element. We obtain a Sylow $q$–subgroup $S_q$ that is normalized by $S_p$; up to conjugation we can suppose that the intersection of $S_q$ and $B'$ is not trivial. By Step 2 we obtain that $S_p\cap B$ normalizes $S_q\cap B'$; by Step 1 we obtain that $K$ and $K'$ arise from the standard abelian construction. Let $B$ (resp. $B'$) be a subgroup of $H$ (resp. $H'$) such that the order of $B$ (resp. $B'$) is not a power of two. If $B$ and $B'$ generate a solvable subgroup $B\cdot B'$ of $G$, then $K$ and $K'$ arise from the standard abelian construction. As in step 3 we can suppose that there exist two different odd primes $p$ and $q$, such that $p$ divides the order of $B$ and $q$ divides the order of $B'$. By Sylow theorems for solvable groups, we obtain that there exists $A$, a subgroup of $B\cdot B'$ with order $p^{\alpha}q^{\beta}$, that contains a Sylow $p$–subgroup and a Sylow $q$–subgroup of $B\cdot B'$. Up to conjugation we can suppose that the intersection of $A$ both with $B$ and with $B'$ is not trivial. The group $A$ does not contain any involution, so applying Step 3 to $A$ we obtain that $K$ and $K'$ arise from the standard abelian construction. If $E$ is not trivial, either $K$ and $K'$ arise from the standard abelian construction or there exists in $G$ an involution $h$ with nonempty connected fixed point set such that $h\O(G)\in E$. If $G$ does not contain any involution with nonempty connected fixed point set, by Step 3 the knots $K$ and $K'$ arise from the standard abelian construction. We can suppose that $G$ contains one involution with nonempty connected fixed point set. We denote by ${\cal E}$ the preimage of $E$ in $G$ with respect to the projection of $G$ onto $O/\O(G)$; by the factor group $G/{\cal E}$ is solvable. Suppose that the group $\cal E$ does not contain any involution with nonempty connected fixed point set. Let $p$ be an odd prime number that divides $n$ and let $q$ be an odd prime number that divides $m$. By the Sylow Theorem for solvable groups, there exists a subgroup ${\cal E}'$ of $G$ that contains ${\cal E}$, with the following properties: - ${\cal E}'$contains a Sylow $p$–subgroup and a Sylow $q$–subgroup of $G$; - the factor group ${\cal E}'/{\cal E}$ has order $p^{\alpha}q^{\beta}$. All the involutions in ${\cal E}'$ are contained in ${\cal E}$, so ${\cal E}'$ does not contain any involution with nonempty connected fixed point set. Moreover, up to conjugation, we can suppose that ${\cal E}'$ contains a covering transformation of order $p$ (resp. $q$) of $K$ (resp. $K'$). By Step 3 the knots $K$ and $K'$ arise from the standard abelian construction. If $E$ is trivial, either $K$ and $K'$ arise from the standard abelian construction or there exists a normal subgroup $N$ of $G$ such that $N$ is solvable and $G/N$ is isomorphic to a subgroup of ${\rm GL}(4,2)$. In this case if $K$ and $K'$ do not arise from the standard abelian construction all prime divisors of $n$ and $m$ are contained in $\{2,3,5,7\}.$ If $G$ does not contain any involution with connected fixed point set, by Step 3, the knots $K$ and $K'$ arise from the standard abelian construction. If $G$ contains an involution with connected fixed point set, by there exists a normal subgroup $N$ of $G$ such that $N$ is solvable and $G/N$ is isomorphic to a subgroup of ${\rm GL}(4,2)$. Finally we prove that, if the intersection group $H\cap N$ contains a nontrivial element of odd order, then $K$ and $K'$ arise from the standard abelian construction. Let $p$ be an odd prime number that divides the order of $H\cap N$, let $q$ be an odd prime number that divides $m$. We can suppose that $p$ is different from $q$, otherwise by Step 2 the knots $K$ and $K'$ arise from the standard abelian construction. By the Sylow Theorem applied to $G/N$, there exists a subgroup $ N'$ of $G$ that contains $ N$, with the following properties: - $N'$ is solvable; - $N'$ contains a Sylow $q$–subgroup of $G$. By Step 4 the knots $K$ and $K'$ arise from the standard abelian construction. The same property holds if $H'\cap N$ contains a nontrivial element of odd order. Since the order of ${\rm GL}(4,2)$ is $7 \cdot 5 \cdot 3^2 \cdot 2 ^6$, the Sylow $p$–subgroups of $G$ are contained in $N$ when $p\neq 2,\,3,\,5,\,7$. So if $K$ and $K'$ do not arise from the standard abelian construction, all prime divisors of $n$ and $m$ are contained in $\{2,3,5,7\}.$ ### Acknowledgements {#acknowledgements .unnumbered} I am deeply indebted to Marco Reni. He was the first that considered the possibility to extend the result obtained by him and Zimmermann [@RZ1] for simple groups to the nonsolvable case but he sadly died in June 2000 and he could not continue his work. Bruno Zimmermann kindly communicated to me some of his ideas and these suggestions are the starting point to write this paper. The author is also grateful to the referee for many helpful comments that improved the article.

--- abstract: 'Assurance cases are often required as a means to certify a critical system. Use of formal methods in assurance can improve automation, and overcome problems with ambiguity, faulty reasoning, and inadequate evidentiary support. However, assurance cases can rarely be fully formalised, as the use of formal methods is contingent on models validated by informal processes. Consequently, we need assurance techniques that support both formal and informal artifacts, with explicated inferential links and assumptions that can be checked by evaluation. Our contribution is a mechanical framework for developing assurance cases with integrated formal methods based in the Isabelle system. We demonstrate an embedding of the Structured Assurance Case Meta-model (SACM) using Isabelle/DOF, and show how this can be linked to formal analysis techniques originating from our verification framework, Isabelle/UTP. We validate our approach by mechanising a fragment of the Tokeneer security case, with evidence supplied by formal verification.' author: - Yakoub Nemouchi - Simon Foster - Mario Gleirscher - Tim Kelly bibliography: - 'FM2019.bib' title: | Mechanised Assurance Cases with\ Integrated Formal Methods in Isabelle --- Introduction {#sec:intro} ============ Preliminaries {#sec:prelim} ============= Running Example: Tokeneer {#sec:tokeneer} ========================= {#subsec:isacm} Modelling and Verification of Tokeneer {#sec:model} ====================================== Mechanising the Tokeener Assurance Case {#sec:tokassure} ======================================= Related Work {#subsec:relatedWork} ============ Conclusion {#sec:conclusion} ==========

--- abstract: 'A theory of short-range correlations in two-nucleon removal due to elastic breakup (diffraction dissociation) on a light target is developed. Fingerprints of these correlations will appear in momentum distributions of back-to-back emission of the nucleon pair. Expressions for the momentum distributions are derived and calculations for reactions involving stable and unstable nuclear species are performed. The signature of short-range correlations in other reaction processes is also studied.' author: - 'C.A. Bertulani' title: 'Short-range correlations in two-nucleon knockout reactions' --- Introduction ============ A primary goal of nucleus-nucleus scattering has been to learn about nuclear structure. This has become even more critical in recent years, when many groups became very active in the investigation of the physics of nuclei far from the stability, mainly using nucleus-nucleus scattering processes at intermediate energies ($E_{lab}\simeq100$ MeV/nucleon). The theoretical complexity of such collisions has given rise to the use of a number of different approximations. The adequate theoretical tool for this purpose is Glauber’s multiple-scattering theory [@Gl59]. It has long been known both for its simplicity and amazing predictive power. One can find copious examples in the literature where the Glauber theory allows for a simple physical interpretation of experimental results as well as their quantitative analysis [@FH80; @HN81; @BHM02]. In fact, fragmentation reactions of the type discussed here have already been successfully analyzed in the framework of Glauber’s theory: in one-nucleon-removal reactions, the momentum distribution of the outgoing fragment has been shown to reflect the momentum distribution of the nucleon which is removed from the surface of the projectile nucleus [@HN81].  However, because of complications involving multiple scattering processes in nucleus-nucleus collisions, a full Glauber multiple scattering expansion is impracticable. Fortunately, the study of many direct nuclear processes, e.g. nucleon knockout, or stripping, elastic breakup (diffraction dissociation), etc, are possible using the optical limit of the Glauber theory, in which the nuclear ground-state densities and the nucleon-nucleon total cross sections are the main input. In fact, this method has become one of the main tools in the study of nuclei far from stability [@HT04]. When departures from the optical limit are observed, multiple nucleon-nucleon collisions and in-medium effects of the nucleon-nucleon interaction and nucleon-nucleon correlations become relevant. Very peripheral collisions, with impact parameters just around the sum of the nuclear radii (grazing collisions), or larger, are well established tools for studying nuclear properties with intermediate energies and relativistic heavy ion collisions [@BB88; @Gl98; @BP99]. These collisions lead to excitation of giant resonances through both electromagnetic and strong interactions. At intermediate energy collisions ($E_{lab}\simeq100$ MeV/nucleon), or higher, the collision time is short and the action of the short-range nuclear interaction can excite the surface region of the colliding nuclei. This excitation can equilibrate forming a compound nucleus, and/or give rise to pre-equilibrium emission or other fast dissipation processes. An interesting reaction mechanism in high-energy peripheral nucleus-nucleus collisions was suggested by Feshbach and Zabek [@FZ77; @Fes80]. This mechanism has been applied in refs. [@BD78; @Kin86; @TD81; @TDM84; @NDT85; @DNT86] to the calculation of pion production in heavy ion collisions from subthreshold to relativistic energies. It is assumed that pions are produced in peripheral processes through the excitation of the projectiles to a $\Delta$-isobar giant resonance. The results of these calculations were compared to inclusive pion production data for incident energies from 50 MeV to 2 GeV per nucleon. As emphasized by those authors, this comparison is not very meaningful at high energy where peripheral processes are expected to contribute very little to the total pion production. However, at subthreshold energies, coherent pion production should dominate the cross section. This mechanism is known as the nuclear Weizsaecker-Williams method. It works as follows. The uncertainty relation associated to the variation of the time-dependent nuclear field on a scale $\Delta z$ leads a relation between the energy, $\Delta E$, and momentum transfer, $\Delta p$:$$\Delta E\simeq\frac{\hbar}{\Delta t}=\frac{\hbar\mathrm{v}}{\Delta z},\ \ \ \ \ \Delta p\simeq\frac{\hbar}{\Delta z}\ \ \ \ \Longrightarrow \ \ \Delta E=\mathrm{v}\Delta p.$$ The last equation on the right is the dispersion relation of a phonon. For typical situations, $\Delta z$ is a few fermis and the nuclear interaction pulse carries several hundred MeV. This relation can also be directly obtained from the collision kinematics. Let $\left( E_{i},\mathbf{p}_{i}\right) $ be the initial momentum of the projectile and $\left( \Delta E,\Delta \mathbf{p}\right) $ the energy-momentum transfer in the reaction. One has$$\mathbf{P}_{f}=\mathbf{P}_{i}-\Delta\mathbf{p,}\ \ \ \ \ \ \ \ \ E_{f}=E_{i}-\Delta E.$$ From these relations one finds$$\frac{\mathbf{P}_{i}.\Delta\mathbf{p}}{E_{i}}-\Delta E=\frac{-\left( \Delta E\right) ^{2}+\left( \Delta p\right) ^{2}+\left( M_{i}^{2}-M_{f}^{2}\right) c^{4}}{2E_{i}}.$$ Neglect the term on the right-hand side, one gets$$\Delta E=\mathbf{v\cdot\Delta p}=\mathrm{v}\Delta p_{z},\label{phonon}$$ where $\Delta p_{z}$ is the momentum transfer along the longitudinal direction. The above relation can only be satisfied for nuclear excitations of very small momentum transfers, even for moderately large energy transfers. This is the case for the excitation of giant resonances. Thus, the nuclear interaction in grazing nuclear collisions is an effective tool to probe giant resonances (for a review see, e.g. ref. [@CF95]). For very large impact parameters (larger than the sum of the nuclear density radii) only the electromagnetic interaction is present, and eq. \[phonon\] (with $v\simeq c$) is just the energy-momentum relation of a real photon. In fact, relativistic Coulomb excitation is another useful tool for investigating giant resonances [@BB88; @BP99]. The phonon-like relation, eq. \[phonon\], is also a tool for studying nucleon-nucleon short-range correlations. The energy in eq. \[phonon\] could hardly be absorbed by a single nucleon since it would carry the momentum $\sim\sqrt{2m\Delta E}$, which is appreciably larger than that of eq. \[phonon\]. However, the phonon could be absorbed by a correlated nucleon-pair, which can have large kinetic energy and small total momentum, when the nucleons move in approximately opposite directions. This mechanism has been exploited by previous authors to study the emission of correlated pairs in relativistic heavy ion collisions [@BCD89; @NN91]. Remarkably, refs. [@FZ77; @Fes80] do not treat properly the nuclear absorption at small impact parameters, leading to very large cross sections for the emission of correlated pairs in peripheral collisions. In many-body physics the word correlation is used to indicate effects beyond mean-field theories. In nuclear physics one distinguishes between short- and long-range correlations. Nuclear collective phenomena such as vibrations and rotations are known to be ruled by long-range correlations. These effects are relatively well known. Short-range correlations is also a subject of intensive studies in nuclear physics (see, e.g. [@Fra81; @Fra88; @Dim00; @BD02; @Tan03; @Ryc04]). The sources of short-range correlations are the strong repulsive core of the microscopic nucleon-nucleon interaction at short internucleon distances. The nucleon-nucleon interaction becomes strongly repulsive at short distances. The phase shifts for $^{1}$S$_{0}$ and $^{3}$S$_{1}$ are positive at low, and become negative at higher energies [@MAW69]. This indicates a repulsive core at short distances and attraction at long distances. In the nuclear medium this repulsive interaction is strongly influenced by Pauli blocking. The search for nuclear phenomena showing short-range correlations effects is one of the most discussed topics in the nuclear structure community. For the nuclear reaction community, the importance of Pauli correlations in high energy nucleus-nucleus collisions has prompted the consideration of effects of dynamical short-range correlations. When one treats nucleus-nucleus collisions at high energies with an optical phase shift function one can include both the center-of-mass correlations and two-body correlations in a straightforward manner to obtain a rapidly converging series for the physical observables. It would be proper at this time to look for fingerprints of short-range correlations in high-energy collisions involving rare nuclear isotopes. Recent experiments on knockout reactions seem to indicate a quenching of the spectroscopic factor relative to shell-model predictions in neutron-rich nuclei [@HT04].  This reduction is thought to be a consequence of short-range correlations which spread the single particle strength to states with higher energies. In fact, systematic studies with the $A\left( e,e^{\prime}p\right) $ reaction have provided ample evidence for this quenching phenomenon [@Pan97]. In this context, two-proton knockout reactions with exotic nuclear beams seem to be a promising tool to investigate short-range correlations in neutron(proton)-rich nuclei [@Baz02]. Indeed, for decades two-proton knockout has been considered a valuable tool to study short-range correlations in proton-nucleus and electron-nucleus processes (for recent work, see e.g. [@Tan03; @Ryc04]). In high-energy nucleus-nucleus collisions, the phonon mechanism, proposed by Feshbach and Zabek, is a useful guide for the investigation of short-range correlations. The plan of this paper is as follows. In this work we treat the effects of short-range correlations on heavy-ion scattering at high energies. In Sec. 2 the Glauber formalism for diffraction dissociation is reviewed. In section 3 this formalism is shown to lead to the same result as the traditional DWBA calculations under the proper conditions. This is an important point, as diffraction dissociation and DWBA approaches are commonly referred to as distinct reaction mechanisms in the literature. In section 4 the role of absorption and Lorentz boosts is discussed. In section 5 the formalism is applied to heavy-ion collisions in the presence of two-body correlations, showing the connection with the Feshbach and Zabek method. The significance of short-range correlations is further discussed. In sec. 6 the formalism is applied to carbon-carbon and $^{11}\mathrm{Li}+^{9}\mathrm{Be}$ collisions. In Sec. 7 some concluding remarks are made. Diffraction dissociation ======================== Let us consider high energy scattering, so that the energy transfer in the collision, $\Delta E$, is much smaller than the kinetic energy of the colliding nuclei, $E$. In most cases, one is also interested in processes for which the fragments fly in the forward direction, i.e. we will also assume that $\Delta\theta\ll1$. In such situations the particle wavefunctions are well described by eikonal waves [@BD04], i.e. a plane wave distorted by an interaction, $V$, so that the $S$-matrix is given by the simple formula $S\left( b\right) =\exp\left[ -(i/\hbar\mathrm{v})\int dZ\ V\left( R\right) \right] $, with $\mathrm{v}$ equal to the projectile velocity and $R=\left( \mathbf{b},Z\right) $ the distance between projectile and target ($V$ is assumed to be spherically symmetric). Extending this approach to account for scattering of bound particles, the initial and final states are given by$$\Psi_{i}=\phi_{i}\left( \mathbf{r}\right) \exp\left( i\mathbf{k}\cdot\mathbf{R}\right) ,\ \ \ \ \ \ \ \ \ \ \ \ \Psi_{f}=\phi_{f}\left( \mathbf{r}\right) S\left( b\right) \ \exp\left( i\mathbf{k}\cdot \mathbf{R}\right) \ ,$$ where $\phi_{i,f}\left( \mathbf{r}\right) $ are the initial and final probability amplitudes (wavefunctions) that a particle in the projectile is at a distance $\mathbf{r}$ from the center of mass. The particle’s $S$-matrix, $S\left( b\right) $, accounts for the distortion due to the interaction. For a projectile with two-body structure (e.g. a core+valence particle)$$\begin{aligned} \Psi_{i} & =\phi_{i}\left( \mathbf{r}\right) \exp\left[ i\left( \mathbf{k}_{c}\cdot\mathbf{r}_{c}+\mathbf{k}_{v}\cdot\mathbf{r}_{v}\right) \right] \nonumber\\ \Psi_{f} & =\phi_{f}\left( \mathbf{r}\right) S_{c}\left( b_{c}\right) S_{v}\left( b_{v}\right) \exp\left[ i\left( \mathbf{k}_{c}^{\prime}\cdot\mathbf{r}_{c}+\mathbf{k}_{v}^{\prime}\cdot\mathbf{r}_{v}\right) \right] \ ,\label{eikw1}$$ where now $\phi_{i,f}\left( \mathbf{r}\right) $ are the initial and final intrinsic wavefunctions of the (core+valence particle) as a function of  $\mathbf{r=r}_{1}-\mathbf{r}_{2}$. The relation between the intrinsic, $\mathbf{r}$, and center of mass, $\mathbf{R}$, coordinates is given in terms of the mass ratios $\beta_{i}=m_{i}/m_{P}$. Explicitly, $\mathbf{r}_{v}=\mathbf{R}+\beta_{c}\mathbf{r}$ and $\mathbf{r}_{c}=\mathbf{R}-\beta _{v}\mathbf{r}$. The core and valence particle $S$-matrices, $S_{c}\left( b_{c}\right) $ and$\ S_{v}\left( b_{v}\right) $, account for the distortion due to the interaction with the target. The probability amplitude for diffraction dissociation is the overlap between the two wavefunctions above, i.e. $$A_{\mathrm{(diff)}}=\int d^{3}r_{c}d^{3}r_{v}\ \phi_{f}^{\ast}\left( \mathbf{r}\right) \phi_{i}\left( \mathbf{r}\right) \delta\left( z_{c}+z_{v}\right) S_{c}\left( b_{c}\right) S_{v}\left( b_{v}\right) \exp\left[ i\left( \mathbf{q}_{c}\cdot\mathbf{r}_{c}+\mathbf{q}_{v}\cdot\mathbf{r}_{v}\right) \right] ,\label{dif1}$$ where $\mathbf{q}_{c}=\mathbf{k}_{c}^{\prime}-\mathbf{k}_{c}$ is the momentum transfer to the core particle, and accordingly for the valence particle. The above formula yields the probability amplitude that the projectile starts the collision in a bound state and ends up as two separated pieces. The $S$-matrices, $S_{c}$ and $S_{v}$ carry all the information about the dissociation mechanism. The delta-function $\delta\left( Z\right) $  in eq. \[dif1\] was introduced to account for the fact that the $S$-matrices calculated in the eikonal approximation only depend on the transverse direction. It is instructive to follow another argument to obtain eq. \[dif1\]. If only the core scatters elastically, whereas the valence particle remains in its unaltered plane wave state, the final projectile wavefunction is given by $$\Psi_{f}^{\mathrm{(scatt)}}=\phi_{f}\left( \mathbf{r}\right) \ \left[ 1-S_{c}\left( b_{c}\right) \right] \ \exp\left[ i\left( \mathbf{k}_{c}^{\prime}\cdot\mathbf{r}_{c}+\mathbf{k}_{v}^{\prime}\cdot\mathbf{r}_{v}\right) \right] .$$ The factor $\left[ 1-S_{c}\left( b_{c}\right) \right] $ is the amplitude for elastic scattering of the core. The same relation can be applied for the valence particle. The diffraction dissociation occurs by subtracting the simultaneous scattering of the core+valence particle, represented by $\left[ 1-S_{c}\left( b_{c}\right) \right] \left[ 1-S_{v}\left( b_{v}\right) \right] ,$ from the independent scattering of core and the valence particle, i.e.$$\widehat{S}_{\mathrm{(diff)}}=\left[ 1-S_{c}\left( b_{c}\right) \right] \left[ 1-S_{v}\left( b_{v}\right) \right] -\left[ 1-S_{c}\left( b_{c}\right) \right] -\left[ 1-S_{v}\left( b_{v}\right) \right] =S_{c}\left( b_{c}\right) S_{v}\left( b_{v}\right) -1.$$ The factor (-1) is not relevant because of the orthogonality of the wavefunctions $\phi_{i}\left( \mathbf{r}\right) $ and $\phi_{f}\left( \mathbf{r}\right) $. Using $A_{\mathrm{(diff)}}=\left\langle \phi _{i}\ \varphi_{\mathbf{k}_{1},\mathbf{k}_{2}}\left\vert \widehat {S}_{\mathrm{(diff)}}\right\vert \phi_{f}\ \varphi_{\mathbf{k}_{1}^{\prime },\mathbf{k}_{2}^{\prime}}\right\rangle $, with $\varphi_{\mathbf{k}_{1},\mathbf{k}_{2}}$ equal to plane waves, we regain eq. \[dif1\]. We thus see that diffractive dissociation (or elastic nuclear breakup) arises from the momentum transfer to each particle due to elastic scattering, subtracting the momentum transfer to their center of mass. The cross section for the diffraction process $\phi_{i}\left( \mathbf{r}\right) \rightarrow\phi_{f}\left( \mathbf{r}\right) $ is given by$$d\sigma=\rho\left( E\right) \left\vert \int d^{3}r_{c}\ d^{3}r_{v}\ \phi _{f}^{\ast}\left( \mathbf{r}\right) \phi_{i}\left( \mathbf{r}\right) \delta\left( z_{c}+z_{v}\right) S_{c}\left( b_{c}\right) S_{v}\left( b_{v}\right) \exp\left[ i\left( \mathbf{q}_{c}\cdot\mathbf{r}_{c}+\mathbf{q}_{v}\cdot\mathbf{r}_{v}\right) \right] \right\vert ^{2},\label{dif}$$ where $\rho\left( E\right) \ $is the density of final states, $\rho\left( E\right) =\delta\left( Q_{z}\right) d^{3}q_{c}d^{3}q_{v}/\left( 2\pi\right) ^{5}$, where $\mathbf{Q}=\mathbf{q}_{c}+\mathbf{q}_{v}$ is the momentum transfer to the center of mass of the projectile. The delta function accounts for the conservation of the longitudinal momentum of the projectile arising from the use of eikonal wavefunctions (i.e. no dependence on the longitudinal c.m. scattering). It is important to notice that the above formula is somewhat different than eq. 8 of ref. [@HEB96] . In that reference the coordinates $\mathbf{r}$,$\ \mathbf{R}$ were used from the start. One can transform the integral of eq. \[dif\] to those variables. The Jacobian of the transformation is equal to one and $d^{3}r_{c}d^{3}r_{v}=d^{3}rd^{3}R$, $d^{3}q_{c}d^{3}q_{v}=d^{3}qd^{3}Q$, where $\mathbf{q}=\beta_{c}\mathbf{q}_{v}-\beta_{v}\mathbf{q}_{c}$ is the momentum transfer to the intrinsic coordinates of the projectile. Thus, in the coordinates $\mathbf{r}$,$\ \mathbf{R,}$  eq. \[dif\] reduces to$$d\sigma=\frac{d^{3}qd^{2}Q}{\left( 2\pi\right) ^{5}}\left\vert \int d^{3}rd^{2}b\ \phi_{f}^{\ast}\left( \mathbf{r}\right) \phi_{i}\left( \mathbf{r}\right) S_{c}\left( b_{c}\right) S_{v}\left( b_{v}\right) \exp\left[ i\left( \mathbf{q}\cdot\mathbf{r}+\mathbf{Q}\cdot\mathbf{b}\right) \right] \right\vert ^{2}.\label{dif2}$$ The above formula reduces to eq. 8 of ref. [@HEB96] if one sets $\beta _{v}=1$ and $\beta_{c}=0$.  In this equation, $\phi_{f}\left( \mathbf{r}\right) $ can be taken as any final state of the projectile. Thus, it is not only appropriate to calculate *diffraction dissociation*, but also *diffraction excitation*. Diffraction excitation occurs when the final state $\phi_{f}\left( \mathbf{r}\right) $ is a bound state. If it is a state in the continuum (diffraction dissociation), then $\phi_{f}\left( \mathbf{r}\right) $ should be set to the unity[^1], since the part of the wavefunction given by $S_{c}S_{v}\exp\left[ i\left( \mathbf{k}_{c}^{\prime}\cdot\mathbf{r}_{c}+\mathbf{k}_{v}^{\prime }\cdot\mathbf{r}_{v}\right) \right] $ already accounts for the proper wavefunction of the projectile. A natural improvement of eq. \[dif\] is to include final state interactions between the core and the valence particle in the coordinate dependence of $\phi_{f}\left( \mathbf{r}\right) .$ Since I claim here that eq. \[dif2\] can also be used for calculating excitation cross sections, it is adequate to show the relation of this approach to the traditional DWBA and semiclassical methods for nuclear excitation in nucleus-nucleus collisions. We will see that the latter are perturbative expansions of the eq. \[dif2\]. DWBA and semiclassical methods ============================== On can factorize the $S$-matrices defined in section 2 for the interaction of the core and valence particle with the target in terms of their phase-shifts $$\chi=-\frac{i}{\hbar v}\int_{-\infty}^{\infty}dZ\ V\left( R\right) \ .\label{eikphase}$$ In the weak interaction limit, or perturbation limit, the phase-shifts are very small so that$$\begin{aligned} S_{c}\left( b_{c}\right) S_{v}\left( b_{v}\right) & =\exp\left[ i\left( \chi_{c}+\chi_{v}\right) \right] \ \simeq1+i\chi_{c}+i\chi _{v}\nonumber\\ & =1+\frac{1}{\hbar v}\int V_{cT}\left( \mathbf{r}_{c}\right) \ dz_{c}+\frac{1}{\hbar v}\int V_{vT}\left( \mathbf{r}_{v}\right) \ dz_{v}.\label{dwba1}$$ The factor 1 does not contribute to the breakup**.** Thus, inserting the result above in eq. \[dif1\], one obtains$$A_{\mathrm{(PWBA)}}\simeq\frac{1}{\hbar v}\int d^{3}r_{c}d^{3}r_{v}\ \phi _{f}^{\ast}\left( \mathbf{r}\right) \phi_{i}\left( \mathbf{r}\right) \left[ V_{cT}\left( \mathbf{r}_{c}\right) +V_{vT}\left( \mathbf{r}_{v}\right) \right] \exp\left[ i\left( \mathbf{q}_{c}\cdot\mathbf{r}_{c}+\mathbf{q}_{v}\cdot\mathbf{r}_{v}\right) \right] ,\label{dwba2}$$ where the integrals over $z_{c}$ and $z_{v}$ in eq. \[dwba1\] were absorbed back to the integrals over $\mathbf{r}_{c}$ and $\mathbf{r}_{v}$ after use of the delta-function $\delta\left( z_{c}+z_{v}\right) $. The above equation is nothing more than the plane-wave Born-approximation (PWBA) amplitude. However, absorption is not treated properly. For small values of $\mathbf{r}_{c}$ and $\mathbf{r}_{v}$ the phase-shifts are not small and the approximation used in eq. \[dwba1\] fails. A better approximation is to assume that for small distances, where absorption is important, $S_{c}\left( b_{c}\right) S_{v}\left( b_{v}\right) \simeq S\left( b\right) $, where the right-hand side is the $S$-matrix for the projectile scattering as a whole on the target. Using the coordinates $\mathbf{r}$ and$\ \mathbf{R}$ , and defining $U_{int}(\mathbf{r,R})=V_{cT}\left( \mathbf{r}_{c}\right) \ +V_{nT}\left( \mathbf{r}_{n}\right) $, one gets $$T_{\mathrm{(DWBA)}}=\hbar vA_{\mathrm{(DWBA)}}\simeq\int d^{3}rd^{3}R\ \phi_{f}^{\ast}\left( \mathbf{r}\right) \exp\left[ i\mathbf{q}\cdot\mathbf{r}\right] \phi_{i}\left( \mathbf{r}\right) U_{int}(\mathbf{r,R})S\left( b\right) \exp\left[ i\mathbf{Q}\cdot\mathbf{R}\right] \ .\label{TDWBA0}$$ In elastic scattering, or excitation of collective modes (e.g. giant resonances), the momentum transfer to the intrinsic coordinates can be neglected and the equation above can be written as$$T_{\mathrm{(DWBA)}}=\left\langle \chi^{\left( -\right) }\left( \mathbf{R}\right) \phi_{c}\left( \mathbf{r}\right) \left\vert U_{int}(\mathbf{r,R})\right\vert \chi^{\left( +\right) }\left( \mathbf{R}\right) \phi_{i}\left( \mathbf{r}\right) \right\rangle \ , \label{TDWBA}$$ which has the known form of the DWBA T-matrix. The scattering phase space now only depends on the center of mass momentum transfer $\mathbf{Q}$. When the center of mass scattering waves are represented by eikonal wavefunctions, one has$$\chi^{\left( -\right) \ast}\left( \mathbf{R}\right) \chi^{\left( +\right) }\left( \mathbf{R}\right) \simeq S\left( b\right) \exp\left[ i\mathbf{Q}\cdot\mathbf{R}\right] \ . \label{chieik}$$ This shows that the PWBA and the DWBA are perturbative expansions of the diffraction dissociation formula \[dif1\]. In DWBA (or in the eikonal approximation, eq. \[chieik\]), $b$ does not have the classical meaning of an impact parameter. To obtain the semiclassical limit one goes one step further. By using eq. \[TDWBA0\] and assuming that $R$ depends on time so that $R=\left( \mathbf{b},z=vt\right) $, the semiclassical scattering amplitude is given by $A_{\mathrm{(semiclass)}}^{\left( i\rightarrow f\right) }=i\int d^{2}b\ a_{\mathrm{(semiclass)}}^{\left( i\rightarrow f\right) }\left( b\right) $ exp$\left( i\mathbf{Q}\cdot\mathbf{b}\right) $, where$$a_{\mathrm{(semiclass)}}^{\left( i\rightarrow f\right) }\left( b\right) =\frac{1}{i\hbar}\ S\left( b\right) \int dtd^{3}r\ \exp\left( i\omega _{if}\ t\right) \phi_{f}^{\ast}\left( \mathbf{r}\right) U_{int}(\mathbf{r,}t)\phi_{i}\left( \mathbf{r}\right) \ ,\label{semi1}$$ where eq. \[phonon\] was used ($Q_{z}Z=\omega_{if}\ t$). The semiclassical probability for the transition $\left( i\rightarrow f\right) $ is obtained from the above equation after squaring it of integrating it over **Q**. One gets $\sigma^{\left( i\rightarrow f\right) }=\int d^{2}b\ P_{\mathrm{(semiclass)}}^{\left( i\rightarrow f\right) }\left( b\right) $, where $P_{\mathrm{(semiclass)}}^{\left( i\rightarrow f\right) }\left( b\right) =\left\vert a_{\mathrm{(semiclass)}}^{\left( i\rightarrow f\right) }\left( b\right) \right\vert ^{2}$, with $b$ having now the explicit meaning of an impact parameter. Thus, $a_{\mathrm{(semiclass)}}^{\left( i\rightarrow f\right) }\left( b\right) $, is the semiclassical excitation amplitude. Equation \[semi1\] is well-known (for example in Coulomb excitation at low energies) except that the factor $S\left( b\right) $ is usually set to one. In high energy collisions it is crucial to keep this factor, as it accounts for refraction and absorption at small impact parameters:  $\left\vert S\left( b\right) \right\vert ^{2}=\exp\left[ 2\chi^{\mathrm{(imag)}}\right] $, where $\chi^{\mathrm{(imag)}}$ is calculated with the imaginary part of the optical potential. The derivation of the DWBA and semiclassical limits of eikonal methods can be easily extended to higher-orders in the perturbation $V$. The eikonal method includes all terms of the perturbation series in the sudden-collision limit. Role of absorption and of Lorentz boosts ======================================== At this point it is interesting to consider the calculation performed by Feshbach and Zabek [@FZ77]. In that work, eq. \[semi1\], or its equivalent PWBA form, eq. \[dwba2\], without a proper account of the strong absorption at small impact parameters (described in eq. \[semi1\] by $S\left( b\right) $), was used to calculate the total cross section for emission of a correlated nucleon pair in peripheral collisions with heavy ions. Also, interactions without imaginary parts were used. As a consequence, they found extremely large cross sections; $\sim1$ barn for $^{16}$O+$^{16}$O collisions at energies $\sim1$ GeV/nucleon. This is certainly inconsistent with perturbation theory. As seen schematically in figure \[SXU\], the product of the $S$-matrix and the interaction potential implies that the reaction occurs in a narrow region at grazing" impact parameters. The width of this region is approximately $\Delta\simeq1-2$ fm. The cross section might be written as $\sigma\simeq2\pi\Delta\left( R_{P}+R_{T}\right) P$, where $P$ is the average probability for this reaction to occur within the impact parameter interval $\Delta$, and $R_{P}$ $\left( R_{T}\right) $ is the projectile (target) radius. For light nuclei  $2\pi\Delta\left( R_{P}+R_{T}\right) \simeq300-600$ mb. Thus, the probability $P$ violates unitarity (perturbation theory is invalid) if cross sections of the order of 1 b are obtained. Ref. [@FZ77] also introduced relativistic corrections to the nuclear potential. This relativistic property is most easily seen within a folding potential model for a nucleon-nucleus collision: $$V\left( \mathbf{r}\right) =\int dr^{\prime3}\ \rho_{T}\left( \mathbf{r}^{\prime}\right) \ v_{NN}\left( \mathbf{r-r}^{\prime}\right) ,\label{fold}$$ where $\rho_{T}\left( \mathbf{r}^{\prime}\right) $ is the nuclear density of the target. In the frame of reference of the projectile, the density of the target looks contracted and particle number conservation leads to the relativistic modification of eq. \[fold\] so that $\rho_{T}\left( \mathbf{r}^{\prime}\right) \rightarrow\gamma\rho_{T}\left( \mathbf{r}_{\perp}^{\prime}\mathbf{,\gamma}z^{\prime}\right) $, where $\mathbf{r}_{\perp}^{\prime}$ is the transverse component of **r’** and $\gamma=\left( 1-v^{2}/c^{2}\right) ^{-1/2}$ is the Lorentz contraction factor, with $v$ equal to the relative velocity of projectile and target. But the number of nucleons as seen by the target (or projectile) per unit area remains the same. In other words, a change of variables $z^{\prime\prime }=\mathbf{\gamma}z^{\prime}$  in the integral of eq. \[fold\] seems to restore the same eq. \[fold\]. However, this change of variables also modifies the nucleon-nucleon interaction $v_{NN}$. Thus, relativity introduces non-trivial effects in a potential model description of nucleus-nucleus scattering at high energies. Colloquially speaking, nucleus-nucleus scattering at high energies is not simply an incoherent sequence of nucleon-nucleon collisions. Since the nucleons are confined within a box (inside the nucleus), Lorentz contraction induces a collective effect: in the extreme limit $\gamma\rightarrow\infty$ all nucleons would interact at once with the projectile. This is often neglected in pure geometrical (Glauber model) descriptions of nucleus-nucleus collisions at high energies, as it is assumed that the nucleons inside firetubes" scatter independently. Assuming that the nucleon-nucleon interaction is of very short range so that the approximation $\ v_{NN}\left( \mathbf{r-r}^{\prime}\right) =J_{0}\ \delta\left( \mathbf{r-r}^{\prime}\right) $ can be used, one sees from eq. \[fold\] that $V\left( \mathbf{r}\right) $, the interaction that a nucleon in the projectile has with the target nucleus, also has similar transformation properties as the density: $V\left( \mathbf{r}\right) \rightarrow\mathbf{\gamma}V\left( \mathbf{r}_{\perp}\mathbf{,\gamma}z\right) $, i.e. $V\left( \mathbf{r}\right) $ transforms as the time-component of a four-vector. In this situation, the Lorentz contraction has no effect whatsoever in the diffraction dissociation amplitudes, described in the previous sections within the eikonal approximation. This is because a change of variables $Z^{\prime}=\mathbf{\gamma}Z$ in the eikonal phases leads to the same result as in the non-relativistic case, as can be easily checked from eq. \[eikphase\]. Of course, the delta-function approximation for the nucleon-nucleon interaction means that nucleons will scatter at once, and Lorentz contraction does not introduce any additional collective effect. This is not the case for realistic interactions with finite range. Thus, nuclear structure studied with high-energy nucleus-nucleus collisions is immensely complicated by retardation effects and is not well understood. Emission of correlated pairs in peripheral reactions ==================================================== Lets us now consider the emission of correlated pairs in peripheral collisions. The projectile is now a three-body system, with notation for the coordinates as shown in figure \[coll\]. Following the same arguments used in section 2, the wavefunction of a three-body projectile in the initial and final states is given by$$\begin{aligned} \Psi_{i} & =\phi_{i}\left( \mathbf{r}_{1},\mathbf{r}_{2}\right) \exp\left[ i\left( \mathbf{K}_{c}\cdot\mathbf{r}_{c}+\mathbf{k}_{1}\cdot\mathbf{r}_{1}+\mathbf{k}_{2}\cdot\mathbf{r}_{2}\right) \right] \nonumber\\ \Psi_{f} & =\phi_{f}\left( \mathbf{r}_{1},\mathbf{r}_{2}\right) S_{c}\left( b_{c}\right) S_{1}\left( b_{1}\right) S_{2}\left( b_{2}\right) \exp\left[ i\left( \mathbf{K}_{c}^{\prime}\cdot\mathbf{r}_{c}+\mathbf{k}_{1}^{\prime}\cdot\mathbf{r}_{1}+\mathbf{k}_{2}^{\prime}\cdot\mathbf{r}_{2}\right) \right] \ ,\end{aligned}$$ where now $\phi_{i,f}\left( \mathbf{r}_{1},\mathbf{r}_{2}\right) $ are the initial and final intrinsic wavefunctions of the correlated nucleon-nucleon pair as a function of  their intrinsic coordinates $\mathbf{r}_{1},$ $\mathbf{r}_{2}$. Assuming that the nucleon mass is much smaller than that of the core, one can replace $\mathbf{r}_{c}\simeq\mathbf{R}$, where **R** is the center of mass of the projectile. Following the same steps as before, a relation similar to eq. \[dif2\] can be obtained for the cross section for the energy absorption by a correlated pair (when final state interactions are neglected):$$\begin{aligned} d\sigma & =\frac{d^{3}q_{1}d^{3}q_{2}d^{2}Q}{\left( 2\pi\right) ^{8}}\left\vert \int d^{3}r_{1}d^{3}r_{2}d^{2}b\ \phi_{f}^{\ast}\left( \mathbf{r}_{1},\mathbf{r}_{2}\right) \phi_{i}\left( \mathbf{r}_{1},\mathbf{r}_{2}\right) \right. \ \nonumber\\ & \times\left. S\left( b\right) S_{1}\left( b_{1}\right) S_{2}\left( b_{2}\right) \exp\left[ i\left( \mathbf{q}_{1}\cdot\mathbf{r}_{1}+\mathbf{q}_{2}\cdot\mathbf{r}_{2}+\mathbf{Q}\cdot\mathbf{b}\right) \right] \right\vert ^{2},\end{aligned}$$ where $\mathbf{Q=K}_{c}^{\prime}-\mathbf{K}_{c}$. If the intrinsic nucleon coordinates are denoted by $\mathbf{r}_{i}^{\prime}=\mathbf{r}_{i}-\mathbf{R}$,  one has $b_{i}=\sqrt{b^{2}+r_{i}^{2}\sin^{2}\theta_{i}+2r_{i}b\sin \theta_{i}\cos\left( \phi-\phi_{i}\right) }$. The above relation can be used for the emission of the nucleon pair. Neglecting final state interactions and assuming that the core is not observed (i.e. integrating over $\mathbf{Q}$), one gets$$d\sigma=\frac{d^{3}q_{1}d^{3}q_{2}}{\left( 2\pi\right) ^{6}}\int d^{2}b\ \left\vert S\left( b\right) \right\vert ^{2}\ \left\vert \int d^{3}r_{1}d^{3}r_{2}\ \phi_{i}\left( \mathbf{r}_{1},\mathbf{r}_{2}\right) S_{1}\left( b_{1}\right) S_{2}\left( b_{2}\right) \exp\left[ i\left( \mathbf{q}_{1}\cdot\mathbf{r}_{1}+\mathbf{q}_{2}\cdot\mathbf{r}_{2}\right) \right] \right\vert ^{2}. \label{dsig5}$$ In order to proceed further one needs a model wavefunction for the correlated pair, $\ \phi_{i}\left( \mathbf{r}_{n},\mathbf{r}_{n^{\prime}}\right) $. The wavefunction used will have the form$$\phi_{i}\left( \mathbf{r}_{1},\mathbf{r}_{2}\right) =\phi_{\alpha}\left( \mathbf{r}_{1}\right) \phi_{\beta}\left( \mathbf{r}_{2}\right) f_{\mathrm{corr}}(\mathbf{r,r}_{c}\mathbf{)}\ ,\label{pairwf}$$ where $\mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}$,$\ \phi_{\alpha}\left( \mathbf{r}\right) =\phi_{nljm}\left( \mathbf{r}\right) $ are single particle wavefunctions with quantum numbers $\alpha=nljm,$ and $f_{\mathrm{corr}}\left( \mathbf{r,r}_{c}\right) $ is a function for the nucleon pair distance **r**, which also depends on a two-particle correlation parameter $\mathbf{r}_{c}$ so that $f_{\mathrm{corr}}\left( \mathbf{r,r}_{c}\right) \rightarrow0$ as $\mathbf{r}_{c}\rightarrow0$. The effective correlation function $f_{\mathrm{corr}}\left( \mathbf{r,r}_{c}\right) $, the so-called Jastrow factor [@Ja55],  is a statistical average of the Pauli correlation function [@BM69] and the correlation function for the dynamical short-range  (e.g., hard core) correlation. As argued in ref. [@GWW58], the true ground-state wave function of the nucleus containing correlations coincide with the independent particle, or Hartree-Fock wavefunction, for interparticle distances $r\geq r_{\mathrm{heal}}$, where $r_{\mathrm{heal}}\simeq1$ fm is the so-called healing distance. This behavior is a consequence of the constraints imposed by the Pauli principle. Nucleons are kept apart at short-distances, while for distances beyond several $K_{F}^{-1}$’s there is little effect. Consequently, nucleon-nucleon collisions at short distances are rare in nuclear matter, and because the strongest part of the interaction is at short distances, the effective force  between the nucleons is much less than in free space. For example, if a nucleon in $^{16} $O felt the cumulative sum of 16 nucleon-nucleon potentials, it would feel a potential of $\sim1400$ MeV; yet empirically it is known that the effective potential felt by the nucleon in the middle of the nucleus is only $\sim40$-50 MeV deep. Although, in general, the correlation function $f_{\mathrm{corr}}(\mathbf{r,r}_{c}\mathbf{)}$ may depend on the isospin and spin quantum numbers of the two-body channel, we will assume for simplicity that it is a plain, state independent, Jastrow factor [@Ja55]. The effects of nucleon-nucleon correlations in nucleus-nucleus collisions have also been studied in several works. For example, in ref. [@Ray79] short-range correlations were shown to play an important role in nucleon-nucleus collisions at intermediate and high energies. The two-particle correlation distance, $\mathbf{r}_{c}$, is a combination of four contributions [@Ray79]$$r_{c}=r_{\mathrm{Pauli}}+r_{\mathrm{SRD}}+r_{\mathrm{PSR}}+r_{\mathrm{CM}},$$ where $r_{\mathrm{Pauli}}$ is due to Pauli exclusion-principle correlations, $r_{\mathrm{SRD}}$ is related to short-range dynamical correlations, $r_{\mathrm{PSR}}$ is due to a combination of the Pauli and the short-range dynamical term, and $r_{\mathrm{CM}}$ is due to center-of-mass correlations [@BF77]. An approximate set of expressions for each of these terms is given by $$\begin{aligned} r_{\mathrm{Pauli}} & =\frac{1}{2}\left( 1-\frac{5}{A}+\frac{4}{A^{2}}\right) \frac{3\pi}{10K_{F}}\frac{1}{1+\frac{8}{5}BK_{F}^{2}},\nonumber\\ r_{\mathrm{SRD}} & =\frac{1}{2}\left( 1-\frac{2}{A}+\frac{1}{A^{2}}\right) \sqrt{\pi}\frac{b^{3}}{b^{2}+8B},\nonumber\\ r_{\mathrm{PSR}} & =\frac{1}{2}\left( 1-\frac{5}{A}+\frac{4}{A^{2}}\right) \frac{3\pi}{10}\left( K_{F}^{2}+\frac{5}{b^{2}}\right) ^{-1/2}\left[ 1+8B\left( \frac{K_{F}^{2}}{5}+\frac{1}{b^{2}}\right) \right] ^{-1}\nonumber\\ r_{\mathrm{CM}} & =\left( 1-\frac{2}{A}+\frac{1}{A^{2}}\right) l_{c},\end{aligned}$$ where $A$ and $K_{F}=\left( 1.5\pi^{2}\rho\right) ^{1/3}\simeq1.36$ fm$^{-1}$ are the target number and the Fermi momentum of the target nucleus, respectively. $b$ is a short-range dynamical correlation, $b\simeq0.4$ fm, $B$ is the finite-range parameter of the nucleon-nucleon elastic $t$-matrix, $B\simeq0.62$ fm$^{2}$ (for collisions around 200 MeV/nucleon), and $l_{c}$ is the effective correlation length", $l_{c}\simeq1.3\ A^{-5/6}$ fm$.$ For proton+$^{12}$C collisions at 200 MeV/nucleon this set of parameters yields, $r_{\mathrm{Pauli}}\simeq0.3$ fm, $r_{\mathrm{SRD}}\simeq0.01$ fm, $r_{\mathrm{PSR}}\simeq0.0016$ fm, $r_{\mathrm{CM}}\simeq0.18$ fm, and $r_{c}\simeq0.5$ fm. This in fact overestimates the correlation distance. A more detailed calculation, using the parameters $B$, $b$ and $l_{c}$ from ref. [@Ray79] shows that $r_{c}$ has an appreciable dependence on the collision energy, as shown in figure \[rcorr\] for protons incident on $^{12}$C. Thus, in nuclear reactions, $r_{c}$ can vary substantially with the collision energy and with mass numbers. The estimates done above show that the main contribution to the correlation distance arises from the Pauli principle. Let us assume a correlation function of the form$$f_{\mathrm{corr}}\left( r\right) =1-\exp\left[ -\frac{\left( \mathbf{r}_{1}-\mathbf{r}_{2}\right) ^{2}}{r_{c}^{2}}\right] .\label{fcorr}$$ This correlation function implies that the pair wavefunction decreases for small relative distances, $r=\left\vert \mathbf{r}_{1}-\mathbf{r}_{2}\right\vert \lesssim r_{c}$. The correlation function $f_{\mathrm{corr}}(\mathbf{r,r}_{c}\mathbf{)}$ goes to one for large values of $r$ and to zero for $r\rightarrow0$. In nuclear structure calculations, the effect of correlation, introduced by the function $f_{\mathrm{corr}}(\mathbf{r,r}_{c}\mathbf{)}$, becomes large when the correlation distance parameter $r_{c}$ becomes large, and vice versa. Here, only the effects of short-range correlations are studied and it would be manifest in momentum distributions of highly energetic nucleons, as discussed in the introduction, and explicitly shown in the next section. It is important to notice that the Gaussian correlation function, eq. \[fcorr\], is unrealistic. Indeed, the short-range repulsion is at the origin of the decrease of the pair wavefunction for small relative distances. At the same time, there will be an increased probability to find the nucleon pair at medium internucleon distances. A two-Gaussian parameterization is needed to quantify this well-known effect of short-range correlations. For simplicity, only the simple parameterization of eq. \[fcorr\] is used in this work. Inserting eqs. \[pairwf\] and \[fcorr\] in eq. \[dsig5\] and integrating over the pair momenta, one gets$$\begin{aligned} \sigma_{\mathrm{SR}} & =\frac{\left( C^{2}S\right) _{lj}\left( C^{2}S\right) _{l^{\prime}j^{\prime}}}{\left( 2j+1\right) \left( 2j^{\prime}+1\right) }{\displaystyle\sum\limits_{m,\ m^{\prime}}}\int d^{2}b\ \left\vert S\left( b\right) \right\vert ^{2}\ \nonumber\\ & \times\int d^{3}r_{1}\ d^{3}r_{2}\ \left\vert \phi_{nljm}\left( \mathbf{r}_{1}\right) S_{1}\left( b_{1}\right) \phi_{n^{\prime}l^{\prime }j^{\prime}m^{\prime}}\left( \mathbf{r}_{2}\right) S_{2}\left( b_{2}\right) f_{\mathrm{corr}}(\mathbf{r,r}_{c}\mathbf{)}\right\vert ^{2}\ .\label{cor}$$ The cross section has been averaged over the initial magnetic quantum numbers of the nucleons. If the correlation function were equal to the unity, the integrand would be the product of the probabilities to remove an uncorrelated nucleon, with quantum numbers $nljm$. The later probability is given by $\int d^{3}r\ \left\vert \phi_{nljm}\left( \mathbf{r}\right) \ S_{i}\left( b\right) \right\vert ^{2}$. The spectroscopic factors in eq. \[cor\] have a complex dependence on the angular momenta of the nucleon pair. The correlations arising from angular momentum coupling have been studied in ref. [@Jef04]. Here we will assume a simple combinatorics so that $\left( C^{2}S\right) _{lj}=n\left( n-1\right) /2$, where $n$ is the number of nucleons in the valence shell. We see from the equations above that the cross section for the emission of a correlated pair is smaller than that for the emission of independent particles, since $f_{\mathrm{corr}}(\mathbf{r,r}_{c}\mathbf{)}\leq1$. Most part of the integrand will have $f_{\mathrm{corr}}(\mathbf{r,r}_{c}\mathbf{)}\sim1$, except for the small region of volume $\mathcal{N}r_{c}^{3}$, where $\mathcal{N}$ is a number of order of one. Conservative estimates (using $r_{c}=0.3-1$ fm), imply that the cross section for emission of a correlated pair could not exceed 100 mb, in contrast to the results obtained in refs. [@FZ77; @Fes80]. We will show this for specific reactions in the following section. Results and discussions ======================= The numerical calculations have been carried out for the systems $^{12}$C+$^{12}$C at 250 MeV/nucleon and $^{11}$Li+$^{9}$Be at 287 MeV/nucleon. In both cases, there are some experimental data available for two-nucleon removal. This also allows for the study of the influence of a halo wavefunction ($^{11}$Li) in the results. The wavefunctions were calculated by using a Woods-Saxon potential with a spin-orbit and Coulomb potential, $$V\left( r\right) =U_{r}\left( r\right) +U_{s}\left( r\right) +U_{C}\left( r\right) ,$$ where$$U_{r}\left( r\right) =V_{r}\left( 1+e^{\rho_{r}}\right) ^{-1},\ \ \ \ \ \ \ \ \ U_{s}\left( r\right) =V_{s}\left( \mathbf{l\cdot s}\right) \frac{\left( 2\ \mathrm{fm}^{2}\right) }{r}\frac{d}{dr}\left( 1+e^{\rho_{s}}\right) ^{-1},$$ $U_{C}\left( r\right) $ is the potential for a uniformly charged sphere with charge $Z-1$ ($Z$, for neutrons) and radius $R_{C}$, and $\rho_{i}=\left( r-R_{i}\right) /a_{i}$. For protons in the $1p_{3/2}$ orbital of $^{12}$C the separation energy is 15.96 MeV (the two-proton separation energy is 27.18 MeV), which can be reproduced with the parameters $V_{r}=-57.41$ MeV, $V_{s}=-6.0$ MeV, $R_{r}=R_{C}=R_{s}=3.011$ fm, $a_{r}=0.52$ fm and $a_{s}=0.65$ fm. The reactions and structure of the two-neutron halo nucleus $^{11}$Li have attracted much interest. It is a Borromean system in the sense that although the three-body system, consisting of $^{9}$Li and two neutrons, forms a bound state, none of the possible two-body subsystems have bound states. Hence the stability of $^{11}$Li is brought about by the interplay of the core-neutron and the neutron-neutron interactions, which must lead to a strongly correlated wave function with the two neutrons spatially close together. For the calculation here we will approximate the $^{11}$Li ground state by an inert $^{9}$Li core coupled to a neutron pair in a $\left( 2s1/2\right) ^{2}$ state, although the most probable configuration is an admixture of neutron pairs in $\left( 2s1/2\right) ^{2}$, $\left( 1p1/2\right) ^{2}$, and $\left( 1d5/2\right) ^{2}$ states [@Si99; @BH04]. However, the former assumption allows for a simpler calculation of the correlated-pair emission. The potential parameters are adjusted to obtain the single-particle wave functions, reproducing the effective neutron separation energies. The two-neutron separation energy is 0.3 MeV. From the systematics in Fig. 6 of [@Han01], the estimated $^{10}$Li average excitation energies is 0.2 MeV for the single-particle state. Taking this value for two-neutron coupling to the $^{9}$Li core, one arrives at an effective neutron-separation energy of 0.5 MeV. This binding energy for the n+$^{10}$Li system can be reproduced with the potential parameters $V_{r}=-42.93$ MeV,$V_{s}=-6.0$ MeV, $R_{r}=R_{C}=R_{s}=3.25$ fm and $a_{r}=a_{s}=0.65$ fm. The single-particle wavefunctions obtained in this way were used in eq. \[dsig5\] to calculate the momentum distributions of the correlated pair. The integrals in eq. \[cor\] were performed using a method similar to that described in the appendix of ref. [@BH04]. The S-matrices (and optical potentials) were calculated by using the t-$\rho\rho$" interaction, as described in refs. [@Ray79; @RHB91]. This is the same approximation used in ref. [@HT04]. In heavy ion physics it is common to define a correlation function by means of$$C\left( \mathbf{q}_{1},\mathbf{q}_{2}\right) =\left( \frac{d\sigma}{d^{3}q_{1}d^{3}q_{2}}\right) \left/ \left( \frac{1}{\sigma}\frac{d\sigma }{d^{3}q_{1}}\frac{d\sigma}{d^{3}q_{2}}\right) \right. ,$$ where the cross sections in the denominator are for the emission of a single nucleon. An accurate measurement of $r_{c}$ requires the measurement of this correlation function for back-to-back (or nearly) pair emission. Until now, heavy ion data refer mainly to small relative momentum transfers. Data would only be interesting for the present purposes if the triggering conditions were changed and if special attention was paid to back-to-back emission. In the present work, $d\sigma/dq_{1}dq_{2}$, instead of $C\left( \mathbf{q}_{1},\mathbf{q}_{2}\right) $, will be used for the study of emission of correlated nucleons. In figure \[contour\] contour plots for $d\sigma/dq_{1}dq_{2}$ are presented for the collision $^{12}$C+$^{12}$C at 250 MeV/nucleon and $^{11}$Li+$^{9}$Be at 287 MeV/nucleon, as a function of $p_{1}=\hbar q_{1}$ and $p_{2}=\hbar q_{2}$. The nucleons are assumed to be emitted back-to-back, the nucleon 1 at 0$^{o}$ and nucleon 2 at 180$^{o}$ with respect to the beam axis, respectively. The upper panels are for the C+C collision, while the lower panels are for the Li+Be collisions. The left (right) panels are for $r_{c}=0.4$ fm ($r_{c}=1$ fm). The numbers in the plot indicate the cross section, $d\sigma/dq_{1}dq_{2}$, in units of 10$^{-5}$ \[mb/(MeV/c)\]$^{2}$. One notices a strong correlation between the nucleon momenta, resulting from the phonon relationship, eq. \[phonon\] . The effect of the phonon dispersion relation is to produce a ridge in the cross section. To obtain a greater physical insight, I will now use the PWBA approximation as in eq. \[TDWBA0\] (with $S(b)=1$), so that, instead of the integrals in eq. \[dsig5\], one needs now to calculate $$T_{\mathrm{(PWBA)}}=\int d^{3}r_{1}\ d^{3}r_{2}\ \phi_{i}\left( \mathbf{r}_{1},\mathbf{r}_{2}\right) \left[ V_{1}\left( \mathbf{r}_{1}\right) +V_{2}\left( \mathbf{r}_{2}\right) \right] \exp\left[ i\left( \mathbf{q}_{1}\cdot\mathbf{r}_{1}+\mathbf{q}_{2}\cdot\mathbf{r}_{2}\right) \right] .\label{DWBApair}$$ Let us assume that the potentials $V_{1,2}$ are given by Gaussian functions, i.e. $V_{1,2}=V_{1,2}^{(0)}\ \exp\left( -r_{1,2}^{2}/\lambda^{2}\right) $ and similarly for the wavefunctions, i.e. $\phi_{\alpha,\beta}=N\exp\left( -r_{1,2}^{2}/\Delta^{2}\right) $. In this case it is straightforward to perform analytically all the integrals in eq. \[DWBApair\]. If one further assumes that the correlation distance, $r_{c}$, is much smaller than the dimensions of the uncorrelated wavefunctions and of the potential, i.e. if $r_{c}\ll\Delta,\lambda$, one gets $$\begin{aligned} T_{\mathrm{(PWBA)}} & =\left( V_{1}^{(0)}+V_{2}^{(0)}\right) \frac{\left( \pi N^{2}r_{c}\Delta/\sqrt{\mathcal{A}}\right) ^{3}}{8}\left[ \exp\left( -\frac{q_{1}^{2}r_{c}^{2}}{4}\right) +\exp\left( -\frac{q_{2}^{2}r_{c}^{2}}{4}\right) \right] \\ & \times\exp\left[ -\frac{\Delta^{2}}{16\mathcal{A}}\left\vert \mathbf{q}_{1\perp}+\mathbf{q}_{2\perp}\right\vert ^{2}\right] \exp\left[ -\frac{\Delta^{2}}{16\mathcal{A}}\left( q_{1z}+q_{2z}-\frac{\omega}{v}\right) ^{2}\right] ,\end{aligned}$$ where $B$ is the separation energy of the pair and $\mathcal{A=}\left( 1+\Delta/\lambda\right) /2$. Therefore, the cross section is given by$$\begin{aligned} \frac{d\sigma}{dq_{1}dq_{2}} & \propto q_{1}^{2}q_{2}^{2}\exp\left[ -\frac{\Delta^{2}}{8\mathcal{A}}\left( q_{1z}+q_{2z}-\frac{\omega}{v}\right) ^{2}\right] \ \nonumber\\ & \times\exp\left[ -\frac{\Delta^{2}}{8\mathcal{A}}\left\vert \mathbf{q}_{1\perp}+\mathbf{q}_{2\perp}\right\vert ^{2}\right] \ \left[ \exp\left( -\frac{q_{1}^{2}r_{c}^{2}}{4}\right) +\exp\left( -\frac{q_{2}^{2}r_{c}^{2}}{4}\right) \right] ^{2}.\label{momdis}$$ Now one can easily understand the physics in figure \[contour\] by identifying the terms of the above equation. The first term is due to conservation of the momentum along the beam direction, which yields the dispersion relation, eq. \[phonon\]. Note that in the derivation of eq. \[dsig5\] it was assumed that the core recoils with the same momentum, i.e. $Q_{Z}\simeq\omega/v=-\left( q_{1z}+q_{2z}\right) $. The second term is due to elastic scattering of the pair on the target in the direction transverse to the beam. In eq. \[momdis\] both the first and the second terms imply that the momentum distribution of correlated pair is such that $q_{1}+q_{2}=\omega/v$, i.e. $q_{1}\simeq-q_{2},$ as expected for small $r_{c}$. Also according to these terms, the distribution is smeared by the range of the independent wavefunctions of the pair, i.e. $\left\langle q_{1,2}^{2}\right\rangle \simeq1/\Delta^{2}$. However, the last term implies a smearing, or spreading, of the momentum distribution by a much larger factor (assuming $r_{c}\ll\Delta$), i.e. $\left\langle q_{1,2}^{2}\right\rangle \simeq1/r_{c}^{2}$. This explains all physics presented in figure \[contour\]. The second exponential term in eq. \[momdis\] plays no role in the results presented in figure \[contour\], since it is identical to one. As discussed above, the location of the ridges in figure \[contour\] is a kinematical property of the phonon absorption mechanism, which is independent of the collision energy. Thus, it should be observable in intermediate energy collisions ($E_{lab}\simeq100$ MeV/nucleon), as well as in relativistic collisions. One also observes that the momentum distributions are narrower for correlated-pair emission from a halo nucleus. This is due to the low binding energy, which yields an extended wavefunction of the nucleons in the halo. This is also seen from the first exponential term of eq. \[momdis\], since the two-proton separation energy for $^{12}$C is 27.16 MeV, while the two-neutron separation energy is 0.3 MeV. The effective value of $\Delta$ is much smaller for the first case, leading to a larger spreading of the momentum distributions. However, the last term in eq. \[momdis\] is still the dominant one leading to a small overall effect on the momentum distribution, as shown next. It would be interesting to try to observe the contribution of the emission of correlated pairs in singles spectra. This can be obtained by integrating $d\sigma/dq_{1}dq_{2}$ over one of the two nucleon momenta. This is shown in figure \[singles\] for C+C and Li+Be collisions, using $r_{c}=0.7$ fm. One observes that the peak in the singles spectra occurs at $p\simeq\hbar/r_{c},$ as expected from the arguments presented above. This should be visible in the spectra of nucleons from knockout reactions as a bump at high nucleon momenta. The position of the bump would be a direct reading of the short-range correlation distance. Notice, however, that such a bump could not be noticeable because it is superposed to a large background of knockout nucleons from stripping reactions. Only by doing a measurement of back-to-back pair emission, this signature of short-range pair-correlations could be assessed. The total cross section for the emission of correlated pairs arising from short-range correlations can be calculated from eq. \[cor\]. Assuming $r_{c}=0.7$ fm, the total cross section for the emission of high-energy correlated pairs in C+C collisions at  250 MeV/nucleon is $\sigma _{\mathrm{corr}}=0.61$ mb. The experimental value for two-proton knockout in this collision is $5.88\pm9.70$ mb. For $^{11}$Li+$^{9}$Be at 287 MeV/nucleon the correlated pair cross section is $\sigma_{\mathrm{corr}}=4.1$ mb. These cross sections are much smaller than those obtained in refs. [@FZ77; @Fes80]. For reasons which were explained in the paragraph preceding eq. \[fold\], the results obtained here are much more reasonable. These cross sections are also much smaller than those for one-nucleon knockout reactions (see, e.g. ref. [@HT04]). They are also only one of the contributions (i.e. only from diffraction dissociation) of the two-proton removal cross section. Another contribution (stripping) has not been considered here. Stripping would not contribute to back-to-back nucleon emission, with nearly zero total momentum of the pair, but is responsible for the largest part of the two-nucleon knockout total cross section. In conclusion, I have shown that when a projectile reacts with a light nuclear target, the short-range correlations contribute to the emission of high-energy nucleons which can be visible in measurements of back-to-back emission of nucleon pairs. More experiments and also the development of a more complete reaction theory are interesting challenges. The theoretical results suggest that the pattern and absolute magnitudes of the partial cross sections can provide specific information on the detailed nature of the states involved. 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